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.a (¢) 1 = 1, if I Aa (¢) I > 1. (Neutral periodic points are also sometimes called indifferent.) superattracting attracting neutral repelling
A periodic point is superattracting if its orbit contains a critical point, so Corollary 1 .2 implies that a map of degree can have at most superattract ing cycles. We will state a much stronger result below (Theorem 1 .35(a)). Remark 1 . 1 3. A neutral periodic point is called rationally neutral if its multiplier is a root of unity, that is, if >.a ( ¢) = 1 for some Otherwise, it is called
Remark 1 . 12.
irrationally neutral.
d
k
2d - 2
k ?: 1.
We next prove a useful formula giving a relation between the multipliers of the fixed points of a rational map. For an application of this formula to p-adic dynamics, see Corollary 5.19 and Proposition 5.20.
1. An Introduction to Classical Dynamics
20
K d 2.
Theorem 1.14. Let be an algebraically closedfield and let rationalfunction ofdegree � Assume that
Then
for all
-
E
be a
P E Fix(¢).
2:::: 1 ,\1p ( ) = 1. 1, ,\p (,\ and 4>
-2 2.
¢(
Per() 2-2
2 - 1.
2+
=
o
n 1
--+
ez )
-l . Proposition 1.24. Let J!D 1 (C) J!D1 (C) be a rational map ofdegree 2: 2, and let F and .:J be the Fatou and Julia sets of, respectively. :
--+
d
1. An Introduction to Classical Dynamics
24
(a) The Fatou set F is completely invariant, i.e., ¢ - 1 (F) = F = ¢(F). (b) The Julia set .:J is completely invariant. (c) The boundary EJ.:J of the Julia set is completely invariant. Proof Sketch. (a) Since ¢ is surjective, it suffices to prove that ¢ - 1 (F) = F. Sup pose first that o: E F, and let ¢((3) = o:. For any point (3' that is close to (3, the Lipschitz property ( 1 .3) of ¢ says that p (o:, ¢((3' ) ) :::; C(¢)p((3, (3' ) . In particular, ¢((3') can be made arbitrarily close to o: by taking (3' sufficiently close to (3. Since o: E F, we know that ¢n (o:) stays close to ¢n (¢(3'), and hence that ¢n+1 ((3) stays close to ¢n+ 1 ((3'). Therefore (3 E F, which proves that ¢ - 1 (F) C F. Next suppose that o: E F. We need to check that ¢(o:) E F. If U is a small neighborhood of o:, then the open mapping property of ¢ implies that ¢(U) is a (small) open neighborhood of ¢(o:). In particular, if (3 is sufficiently close to ¢(o:), then (3 will lie in ¢(U), say (3 = ¢((3') with (3' E U. Then the assumption that o: E F implies that the iterates ¢n ( o:) and ¢n ((3') remain close to one another, and hence the same is true ofthe iterates ¢n - 1 (¢o:) and ¢n - 1 ((3). Therefore ¢(o:) E F, which completes the proof of the other inclusion F c ¢ - 1 (F). (You should fill in the details of this proof sketch and give a rigorous ( J, E ) proof. See Exercise 1 .23.) (b) The Julia set is the complement of the Fatou set, so the complete invariance ofF implies the complete invariance of .:J. ( c) Let J0 be the interior of .:J. The rational map ¢ is continuous, so ¢- 1 ( .:1°) is open, and the complete invariance of .:J shows that it is contained in .:J; hence ¢ - 1 (.JD) c .:r . For the other inclusion, we use the fact that ¢ is an open map, so ¢( .:7°) is open. Again the complete invariance of .:J shows that it is contained in .:J, and the fact that it is open shows that it is contained in J0 • Hence This proves that ¢ - 1 (.:1°) = .:1°, so the interior of .:J is completely invariant. Finally, the fact that .:J and .:1° are completely invariant implies that the set difference EJ.:J = .:J .:1° is also completely invariant. D "
Proposition 1.25. For every integer
n 2 1,
and
= ¢n . It is clear from the definition that F(¢) C F('lj;), since if we know that iteration of¢ maintains closeness of points, then the same is certainly true for ¢n . To prove the opposite inclusion, for each 0 :::; we consider the collection of maps = {¢i 'lj; k : k 2 0}. For any fixed the map ¢i satisfies a Lipschitz inequality ( 1 .3), so the Fatou set F( of contains the Fatou set F( 'ljJ) of 'lj;. Hence
Proof Let 'ljJ
i ) ii,
i
ioo cpn
Remark 1 .34. A quadratic polynomial has only one finite critical point. Thus The orem 1 .33 covers all cases, so we can divide the set of quadratic polynomials into two classes according to the behavior of their finite critical point. Every quadratic polynomial is linearly conjugate to a unique polynomial ofthe form c/Jc(z) = z 2 + c, so we define a portion of the c-plane by M
cpn
{ c E C : ( 0) is bounded for n 2: 1} = {c E C : .J(¢c) is connected}. =
This set M is the famous Mandelbrat set, illustrated in Figure 1 .2. It is a dynamically determined subset of the moduli space of quadratic polynomial maps.
1.5. Properties of Periodic Points
27
Figure 1 .2: The Mandelbrot set M. 1 .5
Properties of Periodic Points
The dynamics of a rational map is influenced not only by the behavior of its critical points, but also by the behavior of its periodic points. The next result describes some ofthe properties of the periodic points and periodic cycles of a rational map. C ( be a rationalfunction ofdegree 2:: (a) The map ¢ has at most nonrepelling periodic cycles in lP'1 (C). If¢ is a polynomial map, then it has at most nonrepelling periodic cycles in C. (b) The Julia set .:7 ( ¢) is equal to the closure ofthe repelling periodic points of¢. (c) Let U C lP' 1 ( q be an open set such that .:7 ( ¢) n U =1- 0. Then there is an integer n 2:: 1 such that ¢n ( u .:! (¢)) = .:7 (¢).
Theorem 1.35. Let ¢(
z) E 2dz) 2 - d- 1 n
d 2.
Proof (a) The sharp bound of - for rational maps is due to Shishikura, see [43, Theorem 9.6]. Much earlier, Fatou gave a weaker bound that is sufficient for many applications, see [95, Theorem III.2.7]. The bound for polynomial maps is due to Douady, see [95, Theorem VI. l .2]. (b) This important result is due independently to Julia and Fatou. See [43, Theo rem 6.9.2] or [95, Theorem III.3. 1]. (c) See [95, Theorem 111.3.2]. D
2d 2
The Fatou set :F( ¢) is an open subset of lP' 1 (C), so it consists of one or more con nected components. It is known that the number of components is equal to 0, 1 , 2, or oo; see [95, Theorem IV.l .2]. If U is a connected component of :F(¢), then the open mapping property of ¢ implies that ¢(U) is also a connected component of :F(¢). In this way we obtain a map
1. An Introduction to Classical Dynamics
28
Components (F( ¢) )
-----+
Components (F( ¢) ) ,
U
f------7
¢( U) ,
and we say that U is aperiodic domain if ¢n (U) = U for some apreperiodic domain if some iterate ¢n ( U) is periodic, and a wandering domain otherwise. The following important result answers a long-standing question ofFatou and Julia.
n ?: 1,
Theorem 1.36. (Sullivan's No Wandering Domains Theorem) A rational map ¢ E C ( z) has no wandering domains.
Proof See [43, Chapter 8], [95, Theorem IV. 1 .3] or [426].
D
The possibility of wandering domains having been eliminated, the periodic com ponents of F(¢) are classified into several types. We begin with the necessary defi nitions, then state the classification theorem. Definition. Let U be a connected component of the Fatou set F(¢), and assume
that U is forward invariant, i.e., ¢(U) •
=
U.
U is parabolic if the boundary of U contains a rationally neutral periodic point ( such that limn ---> oo ¢n(a) = ( for every a E U.
f w I w I < 1} U is a Herman ring if there is an analytic isomorphism f from some annulus { w E C a < I w I < b} to U such that ¢! is a rotation of the annulus. Here a rotation is simply a map of the form w eiew. If U is a periodic connected component of period n ofF(¢), then U is forward invariant for ¢n and we say that U is parabolic, a Siegel disk, or a Herman ring if it •
U is a Siegel disk if there is an analytic isomorphism from the unit disk to U such that ¢! is a rotation of the unit disk. { EC :
•
:
r--+
is of the appropriate type for ¢n .
Theorem 1.37. Let U be a periodic connected component of the Fatou set of a rational map ¢ E C(z). Then Ufits into exactly one of thefollowing categories: (a) U contains an attracting periodic point. (b) U is parabolic. (c) U is a Siegel disk. (d) U is a Herman ring.
Proof See [43, Theorem 7.1] or [95, Theorem IV.2.l]. We mention that it is nontriv ial to prove that Siegel disks and Herman rings exist. In particular, the Fatou set of a polynomial map cannot contain a Herman ring. D 1 .6
Dynamical Systems Associated to Endomorphisms of Algebraic Groups
In this section we study certain rational maps whose dynamical properties are com paratively easy to analyze due to the existence of an underlying group structure.
1.6. Dynamical Systems Associated to Algebraic Groups
29
These maps thus provide a class of examples on which to make and test conjectures, albeit with the caution that they are rather special and in some ways atypical. Addi tional material on this interesting collection of maps may be found in Chapter 6. 1 .6.1
The Multiplicative Group
z) zdn . zd
The dynamics of the polynomial ¢( = is extremely easy to analyze, since However, there is a more intrinsic rea we have the explicit formula ¢n ( z ) = son that is such a nice map; namely, it is an endomorphism of the multiplicative group IC* of nonzero complex numbers. In other words, the map
zd
IC* ---+ IC* , is a homomorphism. It is the existence of the underlying group IC* that makes relatively easy to analyze.
zd
zd
Remark 1 .3 8. In fancier language, the map ¢( z) = is an endomorphism of the algebraic group Gm, where for any field K, the set of points of Gm is the multi plicative group Gm(K) = K*. The full endomorphism ring of Gm is via the identification
.Z
In even fancier language, Gm is a group scheme
Gm =
Spec.Z[X, Y]/(XY - 1 )
called the multiplicative group scheme. It is characterized by the property that for every ring R, the group of R-valued points of Gm is the unit group Gm ( R) � R* . 1 .6.2
Chebyshev Polynomials
The multiplicative group IC* has a nontrivial automorphism given by the reciprocal map z � z- 1 . We can take the quotient ofiC* by this automorphism to obtain a new space that turns out to be isomorphic to IC via the map
[z] �---> z + z- 1 .
The inverse isomorphism takes w E IC to the equivalence class of either of the roots of - zw + 1 = 0. This inversion automorphism commutes with the cfh-power map, so when we take the quotient, the cfh -power map will descend to give a map on the quotient space. In other words, there is a unique map that makes the following diagram commute:
z2
Td
1. An Introduction to Classical Dynamics
30
Z--+Z d
C*
--------7
1
C*
z
{z "" z- 1}
L 1 -1
d
--------7 Z--+Z
c
w ->Td(w)
C*
1
C*
{z "" z-1} 1
c
z
! z+z- 1
Td is characterized by the equation Td (z + z- 1 ) = zd z-d for all z C*. ( 1 .5) It is not hard to see that Td is a polynomial; indeed, it is a monic polynomial of de gree d with integer coefficients called the rfh Chebyshev polynomial. Writing z with t C, we see that the Chebyshev polynomial satisfies Td (2cost) 2cos(dt). Using these formulas, it is not hard to prove that the Julia set of Td is given by .J(Td ) = [-2, 2], the closed real interval from -2 to 2, and to derive many fur ther properties of the Chebyshev polynomials. See Exercises 1 .30 and 1 .3 1 , as well The map
+
E
=
E
eit
=
as Section 6.2. Remark 1 .39. Classically the Chebyshev polynomials are normalized to satisfy the identity = in which case the Julia set becomes = The two normalizations are related by the simple formula ( = �
Td (cos t) cos(dt),
1 .6.3
.J(T ) [-1, 1]. Td w) Tdd(2w).
Rational Maps Arising from Elliptic Curves
There are three different types of (connected) algebraic groups of dimension one. First is the additive group Ga(C) c+ , whose endomorphisms are the maps The dynamical systems associated to endomorphisms of the additive group are relatively uninteresting. Second is the multiplicative group Gm (C) = C*, whose endomorphisms have interesting, but relatively easy to analyze, dynamical properties. The same is true of the Chebyshev polynomials, which come from the restriction of to a quotient ofGm. The third type of one-dimensional algebraic groups consists of a family of groups called elliptic curves. We do not have space to go into the theory very deeply, so we are content to make a few brief remarks here and later expand on this material in Chapter 6. For further material on the analytic, algebraic, and arithmetic theory of elliptic curves, see for example ( 1 , 198, 254, 410, 420, 412]. We note that the reader may omit this section on first reading, since the material is not used other than as a source of examples until Chapter 6. An elliptic curve E (C) is the set of solutions to an equation of the form =
z ----) dz.
z zd z zd t-t
t-t
( x, y) y2 x3 + ax + b =
1.6. Dynamical Systems Associated to Algebraic Groups
31
Adding a point to itself
Addition of distinct points
Figure 1 .3: The addition law on an elliptic curve.
4a3 + 27b2 1P'2 (1C)
together with an extra point 0. We also require that � =1- 0, which means that the cubic polynomial has distinct roots and ensures that the curve E(IC) is nonsingular. Alternatively, E(IC) is the curve in the projective plane defined by the homogeneous equation =
with the extra point being 0 [0, 1 , 0] . There is a natural automorphism on E(IC) given by [ - 1 ] (X, = (X, Z). If P and Q are two points on E(IC), then the line through P and Q will intersect E ( q at a third point R, and we define an addition law by setting P EB Q = [ - 1] R. (If P = Q, we take the "line through P and Q" to be the tangent line to E(IC) at P.) These operations satisfy =
Y, Z)
PEBO = P,
PEB [ - 1] (P)
-
Y
,
= 0, PEBQ QEBP, (PEBQ) EBR = P EB (QEBR), =
so they make E(IC) into an abelian group. (The first three equalities are obvious, while the associativity is somewhat difficult to prove.) An elementary calculation shows that the coordinates of P EB Q are given by ra tional functions of the coordinates of P and Q. In particular, if and are in some field K, and if P and Q have coordinates in K, then P EB Q will also have coordi nates in K. Thus E(K) = E(IC) n 1P' (K) will be a subgroup of E(IC) . Figure 1 .3 illustrates the group law for points in E(IR.). Repeated addition in any abelian group gives an endomorphism of the group, so for each d ;:::: 1 we obtain a map a
2
[d] : E(IC) ------> E(IC),
P � P EB P EB · · · EB P d tenus
called the multiplication-by-d map. Setting [O] (P) = 0
and
we obtain an (injective) homomorphism
[ - d] (P)
=
[ - 1 ] ( [d] P),
b
1. An Introduction to Classical Dynamics
32
Z -----+ End(E(C)), d � [d] . If this map is not onto, then E is said to have complex multiplication, or CM for short. For example, the curve E : y 2 x3 + x has complex multiplication, since it has extra endomorphisms such as [i] : (x, y) ( -x, iy). Most curves do not have CM. The multiplication map [d] clearly commutes with the involution [ -1], so it de scends to a map on the quotient space E (C)/ { ± 1}. The quotient space is very easy to describe, E(C)/{ ± 1} 1P'1 (C), (x,y) � so the multiplication-by-d map descends to give a rational map ¢E, d making both squares in the following diagram commute: E(C) � E(C) =
r---+
�
x,
1 d E(C) E(C) [----J -) 1 { ± 1} {±1}
(xt1
1(xt
JIDl (q � lP'l (q
¢E,d
¢E,d is a rational function char ( 1.6) 1/JE,d (x(P)) x([d](P)) for all P E(C). Notice the close similarity to the defining property (1.5) of the Chebyshev polyno mials. More generally, it turns out that every endomorphism End( E) commutes with [-1], even if E has complex multiplication, so every endomorphism deter mines a rational function ¢E, u on lP' 1 (C). Example 1 .40. Let E : y 2 x3 + ax + b be an elliptic curve as above. Then the duplication map [2] : E(C) ___, E(C) leads to the rational function 2 - 8bx + a2 ¢E,2 (x) x4 -4x2ax 3 + 4ax + 4b Example 1 .41. Let E : y 2 x3 + x be the CM elliptic curve mentioned earlier. Then the map [1 + i] : E(C) E(C) defined by [1 + i](P) = P [i](P) leads to the rational function ¢E,Hi (x) ;i (x + �) . Proposition 1.42. Let E : y2 = x3 + ax + b be an elliptic curve, let d 2: 2 be an lP'1 (q be the associated rational map as above. integer, and let ¢E , d : lP'1 ( q Then The map is an example of a Lattes map. Thus acterized by the formula
E
=
uE
u
=
=
=
EB
___,
=
___,
1.6. Dynamical Systems Associated to Algebraic Groups
33
• • •
L
•
•
•
•
w1 +w2 •
•
• •
Figure 1 .4: A lattice and fundamental domain associated to an elliptic curve.
PrePer(¢E,d)
=
x ( E(C)tors) ,
where we recall that the torsion subgroup E(C)tors of E(C) is the set of elements of finite order (cf Example 0.2). Proof We leave the proof as an exercise; see Exercise 1 .32.
D
¢ ,d lP'1
The dynamics of the rational function E on (C) can be analyzed using the group law on the larger space E(C). The utility of this approach is further enhanced by the analytic uniformization of E(C), which we now briefly describe. A lattice L in E,d
+1
=
¢>'z;;,d ( p(() ) p' (()
=
(1 .8)
dp' ( d().
But we know that p(z) is a meromorphic function on C/ L, or equivalently, it is a meromorphic function on C with the periodicity property p(z +
w) = p(z)
for all
w E L.
' ' p (z + ) p (z) for the derivative. (Notice i 2 the analogy with the function e 1r z, which is holomorphic on C and invariant under the translations z z n for n Z.) We evaluate at = d(, use the fact that d( ±( (mod L) to get p'(d() = p'(±() = ±p' ((), and substitute into (1 .8) to obtain the relation
Differentiation gives the same relation r-+
=
+
E
w
=
z
Exercises
35
" (o:)/2) 2
In other words, if the series expansion of ¢( z) around z = a has the form c/>(z) = a + (z - a:) + A(z - o:) 2 + B (z - o:)3 + . . .
with A =f- 0,
then L(c/>, o:) = B jA2• (e) The polynomial cp(z) = z + Az2 + z3 has fixed points {0, -A, oo}. The point at infinity is superattracting, so Aoo (cl>) = 0. Use the formula in (d) and the summation formula to compute A_ A ( cp), and then check your answer by computing A_ A ( cp) directly.
1.18. Let cp(z) E C(z) be a rational function of degree d � 2. (a) Prove that # Pern (c/>) :::; dn + 1 . (b) Prove that # Pern (c/>) -+ oo as n -+ oo. (c) Prove that Per�* ( ¢) is nonempty for infinitely many values of n. (d) * More precisely, prove that if Per�* ( ¢ ) is empty, then (n, d) is one ofthe pairs
(n, d) E { (2 , 2) , (2, 3), (3 , 2) , (4, 2) } . Let cp(z) E C [z] be a polynomial of degree d � 2.
1.19. (a) Ifm I n, prove that the polynomial cpm (z) - z divides the polynomial cpn (z) - z. (b) Let p, be the Mobius p, function. Prove that for every n � 1, the rational function m �(z) = IJ (¢m (z) - z) "( n / ) mi n
is a polynomial. The polynomial � (z) is a dynamical analogue of the classical cyclo . tomic polynomial fl ml (zm - 1 ) JL ( n / m ) n
(c) Compute the first few polynomials � (z) for the quadratic polynomial cl>c (z) = z 2 + c. (d) Let a be a root of ;',(z). Prove that ¢n (o:) = a . Thus roots of ;', (z) are in Pern (¢) . (e) Let c/>c (z) = z2 + c. Find a value of c such that 2(z) has a root whose exact period is strictly smaller than 2? (t) For n = 3 and n = 4, find all values of c such that � ( z) has a root whose exact period is strictly smaller than n . The roots of � ( z) are said to have formal period n. They behave in many ways as if they have exact period n, although their actual period is smaller than n.
1.20. Find explicit formulas for the fixed points and corresponding multipliers of the func tion ¢( z) = az/ ( z2 + b). VerifY directly the formula in Theorem 1 . 14 (cf. Example 1 . 1 7). Section 1 .4. The Julia Set and the Fatou Set
1.21. Let (SI , PI ) and (S2 , p2 ) be metric spaces. A collection of maps from 81 to 82 is said to be uniformly continuous if for every E > 0 there exists a [J > 0 such that PI (a , {3)
{3)
(a) , cp(f3)) S:: C PI (a , {3) ·
for all a, {3 E 81 and all ¢ E .
40
Exercises
(a) If .P is uniformly continuous, prove that .P is equicontinuous at every point of S1 · (b) If .P is uniformly Lipschitz, prove that .P is uniformly continuous. (c) Let .P = { 4>n } n ;:o: 1 and suppose that for every point a E S1 , the limit ¢(a) = n-->oo lim 4>n (a) exists. If .P is equicontinuous at a E 81, prove that ¢ is continuous at a. Give an example to show that the equicontinuity assumption is necessary. 1.22. Give an example to show that equicontinuity is not an open condition. Thus in order to ensure that the Fatou set is open, the definition of :F( 4>) must include the requirement that it be open. 1.23. Let 4> S S be a surjective continuous map of a metric space with the additional property that 4> maps open sets to open sets. (a) Give a rigorous proof that the Fatou and Julia sets of 4> are completely invariant, that is, ¢-1 (:F) = :F = ¢(:F) and ¢ - 1 (:1) = :1 = 4>(:1). (b) Is (a) true without the assumption that 4> is surjective? (c) Is (a) true without the assumption that 4> maps open sets to open sets? :
--+
1.24. Let ¢( z) E C[z] be a polynomial map, and let :Foo be the connected component of :F( 4>) containing the point oo. Prove that ¢- 1 (:Foo) = :Foo = ¢(:F00). In other words, for a poly nomial map, the connected component of oo is completely invariant for ¢.
Let ¢( z) = 1 2/ z2 be the rational map from Example 1 . 3 1 . (a) Let f(z) = 1 /z . Prove that (¢f)' (0) = 0 and conclude that oo is a critical point of¢. (b) Let f(z) z/(z - 1). Prove that ( q/ )'(0) 0 and conclude that 0 is a critical point of¢.
1.25.
-
=
=
1.26. Let if>(z) E Q (z) be a rational map of degree d 2: 2 with coefficients in Q, and let .J ( ¢; Q) = :1 ( 4>) n lP'1 ( Q) be the set of points in the Julia set whose coordinates are algebraic
numbers. (a) Is .J( ¢; Q) Galois-invariant? (b) Is :1(4>; Q) infinite? More generally, is :1(4>; Q) dense in :1(4>)? (c) ** Does :1 ( ¢; Q) contain points that are not preperiodic? More generally, does :1 ( ¢; Q) contain infinitely many nonpreperiodic points Pi such that the orbits of the Pi are dis joint? Section 1 .5. Properties of Periodic Points
Let a be a periodic point of a rational function 4> E C(z). (a) If a is attracting, prove that a E :F( 4> ). (b) If a is repelling, prove that a E :1 ( 4>).
1.27.
1.28. Let if>(z) = z2 - 2. Prove that the Julia set of 4> is the closed interval on the real axis between -2 and 2. Find the periodic points of¢, compute their multipliers, and show directly that :1(4>) is equal to the closure of Per(¢). (Hint. Write z = eit + e- it 2 cos(t), so ¢(z) = 2 cos(2t). Prove that 4>n (z) = 2 cos(2n t) and use this formula to study the dynamics of¢.) =
Exercises
41
1.29. Let ¢( z) E IC[z] be a polynomial of degree d ;::: 1, and suppose that every a E Pern (4>) has multiplier Aq,(a) f. 1. Prove that the equation t/>n (z) = z has simple roots, and deduce that Pern ( ¢) contains exactly dn + 1 distinct points (including the point oo) . Section 1 .6. Dynamical Systems Associated to Algebraic Groups
1.30. This exercise describes algebraic properties of the Chebyshev polynomials Td (w). (a) Prove that T2 (w) = w2 - 2, T3 (w) = w3 - 3w, and T4 (w) = w4 - 4w2 + 2. (b) Prove that Td(Te (w)) = Tde (w) for all d, e ;:=: 0. d (c) Prove that Td( -w) = ( -1) Td(w). Thus Td is an odd function if d is odd and it is an even function if d is even. (d) Prove that the Chebyshev polynomials satisfy the recurrence relation
where we use the initial values To ( w) = 2 and T1 ( w) = w. (e) Prove that the generating function of the Chebyshev polynomials is given by
l: Td(w)Xd = 00
d=O
2 - wX
1 - wX + X 2 .
(f) Prove that the d1h Chebyshev polynomial is given by the explicit formula
1.31. This exercise describes dynamical properties of the Chebyshev polynomial Td ( w) for a fixed d ;::: 2. (a) Let U be an open subinterval of [-2, 2]. Prove that there exists an integer n ;:=: 1 such that T:J(U) = [-2, 2]. (b) Prove that limn�= Td: (w)
= oo
for all points w E IC not lying in the interval [-2, 2) .
(c) Prove that the Julia set .:J(Td) is the closed interval [-2, 2]. (d) Find all of the periodic points of Td and compute their multipliers. Observe that .:l(Td) is the closure of the (repelling) periodic points. (e) Let F(w) E C(w] be a polynomial of degree d � 2 with the property that the interval [-2, 2] is both forward and backward invariant for F. Prove that F(w) = ±Td(w).
1.32. (a) Let A be an abelian group, let Ators = { P E A : nP = 0 for some n � 1} be the torsion subgroup of A, let d ;:=: 2 be an integer, and let 4> : A --+ A be the map tj>(P) = dP. Prove that
PrePer( t/>) = Ators·
(This is Proposition 0.3, but try reproving it without looking back at our proof.) (b) Let E be an elliptic curve, let d ;::: 2 be an integer, and let 4>E , d : lP'1 --+ lP'1 be the rational map associated to multiplication-by-d on E, as described in Section 1 .6.3. Prove that
PrePer(t/>E , d) = x( E(IC)tors) ·
Chapter 2
Dynamics over Local Fields : Good Reduction The study of the dynamics of polynomial and rational maps over � and C has a long history and includes many deep theorems, some of which were briefly discussed in Chapter 1 . A more recent development is the creation of an analogous theory over complete local fields such as the p-adic rational numbers QP and the completion Cp of an algebraic closure of Qp. The nonarchimedean nature of the absolute value on QP and Cp makes some parts of the theory easier than when working over C or R But as usual, there is a price to pay. For example, the theory ofnonarchimedean dynamics must deal with the fact that Qp is totally disconnected and far from being algebraically closed, while Cp is not locally compact. In this chapter we begin our study of dynamics over complete local fields K by concentrating on rational maps ¢ that have "good reduction." Roughly speaking, this means that the reduction of¢ modulo the maximal ideal of the ring of integers of K is a "well-behaved" rational map ¢ over the residue field k of K. Thus studying the dynamics of ¢ over k allows us to derive nontrivial information about the dynam ics of ¢ over K. In Chapter 5 we take up the more difficult, but ultimately more interesting, case of rational maps with "bad reduction." 2 .1
The Nonarchimedean Chordal Metric
We begin by quickly recalling the definition and basic properties of absolute values, especially those satisfying the ultrametric inequality. Definition. An absolute value on a field K is a map
I · I : K --.. � with the following properties: l a l 2: 0, and lal 0 if and only if a = 0. •
=
43
2. Dynamics over Local Fields: Good Reduction
44 •
l a/31 = I a I · l/31 for all a, /1 E
K.
Ia + /31 :::; Ia I + l/31 for all a, /1 E
K (triangle inequality). A valuedfield is a pair ( K, I I K ) consisting of a field K and an absolute value on K, although we often omit the absolute value in the notation. A map of valued fields i : K L is a field homomorphism that respects the absolute values, for all a E K. Definition. Let (K, I I ) be a valued field. If the absolute value satisfies the stronger estimate Ia + /31 :::; max{ l a l , l/31} for all a, /1 E K, (2. 1) •
·
____.
·
then the absolute value is nonarchimedean or ultrametric. The associated valuation is the homomorphism v
It satisfies
: K*
----"
JR,
v (a) = - log Ia I.
+ /1) � min{ v(a) , v(/1) } . The valuation v is discrete if v ( K*) is a discrete subgroup of JR, in which case the associated normalized valuation (sometimes denoted ordv) is the constant multiple of v chosen to satisfy ordv ( K*) = Z. v(a
Example 2.1. The field Q has the usual real absolute value l a loo = max{ a, - a } .
For each prime p it also has a p-adic absolute value defined as follows. Every nonzero rational number a has a unique factorization of the form a=±
Then
IT
p prime
p ep( a )
with ep(a) E
Z.
I a lp = p -ep( a ) .
The p-adic absolute values are nonarchimedean. Two absolute values are said to be equivalent if there is a constant r > 0 such that l ah = l a l 2 for all a E Ostrowski's theorem says that up to equivalence, the real absolute value and the p-adic absolute values are the only nontrivial absolute values on Q; see [78, 1 .4.2, Theorem 3] or [249, 1.2, Theorem 1]. Example 2.2. Let be a field and let = be the field of rational functions. Then each a E determines an absolute value on K associated to the valuation orda that gives the order of vanishing of f(T) E at T = a. The degree map deg Z is also a valuation. If is algebraically closed, this yields the complete set of absolute values on k(T) that are trivial on up to equivalence. More generally, if is not algebraically closed, there is a valuation corresponding to each monic
K.
k(T)* k
kk
____.
K k(T) k k(T)k,
:
2.1. The Nonarchimedean Chordal Metric
45
k [T].
irreducible polynomial in In this book we are primarily interested in the number field scenario, i.e., the field Ql and its extensions and completions, but the reader should be aware that it is also interesting to study the analogous case of function fields, i.e., the field and its extensions and completions. We now prove the elementary, but extremely useful, fact that if -=1- 1 ;31 , then the ultrametric inequality (2. 1 ) is actually an equality.
k[T]
l o: l
Lemma 2.3. Let K be a field with a nonarchimedean absolute value let jJ E K. Then
o:,
I lv and ·
l o: l v > I;Ji v · The strict inequality I ;Ji v < l o: l v = l (o: + ;3) - ;J iv :S max{l o: + ;Ji v , I;Ji v } implies that the maximum on the right is l o: ;J i v , so we find that l o: l v :S l o: + ;J i v · The opposite inequality is also true, since l o: + ;J i v :S max{l o: l v , I ;J i v } = l o: l v · D Recall that the chordal metric on lP'1 (C), which we now denote by Poo , is defined by the formula
Proof We suppose that
+
for points P1 = [X1 , Y1 ] and P2 X2 , Y2 in lP'1 q. In the case of a field K having a nonarchimedean absolute value · it is convenient to use a metric given by a slightly different formula. =
[
I lv.
] (
I · and let P1 = [X1 , Y1 ] and P2 = [X2 , Y2] be points in lP' 1 K The v-adic chordal metric on lP' 1 K) is Definition. Let K be a field with a nonarchimedean absolute value
( ).
(
lv.
Pv The first thing to check is that Pv is indeed a metric. In fact, it is an ultrametric; that is, it satisfie s the nonarchimedean triangle inequality.
It is clear from the definition that ( P1 , P2 ) is independent of the choice of homo geneous coordinates for P1 and P2 .
Proposition 2.4. The v-adic chordal metric has thefollowing properties. (a) 0 :S P2) :S 1.
(Pl , Pv (b) Pv (Pl ,P2 ) = O ijand only iJP1 = P2. (c) Pv (Pl ,P2 ) = Pv (P2 ,Pl ). (d) Pv (Pl , P3 ) max{pv(Pl, P2 ), Pv (P2 , P3 ) }. :S
2. Dynamics over Local Fields: Good Reduction
46
Proof The lower bound in (a) and parts (b) and (c) of the proposition are obvious from the definition. For the upper bound in (a), we use the nonarchimedean nature of v to compute
I X1 Y2 - XzY1 I v :::; max{I X1 Y2 I v , I XzY1 I v } :::; max{ I X1 I v , I Y1 I v } max{I Xz l v , I Yzl v } .
The proof of (d) requires the consideration of several cases. The following useful lemma makes the proof more transparent by allowing some freedom to change co ordinates. It is the analogue of the fact that the classical chordal metric is invariant under linear fractional transformations that define rigid rotations of the Riemann sphere (cf. Exercise 1 .2). Lemma 2.5. Let
:::;
= {a l o: l v 1 } f : JPl1 JPl1 be a linearfractional transforma f([X' Y]) = eXaX ++ dYbY with a, b, e, d E R and ad - be E R* , i.e., f E P GL2 ( R) Then Pv (f(P1 ), J(P2 )) = Pv (Pl , P2 ) It is crucial that the quantity ad - be is a unit.) Proof Write each point as Pi = [Xi , Yi] with Xi , Yi E R and at least one of Xi or Yi in R*. Then max{ I Xi l v , I Yi l v } = so Pv (Pl , P2 ) = I X1 Y2 - X2 Y1 I v · R EK be the ring of integers of K, and let tion of theform
:
___,
.
(NB.
1,
Further, the identities
d(aXi + bYi) - b(cXi + dYi) = (ad - bc)Xi , c(aXi + bYi) - a(cXi + dYi) = -(ad - bc)Yi, and the fact that max{ I Xi l v , I Yi l v } = 1 and l ad - bc l v = 1 immediately imply that max{l aXi + bYi l v , l cXi + dlil v } = 1 .
Thus
Pv(/(Pl ), j(P2 )) = i(aX1 + bYI)(cX2 + dY2 ) - (aXz + bY2 )(cX1 + dY!)I v = i(ad -be)(X1 Y2 - X2Yl )i v = Pv (Pl , P2 ), where we have again made use of the fact that l ad - bel v = 1.
D
2.2. Periodic Points and Their Properties
47
We resume the proof of Proposition 2.4 and write each point as with E R and at least one of or in R*. Then
Xi , Yi
Xi Yi
Pi = [Xi , Yi]
and
as usual. If we apply the map f = to the three points. This > preserves the chordal distance (Lemma 2.5) and allows us to assume that
I X2 I v I Y2 I v ,
Y/X
IX2 Iv :::; I Y2 I v = 1. Next we apply the map f (Y2 · X - X2 · Y)/Y to the three points. Lemma 2.5 again tells us that the chordal distance is preserved (note that I Y2 I v = 1), and we are reduced to the case that P2 = [0, 1]. Finally, we compute I X1 Y3 - X3Y1 I v :::; max{I X1 Y3 1 v , I X3Yll v } :::; max{IXl l v , I X3 1 v } , which is exactly the desired inequality when P2 is the point [0, 1]. =
0
2.2
Periodic Points and Their Properties
In Section 1 .3 we described various properties of periodic points for rational maps defined over IC. Virtually all of these definitions make sense, mutatis mutandis, when we work over any field with an absolute value. In this section we briefly recall the relevant material. Let K be a field with an absolute value · and let ¢( z ) E K z ) be a nonconstant rational map. The multiplier of¢ at a fixed point o: E K is the derivative
(
II
Aa ( ¢) = ¢' ( ) (See Exercise 1 . 13 for the case o: = oo.) The multiplier is well-defined, independent ofthe choice of coordinates on lP' 1 ; see Proposition 1 .9. E IP' 1 ( ¢, 0: .
More generally, if o:
K) is a point of exact period n for
point of ¢n and we define the multiplier of¢ at o: to be
then o: is a fixed
The multiplier may be calculated using the chain rule as The magnitude of the multiplier determines, to some extent, the behavior of¢ in a small neighborhood of a periodic point o:. The periodic point o: is called superattracting if ¢) = attracting if 1, = 1, neutral (or indifferent) if > 1. repelling if The neutral periodic points are further divided into two types. The rationally neutral periodic points are those whose multiplier is a root of unity. The others are called irrationally neutral.
Aa ( 0, l >..a (¢) I < I Aa (¢) I I Aa (¢) I
2. Dynamics over Local Fields: Good Reduction
48 2.3
Reduction of Points and Maps Modulo
p
One of the most important gadgets in the number theorist's toolbox is reduction modulo a prime. Thus when studying the number-theoretic properties of an object, we reduce it modulo a prime, analyze the properties of the hopefully simpler object, and then lift the information back to obtain global information. A typical example is provided by Hensel's lemma, which under certain circumstances allows us to lift solutions of a polynomial congruence (mod to solutions in Then, using information gathered from many primes, one is sometimes able to deduce re sults for a global field such as Ql. In this section we study Our principal objects of study are maps ¢ the behavior of such maps under reduction modulo a prime. We work over a discrete valuation ring, so we set the following notation: a field with normalized discrete valuation · lv = cv(x) for some > 1, an absolute value associated to R = ::=: the ring of integers of p ::=: 1 the maximal ideal of R. = R = the group of units of R. Rjp, the residue field of R. reduction modulo p, i.e., R a f---t Before studying reduction of maps, we consider the problem of reducing points modulo p. This is as easy in as it is in so we look at the general case. Let
f(x) 0 p) : JID1 JID1 .
Zp.
=
___.
I
K
v : K* Z. v. K. ....,.
c
{a E K : v(a) 0}, = {a E K : v(a) } , * {a E K : v(a) 0}, k =
k, a. pN JID1 , p = [xo , Xt, . . . ,xN ] E JIDN (K) be a point defined over K. We cannot immediately reduce the coordinates of P mod ulo p, since some of the coordinates might not be in R. However, since the coordi ___.
nates of P are homogeneous, we can replace them with p=
c E K*.
c
[cxo, CXt, . . . , CXN]
cxi
for any Choosing to be highly divisible by p, we can ensure that every is in R. However, if we overdo this process and end up with every in the prime ideal p, then when we reduce modulo p, we end up with which does not represent a point of projective space. The trick is to "clear the denominators" of the as efficiently as possible. To do this, we choose an element satisfying
[0, 0, . . . , 0],cxi
Xi 's a E K* (2.2) v(a) = min{v(xo), v(xt), . . . , v(xN )}. For example, a could be the Xj having minimal valuation. Then a- 1 xi E R for every i, so we can reduce these quantities modulo p. We define the reduction of P modulo p to be the point
F = [�, �, . . . ,a�] E JID1 (k).
2.3. Reduction of Points and Maps Modulo p
49
Note that P has at least one nonzero coordinate, since at least one of the num bers is a unit. Hence P is a well-defined point in ( We say that = has been written using normalized coordinates if
lP'1 k).
a- 1 Xi
P [xo, . . . , xN ] min{v(xo), v(xl ), . . . , v(xN )} = 0 , in which case P is simply [xo, . . . , x N ]. Example 2.6. Consider the point P = [ 13 , i �� , J ] E 1P'1 (Qi). We can reduce P 4 modulo 1 1 without any modification, since every coordinate is an 1 1 -adic integer and not all coordinates vanish modulo 1 1 . Thus P = [1, 1 0 , 7, 9] (mod 1 1 ) . However, if we want to reduce P modulo 3, then we first need to divide all of the coordinates by 3, - [ 143 ' 359 ' 4924 ' 24527 ] - [ 141 ' 353 ' 498 ' 2459 ] ' and then P = [2 , 0, 2 , 0] (mod 3). Similarly, in order to compute P modulo 7, we first multiply the coordinates by 49, 2 2 5
5,
p-
p-
-
-
[3
9 24 27 14 ' 35 ' 49 ' 245
] [
]
21 63 24 27 2 ' 5 ' 1 ' 5 '
and then P = [0, 0, 3, 4 (mod 7) .
]
It may appear that the reduction P depends on the choice of that this is not the case.
a. We now check
P = [x0 , . . . ,x N ] E lP'N (K). Then the reduction P is inde pendent of the choice of a satisfying (2.2). Proof Suppose that a and j3 both satisfy (2.2). Then a and j3 have the same valua tion, so a/3 - 1 E R*. This allows us to compute aj3- 1 a- 1 xo,af3�� - 1 XN] -- [�� - 1 a - 1 x1 , . . . ,aj3�-1 a� a -1 xo,a� - 1 x1 , . . . , a� - 1 XN] [� - 1 XN] . j3- 1 Xo,/3� - 1 X1 , . . . ,j3� - [� Hence the reduction of P modulo p is independent of the choice of a satisfying (2.2). Proposition 2.7. Let
D
We next prove an easy and useful lemma that relates reduction modulo p to v-adic distance. Lemma 2.8. Let
P1 and P2 be points in lP'1 (K). Then (P1 , P2 ) P1 P2 ifand only if =
Pv
IP'1 be a rational map as above and write ¢ = [F(X, Y) , G(X, Y)]
with homogeneous polynomials F, G E K[X, Y]. We say that the pair (F, G) is normalized, or that ¢ has been written in normalizedform, ifF, G E R[X, Y] and at least one coefficient of F or G is in R* . Equivalently, ¢ = [F, G] is normalized if F(X, Y) = aoX d + a1Xd - I y + ad l xyd - l ad Y d and
+
· · ·
+
satisfy
{
min v(ao ) , v(a i ) , . . . , v(ad ) , v( bo ) , v( bi ) , . . . , v( bd ) } = 0.
(2.3)
Given any representation ¢ = [F, G], it is clear that one can always find some c E K* such that [cF, cG] is a normalized representation. Further, the element c is unique up to multiplication by an element of R* .
2. Dynamics over Local Fields: Good Reduction
52
Writing ¢ = [F, G in normalized form, the reduction of¢ modulo p is defined in the obvious way,
] ¢;(X, Y) = [F(X, Y), G(X, Y)] = [aoX d + a 1 X d - l y +
adYd, boXd + b1 Xd- I y + + bdYd]. In other words, ¢ is obtained by reducing the coefficients of F and G modulo p . (If the prime ideal is not clear from context, we write ¢P or ¢ mod p.) The fact that at least one coefficient of F or G is a unit ensures that at least one of F and G is a nonzero polynomial, so the reduction ¢ gives a well-defined map ¢ JPl1 ( k) ----) JPl1 ( k). Further, the reduced map ¢ is independent of the choice ofF and G, a fact whose proof we leave to the reader (Exercise 2.4) since it is quite · · ·
+
· · ·
:
similar to the proof of Proposition 2.7.
The mere existence of the reduction ¢ of a rational map ¢ does not imply that ¢ has good properties, as is shown by the following simple example. Example 2. 12. Let E K* and consider the rational map
a
(X, Y) = laX d , Y d] is again a rational a) 0, a = l0, so then is a constant map! Similarly, if v then
is again a constant map, but not to the same point! ! To summarize, the reduction of the map a , = separates into three cases,
ifv(a) < 0. Clearly it is only in the first case that the reduced map is interesting. Keep in mind that our goal is to use the dynamics of ¢ to help us understand the dynamics of ¢. In the above example, if (a) = 0, then it is easy to see that for any P E JPl1 (K), ¢(?) = ¢; (F), and hence by induction, ¢n ( P) = Jn(P). Thus the ¢ orbit of P yields valuable information about the ¢ orbit of P. However, ifv(a) -=f. 0, then ¢ (P) is constant, independent of P, so the ¢ orbit of P contains no information. We formalize this notion in Section 2.5 after a preliminary discussion [1,0]
v
of the theory of resultants.
2.4. The Resultant of a Rational Map 2.4
53
The Resultant of a Rational Map
A rational map ¢ :
IP'1 IP'1 is given by a pair ofhomogeneous polynomials ¢ = [ F (X, Y), G(X, Y)] -+
having no nontrivial common roots. However, if we reduce the coefficients of F and modulo some prime, they may acquire common roots in the residue field. In order to understand this phenomenon, it is useful to have a tool that characterizes the existence of common roots in terms of the coefficients ofF and This tool is called the resultant. Resultants and their generalizations are widely used, both theoretically and computationally, in number theory and algebraic geometry. We give in this sec tion only a brief introduction to the theory of resultants. The reader desiring further information might consult [105, Section 3.3], [1 12, Chapter 3], [259, Section V. l 0], or [436, Sections 5.8, 5.9], while the reader interested in applications to dynamics may wish to peruse only the statements of Proposition 2.13 and Theorem 2.14 and return to the proofs at a later time. Proposition 2.13. Let
G
G.
···
A(X, Y) = aoXn + a1 Xn - 1 Y + + an- 1 xyn - 1 + an Yn , B(X, Y) = boXm + b1 Xm- 1 Y + · · · + bm- 1 xym- 1 + bm Ym
be homogeneous polynomials ofdegrees n and m with coefficients in afield K. There exists a polynomial
A
A
in the coefficients of and B, called the resultant of and B, with the following properties: (a) = ifand only if and have a common zero in (K). (b) If =1- and ifwefactor and as
Res(A, B) 0 A B IP'1 A B aobo 0 A = ao i=IIn (X - aX) and B = b0 j=IIml (X - {J1 Y), 1 then Res( A, B) = a0b� i=IIn j=IIm (ai - /Jj ) · 1 1 (c) There exist polynomials F1 , G1 , Fz , G2 E Z[ao, . . . , an , bo, . . . , bm][X, Y], homogeneous in X and Y of degrees m - 1 and n - 1, respectively, with the property that F1 (X, Y)A(X, Y) + G 1 (X, Y)B(X, Y) = Res( A, B)x m+n - 1 , F2 (X, Y)A(X, Y) + G2 (X, Y)B(X, Y) = Res( A, B)Ym+n- 1 . Notice that in the first equation, the variable Y has been eliminated, and simi larly X has been eliminated in the second equation.
2. Dynamics over Local Fields: Good Reduction
54
( + n) (m + n) determinant ao aoa1 aa2 .a. . .a.n. a ao1 a21 a2 . .n. an Res( A, B) = det bo b1 b2 . . . .ao. . .a.1. bam2 . . . an bo bob1 bb2 .b. . .. .. .. .. .. .. .b.m. b 1 2 m
(d) The resultant is equal to the m
x
Res( A, B)
m
n
bo, . . . , bm .ao, . . . , an
In particular, is homogeneous of degree m in the variables and simultaneously homogeneous ofdegree n in the variables
Proof We begin by showing that the following three conditions are equivalent.
A( X, Y) and B(X, Y) have a common zero in JP>1 (J() . (ii) A(X, Y) and B(X, Y) have a common (nonconstant) factor in the polynomial ring K[X, Y]. (iii) There are nonzero homogeneous polynomials C, D E K[X , Y] satisfying A(X, Y)C(X, Y) = B(X, Y)D(X, Y) with deg(C) m - 1 and deg(D) n - 1. (i)
::::;
::::;
(2.4)
The equivalence of (i) and (ii) follows immediately from the fact that the greatest common divisor of and B in vanishes at exactly the common zeros of and in (What we are really using here, of course, is the fact that the ring of homogeneous polynomials is a principal ideal domain.) It is also clear that (ii) implies (iii), since if and have a common factor F, then we simply choose and using the formulas = F and B = Finally, to prove that (iii) implies (i), we suppose that (2.4) is true. If we factor both sides of (2.4) into linear factors in then has n factors, while has at most n - 1 factors. Therefore shares at least one linear factor with in and hence they have a common zero in We next multiply out equation (2.4), treating the coefficients of and as unknowns. This gives a system of m n homogeneous linear equations in the m n variables c0 , . . and the matrix of this system is (up to changing the sign of some columns and transposing) equal to the matrix given in (d). For example, if = 3 and = 2, then equating coefficients in (2.4) gives the system of linear equations
A B JP>1 (k). A K[K[XX, ,Y]Y] A BA D C D FC. k[X, Y],A(X, A(X, Y) D(X, Y) B(X, Y) Y) k [X, Y], JP>1 (k). C D + + . , Cm-1 , do, . . . , dn - l, deg(A) deg(B)
2.4. The Resultant of a Rational Map
55
coao = dobo, coa + c ao coa2l + c11 a1 == dob dob21 ++ ddd11bo,bb1 ++ dd2bbo,, coa3 + cc1 aa2 == 1 2 d22 b21 , 13 whose associated matrix aoa ao -bo aa21 aa1 -b-b21 -b-bo 1 -b-bo1 -b 2 2 3 a3 -b2
becomes equal to the matrix in (d) if we change the sign of the last three columns and transpose. The general case is exactly the same. To recapitulate, we have shown that equation (2.4) has a nontrivial solution if and only if the system of homogeneous linear equations described by the matrix in (d) has a nontrivial solution, which is equivalent to the vanishing of the determinant of the associated matrix. Hence if we take the determinant in (d) as the definition of the resultant Res( A, B), then the equivalence of (i) and (iii) proven above shows that Res(A, B) = 0 if and only if A and B have a common zero in (K), which proves (a). In order to prove (c), we write
lP'1
and
xj yn- l -j B
for 0 � j < n
as a system of homogeneous equations, ao a 1 a2 . . . an ao a 1 a2 . . . an ao a 1 a2 . . . an ao a 1 a2 . . . an bo b1 b2 . . . . . . . . . bm bo b1 b2 . . . . . . . . . bm bo b1 b2 . . . . . . . . . bm bo b1 b2 . . . . . . . . . bm
x n+m - 1 xn+m - 2 y x n+m - 3 y 2
X"'-l A xm - 2 y A x m - 3y 2 A ym- l A x n- l B x n - 2 yB x n -3y2 B y n-l B
Notice that the matrix M appearing here is exactly the matrix in (d) whose deter minant equals Res( A, B). We multiply on the left by the adjoint matrix Madj of M. (Recall that the entries of Madi are the cofactors of the matrix M, and that the prod uct Madj M is a diagonal matrix with the quantity det ( M) as its diagonal entries.) This yields the following matrix identity, where for convenience we write R(A, B) for Res( A, B):
2. Dynamics over Local Fields: Good Reduction
56 R(A, B)
0 0
0
0 0
R(A, B)
0
R(A, B)
· · ·
0 0 0
xn+m-1 x n+ m - 2 y x n+ m - 3 y 2
xm -1 A xm - 2 y A x m -3 y 2 A ym -1 A xn- 1 B xn- 2 y B x n- 3 y2 B
0
0
0
· · ·
xy n+ m - 2 y n+m-1
R(A, B)
yn-1 B
Res( A, B)xn+m- 1 on the left F1 (X, Y)A(X, Y) G1 (X, Y)B(X, Y) on the righthand side, where F1 and G 1 are homogeneous polynomials of degrees m- 1 and n- 1, respectively, whose coefficients are (complicated) polynomials in the coefficients of A and B. Similarly, the bottom entry shows that Res( A , B)Yn+m- 1 is equal to an expression of the form F2 (X, Y)A(X, Y) G2 (X, Y)B(X, Y).
Examining the top entry on each side, we find that hand side is equal to an expression of the form +
+
This completes the proof of part (c) of the proposition. Finally, we leave the proof of (b) as an exercise for the reader, or see [436, SecD tion 5.9].
We define the resultant of a rational map in terms of its defining pair of polyno mials.
1 -. lP'1 be a rational map defined over a field K with a lP' nonarchimedean absolute value I · lv · Write ¢ [F, G] using a pair of normalized homogeneous polynomials F, G E R[ X , Y]. The resultant of¢ is the quantity Res(¢) = Res(F, G). Since the pair ( F, G) is unique up to replacement by (uF, uG) for a unit u E R*, we see that Res(¢) is well-defined up to multiplication by the 2tfh-power of a unit. In particular, its valuation v(Res( ¢)) depends only on the map ¢. Definition. Let ¢
:
=
The resultant of a rational map ¢ provides an upper bound to the extent that ¢ is expanding in the chordal metric. In particular, a rational map is always Lipschitz with respect to the chordal metric, and if its resultant is a unit, then the map is non expanding. (See also Exercise 2 . 1 0.)
1 lP'1 be a rational map defined over afield K with a lP' nonarchimedean absolute value Then Theorem 2.14. Let ¢
:
->
I · lv ·
2.4. The Resultant of a Rational Map
57
[F(X, Y), G(X, Y)] F 1 , G1 , F2 , G2 R[X, Y] F1 (X, Y)F(X, Y) + G1 (X, Y)G(X, Y) = Res(¢)X2d- l , F2 (X, Y)F(X, Y) + G2 (X, Y)G(X, Y) = Res(¢)Y2d- 1 . Now let P = [x, y] E lP' 1 (K) be a point, which we assume written in normal ized form. We substitute [X, Y] = [x, y] into the first equation and use the nonar chimedean triangle inequality to compute I Res(¢)x2d- l l v = I F1 (x, y)F(x, y) + G1 (x, y)G(x, y) l v ::; max{ I F1 (x, y)F(x, y)l v , I G1 (x, y)G(x, Y) l v } ::; max{I Fl (x,y) l v , I Gl (x,y) l v } · max{I F (x,y) l v , I G(x,y) l v } ::; max{ I F (x, Y) l v , I G(x, Y) l v } ·
Proof Write ¢ = in normalized form. Proposition 2. l 3(c) says that there are homogeneous polynomials E satisfying
A similar calculation using the second equation gives the analogous estimate
I Res(¢)y2d- l l v ::; max{ I F(x, y) l v , I G(x, Y) l v } · Since P is normalized, i.e., max{l x l v , I Y i v } = 1, we find that (2.5) I Res(¢) l v ::; max{ I F(x, y)l v , I G(x, Y) l v } . Notice that this estimate bounds the extent to which F(x, y) and G(x, y ) can be simultaneously divisible by high powers of p. Returning to the proof ofthe theorem, we write P1 = [x 1 , Yl ], P2 = [x 2 , y2 ], and ¢ = [F(X, Y), G(X, Y)] in normalized form. Then the distance from P1 to P2 is while we can use the inequality (2.5) (applied to both P1 and P2 ) to estimate
F (xl , yl) l v , I G(x l , Ydl v } · max{ I Fl (x2 , Y2 ) 1 v , I G(x2 , Y2 ) l v } max{ I F (x1 I,yl)G(x 2 , Y2 ) - F(x2 , Y2 )G(x1 ,yl) v . I Res(¢)! ;
r (P).
P J>n (P) J>r J>m =
=
o
=
o. . . o ------- k iterations
The minimality of m implies that r = 0, and hence m divides n. This proves the assertion about periodic points. Similarly, if is preperiodic, say i (P) 2.6
1i (P)
P
(P),
0
Periodic Points and Good Reduction
Corollary 2.20 tells us that if ¢ has good reduction, then its periodic points reduce to periodic points of ¢. In this section we analyze the reduction map Per(¢) Per(¢) and use our results to study Per(¢). We start with the following theorem, which is an amalgamation of results due to Li (266], Morton-Silverman (3 12, 3 13], Narkiewicz [325], Pezda [355], and Zieve (454]. ____,
JP'1 ( JP'1 ( P lP'1 ( The exact period of P for the map ¢. The exact period of P for the map J;.
d
Theorem 2.21. Let ¢ : K) ____, K) be a rational fimction of degree � 2 defined over a localfield with a nonarchimedean absolute value I · I v · Assume that ¢ has good reduction, let E K) be a periodic point of¢, and define thefollowing quantities: n
m r
p
The order of >.1(P) unity.)
=
((/r)'(P) in k*. (Set r = oo if >.1(P) is not a root of
The characteristic ofthe residuefield k ofK.
Then n has one ofthefo/lowingforms: n=
m
or
n =
mr
or
n =
mrpe .
E/ E E( (
Remark 2.22. Let
K be an elliptic curve defined over a local field, and assume that has good reduction. Then one knows (41 0, VII.3. 1] that the reduction map K) E k) is injective except possibly on p-power torsion, where p is the char acteristic of the residue field k. This is very similar to the statement of Theorem 2.21 . In the case o felliptic curves, it is also possible to bound the power ofp in terms of the ramification index of p in K. We discuss below (Theorem 2.28) analogous bounds in the dynamical setting. ____,
Proof We make frequent use of Theorem 2. 18, which tells us that
P
Recall that we used this relation in Corollary 2.20 to prove that the ¢-period of is divisible by the ¢-period of which, in our current notation, says that m divides n.
P,
2.6. Periodic Points and Good Reduction
63
Replacing ¢ by ¢m and m by we are reduced to th� that is a fixed point of ¢. Having done this, we note that A¢(F) is equal to .n 1 (mod p), so we find that r n. If n = r, the proof is complete. Otherwise, we replace ¢ by ¢r and n In particular, since ¢n(o)
2
=
;:::
=
by njr. By an abuse of notation, we continue to write
2 ¢(z) = 11 + J..z + 1 +A(z) z zB(z) with the understanding that the values of 11, >., A(z), and B(z) may have changed. The principal effect of replacing ¢ by ¢r is that we are now in the situation that
2. Dynamics over Local Fields: Good Reduction
64
>. =
1
(mod p),
i.e., the new value of r is 1, which brings us to the second case that we need to consider. To recapitulate, we have a rational function ¢ satisfying ¢n(o) = 0,
J-l
= ¢(0) = 0 (mod p), and >. = ¢'(0) = 1 (mod p).
We are further assuming that ¢(0) -=1- 0 (otherwise, we are done), so (2.8) becomes +
+
n 1 >. >.2 + + >.n- l 0 (mod p). Thus n is divisible by p. We replace ¢ by ¢P and n by njp. If now ¢(0) = 0, we are done. If not, the same argument shows that n is again divisible by p. Repeating, we continue dividing n by p until finally we reach n = 1. This concludes the proof that the original period n has one of the forms n m or n = mr or n mrpe for some 1. 0 =
=
e
···
=
=
;::::
We have seen that maps with good reduction are nonexpanding. This implies that their periodic points are nonrepelling. If the reduction (/> is separable, we can say even more. (Recall that (/>(z) E is separable if it is not in See Exercise 1 . 10 for details.)
k(z) k(zP ). Corollary 2.23. Let ¢ IP'1 ----. IP'1 be a rational map that has good reduction. (a) Every periodic point of¢ is nonrepelling. :
(b) Ifthe reduction (/> is separable, then ¢ has onlyfinitely many attracting periodic points. Proof (a) Theorem 2. 1 8 tells us that ¢n has good reduction. Let P be a periodic point of¢ of exact period Using Lemma 2.5, we can make a change of coordinates so that P = [0, 1]. Then we can write ¢n (z) in normalized form as
n.
The fact that ¢n has good reduction implies that b0 E R*, since otherwise would be a common root of P and G. Hence
z=0
so I .Ap( ¢) lv :S 1, which shows that P is a nonrepelling point for ¢. (b) Again let P be a periodic point for ¢ of exact period and let m be the period of the reduced point P. Then we have equivalences
n,
2.6. Periodic Points and Good Reduction
65
P is attracting ¢:=::} 1Ap (¢) 1 v = i (¢n) ' ( P) I v < 1 ¢:=::} (¢n ) ' ( P) = 0 ¢:=::} (Jyn) ' (P) = 0
¢' (P) · ¢'(¢?) · ¢' ( ¢2 P) . . . ¢' (Jyn- l P) = o ¢:=::} ( ¢' (P) . ¢'(¢P ) . ¢' ( ¢2 P) . . . ¢' (Jym -1 P) ) n / m = O ¢:=::} (Jym ) ' (P)n / m = 0 ¢:=::} 0J, ( contains a critical point.
¢:=::}
P)
The fact that ¢ is separable implies that a version of the Ri�mann-Hurwitz formula (Theorem 1 . 1 ) is valid; see Exercise 1 . 1 0. Hence the map ¢_has finitely many (pre cisely, at most 2d - 2) critical points, and a fortiori, the map ¢ has only finitely many periodic orbits containing a critical point. In particular, there is a finite list of possible periods for Further, we know that the multiplier AJ, (P) = (Jym ) ' (P) of P is 0, so Theorem 2.21 tells us that n = m. There are thus only finitely many possibilities for the period of P, and since ¢ has finitely many points of any given period, we conclude that ¢ has only finitely many attracting periodic points. D
P.
Remark 2.24. The separability assumption in Proposition 2.23 is necessary, as is
z) zP , ¢(z) Z2 [z]
shown by the example ¢( = all of whose periodic points are attracting. See also Exercise 2. 16. Example 2.25. Let E be a polynomial of degree d 2:: 2 whose leading coefficient is a 2-adic unit. Then ¢ has good reduction. Let P E lP'1 be a periodic point of exact period n 2:: 2. In the notation of Theorem 2.21 , n = mr2e, where m is the period of P in lP'1 (JF2 ) and r is the order of AJ, ( P) in But 2 ) has only three points, and the fact that ¢ is a polynomial means that the point at infinity is not in the orbit of P, so either m = 1 or m = 2. Similarly, we note that !1:"2 has only one element, so r = 1 . It follows that n = 28 for some s 2: 0. Similarly, if E is a polynomial of degree d 2: 2 with leading coeffi cient a 3-adic unit, and if P E lP'1 is a periodic point of exact period n 2: 2, then we find that n = mr3e with 1 � m � 3 and 1 � r � 2. Thus n = 2t · 3u for some 0 � t � 2 and some u 2:: 0. be a polynomial of degree d 2: 2 whose leading coeffi Finally, let ¢(z) E cient is relatively prime to 6. Then ¢ has good reduction at 2 and 3, so the period n of a periodic point P E lP'1 ( satisfies both n = 28 and n = 2t · 3u with t � 2. This proves that n is either 1 , 2, or 4. The examples = with = 0 and - 1 with z = 0 show that n = and n = 2 are possible. Can you find an ¢( ) example with n = 4? See Exercise 2.20 for a stronger version of this example. The above example illustrates how the local result given in Theorem 2.21 can be used to derive strong bounds for the periods of periodic points defined over number fields by applying the theorem to two different primes. We now use the same argu ment to give a general result that, although not the strongest possible bound using these methods, is sufficient for many applications. z
=
z2
¢(z) Z3 [z] (Q ) 3 Z[z] Q)
( Q2 ) lF�. lP'1 (JF
1
¢(z) z2
z
2. Dynamics over Local Fields: Good Reduction
66
lP'1
Corollary 2.26. Let K be a numberfield, let ¢ : --+ lP'1 be a rational map defined over K, and let and be primes of K such that ¢ has good reduction at both and and such that the residue characteristics of and are distinct. Then the period n ofanyperiodic point of¢ in ( K) satisfies
p p q lP'1 n :::; (Np 2 - 1)(Nq 2 - 1), where Np and N q denote the norms of p and q respectively. In particular, the set Per(¢, K) of K-rational periodic points is finite. (For an q
p q
alternative proof of the finiteness of Per(¢, K) using the theory of height functions, see Theorem 3 . 1 2.) Proof Using the obvious notation, we have
mp = (period of J;(P) mod p) :::; #lP'1 (lFp ) = Np + 1, rp = (period of A¢ (P) mod p) :::; #lF; = Np - 1,
and similarly for m q and rq . Let p and q denote the residue characteristics of and respectively. Then Theorem 2.2 1 says that
p q,
Since p and q are distinct primes, it follows that
n :::; mp · rp · mq · rq
:::;
(Np + 1)(Np - 1)(Nq + 1)(Nq - 1),
which is the first part of the corollary. The finiteness of Per(¢, K) then follows from the fact that ¢ has good reduction at almost all primes of K and the fact that it has only finitely many periodic points of any given period n.
0
Remark 2.27. The bound for rational periodic points in Corollary 2.26 depends only
weakly on ¢ in the sense that the bound is solely in terms of the two smallest primes of good reduction for ¢. There are many results in the literature using local and/or global methods that describe bounds for rational periodic points that depend in var ious ways on the rational map. See for example [52, 87, 90, 9 1 , 1 62, 1 7 1 , 3 12, 325, 326, 328, 332, 353, 355, 358, 359, 361 , 454]. However, none of these arti cles achieves the uniformity predicted by a conjecture that we discuss in Chapter 3 (Conjecture 3 . 1 5). This conjecture asserts that for a number field K of degree D and a rational map ¢ E K(z) of degree 2 2, the number of K-rational preperiodic points of ¢ should be bounded solely in terms of D and If K is a discrete valuation ring of characteristic 0, then it is possible to bound the exponent e appearing in the formula n = mrpe in Theorem 2.2 1 .
d
d.
Theorem 2.28. (Zieve [454], see also Li [266] and Pezda [355]) We continue with the notation and assumptions from Theorem 2.2 1 . We further assume that K has characteristic 0 and we let v : K* _..,. be the normalized valuation on K. If the period n ofP E lP'1 K) has the form n = mrpe , then the exponent e satisfies
(
Z
2.6. Periodic Points and Good Reduction
p
67
e- 1 .(f-lak + f.12 vbk ) + Vf.12a� (mod f-l33 ) f.1(1 + >..ak ) + f-l2 v(a� + >..bk ) (mod f-l ). =
=
This yields the recurrences
and
a 1 = 1 b1 =
Starting from and 0, it is now a simple matter to find formulas for and and check them by induction. The end result is
bk
1' (0)
=
�(�A;) + i'2v (�A (� A;)') H-
..i . We know that p), and wethatare assuming that p is unramified in K, so >.. = 1 + cp for >..some1 (modR. Assuming -f:. 0, i.e., that >.. -f:. 1, we compute =
c
E
c
Note that the final congruence is true because every binomial coefficient (�) with ::; < p is divisible by (and we are assuming that -f:. 2). We also observe that the congruence is true even for >.. although the intermediate calculation is incorrect. We perform a similar calculation for the more complicated sum in (2. 1 1 ), but this time we are interested only in the value modulo so we can replace >.. by
1 i
p
p
= 1,
p,
1:
p
Note that for the last step we are using the assumption that 2: 5. Substituting (2. 12) and (2. 1 3) into the iteration formula (2. 1 1) (with k yields
(2. 1 3)
= p)
2.7. Periodic Points and Dynamical Units
69
¢1'(0) J-L(P + ap2 ) + J-L2 vbp (mod 1i), where a and b are in R. The fact that p is unramified in K means in particular that p divides so ¢1' (0) f-LP (mod f-Lp2 ). Now using our assumption that -1- 0, we deduce that ¢P (O) -1- 0, which completes the proof of the theorem. =
J-L,
=
J-L
2.7
(
D
Periodic Points and Dynamical Units
Let be a primitive
pth root of unity. Then the expression ()
(1 - (2 (1
is a unit in the cyclotomic field IQ( , a so-called "cyclotomic unit." Similarly, if and are roots of unity of relatively prime orders, then the difference is an algebraic unit. The crucial fact underlying these constructions is that distinct roots of unity remain distinct when they are reduced modulo primes. It follows that their differences are not divisible by any primes, and hence that they are units. Theorem 2.21 can be used to deduce conditions under which distinct periodic points remain distinct when reduced modulo primes, so it can be used to construct units in a similar fashion. These constructions can be done either using different points in a single periodic orbit or using points of different periods. The following proposition provides us with the information needed to construct units of various kinds. In fact, since Lemma 2.8 tells us that the v-adic chordal metric satisfies
(2
Pv (P, Q) < 1
P = Q,
the proposition actually says something stronger than the simple assertion that certain pairs of points have distinct reductions. Proposition 2.32. Let z) E K z ) be a rational function of degree d 2 2 with good reduction. K) be a point ofperiod for Then (a) Let E
¢(
(
P lP'1 ( n ¢. Pv (¢iP, ¢J P) = Pv (¢i+k p, ¢l+k P) for all i , j, k E Z, where for i < 0 we use the periodicity ¢n P = P to define .¢(F1 ). Then Theorem 2.21 tells us that both n1 and n2 are in the set {m, mr, mrp, mrp2 , mrp3, . . . }. It follows that either n1 divides n2 , or that n2 divides n1, contradicting the assump tion on n1 and n2 . Therefore Pv (Pl , P2 ) = 1. 0
2.7. Periodic Points and Dynamical Units
71
It is a simple matter to use Proposition 2.32 to construct units from periodic points. We begin with the easier case of a polynomial mapping.
z
Theorem 2.33. (Narkiewicz [325]) Let ¢(z) E R [ ] be a polynomial of degree whose leading coefficient is a unit in R. Let a E K be a periodic point of¢ of dexact 2: 2period n (with n 2: 2), and let i, j E be integers satisfYing (i - j, n = 1. 7l
Then
gcd
)
¢i a - ¢1a R* . ¢a - a Proof The assumption that ¢(z) is a polynomial with unit leading coefficient im plies that ¢ has good reduction, since ---- E
Res(aoXd + a1 Xd- I y + + adYd , Yd) ag. =
· · ·
We next observe that every periodic point of ¢ is integral over R, since it is a root of an equation ¢n ( z) - z 0. Finally, we note that the distance between integral points a , (3 E R is given by =
Pv (a , (3 ) = Pv ( [a , 1] , [(3 ,
1])
=
I a - f3iv ·
Using this formula and Proposition 2.32(b), we compute
I
i
¢ a - ¢1a ¢a - a v
I
=
l¢i a - ¢1a lv = Pv (¢i a , ¢1a) = Pv (¢a , a) i ¢a - a iv
1.
0
We create units from periodic points of arbitrary rational maps using a cross-ratio construction.
1 ]Yi P1 , P2 , P3 , P4 JID
Definition. Let E ( K), and choose homogeneous coordinates = [xi , for each point. The cross-ratio of is the quantity
Pi
P1 , P2 , P3 , P4
Notice that ( nates for the points.
K P1 , P2 , P3 , P4 ) is independent of the choice of homogeneous coordi
Theorem 2.34. (Morton-Silverman [3 1 3, Theorem 6.4(a)]) Let ¢ E K(z) be a rational map of degree 2 with good reduction. Let E ( K) be a periodic pointfor ¢ ofexact period and let i and j be integers satisfYing
d 2:n, P JID1 gcd (j, n ) gcd (i - 1 , n ) = gcd (i - j , n ) = 1. =
Then
2. Dynamics over Local Fields: Good Reduction
72
Proof Comparing the definition of the cross-ratio to the definition of the chordal metric, we see that
The assumptions on i and j and Proposition 2.32(b) tell us that
Hence
¢iP)pv (¢P,f! P) ' I (P ,�,P ,�,i p ,�,i P) I v PvPv (P,(P, ¢P)p v (¢' P, cpJ P) = 1
=
""
' '�'
' '�'
' '�'
0
which proves that the cross-ratio is a unit.
We can also construct units using periodic points of different periods. This may be compared with the two different types of cyclotomic units, and where ( indicates a primitive ensures that (m - (n is a unit. n
n1h root of unity and the condition gcd(m, n)
1
Theorem 2.35. Let ¢ E ( ) be a rational map of degree d :2: 2 with good reduc E f and f let tion. Let E Z be integers with be periodic points of exact periods and respectively, and write = in normalizedform. Then
n 1 , n2
K
z
1 (K) lP' n , P n n n , P 1 1 1 2 2 2 n 1 n2 Pi [xi , Yi]
Proof Since we have taken normalized homogeneous coordinates, the chordal met ric is given by
Pv (Pl ,P2 ) lx1Y2 -x2 Y1 I v · The assumptions on n 1 and n2 and Proposition 2.32(c) tell us that Pv (n, P2 ) and hence x 1 Y2 - X 2 Y1 is a unit. =
=
1,
0
Remark 2.36. For further information on the geometric and arithmetic properties of periodic points and the fields that they generate, see Sections 3.9, 3. 1 1, and 4. 1 -4.6. In some sense, periodic and preperiodic points play a role in arithmetic dynamics analogous to the role played by torsion points in the arithmetic of elliptic curves or abelian varieties, albeit the dynamical setting has considerably less in the way of helpful structure. = Example 2.37. We use the polynomial 1 to construct units. First we compute
¢(z) z2 +
2.7. Periodic Points and Dynamical Units
73
¢(z) - z = z2 - z + 1, ¢2 (z) - z = z4 + 2z2 - z + 2 = (z2 - z + 1)(z2 + z + 2), ¢3 (z) - z = z8 + 4z6 + 8z4 + 8z2 - z + 5 = (z2 - z + 1)(z6 + z5 + 4z4 + 3z3 + 7z2 + 4z + 5). It is not surprising that ¢(z) - z divides both ¢2 (z) - z and ¢3 (z) - z, since any root of the former is clearly a root of the latter two; cf. Exercise 1 . 19(a). We have Per�*(¢) = { 1 ± �} and where recall that Per�· ( ¢) denotes the set of points of exact period n for ¢. Further, the six roots of the polynomial constitute a set of the form
3 ( z) (a, 3¢ =2 =1
since we know that the roots of are permuted by into two periodic cycles of period In particular, the splitting field K (3) of is a Galois extension of K of degree at most 18. Applying Theorem 2.33 with i and j gives a unit
3.
u1 is a unit in R[a] by computing NK(a)jK (a2 + a + 1) = Res(3(z), z2 + z + 1) = 1. Similarly, we can apply Theorem 2.33 with i = 2 and j = 0 to obtain a second unit u2 = ¢¢2((a)a) --aa = a2 + a + 2. And replacing a with ¢(a) or with ¢2 (a) in u 1 and u2 gives four more units, u� = ¢(a) 2 + ¢(a) + 1 = a4 + 3a2 + 3, u� = ¢2 (a)2 + ¢2 (a) + 1 = a4 + 2a2 + a + 2, u; = ¢(a) 2 + ¢(a) + 2 = a4 + 3a2 + 4, u� = ¢2 (a)2 + ¢2 (a) + 2 = a4 + 2a2 + a + 3. (Note that in doing these computations, we make use of the fact that ¢3(a) = 0.) We can check that
74
Exercises
H � � � � � � �� a � �� � � � � m � computation yields This is not surprising, since one can check that the polynomial ci>3 (x ) is irreducible over Q and that all of its roots are complex. Thus the global field Q( is a totally complex extension field of degree 6, so its unit group has rank 2. We can use Theorem 2.35 to create more units. Taking n1 = 2 and n2 = 3 and letting 1 = � ( - 1 + ..;=7 ), we have
a)
for 0 :::; i :::; 1 and 0 :::; j
:::;
2.
a, 1) }.
The extension Q( /Q is totally complex of degree 12, so has five independent units. We leave as an exercise for the reader to compute the number of independent units in the set { ui ,j Exercises Section 2 .1 . The Nonarchimedean Chordal Metric
2.1. Let K be a field that is complete with respect to an absolute value v and let ¢( z) E be a polynomial. ThefilledJulia set K (¢) of¢ is the set K(¢)
=
K [z]
{a E K : j ¢n (o:) j v is bounded for n = 1, 2, 3, . . . } .
(Note that this definition applies only to polynomials, not to more general rational maps.) (a) Prove that the filled Julia set K( ¢) is a closed and bounded subset of K. Note that if K is locally compact, for example K =
representation.
Section 2.4. The Resultant of a Rational Map
2.5. (a) Let A(X, Y) = a0 X + a 1 Y be a linear polynomial and let B(X, Y) be an arbitrary homogeneous polynomial. Prove that Res(A, B) = B( -a 1 , a0 ). (b) Let A(X, Y) = aoX 2 + a1XY + a2 Y 2 and B(X, Y) = b0 X 2 + b 1 XY + b2 Y 2 be quadratic polynomials. Prove that
4 Res(A, B) = (2aob2 - a1b 1 + 2a2 bo) 2 - ( 4aoa2 - ai)(4bob2 - bi). (Of course, in characteristic 2 one needs to cancel formula!)
4 from both sides before using this
2.6. With notation as in the statement of Proposition 2.13, prove that the resultant of A and B is related to the roots of A and B by
n m
i=l
2.7. Let
j= 1
n i=l
m
j= l
A(X, Y) = aoXn + a 1 Xn - 1 Y + · · · + an - 1 xyn- 1 + an Yn , B(X, Y) = boX= + b1 X m - 1 Y + · · · + bm - 1 xy=- 1 + brnY=, be homogeneous polynomials of degrees n and and let a, {3, "'(, fl E K be arbitrary. (a) Prove that
m
Res(A(aX + f3Y, "(X + flY), B(aX + {3Y, "YX + flY) ) = (afl - f3"Y) mn Res(A(X, Y), B(X, Y) ) . (Hint. Use Proposition 2. 13(b).) (b) Suppose that m = n = d. Prove that
Res(aA(X, Y) + j3B(X, Y), "YA(X, Y) + fiB(X, Y) ) = (afl - f3"Y)d Res(A(X, Y), B(X, Y) ) . (Hint. Use Proposition 2.1 3(d).)
Exercises
76
(c) Continuing with the assumption that m = n = d, define new polynomials by the formu las
A*(X, Y) = M(aX + /3Y, "(X + 8Y) - f3B(aX + /3Y, 1X + 8Y), B*(X, Y) = -1A(aX + j3Y, "(X + 8Y) + aB(aX + /3Y, "(X + bY). Prove that
Res(A*, B*) = (a8 - f3"f ) d2 +d Res( A, B). (d) Suppose that m = n = d and that Res(A, B) =/= 0, so ¢> = [A, B] is a rational map ¢ IP'1 IP' 1 of degree d, and suppose further that a8 - /3"( =/= 0, so the map f = [aX + /3Y, 1X + 8Y] is a linear fractional transformation. Let ¢* = [A*, B * ]. Prove that ¢* = f - 1 o ¢ o f. Use (c) to prove that ¢* IP' 1 IP' 1 is a rational map of degree d. 2.8. Let f E PGL2 (K) be a linear fractional transformation, and write +b f(z) = az cz + d :
-->
:
-->
in normalized form. (a) Prove that Res(!) = ad - be. (b) Prove that
p ( J ( P ) , J ( Q ) ) :::; 1 Res(f) l - 1 p ( P, Q)
for all P, Q E IP' 1 (K).
(Notice that this strengthens Theorem 2. 14 for maps of degree 1 .)
2.9. Let f E PGL2 (K) be a linear fractional transformation, let cp (z) E K(z) be a rational map, and let ¢1 = r 1 0 ¢ 0 f. (a) IfRes(f) E R * , prove that Res(¢ f) = Res(¢). (b) If Res(!) is not a unit, find a formula or an inequality relating the valuations ofthe three quantities Res(!), Res(¢), and Res( ¢f). 2.10. Let ¢ : IP' 1 --> IP' 1 be a rational map defined over a field K with a nonarchimedean absolute value I · l v , and consider the statement
Pv ( ¢( P1 ), ¢( P2 ) ) Pv ( P1 , P2 ) P1 ,P2 Eil" ( K ) sup1
1
(2. 1 8)
PF IPz
(a) Prove that (2. 1 8) is true for the map cp(z) = azd , where a E R. This example shows that Theorem 2.14 cannot be improved in general. (b) Prove that (2. 1 8) is not true for the map
over the field K = Qp by computing both sides. (You may assume that p =/= 2.)
2.11. Let ¢ : IP' 1 ( q
-->
IP'1 (q be a rational map of degree d.
Exercises
77
(a) Prove that there is a constant C(d), depending only on d, such that (b) Find an explicit value for the constant C (d). Note that this exercise provides an archimedean counterpart to Theorem 2. 14, whose proof may, mutatis mutandis, be helpful in doing this exercise. Section 2.5. Rational Maps with Good Reduction
2.12. Let F(X, Y) and G(X, Y) be homogeneous polynomials of degree D, let j(X, Y) and g(X, Y) be homogeneous polynomials of degree d, and let
A(X, Y) = F( J (X, Y), g(X, Y))
and
B(X, Y) = G( J (X, Y), g(X, Y))
be their compositions. (a) Prove that the resultants satisfy
Res( A, B) = Res(F, G) d · Res(!, g) D
2
.
(b) Use (a) to give an alternative proof ofTheorem 2.1 8(b). (c) For any rational map ¢ : lP' 1 --+ lP'\ define Ov(¢) = v(Res(¢))/ deg(¢), so Ov (¢) is a kind of normalized resultant of ¢. Prove that Ov satisfies the composition formula
Ov (¢ o 1/;) = Ov(¢) + deg(¢)ov (1/J) . 2.13. Show that the good-reduction assumptions in Theorem 2 . 1 8 are necessary by construct ing the following counterexamples: (a) Find a rational map ¢ : lP'1 --+ lP' 1 , which will necessarily have bad reduction, and a point P E lP' 1 (K) such that i{P) -1- ¢(P). (b) �ratio�al �aps ¢ : lP'1 --+ lP' 1 and 1/J : lP'1 --+ lP' 1 such that ¢ has good reduction and ¢ 0 1/J -1- ¢ 0 1/J. (c) Same as (b), except now 7/J is required to have good reduction and ¢ is allowed to have bad reduction. 2.14. Let p 2: 5 be a prime and define rational maps 4 p2 z 2 + p p3 z + 1 ! (a) Prove that ¢ has bad reduction modulo p for all f E PGL2 (Qp). (N.B. We are allow ing f to have coefficients in Qp and/or the determinant of f to be divisible by p.) (b) Prove that 1/;1 has bad reduction modulo p for all f E PCL 2 (Qp) . (c) Prove that the composition 1/J o ¢ has good reduction at p.
¢(z) =
z 2 + p3 z p3 z 2 + p
and
1/;(z) =
2.15. Let ¢ : IP'1 --+ lP'1 be a rational map defined over K. (a) Prove that the map ¢ has good reduction if and only if there is an R-morphism c/JR : JP'k -+ JP'k whose restriction to the generic fiber is equal to the original map ¢ : lP' k --+ lP'k . (b) Assume that ¢ has good reduction. Prove that the reduction ¢ : lP'� --+ lP'� is the restric tion of ¢R to the special fiber oflP'k .
Exercises
78 2.16. Let K/QP be a p-adic field.
(a) Prove that every periodic point of the map rjJ(z) = zP is attracting. (b) More generally, suppose that 'lj;(z) E K(z) has good reduction, and let rjJ(z) = Prove that every periodic point o f rjJ i s attracting. (c) Is (b) true without the assumption that 'lj; has good reduction?
'lj;(zP ).
2.17. Let K/QP be a p-adic field and let rjJ(z) E K[z] be a polynomial with good reduction. Let a E K be a critical point of rjJ, i.e., ¢>' ( a) = 0, and suppose that a E Per( ( z) E
K ( z) of good reduction. 2.18. Let rjJ(z) E K(z) be a rational map of degree d � 2 and suppose that there is a point P E JID1 ( K) and an integer m � 1 such that the following limit exists: T = lim r/Jm ( P) . (2. 19) Prove that rjJm (T) = T, i.e., T E Perm (¢). (This is true for any complete field K, so for n
n �oo
example, it holds for Qp, 0 such that ai = aJ , which shows that a is a root of unity. 0 ifand only if is a root of unity. =
v,
n
n
89
3.2. Height Functions and Geometry
3.2
Height Functions and Geometry
PP
P.
The height of a point measures the arithmetic complexity of We now investigate how the height of changes when we map it to some other projective space. This will allow us to relate geometric properties of maps to the arithmetic information encapsulated by the height function. Remark 3.9. For ease of exposition we restrict attention to heights on projective space, but the reader should be aware that there is a general theory of height func tions on algebraic varieties due to Weil. Height functions provide a powerful tool for converting algebra-geometric relationships into number-theoretic relationships. Theorem 3. 1 1 below provides an example; it converts the geometric information that a map ¢ has degree d into the arithmetic information that ¢( has height that is more-or-less the rfh power of the height of A summary of the general theory of heights on varieties is given in Section 7.3; see [205, Part B] or [256, Chapters 3-5] for further details.
P)
P.
Definition. A rational map ofdegree d between projective spaces is a map
cp : JP'
JP'
N M ¢(P) [fo(P), . . . , fM (P)], where f0, . . . , fM E K[Xo, . . . , XN ] are homogeneous polynomials of degree d with no common factors. (The polynomial ring K[X0, . . . , XN ] is a unique factor ization domain, so it makes sense to talk about common factors.) The rational map ¢ is defined at P if at least one of the values f0(P), . . . , fM (P) is nonzero. The rational map ¢ is called a morphism if it is defined at every point oflP'N (K), or equivalently, if the only solution to the simultaneous equations fo (Xo , . . . , XN ) fM (Xo , . . . , XN ) 0 is the trivial solution Xo XN 0. If the polynomials f0, . . . , fN have coefficients in K, we say that ¢ is defined over K. Our goal is to relate the height of a point P to the height of its image ¢(P). To do this in general, we use an important theorem from algebraic geometry called the ___,
=
= ··· =
= ··· =
=
=
Nullstellensatz. Here Null =
zero,
stellen =
places,
satz =
theorem,
so the Nullstellensatz is a theorem that relates a function to the points at which it vanishes. We give a brief overview of some basic concepts from algebra and alge braic geometry that are needed to understand the statement of the Nullstellensatz. E However, we note that for rational maps in one variable (i.e., maps JP'1 ), the Nullstellensatz may be replaced by a simple argument based on ¢ the fact that the ring is a principal ideal domain; see Exercise 3.8. An ideal I in is homogeneous if it is generated by homogeneous polynomials. The radical ofan ideal I is the ideal :
-+
JP'1
K[XR[o,z.]. . , XN]
¢(z) K(z)
3. Dynamics over Global Fields
90 /J = f E
{ K[Xo, . . . , XN] : r E I for some n � 1}. (N.B. In this definition, the quantity r is the nth power off, not its nth iterate.) The algebraic set attached to a homogeneous ideal I is the set V(I) = { P E JP>N (K) : f(P) = 0 for all f E I}.
The Hilbert basis theorem [259, Chapter VI, Theorem 2. 1 ] implies that I is finitely generated, so V (I) is the set of simultaneous zeros of a finite collection of polyno mials. If V C is an algebraic set, the ideal attached to V is the ideal
pN ( K)
I(V) =
(
). X ] K[ X o, . . . , f N f(P) P
ideal generated by all homogeneous _ polynomials E such that = 0 for all E V
It is clear that if V = W, then I(V) = I(W). The converse is not quite true, since the algebraic set attached to the radical Vi is the same as the algebraic set attached to I. Theorem 3.10. (Hilbert's Nullstellensatz) Let I and J be homogeneous ideals prop erly contained in Then
K[Xo, . . . , XN]. V (I)
= V ( J)
ifand only if /I = /J.
Proof Suppose that VJ = v'J. Let E V(I) and f E J. Then E I for some � so = 0, so = 0. This is true for every f E J, so E V(J). This proves that V(J) c V(J), and the opposite inclusion follows by interchang ing the roles and I and J. Hence V(I) = V(J). This proves the trivial direction
n 1, r(P)
f(P) P
r P
of the theorem. For a proof of the nontrivial converse, see [198, 1. 1 .3A] or [259, Section X.2]. D
We are now ready to prove an important result that says that up to a scalar factor, a morphism of degree d causes the height to be raised to the power.
cfh
pN (K) JP>M (K)
Theorem 3.11. Let cjJ : be a morphism of degree d. Then there --+ are constants cl ' c2 > 0, depending on c/J, such that
for all
P E JP>N (K).
(In fact, the upper bound H (c/J ( is validfor rational maps provided :::; C2H ( we restrict attention to points at which cjJ is defined.)
P)) P
P)d
P = [xo, . . . , xN ] with coordi
Proof We begin with some notation. For any point nates in K and any absolute value v E MK , we write
(This assumes that we have fixed particular homogeneous coordinates for larly, we define the absolute value of a polynomial
P.) Simi
3.2. Height Functions and Geometry
91
f(xo, · · · ' XN ) = io,'""",iN aio . . iN xio . . . XNiN .. to be l f l v = t.o,max . . ,z N l ai0 . . iN l v , and if ¢ = [fo, . . . , fM ] is a collection of polynomials, we let l ¢1 v = max I !J iv · Notice that the height of a point P E IP'N ( K) may now be written in the compact form 1 /[K : and K.
PrePer(¢>, pN (K)) that
Conjecture 3.15. (Morton-Silverman [3 12]) Fix integers d 2 2, N 2 1, and D 2 1. There is a constant C(d, N, D) such thatfor all numberfields KjQ ofdegree __, at most D and all finite morphisms ¢> : ofdegree d defined over K,
pN pN # PrePer( ¢> , JIDN ( K ) ) :::; C(d, N, D). Remark 3.16. There are many results in the literature giving explicit bounds for the size of the sets PrePer ( ¢>, pN ( K) ) or Per ( ¢>, pN (K) ) in terms of ¢, especially in the case N 1. Some of these results use global methods, while others use a small prime of good (or at least not too bad) reduction for ¢. For example, we used local methods in Corollary 2.26 to give a weak bound for # Per(¢>, JID1 ( K)) . For further =
results, see [52, 87, 90, 9 1 , 92, 100, 101, 137, 162, 1 7 1 , 1 9 1 , 192, 193, 194, 195, 227, 33 1 , 265, 3 12, 325, 326, 328, 329, 330, 332, 353, 355, 358, 359, 361 , 454]. Remark 3.17. Very little is known about Conjecture 3. 1 5. Indeed, it is not known even in the simplest case (d, N, D) = 1 , 1), that is, for Q-rational points and denote the degree-2 maps on Specializing further, if we let quadratic map = + c, then the conjecture implies that
JID1 cPc(z) . z2
(2,
cPc : JID1 JID1 __,
Per(cPc, JID1 (Q) ) < but the best known upper bounds for # Per ( cPc, JID1 ( Q) ) depend on There are one-parameter families of c-values for which cPc(z) has a Q-rational periodic point of exact period 1, 2, or 3; see Exercise 3.9 and Example 4.9. And one can show that cPc cannot have Q-rational periodic points of exact period 4 or 5; see [ 1 7 1 , 309]. Poonen has conjectured that cPc cannot have any Q-rational periodic points of period greater than 3. Assuming this conjecture, he gives a complete de sup #
oo ,
c EIQI
c.
scription of all possible rational preperiodic structures for ¢c; see [361]. Remark 3 . 1 8. Another interesting collection of rational maps is the family
cPa,b(z) = az + -zb . These maps have the symmetry property cPa, b ( -z) -ct> a , b (z), i.e., conjugation by the map f ( z) z leaves them invariant. It is known that there are one-parameter families of these maps with a Q-rational periodic point of exact period 1 (in addition to the obvious fixed point at oo), 2, or 4, and that none of the maps cPa, b (z) has a Q rational periodic point of exact period 3. See [286] for details, and Examples 4.69 =
=
-
and 4. 71 and Exercises 4. 1 , 4.40, and 4.41 for additional properties of these maps. Remark 3.19. Conjecture 3.15 is an extremely strong uniformity conjecture. For example, if we consider only maps ¢> : of degree 4 defined over Q, then the assertion that # ¢>, :::; C for an absolute constant C immediately implies Mazur's theorem [292] that the torsion subgroup of an elliptic curve E jQ
1 JID1 JID 1 PrePer ( JID (Q)) __,
3.4. Canonical Heights and Dynamical Systems
97
E. To see this, we observe that Proposition 0.3 tells us Etors = PrePer ([2], E), and hence the associated Lattes map tPE , 2 described in Section 1 .6.3 satisfies X (Etors ) PrePer( tPE, 2 , lP' 1 ). Note that tPE, 2 has degree 4. In a similar manner, Conjecture 3 . 1 5 for maps of degree 4 on JP'1 over number fields implies Merel's theorem [297] that the size of the torsion subgroup of an ellip is bounded independently of that
=
tic curve over a number field is bounded solely in terms of the degree of the number field. Turning this argument around, Merel's theorem implies the uniform bounded ness conjecture for Lattes maps, i.e., for rational maps associated to elliptic curves; see Theorem 6.65. Lattes maps are the only nontrivial family of rational maps for which the uniform boundedness conjecture is currently known. In higher dimension, Fakhruddin [162] has shown that Conjecture 3.15 implies that there is a constant C(N, such that if is a number field of degree at most and if is an abelian variety of dimension N, then
D) K #A(K)tors C(N, D).
A/K
D
::;
He also shows that if Conjecture 3.15 is true over Q, then it is true for all number fields. 3.4
Canonical Heights and Dynamical Systems
It is obvious from the definition of the height that for all o: E Q.
(3. 1 1)
Notice that Theorem 3 . 1 1 applied to the particular map ¢( ) = gives the lessprecise statement = (3. 1 2) + 0(1). Clearly the exact formula (3. 1 1) is more attractive than the approximation (3. 1 2). It would be nice if we could modify the height in some way so that the general for mula (3. 1 2) from Theorem 3. 1 1 is true without the 0(1). It turns out that this can be done for each morphism ¢. To create these special heights, we follow a construction due originally to Tate.
z zd
h(rjy( P)) dh(P) h
Theorem 3.20. Let S be a set, let
d > 1 be a real number, and let and
befunctions satisfying =
h(¢(P)) dh(P) + 0(1)
for all
P E S.
3. Dynamics over Global Fields
98 Then the limit
(3. 13) exists and satisfies:
(a) (b)
h(P) = h(P) + 0 ( 1 ). h(¢(P)) dh(P).
Thefunction
=
h:S
->
IR is uniquely determined by the properties (a) and (b).
Proof We prove that the limit (3. 13) exists by proving that the sequence is Cauchy.
Let n > m 2: 0 be integers. We are given that there is a constant C such that
i h(¢(Q)) - dh(Q )i ::; c for all Q E S. (3. 14) We apply inequality (3. 14) with Q = cpi- 1 (P) to the telescoping sum t �i (h(¢i(P)) - dh(cpi- l (P))) I :n h(cpn(P)) - d� h(¢m(P)) I = t=m+ l ::; =m+ t �i ih(¢i(P)) - dh(¢i- l (P))i i l .
The inequality (3. 1 5) clearly implies that as m, n -> oo, so the sequence d-n h(¢n (P)) is Cauchy and the limit (3. 13) exists. In order to prove (a), we take m = 0 in (3. 1 5), which yields
Next we let n -> oo to obtain
which is (a) with an explicit value for the 0(1) constant. The proof of (b) is a simple computation using the definition of
h'
Finally, to prove uniqueness, suppose that and (b). Then the difference g = satisfies
h - h'
:
S
->
h,
IR also has properties (a)
3.4. Canonical Heights and Dynamical Systems
g (P)
=
g(¢(P)) dg (P).
and
0(1)
These formulas hold for all elements
99 =
P E S, so
for all n 2: 0.
dng (P) is bounded ash which can happen only h(P) h (P) , so is unique. g (P) N be a morphism of degree d 2: 2. The canonical height Definition. Let ¢ pN p function (associated to ¢) is the unique function In other words, the quantity if = 0. This proves that :
n -+ oo,
=
D
'
-+
satisfying
h¢ (P) h(P) + 0(1) and The existence and uniqueness of h¢ follow from Theorem 3.20 applied to the maps =
and
h
since Theorem 3. 1 1 tells us that ¢ and satisfy
h(¢(P)) dh(P) + 0(1) for all P E pN (Q). -n1h(¢n (P)) is not practical for Remark 3.2 1 . The definition h ¢ (P) ---. d lim oo n accurate numerical calculations. Thus even for P E JP> (Q), one would need to com pute the exact value of ¢n (P), whose coordinates have O(dn ) digits. A practical method for the numerical computation of h¢ ( P) to high accuracy uses the decom hq, is described in Sections 3.5 and 5.9. See in particular Exercise 5.29 for a detailed =
=
position of
as a sum of local heights or Green functions. This decomposition
description of the algorithm. The canonical height provides a useful arithmetic characterization of the prepe riodic points of¢.
pN pN be a morphism ofdegree d 2: 2 defined over Q P E PrePer(¢) ifand only if h¢ (P) 0. Proof If P is preperiodic, then the quantity h( ¢n ( P)) takes on only finitely many values, so it is clear that d- n h( ¢n ( P)) 0 as Now suppose that h¢ ( P) 0. Let K be a number field containing the coor dinates of P and the coefficients of ¢, i.e., P E pN (K) and ¢ is defined over K. Theorem 3.20 and the assumption h¢ (P) 0 imply that Theorem 3.22. Let ¢ : and let E (Q). Then
P jp>N
-+
=
-+
=
=
n -+ oo.
3. Dynamics over Global Fields
100
Thus the orbit
0¢ (P) = {P, cj;(P), ¢2 (P), ¢3 (P), . . . } lPN (K) c
is a set of bounded height, so it is finite from Theorem 3.7. Therefore P is a preperi D odic point for ¢. Remark 3.23. Further material on canonical heights in dynamics may be found in Sections 3.5, 5.9, and 7.4, as well as [16, 20, 23, 36, 38, 39, 87, 88, 89, 147, 159, 23 1 , 228, 230, 232, 234, 406, 409, 446, 453]. Theorem 3.22 is a generalization of Kronecker's theorem (Theorem 3.8), which says that 0 if and only if is a root of unity. Kronecker's theorem follows by applying Theorem 3.22 to the £ilh -power map = z d , whose canonical height is the ordinary height The fact that only roots of unity have height 0 leads naturally to the question and of how small a nonzero height can be. If we take the relation substitute in 21 /d , we find that 2 '
h( a) =
a
h.
a=
cj;(z)
h(ad ) = dh(a)
h(2 1fd ) = � h(2) = lo!
so the height can become arbitrarily small. However, this is possible only by taking numbers lying in fields of higher and higher degree. For any algebraic number let
a,
D(a) = [Q(a) : Q] denote the degree of its minimal polynomial over Q.
Conjecture 3.24. (Lehmer's Conjecture [264]) There is an absolute constant �
such that
h(a) 2: �1D(a) for every nonzero algebraic number a that is not a root of unity.
>
0
There has been a great deal of work on Lehmer's conjecture by many mathemati cians; see for example [7, 8, 75, 94, 264, 366, 421 , 422, 424, 445]. (The survey [422] contains an extensive bibliography.) The best result currently known, which is due to Dobrowolski [ 138], says that
� ( loglogD(a) ) 3 h(a) - D(a)- logD(a) The smallest known nonzero value of D(a)h(a) is D(f3o)h(f3o) 0.1623576 . . . , where (30 = 1.17628 . . . is a real root of x 1 0 x 9 - x7 - x6 - x5 - x4 - x3 x + 1 . Theorem 3.22 tells us that h¢ (P) 0 if and only if P is a preperiodic point for ¢. This suggests a natural generalization of Lehmer's conjecture to the dynamical >
=
+
+
=
setting. (See [3 1 7] for an early version of this conjecture in a special case.)
3.4. Canonical Heights and Dynamical Systems
101
N ----) NlP'N be a : JP> morphism defined over a number field K, and for any point P E lP' (K), let D(P) [K (P) : K]. Then there is a constant K K(K, ¢) > 0 such that for all P E lP'N ( K) with P � PrePer( ¢) . There has been considerable work on this conjecture for maps ¢ : lP' 1 ----) lP' 1 that
Conjecture 3.25. (Dynamical Lehmer Conjecture) Let ¢ =
=
are associated to groups as described in Section 1 .6. For example, in the case that ¢ is attached to an elliptic curve E, it is known that
in general [291], D(P) 3 log2 D(P) if j (E) is nonintegral [203], 2 D(P) K_ ( log log D( P) ) 3 if E has complex multiplication [263]. _ D(P) log D(P) Aside from maps associated to groups, there does not appear to be a single exam ple for which it is known that hq,(P) is always greater than a constant over a fixed power of D(P). Using trivial estimates based on the number of points of bounded height in projective space, it is easy to prove a lower bound that decreases faster than exponentially in D(P); see Exercise 3.17.
Remark 3.26. The Lehmer conjecture involves a single map ¢ and points from num ber fields of increasing size. Another natural question to ask about lower bounds for the canonical height involves fixing the field and letting the map ¢ vary. For ex ample, consider quadratic polynomials = + c as c varies over Q. Is it true that is uniformly bounded away from 0 for all c E Q and all nonpreperi odic E Q? In other words, does there exist a constant K > 0 such that
hq,Ja) a
¢c(z) K z2
hq,Ja) 2 K
a
for all c, E Q with
a � PrePer(¢c)?
We might even ask that the lower bound grow as c becomes larger (in an arithmetic sense). Thus is there a constant > 0 such that
hq,Ja) 2 Kh(c)
K
for all c, a E Q with a
� PrePer(¢c)?
This is a dynamical analogue of a conjecture for elliptic curves that is due to Serge Lang; see [202], [254, page 92], or [410, VIII.9.9]. For the quadratic map + the height of the parameter c provides a natural measure of its size, but the situation for general rational maps E is more complicated. We cannot simply use the height of the coefficients of ¢, because the canonical height is invariant under conjugation (see Exercise 3. 1 1 ), while the height of the coefficients is not conjugation-invariant. We return to this question in Sec tion 4. 1 1 after we have developed a way to measure the size of the conjugacy class of a rational map.
z2
c,
¢(z) K(z)
3. Dynamics over Global Fields
102 3.5
Local Canonical Heights
hq;,
The canonical height attached to a rational map ¢ is a useful tool in studying the arithmetic dynamics of ¢. For more refined analyses, it is helpful to decompose the canonical height as a sum of local canonical heights, one for each absolute value on K. In this section we briefly summarize the basic properties of local canonical heights, but we defer the proofs until Section 5.9. The reader wishing to proceed more rapidly to the main arithmetic results of this chapter may safely omit this sec tion on first reading, since the material covered is not used elsewhere in this book. The construction of the canonical height relies on the fact that the ordinary height satisfies + , so it is "almost canonical." The ordinary height of a point is equal to the sum
h(P¢(P)) = [a,=1]dh(P) 0(1) h(P) = h(a) = vEMK L nv logmax{ l a l v , 1},
so for each v E MK it is natural to define a local height function
Av(a) = log max{ l a l v , 1 } .
Av geometrically by observing that for v E M� , Av(a) = -logpv(a, ) where Pv is the nonarchimedean chordal metric defined in Section 2. 1 . One says that Av (a) is the logarithmic distance from a to Unfortunately, the function >-v does not transform canonically, since Av (¢(a)) is not equal to d>.v(a) + 0(1). To see why, note that >-v (¢(a)) is large if a is close to 1 . (Here the word a pole of ¢, while >-v (a) is large if a is close to the point lP' "close" means in terms of the v-adic chordal metric Pv· ) Thus we can hope to find a canonical local height only if we allow an appropriate correction term, as in the We can understand
oo ,
oo.
oo
E
following theorem.
1 lP'1 lP' d 2 = ( ( [ ] mon factors, and let v be an absolute value on K. Then there is a uniquefimction
Theorem 3.27. Let ¢ : ---+ be a rational function of degree � defined over K, write ¢( z ) F z ) / G z ) using polynomials F, G E K z having no com
with thefollowing two properties: (a) For all E with -=/:- oo and ¢(
a lP'1 (Rv ) a
(b) The function
a) -=/:-
oo,
thefunction
a f----+ 5..q;,,v (a) - logmax{ l a l v , 1} extends to a bounded continuous function on all of lP' 1 (Kv ).
5..q;,,v satisfies
(3.16)
3.5. Local Canonical Heights
Thefunction
103
�, v is called a local canonical height (associated) to ¢.
D
Proof We defer the proof until Section 5.9; see Theorem 5.60.
�,FIG. v
Remark 3.28. The local canonical heightfunction constructed in Theorem 3.27 depends on the choice of a decomposition of¢ as ¢ = Ifwe use instead cG for some E K*, so is replaced by then it is easy to see that the new function differs from the old one by a constant,
c
G
cG,
cFI
A<J>,v,cc(a) = A<J>,v,c (a) + d 1 1 log l cl v In the sequel, when we refer to � , v without further specification, we assume that some particular G has been fixed. However, we note that there are situations in which it may be convenient to normalize � , v differently for different see Exercise 3.29. The utility of local canonical heights is that on the one hand, they are defined on lP' 1 ( Kv ) and have various nice metrical and analytic properties, while on the other hand, they fit together to give the global canonical height, as described in the next A
A
_
v;
theorem.
�,v be the local {oo}. (3. 1 7)
Theorem 3.29. Let K be a numberfield, andfor each v E MK , let
canonical heightfunction constructed in Theorem 3.27. Then
hq,(a) = v L EMK nv �<J>,v (a)
for all
a
E
lP'1 (K)
"-
D
Proof We defer the proof until Section 5.9; see Theorem 5.6 1 .
[] Aq,,v (a) = }�� d1n logmax{ l ¢n (a) l v , 1 } . If is a nonarchimedean absolute value and if¢ has good reduction at or more precisely, if ¢ FIG with I Res( F, G) l v = 1, then the local height is given by the simple formula (Exercise 3.30) �q,,v (a) = log max{ l a l v , 1 } . In general there is no simple limit formula to compute � , v (a). However, there is an algorithm that leads to a rapidly convergent series for �q, ,v (a). Roughly, the N1h partial sum of the series approximates �q,, v (a) to within O(d- N ). For details see Exercise 5.29, which describes an efficient algorithm to compute the Green func tion 9q,, and then use Theorem 5.60, which says that the local canonical height and the Green function are related by the formula 5.,v ([x, yl) = 9¢,v (x, y) - log I Y i v ·
Remark 3.30. If cp(z) E K z is a polynomial, then the local height may be computed as the limit (Exercise 3.24) A
• .
v
=
v,
3. Dynamics over Global Fields
104
Remark 3.3 1. Baker and Rumely [26, Chapter 7] give an interesting interpreta tion of the local canonical height function. They construct an invariant measure on Berkovich space and prove that the measure is the negative Laplacian of a suitable extension of the local canonical height function See Section 5.10 for a brief introduction to the geometry/topology of Berkovich space. For the more elaborate machinery required to do harmonic and functional analysis on Berkovich space, in cluding the construction of the invariant measure, we refer the reader to [26, 29] and to the other references listed in Remark 5.77. Remark 3.32 The theory oflocal canonical heights began with Neron's construction on abelian varieties [333], or see [256, Chapter 1 1]. The general theory of global and local canonical heights associated to morphisms on varieties with eigendivisor classes is described in [88].
5-q,,
v.
.
3.6
Diophantine Approximation
The theory of Diophantine approximation seeks to answer the question of how closely one can approximate irrational numbers by rational numbers. The subject now includes a large and well-developed body of knowledge, while at the same time there is considerable ongoing research on many deep questions. In this section we state a famous theorem on Diophantine approximation and show how it is applied to deduce finiteness results for certain Diophantine equations. In Sections 3.7 and 3.8 we apply the theory of Diophantine approximation in a similar fashion to deduce integrality properties of points in orbits (a). It is clear that any irrational number a can be approximated arbitrarily closely by rational numbers, since Ql is dense in R The following elementary result quantifies this observation by relating the closeness of and a to the arithmetic complexity of the rational number We leave the proof as an exercise (Exer cise 3.3 1), or see any elementary number theory text that discusses Diophantine ap proximation.
0q, E .!Pi. xIy xly.
Proposition 3.33. (Dirichlet) Let a E .!Pi. "- Ql be an irrational number. Then there are infinitely many rational numbers satisfYing
xIy
Dirichlet's theorem says that every irrational number can be fairly well approx imated by rational numbers. Some irrational numbers can be much better approxi mated, but it turns out that this is not true for algebraic numbers. A succession of mathematicians (Liouville, Thue, Siegel, Gel'fond, Dyson) derived ever better esti mates for the approximability of algebraic numbers by rational numbers, culminat ing in the following deep theorem of Roth, for which he received the Fields Medal in 1958. Theorem 3.34. (Roth) Fix E > 0 and let a E Q be an algebraic number with a � Ql. Then there exists a constant = ( a) > 0 such that "'
"' c:,
3.6. Diophantine Approximation
105 X
y Q.
for all - E
Proof The proof of Roth's theorem is beyond the scope of this book. A nice expo sition may be found in [393]. For more general versions, see [205, Part D] and [256, Chapter 7]. 0 We apply Roth's theorem to prove an important result of Thue on the repre sentability of integers by binary forms.
Z[X, Y]Y) behasa homogeneous polynomial of Z. G(X, Y) G(X, at least three distinct roots (3. 1 8) G(X, Y) B has onlyfinitely many integer solutions (X, Y) E Z 2 . Proof We begin by factoring G(X, Y) into irreducible factors in Q[X , Y], say E degree and let B E Assume that in (C). Then the equation
Theorem 3.35. (Thue) Let
1lP' d
=
G(X, Y) cG(X, Y) G(X, Y) b)
c,
Replacing by and B by cB for a sufficiently large integer we may assume without loss of generality that have integer coefficients.' If r � 2, that is, if has more than one irreducible factor, then any solution ( a , to (3. 1 8) has the property that divide Since B and has only finitely many distinct factors, we are reduced in this case to showing that there are only finitely many solutions to the pair of simultaneous equations
G1 , . . . , Gr G1 (a, b) G2 (a, b)
B.
and G 1 = B1 and G2 B2 have no common components, so they intersect in only finitely many points. Next suppose that r = 1, so = If B is not equal to B�1 = B has no solutions. Otherwise, we can take for some integer B 1 , then roots of both sides and reduce to the case that is irreducible in Then has only simple roots, and by assumption there are at least three such roots. We factor in qx, say
This is clear, since the plane curves
1 (X, Y)e1• G(X, Y)G(X, Y) GG(X, Y) G(X, Y) G(X, Y) Y],
with distinct algebraic numbers by yd to obtain
=
Q[X, Y].
a1 , . . . , ad. Now divide the equation G(X, Y) b =
(3. 1 9) 1 0r we can invoke Gauss's lemma, which implies that the factorization of G(X, Y) in IQ[X, Y] can already be achieved in Z[X, Y].
3. Dynamics over Global Fields
106
We observe that if (x, E 'l} is a large solution to G(X, Y) = then the righthand side of (3 . 1 9) is small, so at least one of the factors on the lefthand side must also be small. However, since o: 1 , . . . , o:d are distinct, the rational number xly can be close to at most one of o: 1 , . . . , ad . Hence we can find a constant C = C ( G, such that
b,
y)
min 1 �i� d
��y -a:·t I -< !!_ IYi d
b)
for all (x,
y) E
'l}
satisfying G(x,
. , a:
y) = b.
(3.20)
In the other direction, Roth's Theorem 3.34 applied to each of o: 1 , . . . , o:d tells us that for any E > 0 there is a constant "' = "'( E, o: 1 , . . d ) > 0 such that
"' -- o:t I > IX l�i� . d y IYI 2+E ·
mm
X
for all - E Q. y
Combining (3.20) and (3.21), we find that
(3.2 1)
IYi d - 2 - E � cI"' ·
By assumption, 3, so this shows that there are only finitely many possible values for Finally, we observe that for each the equation G(x, implies that there are at most possible values for x. Therefore the equation G(x, has only finitely many integer solutions D
y.
d
d 2::
y,
y) = b y) = b
Siegel observed that Thue's theorem can be reformulated in terms of integral values of rational functions. Theorem 3.36. (Siegel) Let ¢(z) E Q(z) be a rationalfunction with at least three distinct poles in lP' 1 (C). Then
{o: E Q : ¢(o:) E Z} is a finite set. Remark 3.37. Note that Theorem 3.36 need not be true if the rational function ¢ has fewer than three poles. A simple example with two poles is the function ¢(z)
= (z2F(z) - D)d '
where D > 1 is a squarefree integer and F ( z) E Z [z] is a polynomial of degree If we now take any solution ( u, v) E Z2 to the Pell equation u2 - Dv2 = 1, then ¢( ulv) = v2 d F( ulv) E Z. The Pell equation has infinitely many solutions, so there are infinitely many rational numbers u v E Q with ¢( u v) E Z.
I
Proof Write
2d.
I
¢ [F(X, Y), G(X, Y)] using homogeneous polynomials F(X, Y), G(X, Y) E Z[X, Y] of degree d having no common factors. For any fraction o: E Q written in lowest terms, we have =
= alb
3.6. Diophantine Approximation -+. (
107
F(a,b) a ) '�' G(a,b)' so ¢ ( ) E Z if and only if G( a, b) divides F( a, b). Let R Res(F, G) be the resultant of F and G, which is a nonzero integer since F and G have no common factors. Proposition 2. 13 says that there are homo geneous polynomials h , 91 , h, 92 E Z [X, Y] satisfying d h (X, Y)F(X, Y) + 91 (X, Y)G(X, Y) RX 2 - 1 , h(X, Y)F(X, Y) + 92 (X, Y)G(X, Y) RY2d- 1 . Substituting (X, Y) (a, b) into these equations, we see that if G(a, b) divides F(a,b), then G(a,b) also divides both Ra2 d- 1 and Rb2d- 1 . However, a and b are relatively prime, so we have proven that ¢(a/b) E Z implies that G(a, b) divides R. It is important to emphasize that the resultant R depends only on F and G and that it is nonzero. Hence { � E Q : ¢ (�) E Z } DUR { � E Q : G(a, b) = D } . (3.22) =
a
=
= =
=
C
I
Thue's Theorem 3.35 tells us that each set on the righthand side of (3.22) is finite, which completes the proof of Theorem 3.36.
0
Remark 3.38. Roth's Theorem 3.34 is not effective, in the sense that it does not pro vide a method for computing an allowable constant 11, ( E, ) in terms of E and Thus our proofs of Thue's and Siegel's theorems (Theorems 3.35 and 3.36) are also inef fective. Baker's theorem on linear forms in logarithms gives an effective, although weak, version of Roth's theorem, from which one can derive effective versions of Theorems 3.35 and 3.36. This means that our finiteness result on integral points in or bits (Theorem 3.43) proven in the next section can be made effective, but the stronger integrality statement (Theorem 3.48) cannot, since it relies on the full strength of Roth's theorem. Let C be a smooth projective curve defined over Q and let ¢ : be a nonconstant rational function on If has genus 2: 1, Siegel proved that = {P E C(Q) : ¢(P) E is a finite set. For 2: 2, this is superseded by Faltings' theorem that C(Q) is finite, but for elliptic curves 1), the set is often infinite. In this case, Siegel greatly strengthened the qualitative statement that is finite by showing that the coordinates of the points in C(Q) have numerators and denominators of approximately the same size. The precise theorem, which we generalize to the dynamical setting in Section 3.8, is as follows.
C IP'N
C.p(Z) C.p(Z)
C. C Z}
a
a.
9 9
C --+ IP'1 C(Q)
(9
=
Theorem 3.39. (Siegel [403, 404]) Let E jQ be an elliptic curve, let ¢ E Q(E) be
a nonconstant rationalfunction on E, andfor each rational point P E E(Q), write
3. Dynamics over Global Fields
108
¢(P) = [a(P), b(P)] E P1 (Q)
Then
h(
a(P), b(P) E Z and gcd(a(P), b(P)) = 1. (P) I = 1 . lPEiEm(Q) loglogjj ab(P)j 0. For each v E S, extend v to in some fashion, and choose an algebraic number E Then there is a constant > 0, depending such that on S, E, and {
Theorem 3.40. (Roth) Let E
K,
K
av R.
av }vES· IT min {l z - avi v , l } nv vES
2::
K
z K.
for all E
HK (:)2+'
K[X, K.
Theorem 3.41. (Thue-Mahler) Let G(X, Y) E Y] be homogeneous of de gree d with at least three distinct roots in lP'1 (C), and let B E Then there are only finitely many (X, Y) E satisfying G(X, Y) = B. Theorem 3.42. (Siegel) Let ( ) E ( ) be a rationalfunction with at least three distinct poles in Then there are onlyfinitely many E satisfying )E
R�
R.
3. 7
¢z Kz
a K
¢(a Rs.
Integral Points in Orbits
In this section we prove that the orbit of a rational point by a rational map contains only finitely many integers, except in those cases in which that statement is clearly false. For ease of exposition, we work with and Z, but the result holds quite gen erally for rings of S-integers in number fields; see [41 1] or Exercise 3.38. We begin with a definition. Definition. A wandering point for a rational map is a point E whose forward orbit is an infinite set. Thus every point is either a wandering point or a preperiodic point.2
Q
Oq:,(P)
¢ !ID1 !ID1 :
-->
P !ID1
2 The reader should be aware that in topological dynamics, especially with respect to invertible maps, the standard definition says that a point is wandering if it has a wandering neighborhood. In other words, a point P is topologically wandering if there is a neighborhood U of P and an integer no such that ¢>i (U) n ¢>1 (U) 0 for all i > j 2 no. Our definition coincides with the topological definition if we use the discrete topology. =
109
3.7. Integral Points in Orbits Theorem 3.43. Let ¢( z) E Q( z) be a rational map ofdegree d
2: 2 with theproperty that ¢2 (z) tic Q[z]. Let o: E Q be a wandering pointfor ¢. Then the orbit
contains onlyfinitely many integer points. Theorem 3.43 is an immediate consequence of the following elementary geomet ric result combined with Siegel 's Theorem 3.36 concerning integral values of rational functions. Proposition 3.44. Let ¢ E q z) be a rational function of degree
d 2: 2 satisfying ¢2 ( z) tic C [z ], so Theorem I . 7 implies that no iterate of¢ is a polynomial map. Then
Ifd 2: 3, then #¢ -3 (oo) 2: 3. Proof We give a pictorial proof using the Riemann-Hurwitz formula. Suppose that ¢ is a rational map with #¢- 3 (oo) ::; 2. There are four possible pictures for the backward orbit of oo, as illustrated in Figure 3 . 1 . • •
• •
Q --+- P
--+-
-.....
__.,. --+-
--+-
•
--+-
--+-
Q Q'
•
•
Q --+- p
--......_ __.,.
p
Q --+- P Q'
--+-
P'
oo
(A)
--+-
00
(B)
--+-
00
(C)
00
(D)
--+-
--......_ __.,-
Figure 3 . 1 : Backward orbits containing few points The weak form of the Riemann-Hurwitz formula (Corollary 1 .3) says that
2d - 2
=
L (d - #¢ -1 (P)) .
(3.23)
PEIP'1
We apply (3.23) to each of the four pictures in Figure 3 . 1 . Before computing, we need to determine to what extent the points in the four pictures are distinct. For (A)
3. Dynamics over Global Fields
110
=
and (B), the three points P, Q, and oo must be distinct, since P oo would mean 2 that ¢ is a polynomial and Q oo i- P would mean that is a polynomial. Similarly, in (C) we cannot have oo, so relabeling the points in (C) and (D) if necessary, we may assume that P, Q, and oo are distinct in all four cases. We now use the Riemann-Hurwitz formula to estimate
{
=P =
¢
2d - 2 2': (d - #¢- 1 (oo) ) + (d- #¢- 1 (P)) + (d - #¢- 1 (Q)) (d - 1) + (d- 1) + (d- 1) = 3d - 3 in case (A), case (B), - (d(d -- 1)1) ++ (d(d -- 2)1) ++ (d(d -- 2)1) == 3d3d -- 44 inin case (C), (d - 2) + (d - 1) + (d - 1) = 3d- 4 in case (D). Thus case (A) yields d ::::; 1, a contradiction, while the other three cases give d ::::; 2. This proves that if d 2': 3, then #¢- 3 ( ) 3. Finally, if d = 2, then the Riemann Hurwitz formula tells us that ¢ has only two ramified points, and by inspection we see that those points already appear in the pictures for (B), (C), and (D). Since #¢- 3 ( oo) = 2 in each case, it follows that #¢ - 4 ( ) 3. (See Exercise 3.37 n for an explicit lower bound for #¢ - ( ) in terms of d and n.) 0 Proof of Theorem 3.43. Proposition 3.44 tells us that #¢-4( ) 3. (If d 2': 3, we could even take ¢- 3 .) The condition #¢-4(oo) 2': 3 can be rephrased as saying that _
oo
2':
oo
oo
2':
oo
2':
the rational function ¢4 has at least three distinct poles, so we may apply Siegel's Theorem 3.36 to conclude that the set (3.24) is finite. Consider the set of integers n such that the nth iterate of¢ applied to a is integral, It clearly suffices to prove that N is finite. If n E N with n 2': a
n- ( )
a
4, then
so we see that ¢ 4 o: is in the finite set (3.24). However, for any fixed {3 in the set (3.24), there is at most one iterate of a that is mapped to {3 (remember that a is a 0 wandering point). This completes the proof that n is finite.
Oq,(o:) Z
Theorem 3.43 says that if 2 ( z is not a polynomial, then the orbit con tains only finitely many integral points. It is natural to ask how large 0q, (a n can be. The answer is that it may be arbitrarily large, as shown by the following construction. Example 3.45. In order to create a rational function whose orbit contains a large number of integral points, we simply take a finite (reasonably random) sequence of integers z0 , z2 , Zk - l and treat the equations
¢ )
• • •
,
Oq,(o:) ) Z
111
3.7. Integral Points in Orbits
q,n+l (zn ) Zn+ l for n = 0, 1, =
. .
.,k- 1
as a system of homogeneous linear equations for the 2d + 2 coefficients of ¢. If the degree of ¢ satisfies 2d + 1 2: k, then there should be a nontrivial rational solution. We carry out this procedure with d = 2 to find the rational function
¢(z)
=
2
899x - 2002x + 275 33x2 - 584x 275
+
0
such that the orbit of contains quite a few integer points:
0 _i___. 1 _i___. 3 _i___. -2 _i___. 5 _i___. -7
Of course, as Theorem 3.43 predicts, the list of integer points must end, and indeed the next few iterates make it seem likely that there are no further integral points in the orbit ofO. Thus -
7
¢ 2912997 ¢
--+
29106319988831036
--+
19738601 187619760924349266091969763359
Example 3.45 (see also Exercise 3.41) shows how to construct orbits with many integer points by using rational maps of large degree. The following trick, adapted from work ofChowla [103] and Mahler [285] on elliptic curves (see also [405]), says that may be arbitrarily large even for rational maps of a fixed degree.
0¢ (a:) n Z Proposition 3.46. For all integers N 2: 0 and d 2: 2 there exists a rational map ¢( z) ¢E2 (z)Q( z)t/c with C[z].thefollowingproperties: 0 is a wandering pointfor ¢. 0, ¢(0), ¢2 (0) , ¢3 (0), , N (0) Z. Proof Let 'f/; ( z) E Q(z) be any rational map of degree d for which 0 is not a pre periodic point. For each 0 -::; n -::; N, write • •
.
•
.
E
.
¢(z) B'lf;(z/B). =
Hence
···
function q,n (z) BB'lf;nb0b(z/1B), sobNfor. Consider 0 -::; n -::;theNrational we have
as a fraction in lowest terms, and let = Clearly
q,n (o) E Z for all 0 -::; -::; N.
=
D
n
O¢ (a:)nZ
Although Proposition 3.46 shows how to make arbitrarily large, it is in some sense a cheat. What is really happening is that we are clearing the denominators of rational points in an orbit. We can use the following notion of minimal resultant to rule out this behavior.
3. Dynamics over Global Fields
112
Fq,, Gq, Z[z] 1. ¢(z)Fq, Ql(z),Gq, rj;(z) ¢(z) = Fq,(z)/Gq,(z), ± 1, Res(¢) = I Res(Fq,,Gq,) l .
Definition. For a rational map E write as where E have integer coefficients and the greatest common divisor of all of their coefficients is Then and are uniquely determined by ¢ up to multiplication by and we define the resultant of¢ to be the quantity
(See Section 2.4 for the definition and basic properties of the resultant of two polyno mials. See also Section 4. 1 1 for a general discussion of minimal resultants of rational maps over number fields.) We can use the resultant to rule out the denominator-clearing trick of Proposi tion 3.46, and having done so, we conjecture that the number of integer points in an orbit is bounded solely in terms of the degree of the map. (See Theorem 6. 70 for a special case.) This is a dynamical analogue of a conjecture of Lang [254, page 140]; see also [202, 407].
rj; ( z) E Ql(z) be a rational map ofdegree d 2: 2 with ¢2 (z) � 1 Ql[z], and let a E 1P' (Ql) be a wandering pointfor ¢. Assumefurther that ¢ is affine minimal in the sense that Res(¢) = /EPGL min Res(¢1). 2 (1Qi) Conjecture 3.47. Let
f (z)=az + b
(In other words, we cannot make the resultant smaller by conjugating by an affine linear transformation + b.) Then there is a constant C C (d) depending only on the degree of¢ such that
az
=
#(Oq,(a) Z) 5: C. n
3.8
Integrality Estimates for Points in Orbits
Oq,( a)
Theorem 3.43 says that, except for the obvious counterexamples, an orbit con tains only finitely many integer points. In this section we prove a dynamical analogue of Siegel's Theorem 3.39, which asserts that the numerators and denominators of cer tain rational numbers have approximately the same number of digits. Siegel's proof of Theorem 3.39 uses the existence of the multiplication-by-m maps E E on an elliptic curve. These finite unramified maps have the effect of significantly in creasing the height of points while leaving distances relatively unchanged. We adapt Siegel 's argument to the dynamical setting, but since our maps are on 1P'1, they are always ramified. This causes some additional complications, since distances shrink significantly near ramification points.
[m]
:
--+
map with the ¢2 (z) Ql[z] 1/¢2 (1/z)¢( z)� Ql[E zQl(]. Letz) abeEaQlrational be a wandering point
Theorem 3.48. (Silverman [41 1]) Let rj_ and
property that for ¢, and write
3.8. Integrality Estimates for Points in Orbits
1 13
as a fraction in lowest terms. Then (3.25) ln--->imoo loglog lIabnn Il 1. Remark 3.49. It is clear why we must assume that ¢2 (z) � Ql[z] in the statement of Theorem 3.48, but it may be less clear why we also require that 1/¢2 (11 z) � Ql[z]. Letting f(z) 11 z, we observe that 1/¢k (1lz) u - l 0 cpk 0 f)(z) u- l 0 ¢ 0 f) k (z) (cpf) k (z). So if 1/¢2 (1lz) E Z[z], then (¢f) 2n (z) E Z[z] for all 2: 1. Now consider the orbit of 1lb for integer b. We have =
=
=
=
=
n
a=
an
(¢f)2n (b)
a2n 1 b2n (¢f) 2n (b) Z. ¢(z) 1 I¢2 ( 1 Iz) cp(z) z + 1lz 1 cpn ( l ) an lbn
E Hence is an integer, so = and The quantity the limit in (3.25) for even values of n is 0. Thus the assertion of Theorem 3.48 is not true for rational maps such that is a polynomial. Example 3.50. To illustrate Theorem 3.48, we take and a = and list the first few values of in Table 3 . 1 . =
=
=
an) bn log( log(bn) 0 1 1 1 0.69315 2 1 5 2 2 1.60943 3 29 10 1.46239 4 290 1.20759 941 969581 272890 1.10128 5 6 264588959090 1.05110 1014556267661 7 1099331737522548368039021 268440386798659418988490 1.02613 an
n
(¢) for cp(z) z + 11z, writing ¢n (1) an Ibn . Notice how both the numerator and denominator of ¢n (l) grow extremely rapidly. Of course, the elementary height estimate (Theorem 3. 1 1) tells us that the maximum of l an l and I bn I grows this rapidly, but the fact that they both grow at ap proximately an equal rate lies much deeper and is the content of Theorem 3.48. And to illustrate the speed with which the fractions grow, even for a very simple map of degree 2 such as ¢(z) z + liz, here is the exact value of ¢9 (1), which is the last value that fits on one line using very small (5 point) type: Table 3. 1 : Orbit 0 1
=
=
=
3. Dynamics over Global Fields
1 14
1 726999038066943 72485 75086385863865042815392 793 76091034086485 11 2150 1213389899778415 73308941492781 377908 70974605039248 107160960958052 74361225692614241 3 1 1 1 204802346 7330784 739529329885668846964890 •
This ninth iterate has logarithmic ratio
log( log
a ) log(17269990380 . . . 492781) (bnn) = log(37790870974 . . . 964890)
�
1.00690.
Proof The idea underlying the proof of Theorem 3.48 is fairly simple. Choose some E > 0, and suppose that (3.26) l an l I bn I I+' for infinitely many 0. This means that cpn (a) = an /bn is very large, so cpn (a) is close to It follows that a is quite close to one of the points in the inverse image ¢ - n ( ) say a is close to (3 E ¢ - n ( ) But a is in Q, so one can hope that if is sufficiently large, then a and (3 are so close to one another that they contradict Roth's Theorem 3.34. Unfortunately, this naive approach does not work, because the point (3 depends on so the constant in Roth's theorem changes with each new value of A more sophisticated idea is to use a fixed (large) integer m and apply Roth's theorem to the rational point cpn -m (a) and a nearby point in ¢ - m ( ) Note that n ;=::
;=::
oo.
oo .
oo ,
n
n,
n.
oo .
cpn-m (a)
cpn (a).
so the height of is much smaller than the height of > 6/t: (we will see later why this is a good We fix an integer m satisfying choice for m) and we let (3 be the point in ¢ oo ) that is closest to How close is to It turns out that this depends on the ramification index of¢ at various points. If we make the (incorrect) assumption that ¢ is everywhere unramified, then ¢ preserves distances up to a scaling factor, which allows us to make the following estimates, where the constants . . . may depend on ¢, and m, but they do not depend on n:
dm -m (
(3 a?
cpn-m (a).
cl , c2 , a, bn I = l c/Jn (a) l - 1 � ;=: : I l an l an n C1 p(¢n (a), ) definition of p, since I cpn (a) I > 1, = CClp(cpp(c/Jn-m(a),(a),(3), cpm ((J)), since ¢m ((3) = assuming ¢ is unramified, 2n C3 l c/J - m (a) - fJI , definition of p, c4 3 ' Roth's Theorem 3.34 (with exponent 3), -> H(cpn-m(a)) Cs 3/d ' property ofheights (Theorem 3.1 1), n H(cp (a)) "' Cs �
�
�
�
oo ,
oo,
3.8. Integrality Estimates for Points in Orbits
115
since m satisfies
dm > 6IE.
Taking logarithms, we have proven that log i an
I -2 log(C5) :S:
for all n satisfying (3.26), i.e., ian
E
I 2: I bn II+' .
We reiterate that the constant C5 depends only on ¢, a, and m ; it does not depend on n. Hence if n satisfies (3.26), then an , and also bn , are bounded. It follows that the ¢-orbit of a contains only finitely many points satisfying (3.26), and therefore (3.27) h.n--->m supoo log1og ianl bn II :S: 1 + E. Repeating the argument with the rational map 1 /¢( 1 I z) and the initial point 1 I yields
a
(3.28) Since (3.27) and (3.28) hold for all > 0, we conclude that the limit in (3.25) is This would complete the proof of Theorem 3.48 except for the unfortunate fact that rational maps IP'1 IP'1 are always ramified. Further, our proof sketch made no mention of the assumption that ¢2 ( z) is not a polynomial, and the theorem is false for polynomials! In order to fix the proof, we begin by studying how ramification affects the distance between points.
E
1.
---->
Lemma 3.51. Let ¢ p : JP>1 ( q x JP>1 ( q
for Q E IP'1 (C), let
:
IP'1 (C) IP'1 (C) be a rational map of degree d let IR be the chordal metric as defined in Section 1 . 1, and
2: 2,
---->
---->
eq
( ¢)
=
max
Q'E¢-l (Q)
eq' (
¢)
(3.29)
be the maximum ofthe ramification indices ofthe points in the inverse image of Q. Then there is a constant C = C( ¢, Q), depending on ¢ and Q, such that
min
Q'E ¢ -l (Q)
p(P, Q ' )eq ()
:S:
Cp ( ¢(P), Q)
for all P E IP'1 (C).
(3.30)
In other words, if ¢(P) is close to Q, then there is a point in the inverse image ofQ that is close to but ramification affects how close.
P,
Proof We dehomogenize using a parameter z such that Q -1 oo and oo � ¢- 1 (Q). This means that we can write Q = f3 and P = a and that we are looking for the (3' E ¢- 1 (/3) that is closest to a . Writing ¢(z) F(z)IG(z) as a ratio ofpolynomials, the set ¢- 1 (/3) is precisely the set of roots of the polynomial F ( z) - f3G ( z). If we factor this polynomial over C as =
3. Dynamics over Global Fields
116
/31 , . . . , f3r , /3i E ¢- 1 (/3) has ramification index ei . For notational eQ (¢) maxei . We may assume that ¢(P) is quite close to Q, i.e., that p( ¢(P), Q) is small, since otherwise, the fact that the chordal metric satisfies p � 1 lets us choose a C for which the inequality (3.30) is true. In particular, P =I and writing a z(P), we may assume that a is quite close to at least one of the points f3k in ¢- 1 (/3). For example, we may require that a satisfy with distinct then convenience, we write
=
e =
oo,
=
and This implies in particular that
i
for all =I k. Hence ••
I F(a) - /3G(a)i l b ll a - f31 1e1 l a - f32 le2 · I a - f3r ler l b ll a - f3k l ek i#II I a - /3i l ei ::;:: l b ll a - f3k l ek if'IIk � l f3k - /3i l ei C1 l a - f3k l ek , where the constant C1 is positive and depends only on ¢ and /3. Further, the exponent satisfies e k � so we obtain the estimate =
=
=
e,
G(a)¢(a) F (a)/G(a)
/3
The fact that is close to and the assumption that = implies that is bounded away from 0, so dividing by yields
G(a)
G (/3) =I 0
This in turn implies the same estimate for the chordal metric, since we can estimate using and we can estimate using Therefore
l ¢(a) l
l/31
lal
l !3k l ·
0
Next we show that if we stay away from totally ramified periodic points, then iteration of ¢ tends to spread out the ramification.
3.8. Integrality Estimates for Points in Orbits
117
lP'1 lP'1 be a rational map ofdegree at least 2 and let Q E 1P'1 be a point such that Q is not a totally ramifiedfixedpoint of q}. Then lim eQ ( qr) 0. m-+oo (deg 0
where as usual depends only on ¢ and m. We are now ready to repeat the calculation from page 1 14, but this time done rigorously to account for the fact that has ramification. As usual, all constants may depend on ¢ and m, but are independent of n:
¢
1
i an l ' 2: p(¢n (a),oo) Wn) from (3.34), 2: C3 p(¢n -m (a),f3t "" fro m (3.3 6), ) C4 j ¢n-m (a) - (Jj e"" ( - H(¢n-m(a)) 3eoco Theorem 3.1 1, which says that >- H(¢n (a))c63eoco (¢m)/dm from h(P) d-m h(¢m (P)) + 0(1), >- H(¢nc6(a)) '/2 from (3.3 5), c6 l an l ' / 2 Hence i an I :S ci 1 ' is bounded for n E N(¢, a, (3), and the same is true of l bn l since l bn i H ' l an l , so ¢n (a) takes on only finitely many values for 2:
=
:S
E,
121
3.8. Integrality Estimates for Points in Orbits
E N(N(¢,¢,a,a,
a oo.
E , /3). But by assumption, is a wandering point for ¢, so we conclude that E, (3) is finite in the case that (3 ¥Suppose now that (3 Then a similar, but elementary, argument not requiring Roth 's theorem yields
n
=
oo.
from (3.34), from (3.36), definition of p, definition of height, where note that Cs m ) >- H(q;n -m(a) boon-mis periodic, ::/- 0, since a is wandering and too ( >- H (q;n (a)Cgroo (m J/dm from Theorem 3.1 1 , Cg :>- -H(q;---;-: from (3.35), n (a)r/6 Cg ai n i '/6 As above, we conclude that an and bn are bounded, and hence that N( ¢,a, oo) is finite. We have now proven that N(¢, a, (3) is finite for all (3 E q; - m (oo), and hence that N( ¢,a, ) is finite. The finiteness of N( ¢,a, ) is equivalent to the statement that log--':l an--'--- l 1 is Galois equidistributed with respect to if the sequence of measures conve�ges weakly3 to '-----'
=
p
=
u.
QEC(P/K)
Oq
p,
/-tP,
1-l ·
p,
We are now ready to state a dynamical equidistribution conjecture for Galois orbits of preperiodic points on and more generally for points of small height.
lP'N , Conjecture 3.60. (Dynamical Galois Equidistribution Conjecture) Let K/Q be a N be a morphism ofdegree d � 2 defined over K, and numberfield, let ¢ lP'N lP' N let P1 , P2 , P3 , . . . E lP' (K) be a sequence of distinct points such that no infinite subsequence lies entirely within a preperiodic subvariety of lP'N . (a) If H , P2 , P3 , . . . E PrePer( ¢) , then the sequence { Pi } i ;::: l is Galois equidis tributed in lP'N (C) with respect to the canonical ¢-invariant probability mea sure 1-l¢ · (b) /f limi __. oo h,p(Pi ) 0, then the sequence {Pi } i ;::: l is Galois equidistributed in lP'N (C) with respect to the canonical ¢-invariant probability measure It¢ · :
--->
=
It is clear that Conjecture 3.60(b) implies Conjecture 3.60(a), since preperiodic points have canonical height equal to 0. A version of the conjecture is known pro vided that the sequence of points satisfies a somewhat stronger Zariski density con dition as in the following theorem.
3Recall that a sequence of measures J.L; on a compact space X converges weakly to J.L if for every Borel-measurable set U, the sequence of values J.L;(U) converges to J.L(U) as i ---+ oo.
129
3.1 1. Ramification and Units in Dynatomic Fields
Theorem 3.61. (Yuan [450]) Let ¢ : JIDN ----+ JIDN be a morphism of degree d 2: 2 defined over and let . . . E JIDN ( K) be a sequence ofpoints satisfying the following two conditions: (a) Every infinite subsequence of is Zariski dense in JIDN . (b) ----+ 0 as i ----+ oo. (In the terminology of [453], sequences with property (a) are called generic and sequences with property (b) are called small.) Then the sequence is Ga lois equidistributed with respect to the canonical ¢-invariantprobability measure p,q, on JIDN (C).
K
P1 , P2 , P3 ,
hq,(Pi )
{Pi } i>l
{Pi } i> l
Proof The proof is beyond the scope of this book. See Yuan [450] for a gen
eral version over archimedean and nonarchimedean base fields and algebraic dy namical systems on arbitrary projective varieties. Earlier results and generalizations are given by Autissier [15, 16], Baker-Ih [24], Baker-Rumely [28], Chambert Loir [98], Chambert-Loir-Thuillier [99], Favre-Rivera-Letelier [169] and Szpiro 0 Ullmo-Zhang [432]. The classical Bogomolov conjecture, which says that sets of points of small height on abelian varieties lie on translates of abelian subvarieties, was proven by Ullmo [435] and Zhang [452]. We state a dynamical analogue. JIDN be a Conjecture 3.62. (Dynamical Bogomolov Conjecture) Let ¢ : JIDN morphism of degree d 2: 2 defined over a numberfield and let X C JIDN be an irreducible subvariety that is not preperiodic. Then there is an > 0 such that the
K
set is not Zariski dense in X.
----+
E
{P E X (K) : hq,(P) < c}
Notice that Conjecture 3.62 implies Conjecture 3.58, since the set of points with < includes all of the preperiodic points. Finally, in closing this section, we mention that canonical invariant measures have been constructed on Berkovich spaces; see Remark 5.77 and the references listed there.
hq,(P)
3.1 1
E
Ramification and Units in Dynatomic Fields
Kq,,n
In Section 3.9 we used periodic points to construct field extensions and studied their Galois groups. We now take up the question of the arithmetic properties of these algebraic number fields. In general, the three basic questions that one would like to answer about a given number field are these: Where is it ramified? What is its ideal class group? What are its units? In addition, one wants to know how the Galois group acts on ideal classes and on units. In this section we provide partial answers to the question of ramification and units for dynatomic fields. We first recall the classical case of cyclotomic extensions, which provide a model for the dynatomic theory. Let ( is a primitive root of unity and E Gal(Q/Ql). Then
n1h
a
130
3. Dynamics over Global Fields
u(
() (j(a) =
for a unique j ( u)
This defines an isomorphism
j : Gal (Q(()/Q)
____,
E ('ll/n'll) * .
( 'll/ n'll ) * ,
u � j (u) ,
expressing the action of u on Q( () as a polynomial action ¢( Now let p be a prime not dividing let p be a prime ofQ(() lying above p, and let Gal(Ql/Q) be the corresponding Frobenius element. The definition of Frobenius says that
z) zl. E =
n,
O" p
(3.44)
nth
However, the roots of unity remain distinct when reduced modulo p, so the con gruence (3.44) implies an equality in Q((), O" p
( ()
=
(P .
This exact determination of the action of Frobenius as a polynomial map on certain generating elements ofQ(() is of fundamental importance in the study of cyclotomic fields. To some extent, we can carry over the analysis of Q( () to dynatomic fields, al though the final results are not as complete as in the classical case. Let p be a prime ideal of the ring of integers of let p be the residue characteristic of p, and let q be the norm of p. We assume throughout that p f Choose a prime ideal s,p in lying above p. Assuming that p does not ramify in the associ ated Frobenius element uP is determined by the condition
K�,n '
K�,n (P)
up (a)
=
n. K� (P), ,n
aq
K�,n (P).
(mod !,p)
for all a in the ring of integers of On the other hand, by construction the action of uP on E Per�* ¢) is given by by formula
P
(
next proposition allows us to characterize the primes at which the extension K�,The (P)/ n K�,n may be ramified. Proposition 3.63. Let K be a number field, let ¢ E K ( z) be a rational map of let P E Per�* ( ¢, K) be a point in IP'1 ( K) of exact period n, and let p degree d be a prime of K satisfying thefollowing three conditions: 2 2,
¢ has good reduction at p. p does not divide n. then p does not divide
If>.cf>(P) =1- 1, A¢ (P) - 1. Then P mod p has exact period n. In particular, the set { p P mod p has period strictly less than n } is a finite set ofprimes ofK. :
(3.45) (3.46) (3.47)
3.11. Ramification and Units in Dynatomic Fields
131
Proof Let p be a prime of satisfying (3.45), (3.46), and (3.47)0 Let m be the exact period of P and let r be the order of A¢(F) in lF�o Theorem 2o21 tells us that either n = m or n = mr, since (3.46) rules out powers of p appearing in no If = 1, then also A¢(F) = 1, so r = 1 and n = mo On the other hand, if i= 1, then (3.47) tells us that
K
A.q,(P) A.q,(P)
But A J, (Pt = i by definition of r, so cannot equal ro Hence n = m m all cases, which shows that P mod p has exact period n for all primes p satisfy ing (3.45), (3.46), and (3.47)0 This proves the first part of the proposition, and since each of the three conditions is satisfied for all but finitely many primes, the second � � �R D
n/m
be a number field, let c/>( E ) be a rational map of degree 2, and let be the n1h dynatomic field for c/>o Let be the set of primes p of such that either c/> has bad reduction at p or p divides n or p divides the quantity - 1) 0 (3.48) II Corollary 3.64. Let
d 2: K
KK n,
z) K(z
PEPer�* ( ) >.q, (P)# l
S
(>..q, (P)
Kn ,q,/K is unramified outside ofSo Proof The field extension Kn , / K is generated by the points of Per�* (c/>)0 Propo sition 3063 tells us that if p � S, then those points remain distinct when reduced modulo primes lying over po Hence the extension Kn , / K is unramified at po Example 30650 We continue studying the quadratic polynomial c/> (z) z2 1 from Example 3055 (see also Example 2037)0 The polynomial c/> (z) has everywhere good reduction, so if we assume for the moment that none of its periodic points have Then
D
=
+
multiplier equal to 1, then the quantity (3.48) in Corollary 3 0 64 is equal to
II
=
II
((c/>n )'(a) - 1) = Res(c/>� (z), (c/>n )'(z) - 1) 0 Denoting this resultant by b.n ( c/>), it is not hard to compute the values of b.n ( c/>) for small values of no We obtain b.2 (cP) = 72 , b.3 (c/>) = (33 0 1 1) 3 , 4 b.4 (c/>) = (32 0 1 1 ° 13 ° 41) , b.5 ( cP) = (334 0 7 83 331 140869)5' b.6 (c/>) (3 0 5 ° 7 23 ° 73 ° 223 ° 2251 ° 347495839) 6 0 �(a) = O
(A.q,(a) - 1)
O andv(y) > 0} {v E M� : v(x) < O or v(y) < 0}, and similarly for Sq. Then xy' - x'y is an (S¢ Sp U Sq)-unit in the compositum K , m K, n · Theorems 3.66, 3.67, and 3.68 allow us to construct many S-units in dynatomic fields K, n and their composita K, m K, n · Further, since Gal(K¢, n / K) frequently contains an element whose action on points P E Per�* ( ¢) is characterized by the equation CJ¢ (P) ¢(P), we obtain a partial description of the action of the Galois group on the dynamical units. For example, the units in Theorem 3.67 transform via Sp
U
=
U
CJ1>
=
We compare this to the construction of cyclotomic units in cyclotomic fields. Let be a prime power and let ( be a primitive root of unity. Then the cyclotomic field ( contains the units
q1h
qQ( )
for 2 :::; The Galois group
i q ; 1 with gcd(i, q). :=:;
Gal(Q(()/Q) is the set of elements characterized by (() ( with 0 < t < and p f t. CJt
CJt
=
q
t
The action of the Galois group on the cyclotomic units is given by the explicit for mula (it 1 = . 1 (- 1 ( Further, the cyclotomic units generate a subgroup of finite index in the full group of units and the index of this subgroup is related to the class number ofQ ( ( ) . The situation for dynatomic fields is not nearly as complete. One problem is that the dynatomic fields tend to have very large degree over so the dynamical unit theorems cannot produce enough units to give a subgroup of finite index in the full unit group. Further, the Galois group is usually huge, and we have However, an explicit description only of the subgroup generated by the element since for general number fields there is no known way to systematically produce any units with any explicit Galois action, the dynatomic construction might be said to fall under the heading of "half a loaf is better than none." (J
t
Z[(]*,
K,n
(�)
-
t -
Gal(K¢,n /K)
K,
CJ4>.
134
3. Dynamics over Global Fields
Example 3.69. Consider the rational map
After some algebra, we find that
¢(z) = z2 - 4.
¢( z) - z = z2 - z - 4, ¢2 (z) - z = z2 + z - 3, ¢(z) - z ¢3 (z)'----:-¢(z) ---:-'--- zz = (z3 - z2 - 6z + 7)(z3 + 2z2 - 3z - 5). It is not hard to check that ¢( z) -z divides ¢n ( z)-z for all 1 (Exercise 1 . 1 9(a)), but the further factorization of ¢3 ( z) - z into a product of cubics is less common; see Exercise 3.49. If we let a, (3 E C satisfy a3 - a2 - 6a + 7 = 0 and n 2
then we have
Peri* (¢) = { 1 2vTI}, Per;*(¢) = { -1 � vTI }, Peri*(¢) = {a,¢(a),¢2 (a),(3,¢((3),¢2 ((3) } , where we recall that Per�* ( ¢) denotes the set of points of exact period for ¢. Let K IQ(a). The polynomial z 3 - z 2 - 6z + 7 is irreducible over IQ, but it factors completely in K since its roots are a, ¢(a), and ¢2 (a). Hence K is Galois over IQ with Galois group generated by the map determined by CY(a) = ¢(a). Notice that the discriminant of z 3 - z 2 - 6z + 7 is 19 2 , again confirming that its roots generate a cyclic cubic Galois extension. Further, Z[a] must be the full ring of integers RK of K, since ±
n
=
rY
IQ, so Disc(K/IQ) > 1.) Then the fact that 19 is prime forces RKWe= Z[apply a]. Theorem 3.66 with i = 2 and j = 1 to obtain the unit
(Note that K =1-
NK;Q (ui) = 1, confirming that u1 is a unit. Similarly, taking
It is easy to check that i and j 0 gives the unit
=2
=
Exercises
135
NK;Q (u2 ) =
In this case, -1. The field K is totally real of degree so its unit group RK: has rank 2. It is not hard to see that the two units and are independent, so they at least generate a subgroup of finite index. (In fact, { generates the full unit group RK:, but we leave the verification of this fact to the reader.) We can compute the action of the Galois group on and
3, u1 -1,u2u , u } 1 2
u1 u2 , a(ul ) = a(a2 + a - 4) = ¢(a) 2 + ¢(a) - 4 = a4 - 7a2 + 8 = -a + 1, a(u2 ) = a(a2 + a -3) = ¢(a) 2 + ¢(a) - 3 = a4 - 7a2 + 9 = -a + 2.
These new units are related to the original units by and
points of period 2 and 3 for ¢, we can create units in larger fields. Thus let Lwith=Using K( v'l3 ) Then L contains both a and the points in Per� * (¢), so Theorem 3.68 n1 = 2 and n2 = 3 says that u3 = 1 - a with r = -1 +2 v'l3 is a unit in the ring of integers of L. If we take the norm from L down to K, we find that N L K (u3 ) = u2 is one of the units that we already discovered. Similarly, - ¢(a))we compute = -a + 2.the norm of from L down to Q('y), we find using NL/IfK ('yinstead u3 12 + 1 - 3 = 0 that v'l3 3 2 3 = 7 = . 1 1 + = 6r a) ( -r+ NL/Qhl 'Y 2 .
----
1
-�
This unit and - 1 generate the unit group of the ring of integers of Q( v'13). For additional information about cyclic cubic extensions generated by periodic points of polynomials, see [306), and for a general analysis of units generated by 3-periodic points of quadratic polynomials, see [3 13, Section 8]. Exercises Section 3. 1 . Height Functions
3.1. Show that the constant C(d, N, D) in Conjecture 3 . 1 5 must depend on each of the quan tities d, N, and D by giving a counterexample if any one of them is dropped.
3.2. Let
v(B) = #{ P E lP'N (Q) : H (P) :S: B}.
(a) Find positive constants c 1 and c2 such that
c1 B N + l
:S:
v(B) :::; c2 B N + l
for all B � 1 .
136
Exercises
(b) For N = 1, prove that
v(B) = 12 . lim 71'2 B�= B2 (c) More generally, prove that the limit limB�= v(B)/ B N+ 1 exists and express it in terms of a value ofthe Riemann (-function.
3.3. Prove that
2
+ # { P E lP'N (Q) H (P) ::; B and D(P) ::; D } ::; (1 2D) N 2 N D B N D ( D 1 l . :
Aside from the constant, to what extent can you improve this estimate? In particular, can the exponent of B be improved?
3.4. Let F(X) factor F(X) as
= a0X d + a 1 x d - 1 + · · · + ad
E
Q[X] be a polynomial of degree d, and
F(X) = a0(X - a: 1 )(X - a:2 ) . . . (X - a:d )
over the complex numbers. Prove that
T dH ( a: 1 ) · · · H ( a:d ) ::; H ( [ao, a 1 , . . . , ad ]) ::; 2d H ( a: 1 ) · · · H ( a:d ) · (Hint. Mimic the proof of Theorem 3 . 7 for the upper bound. To prove the lower bound, for each v pull out the root with largest la:i lv and use induction on the degree of F.) Can you increase the 2 - d and/or decrease the 2d ? 3.5. Prove that the number of (N + 1)-tuples ( io,
io, . . . , i N
:2:
0
and
' i N ) E zN + 1 of integers satisfying io + . . . + i N = d
0 0 0
is given by the combinatorial symbol ( N.:id ) . Note that this is equal to the number of mono mials of degree d in the N + 1 variables x0, . . . , x N . Section 3.2. Height Functions and Geometry
lP'2 be the rational map ¢(X, Y, Z) = [X2 , Y2 , X Z]. Although ¢ is not defined at [0, 0, 1], we can define Per(¢, lP'2 ) to be the set of points satisfying cpn (P) = P for some n ;::: 0 and ¢i ( P) -1- [0, 0, 1] for all 0 ::; i < n. Prove that Theorem 3.12 is false by showing that Per(¢, lP'2 (Q)) is infinite. What goes wrong with the proof? Try to find a "large" subset S oflP'2 such that Per(¢, S(K)) is finite for every number field K.
3.6. Let ¢ : lP'2
-->
3.7. Let K/Q be a number field, let P = [x0, . . . , X N ] E lP'N (K), and let b be the fractional ideal generated by xo, . . . , x N . Prove that
3.8. Let K/Q be a number field and let ¢(z) E K(z) be a rational map of degree d Recall that the height H ( ¢) is defined by writing
:2: 2.
¢ = [F(X, Y), G(X, Y)] = [a0X d + a 1 X d - 1 + · · · + ad Y d , b0X d + · · · + bd Y d ] and setting H (¢) = H ([ao, . . . , ad , bo, . . . , bd l ) . (See (3.4) on page 9 1 .) Prove that there are positive constants c1 (d) and c2 (d) such that for all P E lP' 1 (K). Find expressions for c 1 and c2 in terms of d. This gives an explicit version of Theorem 3. 1 1 for lP'1 .
Exercises
137
3.9. Let cPc(z) = z2 + c.
(a) Prove that there are infinitely many c E Q such that cPc has a Q-rational fixed point. (b) Prove that there are infinitely many c E Q such that cPc has a Q-rational point of exact period 2. (c) Prove that there are infinitely many c E Q such that cPc has a Q-rational point of exact period 3. (d) Prove that there are no c E Q such that 4>c has a Q-rational point of exact period 4.
3.10. Let d
� 2 and let cPd (z) = zd . Prove that there is an absolute constant c such that for
all number fields K/Q of degree n we have
# PreFer( cPd , lP'1 (K) )
::;
c [K : Q] log log ( [K : QJ ) .
Prove that aside from the constant, this upper bound cannot be improved. (Hint. You will need the fact that the Euler totient function S and h : S ---> lR are continuous maps. Prove that h : S ---> lR is also a contin uous map. (Hint. Show that the functio!!s hm ( P) = d- m h ( cpm ( P)) for m = 1 , 2, 3, . . . are continuous and converge uniformly to h.) 3.15. Let h4> be the canonical height for ¢(z) = z2 + c. For any given c E Q, there is a minimum nonzero value for h¢(z) as z ranges over nonpreperiodic points in Q. Find that minimum value for: (a) c = 0.
Exercises
138
(b) c = -2. (c) c = - 1 . (d) * * arbitrary c E Z. (e) ** arbitrary c E Q. (Hint. Try to do (a), (b), and (c) directly, but we note that it is easier to do them using the theory oflocal canonical heights for polynomials; see Exercises 3 .24 and 3.28. For (d) and (e) the goal is to describe the minimum value of hq, in terms of c.) 3.16. Let ¢( z) = z2 - 1 . Analyze the canonical height for points in the following sets. (a) Bq,( oo) , the attracting basin of oo, where we recall (Exercise 2. 1 ) that the attracting basin of oo for a polynomial rjJ is the set
{
Bq,(oo) = a E C : (b)
lim
n�oo
rjJn (a) = oo
}
U
{oo}.
:Fq, " Bq, ( oo). (Note that all ofthe points in this set are eventually attracted to the attract
ing 2-cycle 0 ...!'.... - 1.) (c) Jq,. (See also Exercise 3.27.)
lPN --+ lPN be a morphism defined over a number field K, and for any point N E lP (K), let D( P) = [K( P) : Q]. Prove that there is a constant C = C(¢) > 0 such P
3.17. Let rjJ
:
that for all P
E lPN ( K ) with P tf. PrePer( rjJ).
(Hint. Use the estimate for the number of points of bounded degree and height given in Exer cise 3.3.) This is a very weak version of the dynamical Lehmer Conjecture 3 .25.
3.18. ** Let c E Q with c =I 0, -2 and let rjJ(z) and N = N(c) > 0 such that A
K
hq, (a) 2: [Q(a) : Q]N
= z2 +c. Prove that there exist K = K( c) > 0
for all a
E Q with a tf. PrePer( rjJ).
(This is not currently known for any value of c other than c = 0 and c = -2.) 3.19. Let a
E Q * and let f(x) = aoxd + a 1 xd- 1 + · · · + ad E Z[x]
be the minimal polynomial of a, normalized so that ao > 0 and Factor f (x) over C as f (x) = a0 IJ(x - a i ). Prove that
gcd(ao, . . . , ad )
1.
d 1 H(a) d = ao IT max { 1 , l ai l } = logl f ( e21r iO ) I dB . 0 i= 1 The quantity H(a) d , or equivalently HQ( a ) ( a) , is also known as the Mahler measure ofa and denoted by M(a). Lehmer's conjecture (Conjecture 3.24) can be stated in terms of Mahler measure: There is an absolute constant K > 1 such that if a E Q* is not a root of unity, then M(a) > K.
1
139
Exercises
3.20. Let ¢(z) E Q(z). Write a program to estimate hq,(a) directly from the definition. Use your program to compute the following heights to a few decimal places. (See also Exer cises 3.30 and 5.3 1 .) (a) ¢(z) = z2 - 1 and a = � . (b) ¢(z) = z 2 + 1 and a = � · (c) ¢(z) = 3z2 - 4 and a = 1.
¢( ) = z + 1 and a = l . z 3z2 - 1 and a = 1. (e) ¢(z) = z2 1 3.21. Let ¢(z) = z 2 - z + 1. The ¢-orbit o f the point 2 is called Sylvester's sequence [428], (d)
z
-
_
CJq,(2) = {2, 3, 7, 43, 1807, 3263443, 10650056950807, 1 13423713055421844361000443, . . . }.
(a) Prove that Sylvester's sequence satisfies
(b)
¢n + 1 (2) = 1 + ¢0 (2)¢ 1 (2)¢2 (2)¢3 (2) . . . ¢n (2). A rough approximation gives hq,(2) 0.468696, so ¢n (2) is approximately equal to �
e0·468696n 2 • Prove a more accurate statement by showing that
n = 0, 1, 2, . . . ,
is positive, strictly increasing, and converges to �· (Hint. First conjugate ¢(z) to put it into the form z 2 + c. ) 2 (c) Deduce that there is a real number H such that ¢n (2) is the closest integer to Hn for all n 2: 0. Note that this is far stronger than the general height estimate
3.22. Let 8 > 0, let d 2: 2 be an integer, and let numbers with the property that xo
> 1 +8
( n )n::;. x
o
be a sequence of positive real
and
(a) Prove that the sequence Xn is strictly increasing and that Xn (b) Prove that the limit
---.. oo
as n
---..
oo .
n �oo
exists. (Hint. Take logs and use a telescoping sum to show that the sequence is Cauchy.) (c) Prove that 8 2 for all n 2: 0. H n - Xn I ::;
I
3.23. Let d 2: 2 and let ¢(z) E
_
d 1
Z[z] be a monic polynomial of degree d, say ¢(z) = zd + azd - 1 + E Z[z]. Prove that for every E > 0 there is a constant C = C(¢, E) such that for all a E Z satisfy ing Ia! 2: C we have · · ·
for all n 2
0.
Exercises
140
Section 3.5. Local Canonical Heights The general theory of local canonical heights is developed in Section 5 .9. However, the the ory becomes much simpler if c/> is a polynomial, because the local canonical height then has a simple limit definition similar to the limit used to define the global canonical height. Exer cises 3.24-3 .30 ask you to develop some of the theory of local canonical heights for polyno mials. 3.24. Let ¢(z) E K[z] be a polynomial. Prove that the limit
(3 . 49) exists and that the resulting function has the following two properties: (a) For all a E Kv, (b) The function
{
a �---> .5.¢,v (a) - log max lalv, 1 }
is continuous on Kv and has a finite limit as lalv --+ oo. Hence the function defined by (3.49) is a local canonical height as described in Theorem 3.27. Prove that the (global) canonical height is equal to the sum of the local heights, h¢(a) =
L
nv 5.¢,v (a) .
v EMK
3.25. Let ¢(z) E K [z] be a polynomial and let v be an absolute value on K. Prove that the local height 5. ,v as defined by (3.49) has the following properties. (a) .5. ¢,v (a) � O for all a E Kv . (b) 5.,v (a) = 0 i f and only if I ¢n (a) I v i s bounded, equivalently, i f and only if a i s i n the v-adic filled Julia set lCv ( c/>) of c/> (see Exercise 2. 1 ) . 3.26. Let c/>(z) E C [z] be a polynomial with complex coefficients and let Bq,(oo) be the attracting basin of oo for ¢. (See Exercises 2 . 1 and 3 . 1 6 for the definition of B¢ ( oo) .) Prove that the local height 5.q, : C --+ IR
as defined by (3.49) has the following properties: (a) .5. is a real analytic function on B¢( oo) '-- { oo }. (b) The function .5. is harmonic on B¢ ( oo) '-- { oo}. In other words, writing z = x + iy, the function .5. is a solution to the differential equation
on the open set Bq, ( oo) '-- { oo}. It satisfies the boundary conditions that .5. vanishes on the boundary of B¢ ( oo) and has a logarithmic singularity at z = oo, i.e., 5.¢ (z) - log lzl
is bounded as z --+ oo.
(c) .5.q, is the unique function that has the properties described in (b). In classical terminology, the function .5. is the Green function for thefilled Julia set JC( ¢) = 1P' 1 ( 1C) '-- B¢(oo).
141
Exercises 3.27. Let ¢>(z) = z2 - 1 . Analyze the local canonical height � , v for points in: (a) the attracting basin B (oo) of oo. (b) :F
lR be the associated chordal metric. (See Sections 1 . 1 and 2. 1 for the definition of the chordal metric when v is archimedean and nonarchimedean, respectively.) Generalize Lemma 3.51 by showing that for every Q E lP' 1 ( Kv) there is a constant Cv = Cv ( ¢, Q) such that
min
Q' E- ' (Q)
Pv(P, Q'tQ() :::; Cv Pv (¢(P), Q)
Here eQ (¢ ) i s as defined in Lemma 3 .5 1 .
for all P E
lP'1 (Kv ).
145
Exercises
3.44. Let K be a number field and let ¢ : lP'1 --+ lP' 1 be a rational map of degree d � 2 defined over K. Prove that there is a finite set of absolute values S c MK such that for all v rf_ S and all Q E lP' 1 (K), the constant Cv(¢, Q ) in Exercise 3.43 may be taken equal to 1. 3.45. Let cp (z ) E Q(z) be a rational map of degree d � 2 with ¢2 (z ) rf_ Q [z], let a write cp n (a) = an /bn E Q as a fraction in lowest terms as usual. Prove that
E Q, and
3.46. Let K/Q be a number field, let v E MK be an absolute value on K, and let ¢( z) E K ( z) be a rational function of degree d � 2. Suppose that a E lP' 1 ( K) is a wan dering point for ¢ and that 'Y E lP' 1 ( K) is any point that is not a totally ramified fixed point of ¢2 . Prove that . - log Pv ( cpn (a) , "() hm = 0.
n ->oo
h ( cpn(a))
Taking first 'Y = oo and then 'Y = 0, explain how to use this result to generalize Theorem 3.48 to number fields. Section 3.9. Periodic Points and Galois Groups 3.47. Let P Prove that
E lP' N (K) be a point in projective space and let K(P) be its field of definition. K(P) = fixed field of { (]'
E Gal(K/ K) (J'(P) = P}. :
In mathematical terminology, this says that the field of moduli of P is a field of definition for P. We discuss fields of moduli and fields of definition for (equivalence classes of) rational maps in Chapter 4. Section 3.1 1 . Ramification and Units in Dynatomic Fields 3.48. Let cp (z ) =
z2 + c and let
be the polynomial whose roots are periodic points of period 3. (a) Prove that 3 (z) E Z[c, x] is a polynomial in the variables c and z and that it has integer coefficients. (b) Let a be a root of 3 ( z) and assume that the field Q( c, a) is an extension of Q( c) of degree 6 (i.e., 3 (z) is irreducible in Q( c) [z]). Theorem 2.33 implies that and are units. Compute these units explicitly as elements of Q[c, a]. (c) Prove that there is a field automorphism (]' : Q( c, a) --+ Q(c, a ) characterized by the fact that it fixes Q( c) and satisfies (]'(a) = ¢ ( a) = a 2 + c. (Note that in general, Q( c, a) will not be a Galois extension of Q(c) .) Prove that (]'3 is the identity map. (d) Compute the units (J'(u1 ) and (J'(u2 ). (e) Compute the units (J' 2 (u l ) and (J' 2 (u2 ). (f) Express (]'( u 1 ), (]'(u2 ), (]'2 ( ut), and (]'2 ( u2 ) in the form ±ul u�.
146
Exercises
3.49. Let ¢( z) = z2 + c and let qJ;; ( z) be as in the previous exercise. (a) Prove that qJ3 (z) factors into a product of two cubic polynomials in the ring Q(c) [z] if and only if c has the form c = - (e2 + 7)/4 for some e E Q(c). (b) Suppose that c = - (e2 + 7)/4. Show that there is a polynomial 9e (z) E Q[e, z] such that the factorization qJ3 (z) has the form ge(z)g-e(z). Compute the discriminant of ge (z) and verify that it is a perfect square in Q[e]. Conclude that if ge(z) is irreducible over Q(e), then its roots generate a cyclic Galois extension Ke of Q( e). (c) Let a be a root of 9e (z). Use the results of the previous exercise to construct units in Q( e) (a). Analyze these units for some small values of e, say e = 1, 5, 7, 9. (Note that we investigated the case e = 3 in Example 3.69.) 3.50. ** Let ¢( z) be a generic monic polynomial of degree d � 2, i.e., ¢( z) is a polynomial of the form zd + a1zd -l + · · · + ad , where a1 , . . . , ad are indeterminates. Let p be a prime and let a E Per;• ( ¢) . Theorem 3.66 says that the elements
Ui ,j =
¢i (a) - n (z) - z. ¢k (a) = a
z) - z
Let ¢( E be a polynomial. Then the fixed points of¢ are the roots of¢( (plus the point at oo) , and more generally, the points of period n for ¢ are the roots may have roots of period smaller of However, the polynomial than n, since if and kin, then also It is natural to try to eliminate these points of strictly smaller period and focus on the points of exact period n.
cj>n (z) - zcj>n (a) = a.
n-1 z include all nth roots of unity, not only the primitive ones. The nth cyclotomic poly
Example 4. 1 . We recall an analogous situation. The roots of the polynomial
nomial is defined using an inclusion-exclusion product,
nth cyclotomic polynomial =
IJ (zk - 1)�"(n/k) . kin
(4. 1 )
It is the polynomial whose roots are the primitive nth roots of unity. Here J.L is the Mobius function defined by 1 and
J.L
J.L
J.L(1) = (P1 · · · Prer ) -- { 0(-1Y
if e , = · · · = er = 1, . tfany ei 2: 2.
q
(4.2)
See [2 1 6, Section 2.2] or [1 1 , Chapter 2] for basic properties of the Mobius function and the Mobius inversion formula. It is easy to check that the product (4 . 1 ) is a 1 are distinct and the polynomial, using the fact that the (complex) roots of following basic property of the function (Exercise 4.2):
J.L
L
k in
J.L
( �) =
{
zn -
1 0
�f n = 1,
1
f n > 1.
(4.3 )
Taking our cue from the example provided by the cyclotomic polynomials, we might define the nth dynatomic polynomial by the formula
n (z) = IJ(¢k (z) - zt(n/k) . kin However, it is not clear that n (z) is a polynomial, since cj>n (z) may have multiple roots, as shown by the following example. Example 4.2. Let cj>(z) be the polynomial 2
Then
¢(z) = z - 43 .
4.1. Dynatomic Polynomials
149
Thus ¢2 z vanishes with multiplicity 3 at the point �, and although it is true that the ratio �(�)�; is a polynomial, it is somewhat distressing to observe that its root is a fixed point of ¢, not a point of primitive period 2. For simplicity, the preceding discussion dealt with polynomial maps We now tum to general rational maps E and develop tools that are useful for studying their periodic points. Let ¢( be a rational function of degree d and write ¢= using homogeneous polynomials and Then the roots of the polynomial
z -
( z) -
=
¢(z). ¢(z)z) K(z) [F(X, Y), G(X, Y)] F G. YF(X, Y) - XG(X, Y) in JP> 1 are precisely the fixed points of ¢. If we count each fixed point according to the multiplicity of the root, then ¢ has exactly d + 1 fixed points. More generally, we can apply the same reasoning to an iterate cpn of ¢ and assign multiplicities to the n-periodic points.
z) E K( z) be a rational function of degree d, and for any n 2 0, cpn [Fn (X, Y), Gn (X, Y)] with homogeneous polynomials Fn , Gn E K[X, Y] of degree dn . (See Exercise 4.9 for a formal inductive definition of Fn and Gn . ) The n-periodpolynomial of¢ is the polynomial ll>¢,n (X, Y) YFn (X, Y) - XGn (X, Y). Notice that ll> ¢ , n (P) 0 if and only if cpn (P) P, which justifies the name as signed to the polynomial ll>¢, · Definition. Let ¢(
write
=
=
=
=
n The nth dynatomic polynomial of¢ is the polynomial'
kin
kin where is the Mobius function. If ¢ is fixed, we write ll> n and 1!>�. If ¢ (z) E K[z] is a polynomial, then we generally dehomogenize [X, Y] [ z , 1] and write ll> n (z) and ll>� (z). All of the roots P of ll> ¢ n (X, Y) satisfy cpn (P) P, but we saw in Example 4.2 that ll> ¢ , n (X, Y) may have roots whose periods are strictly smaller than n. Following JL
=
,
=
Milnor [302], we make the following definitions. 1
See Theorem 4.5 for the proof that ¢, n is indeed a polynomial.
150
4. Families of Dynamical Systems
¢( ) E K ( ) be a rational map and let P E lP' 1 be a periodic point for ¢. P has period if n (P) = 0. P has primitive (or exact) period if n (P) = 0 and m (P) # 0 for all P has formal period if � (P) = 0.
Definition.
Let
z
z
n
•
n
•
m < n.
n
•
We set the notation
Pern (¢) = { P E lP'1 : n (P) = 0} , Per�(¢) = { P E lP'1 : �(P) = 0} , Per�* (¢) = { P E lP'1 : n (P) = 0 and m (P) # 0 for all 1 � } . Thus Pern (¢) is the set of points of period Per�(¢) is the set of points of formal period and Per�* ( ¢) is the set of points of primitive (or exact) pe riod Sometimes we treat these as sets of points with assigned multiplicities, e.g., if P E Per�(¢), the multiplicity of P is the order of vanishing of � at P. m < n
n,
n,
n.
It is clear that
primitive period n
===>
formal period n
period n,
===>
but neither of the reverse implications is true in general.
q,, n is homogeneous of degree dn + 1, so counted with multiplicity, the map ¢ has exactly + 1 points of period n. And if we let
Remark 4.3. The polynomial
dn (4.4) ( ) = deg(¢ ,n (X, Y)) = I > (�) (dk 1) , kin then counted with multiplicity, the map ¢ has exactly ( ) points of formal pe riod The number ( ) grows very rapidly as d or increases. See Exercise 4.3. The first few period and dynatomic polynomials for ¢( ) = z2 + are listed in Table 4. 1 on page 1 56. Notice how complicated n and � are, even for small values of For example, 6 has degree 54 as a polynomial in and degree 27 as a vd n
+
vd n
n.
vd n
n
c
z
z
n.
polynomial in c. Remark 4.4. Rather than using the homogeneous polynomials and it is some times more natural and convenient to consider instead the associated divisors in especially in generalizing the theory to higher-dimensional situations. This is the approach taken in [3 13], where for any (nondegenerate) morphism ¢ : V V of smooth algebraic varieties, the 0-cycle is defined as the pullback of the graph of
-------t
+c
-------t
c,
---->
---->
-------t
c,
f------t
which is inverse to (4. 1 3), a fact that can also be checked directly with a computer algebra system. 4.2.2
Dynatomic Curves as Modular Curves
The curve Y1 ( n) and its completion X1 ( n) are modular curves in the sense that their points are solutions to the moduli problem of describing the isomorphism classes of pairs (¢, a), where ¢ is a polynomial of degree 2 and a E A1 is a point of formal period n for ¢. Here two pairs (¢1 , at) and (¢2 , a2 ) are PGL2 -isomorphic if there is a linear fractional transformation E PGL2 satisfying
f
and
In order to state this more carefully, we define
k[z],
2,
R}
¢E deg(¢) aE (¢, a) .· a has formal period n for ¢ . Formal(n) . PGL 2 -tsomorph"tsm We have demonstrated tltat the elements ofFormal(n) are in one-to-one correspon dence with the points of Y1 ( n). But much more is true: the correspondence is alge braic in an appropriate sense. Before stating this important result, we must define what it means for a family of maps and points to be algebraic. =
{
=
4.2. Quadratic Polynomials and Dynatomic Modular Curves
159
Definition. Let V be an algebraic variety. An algebraic family ofquadratic polyno mials over V with a markedpoint offormal period n consists of a quadratic polyno
mial
'ij; ( z) = Az2 + Bz + C, A, B, C E K [V] , whose coefficients A, B, C are regular functions on V and such that A does not vanish on V( K ), and a morphism A V such that for all P E V( K ), the point A(P) is a point of formal period n for the quadratic polynonomial '1/Jp (z) = A(P)z2 + B(P)z + C(P) E K [z] . The family is defined over K if the variety V and morphism A are defined over K and the functions A, B, C are in K[V] . :
----+
P}
Example 4. 10. The pair
and is an algebraic family of quadratic polynomials over point of formal period 2.
lP'1
A(t) = t - 1 "
{0, oo} with a marked
Theorem 4.1 1. Let K be a field ofcharacteristic differentfrom 2. (a) The map n) --+ Formal(n) , C, CY ) r---+ C, CY ) ,
(z2 +
(
Y1 (
is a bijection ofsets.
(4. 14)
(b) Let V be a variety and suppose that the points ofV algebraically parameterize
'1/J
a family of quadratic polynomials together with a marked point period n. Then there is a unique morphism ofvarieties
A offormal
with the property that
ry(P) = ('1/Jp (z), A(P)) E Formal(n)
for all
Y1
P E V(K),
(4. 1 5)
where we use (4. 1 4) to identifY Formal(n) with (n). (c) If the family is defined over the field K, then the morphism 77 is also defined over K. Proof (a) We have shown this earlier in this section. (b) By definition '1/J has the form
'ij; ( z) = Az2 + Bz + C, A,B,C E K [V] , with A not vanishing on V and A a morphism A V A1 such that for all P E V( K ), the point A(P) is a point of formal period n for the quadratic polynonomial 'lj;p (z) A(P)z2 + B(P)z + C(P) E K [z] . :
=
----+
4. Families of Dynamical Systems
160
For any point
fp
P E V(K) , let fp(z) be the linear fractional transformation - B(P) . fp(z) = 2z2A(P)
A
Note that is well-defined for every P E V( K ), since we have assumed that is nonvanishing on V. We define a map TJ from V to A2 by the formula
- � B(P) 2 { apCp = A(P)C(P) A(P).A(P) � B(P).
� B(P), (4. 16) TJ(P) = (cp,ap) with + Note that TJ V A2 is a morphism, i.e., it is given by everywhere-defined algebraic functions on V. We are now going to verify that the image of is the curve Y1 ( ) The computation that we performed in deriving formula (4. 12) shows that '1/;{: (z) = z2 + cp. To ease notation, we let cj;p (z) z 2 cp. We also let n , P , � , P • W n , P , and w�,P be the period and dynatomic polynomials for cj;p and '1/Jp , respectively. Note that the period polynomials are related by \ll��p (z) f? 1 o('l/J? (z)-z)ofp = (!? 1 o'lj;p fp t (z)-z ¢rp (z)-z n, P (z). =
:
-+
+
TJ
n .
+
=
=
=
=
Hence the dynatomic polynomials also satisfy (w� , P) fp =
n,P ·
(4. 17)
.A(P) Per�('lj;p ) .A(P) is a f? 1 (.A(P)) = A(P).A(P) + � B(P) is a root of � ' P • and thus is in Pe � ( ¢ p). This proves that the image of the map defined by (4. 1 6) is contained in Y1 ( ) so is a morphism from V to Y1 ( ) Further, this map TJ respects the identification of Y (n) with Formal(n) from (a), since it takes P to a pair (cp,ap) that is isomorphic to the pair ('1/Jp (z),.A(P)) via the conjugation fp E PGL2 ( K ) . Finally, it is clear from the construction that the map is uniquely determined as a map (of sets) from V ( K ) to Formal( ) so it is the unique morphism V Y1 ( ) satisfying (4.1 5). ( ) The definition (4. 16) of shows immediately that is defined over K, since all of A, B, C, and .A are assumed to be defined over K. D Remark 4.12. In the language of algebraic geometry, Theorem 4. 1 1 says that Y1 ( ) is a coarse moduli space. In fact, the curve Y1 (n) is actually a fine moduli space for all 2: 1; see Exercise 4. 18. The underlying reason is that there are no nontrivial We are given that E , which is equivalent to saying that root of w�,P· It follows from (4. 17) that
r
TJ
n ,
n .
I
TJ
n ,
c
TJ
TJ
-+
n
TJ
n
n
elements of PGL2 that fix a quadratic polynomial and its points of formal period n, i.e., the moduli problem has no nontrivial automorphisms.
161
4.2. Quadratic Polynomials and Dynatomic Modular Curves 4.2.3
The Dynatomic Modular Curves
Y1 (n),
X1 (n) and X0 (n)
Y1 (n)X1via(n),thehasmapthe interesting property that the ratio (y, z) (y, z2 y) (y,
f---+
B+
B.
+
+ c
=
=
+ c.
=
+
B(
+c
+
+ c.
=
c)
+ c)
+
0
+ c+
163
4.2. Quadratic Polynomials and Dynatomic Modular Curves
Remark 4. 1 5. Morton [307] describes an alternative method for finding an equation
for the curve and the map The roots of c, 0 can be grouped into orbits, say . . , O:r are representatives for the different orbits, so
Yo (n)
Y (n) Y0 (n). o: 1 , . 1
� ( z)
---+
=
The points in each orbit all have the same multiplier, and we define a polynomial T
\p O:i
On ( ) II ( ( )) Then one can show that On ( ) E Z [ ] and that the equation C, X =
X
i= 1
c, x
-
.
c, x
Y0(n). Using this model, the natural map Y1 (n) Yo (n) Y1 (n) ----+ Yo(n), (y,z) 1------t (y, (c,D�)'(z)).
gives a (possibly singular) model for from to is
See Morton's paper [307] and Exercise 4. 13. Remark 4.1 6. The dynatomic modular curves and defined in this section are analogous to the modular curves that appear in the classical theory of elliptic classifies isomorphism classes of curves. Briefly, the elliptic modular curve pairs (E, where E is an elliptic curve and E is a point of exact order Similarly, the elliptic modular curve ( n) classifies pairs E, where E is an elliptic curve and c E is a cyclic subgroup of order The group acts on via
Y1 (n) Yo (n) e1 (n) Y 1 P), n. e1 P E ( C), Y 0 n. (Z/nZ)* Y1e1 (n) C m (E, P) (E, [m]P) for m E (Z/nZ)*, and the quotient ofY1e1 (n) by this action is Y011 (n). The reader should be aware that standard terminology is to write Yo(n), Y1 (n), Xo(n), X1 (n) *
=
for elliptic modular curves. But since in this book we deal exclusively with dy natomic modular curves, there should be no cause for confusion. In situations in which both kinds of modular curves appear, it might be advisable to use identifying superscripts such as to distinguish them. and We showed earlier that and 1 , are all (irreducible) curves of genus 0, and similar explicit computations show that has genus 2 and has genus 14. See [171, 305, 309] and Exercise 4.20. The geometry of and for general is described in the following theorem, which is an amal gamation of results due to Bousch and Morton.
X0(n)
Y1dyn (n) Y1e1 (n) X1 ( ) X1 (2), X1 (3)X (4) 1 n
XX11 (n)( 5)
164
4. Families of Dynamical Systems
Y1 (n) defined by the equation Y1 (n) : if?� ,q, (Y, ) 0 is nonsingular. (b) The dynatomic modular curves X1 ( n) and Xo ( n) are irreducible. (c) The projection map ( z, c) c, exhibits X1 ( n) as a Galois cover of lP'1. The Galois group is maximal in the sense that it is the appropriate wreath product, cf Section 3.9. (d) Let r.p denote the Euler totient function (not to be confused with the rational map ¢), let be the Mobius function, and let "' be thefunction (�) 2k . "' (n) � Lfl kjn (This is essentially half the degree of if? ¢ , n ; cf Remark 4.3.) Then the genera ofX1 ( n) and Xo ( n) are given by the formulas genusXr(n) n ; "' (n) - � jnL, Rc. (See Exercise 4.15.) If c � M, then one can prove that has an analytic continuation to C " M. The isomorphism () in the theorem is the inverse of the map
'1/Jc(z)
=
'1/Jc
See [ 142, 143, 141], [43, §9. 10], or [95, VIII §§3,4] for details.
lzl
D
Remark 4.24. An immediate consequence of Theorem 4.23 is the connectivity of the Mandelbrot set, since the theorem implies that IP'1 (C) " M is simply connected.
168
4. Families of Dynamical Systems
The uniformization map () from Theorem 4.23 can be used to give an analytic description of the Misiurewicz points.
x 2x 1}
"M
Theorem 4.25. Let () be the isomorphism {w E C : lwl > ---> C described in Theorem 4.23. Consider the doubling map 1-t on Q/7!... Let t E Q/7!.. be preperiodic, but not periodic, for the doubling map. Let m and n be the smallest positive integers for which we can write t in theform
(The fraction need not be in lowest terms. See Exercise 4.1 6.) Then the limit Ct = lim
r->1 + fJ(re21rit ) ) in M, although distinct values of exists and is a Misiurewicz point of type ( t E Q/7!.. may yield the same Misiurewicz point. m, n
Remark 4.26. For a given t E Q/7!.., the "spider algorithm" [213] can be used to com pute Ct numerically. The spider algorithm is mainly topological and combinatorial in nature, although the limiting process that yields Ct E C is analytic. 4.3
The Space Rat d of Rational Functions
{Az2 Bz
The set of quadratic polynomials + + C} has dimension three, since it may be identified with the set of triples (A, B, C) with A =1- In fancier language, the space of quadratic polynomials is equal to the algebraic variety { (A, B, C) E
0.
A3 : A =I- 0}.
We have seen in Section 4.2. 1 that every quadratic polynomial can be conjugated to a polynomial of the form + c, and that polynomials with different c values are not conjugate to one another. Thus the space of conjugacy classes of quadratic polynomials has dimension 1. It may be identified with the variety In the next few sections we study analogous parameter spaces for more general rational maps and their conjugacy classes. We begin in this section by explaining how the set of rational maps of degree d has a natural structure as an algebraic variety and how the natural action of the algebraic group on is an alge braic action. Then in Section 4.4 we discuss (mostly without proof) how to take the quotient of to construct the moduli space of conjugacy classes of rational maps of degree d. We continue in Section 4.5 by describing a natural collection of
z2
Ratd Ratd
A1 .
Md
PGL2 Ratd
algebraic functions on M d that are created using symmetric functions of multipliers of periodic points. These functions can be used to map Md into affine space. Finally,
A2 .
in Section 4.6 we use these functions to prove that M 2 is isomorphic to A rational map ¢ : of degree d is specified by two homogeneous polynomials
!fD1 ---. !fD1
4.3. The Space
Ratd of Rational Functions
169
G Res(F, G) [uP, uG] = [F, G], a= (X Y) =
such that F and have no common factors, or equivalently from Proposition 2.13, such that the resultant does not vanish. Thus a rational map of degree d is determined by the 2d + 2 parameters a0 , a1 , . . . , ad, bo , b1 , . . . , bd . However, if is any nonzero number, then so the 2d + 2 parameters that deter mine ¢ are really well-defined only up to homogeneity. This allows us to identify in a natural way the set of rational maps of degree d with a subset of projective space. To ease notation, for any (d + I)-tuple (a0 , . . . , ad ), let Fa , aoXd + a 1 Xd -l y + · + adYd
u
··
be the associated homogeneous polynomial. Similarly, if and b are ( d + )-tuples, we write b E lP'2d+ l for the point in projective space whose homogeneous coor dinates are
[a,[ao,] . . . , a , bo, . . . , ]. d bd
Ratd.
1
1 The set of rational functions ¢ lP' 1 lP' of degree d i s denoted 2 It is naturally identified with an open subset ofJID d+ 1 via the map b E JID2 d+ 1 Res( Fa , Fb) -:/:- � b
Definition.
by
a
{[a, ]
:
----+
[a, 0}] [FRat a, Ph]d , ·
:
f--+
Ratd,
which a priori is merely a set, thus has the The collection of rational maps structure of a quasiprojective variety. In fact, Rat d is an affine variety, since it is the complement of the hypersurface Res( Fa , Fb) in the projective space JID2 d+ l .
=0
an affine variety defined over Q. The ring of Q[Ratd] ofRatRatd isd isgiven explicitly by [ Res(�a�F�) d + · · + d + · · · + ]d = 2d . Q[Ratd] = Q l Equivalently, Q[Rat d ] is the ring ofrationalfunctions ofdegree 0 in the localization ofQ[ ao, d ] at the multiplicatively closed set consisting ofthe nonnegative powers ofRes(Fa, Fb). Proof We remind the reader that in general, if F E K[X0 , . . . , Xr ] is a homoge neous polynomial of degree then the complement of the zero set of F, Proposition 4.27. The variety regularfunctions
. . . ai d lJio b)t . . . bjd ai0o ah 1
:
io
·
i + Jo
a1 , . . . , ad, bo , b1 , . . . , b
n,
is an affine variety of dimension r. (See [ 1 98, Exercise 1.3.5].) Explicitly, each ratio nal function with io + i 1 +
· · · + ir =
n
is a regular (i.e., everywhere well-defined) function on V. There are functions, and together they define an embedding
(r�n) such
170
4. Families of Dynamical Systems
V
V
of into affine space. The affine coordinate ring of is the ring of polynomials in these fi ir , "0"1 · ·"· "r ] . + z. 1 + · · ·+z. r =n . o...
K[V] K [f• =
"
to
In the language of commutative algebra,
K[V] K[Xo, . . . , Xn , 1/ F] (o) is the set of rational functions of degree 0 in the localization of K[X0 , . . . , Xn ] at the multiplicatively closed set (Fi k�_ 0 . Applying this general construction to =
D
gives the results stated in the proposition.
(
Remark 4.28. The geometry ofRatd q, especially near its boundary, presents many interesting problems. See [121, 122, 369, 370].
.
Example 4 29 . Let
a0X2 + a1 XY + a2Y2 and b0X2 + b1XY + b2Y2 . Then the
be the resultant of collection of 84 functions
84 Rat2 Rat2 A .
gives an embedding of into Of course, this is not the smallest affine space into which can be embedded. Projecting onto appropriately chosen hy perplanes, there is certainly an affine embedding of the 5-dimensional space into see [ 198, Exercise IV.3 . 1 1]. Example 4.30. The set of rational functions of degree 1 is exactly the set of linear fractional transformations,
A11 ;
Rat2
Rah = PGL2 = {[o:X + (3Y, "(X + 8Y] : o:8 - f3"! f. 0} !P'3 . We note that PGL2 is not merely a variety, it is a group variety, which means that the maps PGL2 PGL2 PGL2 , and PGL2 PGL2 , ( h , h) !----+ h h f f- 1 ' c
x
______,
______,
!----+
defining the group structure are morphisms.
Ratd of Rational Functions 171 1 . Further, as we vary Each point of Ratd determines a rational map lP' 1 lP' the chosen point in Rat d , the rational maps "vary algebraically." We can make this vague statement precise by saying that the natural map ¢ lP'1 Rat d 1P'1 x Ratd , ([X,Y], [a,bl) ([Fa(X,Y),Fb(X,Y)j, [a,bl) , (4.20) 4.3. The Space
--->
x
:
______,
f-----*
is a morphism of varieties. The following definition is useful for describing families that vary algebraically.
Definition.
Let V be an algebraic variety. The projective line over V is the product
lP'� lP'�
A morphism 'ljJ over V is a morphism that respects the projection to V, i.e., the following diagram commutes, where the diagonal arrows are projection onto the second factor: :
--->
'\.
v
/
Then 'ljJ can be written in the form 'ljJ
=
[F(X, Y), G(X, Y)],
where K ( V , are homogeneous polynomials with coefficients that are rational functions on V. The degree of'l/J is the degree of the homogeneous polyno mials and Any morphism V W of varieties induces a natural morphism lP'� lP'�, which, by abuse of notation, we also denote by A. Thus
F, G E )[X Y] F G.4 A A(P, t) (P, A(t)) :
--->
=
--->
for
(P, t) E lP'� .
With this notation, the map (4.20) says that there is a natural morphism
Ratd.
over The next proposition says that this ¢ is a universal family of rational maps of degree d. Proposition 4.31. Let V be an algebraic variety and let
'1/J
:
lP'� ------t lP'�
be a morphism over V ofdegree d. Then there is a unique morphism
4More generally. if 'ljJ : II'� -+ II'� is a morphism of S-schemes. then 'lj!*OpN (1) 5 is the degree of 'ljJ.
�
OpN (d) and d s
172
4. Families of Dynamical Systems
..\ V :
such that the induced map ..\
:
JID�
--+
J!D l
--+
Ratd
J!Dkatd fits into the commutative diagram )->q,J ---+ Ratd ----
f't Ratd
(-)
In terms ofelements, (¢/) = (¢) for all ¢ E
Ratd and all f E PGL2.
4. Families of Dynamical Systems
176
(b) The map on complex points is surjective and eachfiber is the full PSL2 (C)-orbit of a single rational map. Thus there is a bijection ofsets
(c) The variety M d is a connected, integral (i.e., reduced and irreducible), affine variety ofdimension 2d - 2 whose ring ofregularfunctions is the ring ofPSL2 invariantfunctions on Ratd, (d) Let VjC be a variety and let T : Rat d V be a morphism with the property that T(tjyf) T(tjy) for all tjJ E Rat d (C) and all E PGL2 (C). Then there is a unique morphism T : M d V satisfying t( (¢)) T( tjJ ) . ____,
=
____,
f
=
ProofSketch. A full proof of Theorem 4.36 (see [416]) uses the machinery of geo
metric invariant theory [322] and is thus unfortunately beyond the scope of this book. Geometric invariant theory tells us that there is a certain subset of!fD2 d+ l , called the stable locus, on which the conjugation action of PSL2 is well behaved. The main part of the proof is to use the PSL2 -invariance of the resultant to verify that
is a PSL2 -invariant subset of the stable locus of !fD2d+ l . Then the existence of the quotient variety M d with affine coordinate ring equal to the ring of invariant func tions in Ql[Ratd ] follows from general theorems of geometric invariant theory [322, Chapter 1]. Further, the fact that M d is connected, integral, and affine follows im mediately from the corresponding property ofRat d [322, Section 2, Remark 2]. The dimension of M d is computed as dim M d
=
dim Rat d - dim PSL2
=
(2d + 1) -
3
=
2d - 2.
This proves (a), (b), and (c). Finally, (d) follows directly from the description of M d and Ql[M d ] in (c), since the morphism T induces a map T* : C[Ratd ], and the assumption that T satisfies T( ¢f) = T( tjJ) implies that the image of T* lies in D the ring C[Rat d]PGL2 (1C) C[M d] .
Ov
____,
=
Remark 4.37. Theorem 4.36 says that the quotient Rat d / PSL2 is an algebraic va
riety. Milnor [301, 302) shows that Ratd (C)/ PGL2 (C) has a natural structure as a complex orbifold, which roughly means that locally it looks like the quotient of a complex manifold by the action of a finite group. Thus its singularities are of a fairly moderate type, although they can still be quite complicated. However, for rational maps of degree 2, we will see in Section 4.6 that not only is M 2 nonsingular, it has a particularly simple structure.
4.4. The Moduli Space
M d of Dynamical Systems
177
Md Q[RatdjP8L2• ofQ[Ratd] · Ratd Md k Ratd (k)
Remark 4.38. According to Theorem 4.36(c), the affine coordinate ring of
is the ring ofPSL2 -invariant functions However, this is the same as the ring of PGL 2 -invariant functions, since it suffices to check for invariance by the action of PGL2 (C) PSL2 (C) on an element by the conjugation ac is the quotient of Remark 4.39. The moduli space tion of the group PSL2 . In particular, if is any algebraically closed field, then the is exactly the collection of cosets of set of points by the conju gation action of PSL2 ( K ) PGL 2 (K). However, if K is not algebraically closed, then the natural map =
M d (K)
=
Md (K), at
is generally neither injective nor surjective. The correct description of least if K is a perfect field, is for every T E I K) there is an (K) (¢) E ( K ) ·. E PGL2 (K) such that T(¢) = ¢fT
} Md - { Md fr _Gal(k The field of moduli of a rational map ¢ E Rat d (k) is the smallest field L such that ( ¢) E M d ( L). Fields of moduli and related questions are studied in detail in Section 4. 1 0. In particular, see Example 4.85 for a map ¢ whose field of moduli is Q, yet ¢ is not PGL 2 (C)-conjugate to any map in Ratd (Q). The moduli space M d classifies rational maps up to conjugation equivalence, just as we want, but it has the defect that it is an affine variety. It is well known that if possible, it is generally preferable to work with projective varieties. How might we naturally complete M d by filling in extra points "at infinity"? Note that we should not do this in an arbitrary fashion. Instead, we would like these extra points to corre _
·
spond naturally to degenerate maps of degree d. One possibility is simply to start with all of IP'2 d+ l and take the quotient by the conjugation action of PSL2. Unfortunately, there is no natural way to give the quo tient JP>2 d+ 1 I PSL2 any kind of reasonable structure. For example, it is not a variety. So IP'2d+ l is too large. Ideally, we would like to find a subset S c JP>2 + l with the following properties: 1 . S contains 2. PSL2 acts on S via conjugation. 3. There is a variety T and a morphism S ----+ T that induces a bijection
d
Ratd .
4. The quotient variety T is projective. Geometric invariant theory gives us two candidates for S. The smaller candidate is the largest variety satisfying (3), but its quotient T may fail to be projective. The larger candidate has a projective quotient, but the map S(C)I PSL2 (C) ----+ T(C) in (4) may fail to be injective. The following somewhat lengthy theorem describes the application of geometric invariant theory to our situation, that is, to the conjugation action of PSL2 on IP'2 + 1 .
d
4. Families of Dynamical Systems
178 Theorem 4.40. There are algebraic sets
Ratd
C
Ratd
C
Ratds
C
lP'2d+ 1
with thefollowing properties: (a) The conjugation action ojPSL2 on Ratd extends to an action ojPSL2 on Ratd and Ratd8• (b) There are varieties M8 and M88 and morphisms ( · ) : Ratd -+ Md
(4.27)
and
that are invariantfor the action ojPSL2 on Ratd and Ratd8• The varieties M8 and M88 and morphisms (4.27) are defined over IQl. (c) Two points and in Ratd(C) have the same image in Md (C) ifand only if there is an f E PSL2 (C) satisfying
[a, b] [a', b']
[a', b'] = [a, b]f.
Thus as a set, Md(C) is equal to the quotient ofRatd (C) by PSL 2 (C).
[a, b] [a', b'] {[a, b] f : f E PSL2 (C) } {[a' , b'] f : f E PSL2 (C) } # 0. Equivalently, they have the same image if and only if there is a holomorphic map f : {t E C : 0 < l t l < 1} ---> SL2 (C) such that [a, b]f' = [a' , b'] . tlim --->0 (e) Md is a quasiprojective variety and Mds is a projective variety. (f) (Numerical Criterion) A point [a, b] E lP'2d+ l (C) is not in Ratds if and only if there is an f E PSL (C) such that [a', b'] = [a, b]f satisfies
(d) Two points and in Ratd8 (C) have the same image in Md8 (C) if and only ifthe Zariski closures oftheir PSL2 (C)-orbits have a point in common, n
2
d-1 d+1 and bi 0 for all (4.28) a� = 0 for all Similarly, [a, b] is not in Ratd ifand only if there is an f E PSL2 (C) such that [a', b'] = [a, b]f satisfies d+1 d-1 (4.29) and bi 0 for all i < ai = 0 for all < i ::;
2-
i
2-
I
=
=
i :S
2-.
2- .
(g) Md is isomorphic to Mds if and only if d is even.
Proof. The proof of this theorem is beyond the scope of this book. However, we note that half of (g), namely Md 2::: Mds for even d, follows directly from the numerical
criterion in (f), since for even d the criteria (4.28) and (4.29) are the same. See [416] for a proof of a general version of Theorem 4.40 over Z. See also [301, 302] for a similar construction over C. D
179
4.5. Periodic Points, Multipliers, and Multiplier Spectra
Remark 4.41. In Theorem 4.40, the set denoted by Rat;i is called the set of stable rational maps, and the set denoted by Rat;18 is called the set of semistable rational maps. Note that points in these sets need not represent actual rational maps of de
IP'1 .
Ratd
gree d on The intuition is that points in Rat;i and Rat;J8 that are not in correspond to maps that want to be of degree d but have degenerated in some rea sonably nice way into maps of lower degree. Remark 4.42. Theorem 4.40(c) says that the stable quotient M;l(C) has the nat ural quotient property, since its points correspond exactly to the PSL2 (C)-orbits of points in Rat;i(C). Quotient varieties with this agreeable property are called geomet ric quotients. The semistable quotient M;18 (C) has a much subtler quotient property. According to Theorem 4.40(d), points in Rat;18 (C) with distinct PSL2 (C)-orbits give the same point in M;is (q if their orbits approach one another in the limit. Quotients of this kind are called categorical quotients. As compensation for the less intuitive notion of categorical quotient, Theorem 4.40(e) tells us that M;18 is projec tive, so M;18 ( q is compact. Finally, Theorem 4.40(g) says that if d is even, then M;i and M;is coincide, so in this case, the moduli space M d has a projective closure with a natural (geometric) quotient structure. Remark 4.43. Applying the full machinery ofgeometric invariant theory to the action of PSL2 /Z on it is possible to prove versions of Theorems 4.36 and 4.40 over Z. In other words, there is a filtration of schemes over Z,
JP>id+ l ,
such that the group scheme PSL2 /Z acts on each of these schemes and such that the quotient schemes M C M;i C M88 exist in a suitable sense. In particular, Theorems 4.36 and 4.40 are true with Q replaced by the finite field lF The proof is similar to the proof over Q, but requires Sheshadri's theorem that reductive group schemes are geometricallly reductive. See [416] for details.
d
4.5
p·
Periodic Points, Multipliers, and Multiplier Spectra
The moduli space M d of rational functions modulo PSL2 -equivalence is an affine variety whose ring of regular functions
Ratd
consists of all regular functions on that are invariant under the action of PSL2 , or equivalently under the action of PGL2 (C); see Remark 4.38. Abstr�ct invariant theory, as described in the proof sketch of Theorem 4.36, says that there are many such functions. In this section we use periodic points to explicitly construct a large class of regular functions on Md .
4. Families of Dynamical Systems
180
be a rational function of degree d defined over the complex numbers. Associated to each fixed point P E Fix(¢) is its mul tiplier Ap ( ¢) E .1 - l (P) ( ¢1 ) =
and
Ap (¢) for all P E Fix(¢).
Md,
Thus in some sense the function ¢ >.q,(P) is a function on where P is a fixed point of¢. Unfortunately, this is not quite correct, since there is no way to pick out a particular fixed point P for a given map ¢. Recall that a rational map ¢ of degree d has d + 1 fixed points counted with appropriate multiplicities, say f-+
The points in Fix(¢) do not come in any particular order, so the set of multipliers for the fixed points, is an unordered set of numbers, but as a set, it depends only on {¢), the PGL2 equivalence class of ¢. Hence if we take any symmetric function of the elements in this set of multipliers, we get a number that depends only on { ¢). The elementary symmetric polynomials generate the ring of all symmetric func tions, so we define numbers
by the formula
d+ l T + Ap(¢) ) I�>·i (¢)Td+l - i . ( IT PEFix(c/>) i=O In other words, the quantity O"i ( ¢) is the elementary symmetric polynomial of the multipliers Ap1 ( ¢) , . . . , Apd+l ( ¢). From this construction, it is clear that O"i ( ¢! ) = O"i (¢) for all f E PG12 • Further, if we treat the coefficients of¢ [Fa, Fb] as indeterminates, then the fixed points Pi and the multipliers >.pi ( ¢) of ¢ are algebraic over Q( . . . , bd ) and form a com plete set of Galois conjugates, from which it follows that symmetric expressions in the A pi ( ¢ ), for example the functions O"i ( ¢ ), are in the field Q( . . . , bd ). With a bit more work, which we describe in greater generality later in this section, one can =
i1h
=
ao ,
ao ,
show that the multipliers Api ( ¢) are integral over the ring
Q[Md] ·
and hence that O"i(¢) E Thus symmetric polynomials in the multipliers of the fixed points of¢ are regular functions on the moduli space
Md.
181
4.5. Periodic Points, Multipliers, and Multiplier Spectra
Example 4.45. We illustrate the construction of Example 4.44 for rational maps of
degree 2. As usual, we write
The map rjJ has three fixed points P1 , P2 , P2 , and after much algebraic manipulation one finds that the first elementary symmetric function of the multipliers,
is given explicitly by the horrendous-looking formula
Notice that the denominator of a1 rjJ) is so a1 rjJ) is in It is far less obvious that this expression for a1 rjJ) is PGL2 -invariant. One can verify directly that a1 rjJ) = a1 rjJf ) by checking that a1 rjJ) does not change when a0, . . . , b2 are replaced by the expressions a� , . . . described in Example 4.34. We leave this task to the interested reader who has, we hope, access to a suitably robust computer algebra system. We have used the set of multipliers of the fixed points of a rational map rjJ to create PGL 2 -invariant functions on More generally, we can use the multipliers associated to periodic points of any order to create such functions. Recall from Section 4. 1 (page 149) that for any rjJ E we write
(
( ( Res(Fa, Fb), ( , b; (
(
IQ[Ratd].
Ratd.
Ratd r/Jn = [F¢,n (X, Y), G¢,n (X, Y)] with homogeneous polynomials F¢ , n , G¢ , n E K[X, Y] of degree dn . (See also Exer cise 4.9.) Then the set Pern ( rjJ) of n-periodic points of rjJ are the roots of the n-period polynomial ¢,n (X, Y) YF¢,n (X, Y) - XG¢,n (X, Y), and the set Per� ( rjJ) of formal n-periodic points of rjJ are the roots of the n1hdynatomic polynomial of rjJ, *,n (X' Y) = ll(YF,k (X' Y)-XG,k (X ' Y))11(n/k) = II ,k (X' Y)ll(n/k) kin kin where we proved in Theorem 4.5 that ¢ , n is a polynomial. The polynomial ¢ , n is homogeneous of degree dn + 1. For the purposes of this section, it is convenient to let Pern ( rjJ) be a "set with multiplicity" in the sense that a point appears in Pern ( rjJ) according to its multiplicity as a root of , n . Similarly, we denote the degree of ¢ , n by vd ( n) (see Remark 4.3) and we assume that points appear in Per� ( rjJ) according to their multiplicity as roots of ¢ , n . =
'
182
4. Families of Dynamical Systems
Definition.
Let ¢ E
Ratd. The n-multiplier spectrum of¢ is the collection ofvalues An (¢) = { Ap(¢) : P E Pern (¢) } .
The formal n-multiplier spectrum of¢ is the analogous set of values A�(¢) = { Ap(¢) : P E Per�(¢) } . In both sets, the multipliers are taken with the appropriate multiplicity. Example 4.46. Let ¢(z) = zd with d 2: 2. Then Per (¢) = U consists of the points and the ( dn - 1 ) 1h roots ofnunity. It is easy to check we have that >..o (¢) = Aoo (¢) = and for ( E
0, oo, 0,
{O,oo} J.t. dn _ 1
J..tdn _ 1
dn - 1
copies _____..._.
Hence
An (¢) = { O, O, dn, dn, . . . , dn . And if n 2: 2, then A� ( ¢) consists of rp( dn - 1) copies of dn, where rp (m) is the Euler totient function (not to be confused with the rational function ¢( z) ). Example 4.47. Let ¢(z) = z2 + bz. Then Per1 (¢) = Next we compute
{0, 1 - b,oo}
and
}
AI (¢) = {b, 2 - b,
0}.
cp2 (z) - z 2 -�."'' 2 = = z + (b + 1)z + b + l. ¢(z) - z *
The two points of formal period 2 are the roots of ¢ , 2 , *
(
Per2 ¢) -
{
}
- (b + 1) ± J(b + 1) 2 - 4 (b + 1) . 2
(4.30)
Letting o: and (3 denote these two values, we substitute them into to compute their multipliers, which tum out to be identical, Thus
A2 ( ¢) = { 4 + 2b - b2 , 4 + 2b - b2 } . The multiplier spectra An (¢) and A�(¢) depend only on the PGL2 -equivalence class of ¢, so we can use them to define functions on
M d.
4.5. Periodic Points, Multipliers, and Multiplier Spectra
183
Ratd and n 2:: 1. Define quantities O"in) (¢) for 0 :S i :S dn + 1 dLn+ l in) rjJ dn+ l -i .\) + (T = II i=O O" ( ) T . .A E An() Similarly, define quantities ;�n ) ( ¢) for 0 :S i :S vd (n) by vd(n) II (T + .\) = L (;.�n ) ( rjJ) Tdn +l - i . i=O .A E A;', ( Example 4.48. Continuing with Example 4.46, let ¢(z) = z d . Then II (T + .\) = T2 (T + dn ) dn - l and II (T + .\) = (T + dn )'PWl . .A E An () .A E A;'. ( ¢ ) Example 4.49. Continuing with Example 4.47, let ¢(z) = z 2 + bz. We computed AI (¢) = { b, 2 - b, O}, so II (T + .X) = (T + b)(T + 2 - b)T = T3 + 2T2 + (2b - b2 )T, .A EAt ( Let ¢ E by the relation Definition.
4>)
4>)
which gives
(jil l
0"�1 ) 2b 0"�1 ) 0. computed in (4.30), we find that
= b2 ' = Similarly, using the set A2 ( ¢) II (T + .X) (T + 4 + 2b - b2 ) 2 ' E ¢ =
2,
=
.A A; ( )
so
and
Theorem 4.50. For E n and i in the appropriate range, let and (;.� be the symmetric polynomials ofthe n-multiplier spectra of¢. (a) The/unctions
¢ Ratd, 2:: 1,
n) (¢)
and
Q[Ratd],
O"in) (¢) (4.31)
R ( ,. . )
are in i.e., they are rational functions in the coefficients ao , . , bd of the map ¢ = [Fa, Fb] with denominators that are a power of es Fa Fb . (b) Thefunctions (4.3 1) are PGL2 -invariant, and hence are in the ring of regular functions of the dynamical moduli space
Q[Md]
M d. �\ Proof We sketch the proof for O"in ) (¢) and leave ; n ¢) as an exercise for the reader (Exercise 4.26). (a) We write
4. Families of Dynamical Systems
184
0 there is a constant Cc such that u
for all d. In particular, the multipliers of a rationalfunction ¢ E Ratd determine the conju gacy class of¢ only up to De ( d� ) possibilities. -c
Proof We will prove this in Chapter 6 using Lattes maps associated to elliptic curves
with complex multiplication; see Theorem 6.62.
0
Following Milnor [300], we define the multiplier spectrum of a rational map ¢ to be the function A ( ¢) that to each positive integer n assigns the set An ( ¢). In other words, A(¢) is the set-valued function
Definition.
A(¢)
:
N
---+
Sets with Multiplicities,
4. Families of Dynamical Systems
188
Two rational maps are said to be isospectral if they have the same multiplier spec trum. Then another way to state McMullen's Theorem 4.53 is to say that aside from the flexible Lattes maps, the multiplier spectrum determines the rational map up to finitely many possibilities. The fact that the flexible Lattes maps form nontrivial isospectral families is proven in Section 6.5.
.1 - 1)22 (,\1 ,\2 - 1)(,\1 ,\3 - 1), (.-\2 - 1) = (,\2 ,\1 - 1)(.-\2 ,\3 - 1), (4.37) 2 (.-\3 - 1) = (,\3,\1 - 1)(,\3 ,\2 - 1). We now start with a rational map 2 a 1 z a2 ¢(z) aoz boz2 b1 z b2 and change coordinates in order to put ¢ into the desired form. ( a) For this part we assume that .-\ 1 .-\2 f- 1, so (4.37) tells us that .-\ 1 f- 1 and the fixed points associated to ,\ 1 and ,\2 have multiplicity 1 , so in .-\particular 2 f- 1. Thus they are distinct and we can find an element of PG 12 (q that moves them to 0 and respectively. After this change of variables has been made, the rational =
=
+
+
+
+
oo,
map ¢ satisfies ¢(0) = 0 and ¢( oo ) = oo, so it has the form with
ao
Since f- 0, we can dehomogenize by setting yields and
a0 1, and then a simple calculation =
Thus f has the form
¢(z) = zA2 2 Z b2 .-\b21 z with Finally, replacing ¢(z) by b2 1 ¢(b2 z) yields the desired form (4.35), and we calculate Res(z2 .A1 z, .A2 z + 1) = det .-\012 ,\11 001 1 - .-\1 .-\2 , ,\2 +
+
+
=
which completes the proof of (a). (b) The proof of this part is similar. We begin by moving the fixed point associated to to oo, so the map ¢ has the form
.-\ 1
with
.-\ 1 ,\ 2 1 ao. a0 1 b1
= combined with (4.37) tells us that The assumption that we have = Dehomogenizing = puts ¢ into the form
.-\1 = ,\2 = 1, so
4. Families of Dynamical Systems
192
Next we replace ¢(z) by ¢(z - b2 ) + b2 , so now ¢(z) looks like z 2 + a 1 z + a2 with ¢(z) z (Of course, the values of a 1 and a2 have changed.) Finally, replacing ¢(z) by ¢ ( y'a2 z) I y'a2 gives ¢ ( z) the desired form 1 ,�, z2 + a 1 z + 1 (4.38) '�-' ( z ) - z + a1 + - . z z We can compute the value of a 1 by observing that ¢ has a double fixed point at oo and that its other fixed point is at -a1 1 . (If a 1 = 0, then there is a triple fixed point at oo.) Thus the third multiplier is =
_
_
so we find that a 1 �. (This is also correct if a 1 of a 1 into (4.38) completes the proof of (b). =
=
0.) Substituting this value 0
We resume the proof of Theorem 4.56(b,ii). Let ¢1 and ¢2 be rational maps of degree 2 satisfying u ( ¢1 ) = u ( ¢2 ). Thus and and then the relation 0'3 0'1 - 2 from (a) implies that also 0'3(¢ 1 ) = 0'3(¢2 ). It follows that the set of multipliers of the fixed points of ¢1 and ¢2 are the same, say {>q , >.2 , >.3}, since they are the three roots (counted with multiplicity) of the polynomial T3 - 0'1 T2 + 0'2 T - 0'3. We consider two cases. Suppose first that >. 1 >.2 # 1. Then Lemma 4.59(a) says that ¢1 and ¢2 are 2-equivalent to the function ( z2 + >. 1 z) I ( >.2 z + 1), so they are PGL2 -equivalent to each other, and similarly if >. 1 >.3 # 1 or if >.2 >.3 # 1. We are left to consider the case >. 1 >.2 = >. 1 >.3 >.2 >.3 1. Then Lemma 4.59(b) tells us that >. 1 >.2 A3 1 and that ¢ 1 and ¢2 are both PGL2 -equivalent to the rational map z + z - 1 . This completes the proof of (b,ii), so we turn to (b,iii), the surjectivity of u. Given (8 1 , 82 ) E A2 (C), we set 83 = 8 1 - 2 and factor the polynomial =
=
=
=
=
=
Notice that the condition 83 = 8 1 - 2 gives the familiar relation (4.39)
193 M2 of Dynamical Systems of Degree 2 which in implies the usual formal identities (4.37) for -\ 1 , -\2 , -\ 3 . Suppose first that some Ai is not equal to 1, say ,\1 =/=- 1. Then ,\ 1 ,\2 =/=- 1 from (4.37), so we can set ¢(z) = zA22+Z +-\11z , since Res(z 2 + -\ 1 z, -\2 z + 1) = 1 - -\ 1 ,\2 from Lemma 4.59(a). The fixed points of ¢ are o: 1 = 0, o: 2 = oo, and 0:3 = i=�� . Its multipliers at o: 1 and o:2 are easily computed, 4.6. The Moduli Space tum
and
and the multiplier at
o:3 is a2 ( 82 . ,\1 = ,\2 = ,\3 1, a1 (¢) = 81 ¢(z) = z + z� 1
and ¢) = where we use (4.39) for the middle equality. Hence = which corresponds to the val We are left with the case ues 3. It is easy to check that the rational map has a triple-order fixed point at oo and that all three multipliers are equal to 1,
8 1 = 82 = so(c)u(¢)We briefl = (3,y3).sketch the proof, which uses basic methods from algebraic geometry. We refer the reader to [416, §5] for further details. The first step is to verify that the map u : M 2 ----> A2 is proper. This follows easily from the valuative criterion [ 198, 11.4. 7] using the fact proven in (b) that every fiber u� 1 ( t) consists of a single point. (Roughly speaking, a morphism of varieties over C is proper if its fibers are complete.) Next one checks that is finite, which can be proven using the fact that both M 2 and A2 are affine varieties (cf. [416, Lemma 5.6]). Alternatively, one can show in general that a proper quasifinite map ([299, that A2 and the bijectivity of u on complex points to prove that is an isomorphism (cf. [41 6, Lemma 5.7]). Finally, it is clear from the explicit formulas for a 1 and a2 in Remark 4.57 that u is defined over Q. (d) This is immediate from (c), which says that Q[M 2 ] = Q[ a 1 , a2 ], since we already know from Theorem 4.50(b) that ai n) and ;; n ) are in Q[M 2 ]. is a consequence of the fact that the isomorphism u : M 2 A2 in (c) is (e)definThis ed over Q. u
is finite
I, Proposition 1 . 1 0]). Then one uses the fact
is nonsingular
u
---->
0
Remark 4.60. Regarding the proof sketch of Theorem 4.56(c), we observe that a
W
morphism of irreducible varieties V may be bijective on complex points, yet not be an isomorphism. A simple example of this phenomenon is the cuspidal cubic map ---->
u : M 2 AA22. Itis does an isomorphism, it is crucial that u not appear to be known in general
Thus in the proof that ----> onto the nonsingular variety maps whether is nonsingular.
MM2 d
4. Families of Dynamical Systems
194
M2 2 A (a1 , a2 ) M2 M� M M�8, M2 • 2. Theorem 4.61. Let M 2 M� M�8 be the completion ofM 2 constructed using geometric invariant theory in Theorem 4.40. Then the isomorphism of dynamical systems The content of Theorem 4.56 is that the moduli space of degree 2 is isomorphic to and an explicit isomorphism is provided by the pair of functions created from the multipliers of the fixed points of a rational map. From our general theory (Theorem 4.40), the space sits naturally inside two larger spaces and but since d 2 is even, these two spaces coincide We conclude this section with a description of and will be denoted by =
=
extends to an isomorphism ii mutes:
=
:
M 2 IP'2 such that the following diagram comM 2 � A2 x (x t,y) 1 1 [ , , 1] -+
M2 (C) M2 (C) cPA,B (X, Y) [AXY, XY + BY2],
The points in that are not in correspond to degenerate maps of degree 2 that may be informally described as maps of theform
(4.40)
=
The point [A, B] is uniquely determined up to reversing A and B. D
Proof See [302] and [416, Theorem 6. 1 and Lemmas 6.2 and 6.3].
Remark 4.62. We expand briefly on what it means to say that the points in the
(M2 " M2 )(C)
set correspond to the maps cPA,B given by (4.40). Let U be a neighborhood of 0 and let
C
A1
be a rational map that is a morphism away from 0. We denote the image of t E U by 4>t to help remind the reader that 4>t is itself a map, i.e., 4>t Consider the composition :
IP'1 IP'1 . -+
which by abuse of notation we again denote by 4>. It is a rational map, and since U c and M2 is complete, we see that it is a morphism from U to In partic ular, the point 4>0 is a well-defined point in If 4>o E there is nothing to say, so we suppose that 4>0 � Then the second part of Theorem 4.61 means that possibly after choosing a smaller neighborhood of 0, there exists a morphism
A1
M2 (C). M2 (C).
M2 (C), M 2 .
4.7. Automorphisms and Twists f
195 :
U
------+
PGL2
such that the conjugate map ¢! has the form with a0 , . . . , b2 regular functions on U and satisfying ao (O)
= a2(0) = bo (O)
=
0,
b 1 (0)
=
1,
and
In other words, the degeneration of ¢! at t 0 has the form [AXY, XY + BY 2] . Further, the map ¢ determines the point [A, B ] = a 1 (0), b2 (0)] E lP'1 () = ¢. We have proven that ¢! = ( ¢f ) for all E Gal( K IK), which shows that ¢! ( z) E K(z)In andorderthusto completes the proof that (b) implies (a). prove the cohomological description of Twist( ¢1 K), we first use Proposition 4.77, which says that there is a natural embedding of Twist( ¢1 K) into the cohomology set H 1 (Gal(KI K), Aut(¢)) (cf. also Remark 4.78). Then the equivalence of(a) and (b) tells us that an element of H 1 ( Gal(KIK), Aut( c;b)) comes from an element of Twist( ¢1 K) if and only if it becomes trivial when mapped to D H1 (Gal(KIK), PGL2 (K)).
Hence is the 1-coboundary associated to the element This proves that (a) implies (b). Next suppose that is an 1-cocycle that is a 1-coboundary. This means that there is an with the property that for all We claim that ! is a twist of with 1-cocycle What we need to check is that ¢! is defined over i.e., that ¢! since once we know that, it is clear from the definitions that is its associated 1-cocycle. Let Then
g
(J
0'
0'
0'
0'
Remark 4.80. A formal argument with commutative diagrams shows that
Br(K)
K.
where is the Brauer group of The Brauer group plays an important role in class field theory and many other areas of number theory and arithmetic geom etry. For example, so has only two elements, and the same is true for finite extensions of !Qp. The Brauer group of a number field is more complicated. Example 4. 81. Let be a field of characteristic not dividing n, and let be a rational map whose automorphism group is
Br(!Qp) = IQIZ, Br(!Qp)[2]
K
¢( z) E K( z)
Aut(¢) = { (z : ( E JLn }, where we recall that JLn K denotes the set of nth roots of unity. Then there is an isomorphism (4.49) K*IK* n Twist(c;I>IK), c
�
4.9. Twists of Rational Maps
205
(- 1 ¢((z) ( J.tn · ¢(z) 1 z---+ (z, ¢(z) z'lj;(zn )¢(z) z'lj;(bz'lj;n( z)), K(z) K(z). z -
Note that by assumption we have for all E In particu lar, the function so it has the is invariant under the substitution form for some E (Exercise 4.37). Thus the twist of given in (4.49) is equal to so it is indeed in Since we are assuming that the automorphism group of ¢ is isomorphic to not merely as an abstract group, but also in terms of the way in which acts on Aut(¢) and on a standard result in Galois cohomology9 says that there is an isomorphism =
=
J.tn ,
¢(z) Gal(RIJ.tK)n ,
(en--+ ) a ( V'b
) . (4.50) V'b This allows us to identify Twist (¢I with a subset of In order to show that they are isomorphic, Theorem 4.79 says that we must show that every cocy cle in (4.50) becomes a coboundary when we consider it as a cocycle with values in But this is clear, since with our identification of with Aut(¢), the cocycle in (4.50) is equal to
b
1----t
K* IK* n . J.tn z with f(z) V1J E PGL2 (K).
K)
PGL2 (K).
=
Note that this example generalizes Example 4.75, which dealt with the case 2. Example 4.82. Let E be a rational map with automorphism group Aut(¢) (See Exercise 4.35.) The Galois group acts triv ially on Aut(¢), so we have n =
=
z,{ z- 1 }.¢(z) K(z)
Gal(KI K)
The isomorphism is given explicitly by associating to any b E (j 1-----)
{ zz-1
if a ( v'b ) if ( v'b) a-
= =
v'b, -v'b.
K* / K * 2
the cocycle
(4.5 1)
To ease notation, we let (3 v'b and let g be the cocycle described by (4.5 1 ). Then Theorem 4.79 says that g comes from a twist of ¢ if and only if there is some E satisfying 9a ). Using the fact that g E we are looking for an E satisfying 9 This statement is equivalent to H1 (Gal( K / K), K * ) 0, which is a version of Hilbert's Theo
a (z)
=
f PGL2 (K) PGL (K) fo-(f- 1 f 2 =
=
rem 90. Using this and taking Galois invariants of the Kummer sequence 1
----+
1-Ln
----+
yields the cohomology long exact sequence
K*
0: �----+ 0: n
------+
K*
----+
1
{ z, z - 1 } ,
206
4. Families of Dynamical Systems
if a( j3) j3 , a(f)(z) { f(z) 11 f(z) if a(j3) -j3. It is not hard to construct such an f, for example f(z) -j3j3zz++11 Note that det ( !!3 � ) 2j3 =1- 0, so f is in PGL2 (K). Hence every gives a twist of ¢, and indeed we can write the twist associated to b j32 E K* IK* 2 explicitly using f, ) ¢ + j3 j3 ( !;:: 1 ¢f (z) ( )+1 For example, let Md ( z) z d be the t:fh-power map. Then a judicious use of the binomial theorem yields a formula for the b-twist Mdb} of Md, " " Mj \ z) � C�) b' z �� ( � ) b' z + ' 2k 1 In particular, the first few b-twists of Md are 1 + 3bz2 ' (b) M3 (z) 3z + bz3 =
=
=
=
g
=
=
=
_d.
'I'
.
f]z+l -f]z+l
=
"
�
·
_
4.10
Fields of Definition and the Field of Moduli
If¢( z) is a rational map whose coefficients lie in a field K, then it is interesting to study the orbits of points whose coordinates lie in K or in finite extensions of K. The smallest field K containing the coefficients of ¢ can generally be determined by inspection. For example, the map ¢(z) z 3 + 1 is clearly in Q(z) and the map ¢(z) z 3 i is just as clearly in Q(i)(z). However, if we make a change of variables f ( z) iz in the latter map ¢( z) z3 i, we find that ¢! (z) r 1 (¢(f(z))) -i¢(iz) - i( -iz3 i) - z 3 + 1 E Q(z). Thus the map ¢(z) z 3 + i is really a Q(z) map that has been altered by an injudicious change of variables. Definition. Let K be a field of characteristic 0, let ¢( z) E k ( z) be a rational map with coefficients in an algebraic closure of K, and let K' IK be an extension field. We say that K' is afield ofdefinitionfor ¢ ifthere is a linear fractional transformation f(z) E PGL2(K) such that =
=
+
=
=
=
=
=
+
+
=
=
In other words, K' is a field of definition for ¢ (or an FOD for short) if, after a change ofvariables, ¢ has coefficients in K'.
4.10. Fields of Definition and the Field of Moduli
207
As in Section 4.9, we can use Galois theory to investigate the possible fields of definition of a given rational function ¢( z) E k ( z) . We let Gal( kI K) act on k ( z) in the natural way by applying CJ E Gal ( k IK) to the coefficients of ¢ E k ( z), CJ ( ¢) = CJ
(
ao + a 1 z + · · · + adz d bo + b1z + · · · + bdz d
)
=
CJ(ao) CJ(a 1 )z + · · · + CJ(ad)zd CJ(bo) + CJ(b l )z + · · · + CJ(bd)zd ·
+
Then Galois theory (and Hilbert's Theorem 90) tell us that
(
)
¢ E K' (z) for the field K' = fixed field of { CJ E Gal(k I K) : CJ( ¢) = ¢} . Suppose instead that ¢ is merely equivalent to a map whose coefficients are in some field K'. That is, suppose that there is a linear fractional transformation f E PGL2 (k) such that ¢! E K'(z). Then CJ(¢f) = ¢! for all CJ E Gal(KIK'), so f- 1 ¢! = ¢! = CJ(¢f) = CJ( j - 1 ¢! ) = CJ( j- 1 )CJ(¢)CJ(f) = CJ( j ) - 1 CJ(¢)CJ(f). Solving for CJ( ¢) yields CJ(¢) = CJ(f)f - 1 ¢fCJ(f - 1 ) = (f(J(f- 1 )r 1 ¢(JCJ(f - 1 )) = ¢fa(r 1 J . Thus CJ(¢) is equal to ¢ conjugated by the map jCJ(f - 1 ) E PGL2 (K), so in partic ular CJ( ¢) and ¢ are equivalent. We have proven that
(defi K' is a field of ) (afor9aevery CJ Q_al(k IK') there exists ) nition for ¢ E PG 12 ( K) such that CJ ( ¢) = ¢g" E
===}
·
(4.52)
Turning this around, we start with an arbitrary rational map ¢(z) E K(z) and study the automorphisms CJ E Gal( k IK) for which u( ¢) is equivalent to ¢. Definition. Let rjJ(z) E K(z) be a rational map. We associate to rjJ a subgroup Gq, of Gal(k IK) and a field Kq, defined by Gq, = {CJ E Gal(KIK) : u(¢) = ¢9" for some ga E PGL2 (K) }, Kq, = fixed field of Gq, = { a: E k : CJ(o:) = a for all CJ E Gq,}. The field Kq, is called the field ofmoduli (FOM) of¢. Remark 4.83. The group PGL2 (K) acts on the space of rational maps via the usual conjugation action, ¢! = f - 1 ¢f. If we consider as usual the equivalence class consisting of all maps that are conjugate to a particular map ¢, then Gq, is the sub group of Gal(kI K) consisting of elements that map the set [¢] to itself. Equiv alently, if ¢ has degree d, let (¢) be the image of ¢ in the moduli space M d = Ratd I PSL 2 that we defined and studied in Section 4.4. The space M d is defined over Q (Theorem 4.36) and the field of moduli of ¢ is exactly equal to the field generated by the coordinates of the point (¢) E Md.
208
4. Families of Dynamical Systems
We begin by proving some elementary properties ofFOM and FOD, in particular, the important fact that the field of moduli is contained in every field of definition. Proposition 4.84. Let
¢(z) E ofGal(K K(z). I
(a) The set Gq, is a subgroup K). (b) Let K' be afield of definition for ¢. Then Kq, FOM , we may assume that K is the field of moduli of ¢ This means that for every E Gal(K I K) there exists a ga E PGL2 (K) satisfying .
a
Note that the assumption (4.53) that Aut(¢) 1 implies that ga is uniquely de termined by The next proposition describes some of the properties of the map a I-t
=
ga . a.
z)=
d
K ( be a rational map of degree 2 2 with field of moduli K and satisfying Aut(¢) 1, andfor each E Gal( K I K) write ¢9rr as above. (a) The map
Proposition 4.86. Let ¢ E
a
a(¢) =
is a 1-cocycle, i.e., it satisfies for all a, T E Gal(K I K).
(b) K is afield ofdefinition for ¢ ifand only if is a 1-coboundary, i.e., ifand only
f
ifthere is an E PGL2(K) such that
g
a
for all E Gal(K I K).
4. Families of Dynamical Systems
210
Proof
a,
(a) Let T E Gal(KI K). Then
a ( (¢)) = a (rjJ9T ) a(r/Jt(gr ) = rjJ9 1 defined over k such that the 1cocycle Gal(KIK) ---+ PGL2 (k), a � ia(i- 1 ), is equal to the 1-cocycle g. Hence C is a twist ofJP> 1 IK, and C is the trivial twist ofJP>1 IK ifand only if g is a 1-coboundary. Proof We construct the curve C by describing its field of rational functions. Note that the field of rational functions for the curve JP>1 is the field k ( z) and that the Galois group Gal( kI K) acts naturally on k ( z) by acting on k and leaving z fixed. Now consider another field of rational functions in one variable K(w) , but this time with a "twisted action" of Gal(kI K). We define this twisted action by let ting Gal(k I K) act on k as usual, but setting a(w) = g;; 1 (w). In other words, if g17(z) = (az + b)l(cz + d), then a(w) = g;; 1 (w) = -dwcw-+ba , 10
:
---+
and for any rational function
we have
+ a(at)g;; 11 (w) + a(a2 )g;;11 (w)22 + + a(ad )g;;11 (w)dd . a(F)(w) = a(ao) a(bo) + a(bt)g;; (w) + a(b2 )g;; (w) + + a(bd)g;; (w) It is clear that a(F + G) = a(F) + a( G) and that a(FG) = a(F)a( G). However, to be an action, we must also have (ar )(F) = a( r (F) ). We use the cocycle relation to verify this condition, a(r(w)) = a(g; 1 w) = a(g; 1 )a(w) = a(g; 1 )g;; 1 (w) = g;;) (w) = (ar)(w). We now look at the subfield of k (w) that is fixed by the twisted action, = {F E K(w) : a( F) = F for all a E Gal(KIK)}. · ··
· · ·
K
We prove that the field K has the following properties:
100ne says that g is a continuous 1-cocycle for the profinite topology on Gal( K/ K) and the discrete topology on PGLz (K).
4. Families of Dynamical Systems
212
(i) K n k = K. (ii) K has transcendence degree 1 over K. (iii) kK
=
k(z).
To verify (i), let a E K n k . Then a E k , so the action of Gal( k I K) on a is the usual Galois action. On the other hand, the fact that a E K means that the Galois action is trivial. Hence a E K. This shows that K n k c K, and the opposite inclusion is obvious. Next let Ll K be a finite Galois extension with the property that g7 1 for every T E Gal( k I L ) . This implies that the action of an element (} E Gal(k I K) on w depends only on the image of(} in the quotient group =
Gal(LIK)
=
Gal(KIK)I Gal(KIL) ,
since ifT E Gal(KIL), then ((J T ) ( ) W
= (}
(gT ( )) W
=
(} ( ) . W
It follows that the polynomial P(T)
=
II
(T - A(w))
=
II
(T - g): 1 (w)) E k (w) [T] (4.54)
.XEGal(L/K)
.XEGal(L/K)
is well-defined. Further, we claim that the coefficients of P(T) are in K. To see this, for any (} E Gal(kI K) we note that (J(P) (T)
=
II
(T - (JA(w))
=
P(T) ,
.XEGal(L/K)
since the effect of replacing A with (}A is simply to permute the order of the factors in the product. We have now constructed a polynomial P(T) E K[T], and we observe that w is a root of P(T). Hence by definition, the extension K( w) I K is an algebraic extension. Since the element w is transcendental over K, this proves that K has transcendence degree at least 1 over K. On the other hand, K is contained in the field k (w), and it is clear that k ( w) I k has transcendence degree 1 over K. Therefore K has tran scendence degree exactly 1 over K, which proves (ii). Finally, consider the splitting field .C over K of the polynomial P(T). As already noted, we have w E .C. The definition of P(T) shows that there is a natural surjection Gal(LI K) -----* Gal(.CIK), which implies in particular that .C LK. Hence w E LK, which proves that K(z) c KK. This gives (iii), since the other inclusion is true from the definition of K. We now have the tools needed to complete the proof of Proposition 4.87, but we pause briefly for an example illustrating the general construction. =
213
4.10. Fields of Definition and the Field of Moduli
Example 4.88. Define a map by the rule
g
= i, ga (z) = { z- 11z ifcr(i) . f (T ( z ) - -z. 1
"
.
It is easy to check that is a 1-cocycle, and indeed it is the 1-cocycle described in Example 4.85. Fix an embedding of Ql into C and let Gal(QliQ) denote complex conjugation. The twisted action on Ql( is given by
g
p(a) =
pE
w)
a E Ql and p(w) = -1lw. The field is the subfield of Ql( w) consisting of elements that are fixed by the twisted action. The coefficients of the polynomial P(T) defined by equation (4.54) /(
a
for
give us elements in JC,
P(T) = (T - w)(T - p(w)) = (T - w) (T + � ) = T2 - (w - � ) T - 1. This yields one interesting element in JC , namely u = w - w - 1 , and we observe that the quantity v = i( w + w - 1 ) gives another element in JC. It is not hard to show that and v generate JC, = K(u, v) = K (w - �' i (w � )) . (See Exercise 4.45.) Of course, u and v are not independent, since 2 u2 v 2 = (w - �) + (i (w + �)) = -4. The field is the function field of the curve C u2 + v 2 = -4. Notice that C is defined over Q, and C is Ql-isomorphic to JP 1 , but C is not «;))-isomorphic to JP 1 , since C(Q) = 0. From our general theory, the fact that C is a nontrivial twist of JP1 IQ is equivalent to the fact (proven by a direct calculation in Example 4.85) that the 1-cocycle g is not a 1-coboundary. u
/(
+
+
2
J(
:
Resuming the proof of Proposition 4.87, we have constructed a field J( that is the function field of a curve CI K, and we have an isomorphism
= k(w) � k(z), w z, that induces a K-isomorphism C ---+ JP1 . In other words, the functions w on C and z on JP 1 are related by the formula w = z i. The curve C is a twist of lP 1 I K, and its associated cocycle is given by cr ---+ icr(i- 1 ). We compute KJC
+------+
i
:
o
214
4. Families of Dynamical Systems
since a(z) = z, z o a(i) = a(z o i) = a(w) since w = z o i, 1 by definition of the twisted action on k ( w), = g;; (w) dw - b where 9a (z) = az + b E PGL2 (K),-cw dz o +i -a b since w = z o i,cz + d -cz o i + a = Z 0 9a- 1 0 z. Thus a(i) = g;; 1 o i, which proves that 9a = ia(i - 1 ) is the 1-cocycle associated to the k-isomorphism C IP'1 . This completes the proof that the algebraic curve C is a twist of IP' 1 IK whose associated 1-cocycle is Finally, Proposition 4.77 tells us that C is the trivial twist --
.
i
:
--+
g.
D
if and only if its associated 1-cocycle is a 1-coboundary.
=
K(z)
Returning to the question of FOD FOM, let ¢(z) E be a rational func tion with field of moduli K and trivial automorphism group. We have constructed a 1-cocycle Ga K) PGL2 (K) that is associated to ¢ (Proposition 4.86) and a twist C,p of K that is associated to the 1-cocycle (Proposition 4.87). We have also proven the following chain of equivalences:
g : l(kI
--+
IP'1 I
g
K is a field of definition for ¢ ¢==} is a 1-coboundary ¢==} Cq, is K-isomorphic to IP' 1
g
(Proposition 4.86), (Proposition 4.87).
It remains to find a way of determining whether a twist of IP' 1 is K-isomorphic to We will use the Riemann-Roch theorem to provide two sufficient conditions for resolving this problem. For the convenience of the reader, we recall the general statement of the Riemann-Roch theorem, although we will need it only for curves of genus 0.
IP'1 .
I
Theorem 4.89. (Riemann-Roch Theorem) Let C K be a smooth projective curve ofgenus defined over K. (a) There is a divisor on C ofdegree that is defined over K. (b) Let D be a divisor on C that is defined over K and assume that the degree ofD satisfies deg(D) 2: 1. Then there is afunction f E K(C) satisfying
g
2g -
2g - 2
div(f)
+ D 2: 0.
Corollary 4.90. (Riemann-Roch in Genus 0) Let C K be a smooth projective curve ofgenus 0 defined over K. (a) There is a K-rational divisor ofdegree on C.
2
I
215
4.10. Fields of Definition and the Field of Moduli
(b) Let D E Div(C) be a divisor defined over K and satisjjling deg(D) there is afunction f E K(C) with div f D 2 0.
( )+
2
1. Then
Proof The proof of the Riemann-Roch theorem over an algebraically closed field
is given in most introductory texts on algebraic geometry, such as [198, IV. 1 .3], or see [255, Chapter I] for an elementary proof due to Weil and [41 0, II, §5] for an overview. The divisor of degree 2g - 2 in (a) is a canonical divisor, i.e., the divisor of any K-rational differential form such as df for any nonzero f E K (C). In particular, for curves of genus 0 we get a K-rational divisor D of degree -2, so -D is a K rational divisor of degree 2. D Proposition 4.91. Let C be a twist ofiP'1 I K. Thefollowing are equivalent:
(a) C is the trivial twist of IP' 1 I K, i.e., C is K-isomorphic to IP'1 . (b) C(K) is nonempty, i.e., C has a point with coordinates in K. (c) There is a divisor D ofodd degree such that D is defined on IP'1 =
I: ni (Pi )
(R) I:n (a(P )) I:n (P ) as a i i i i
over K, i.e., for all a E Gal(KIK) we have formal sum ofpoints.
=
Proof If C is the trivial twist of!P'1 I K, then there is a K-isomorphism j C IP' 1 . In particular, j C K) IP' 1 ( K) is a bij ection, so C K) is certainly nonempty.
( This proves that (a) implies (b). :
--+
P
:
(
--+
(P)
Next, it is clear that (b) implies (c), since if E C(K), then the divisor D has odd degree (one is an odd number!) and is clearly defined over K. Finally, suppose that the degree deg(D) of D is odd and that D is defined over K. We also use the fact that C and IP' 1 are -isomorphic, so in particular C is a curve of genus zero. It follows from the Riemann-Roch the orem (Corollary 4.90(a)) that there is a K-rational divisor on C having degree -2, say
n=
=
= I: ni
= I:Rni (Pi )
Consider the divisor E
= D + -n -2-1 D' = (PI ) + (P2 ) + + (Pn ) - -n -2-1 ((QI ) + (Q2 )). ···
The divisor E is defined over K and has degree 1, so the Riemann-Roch theorem (Corollary 4.90(b)) tells us that there is a rational function 'if; on C such that 'if; is defined over K and 'if; has degree In other words, 'if; is a map 'if; C IP' 1 of degree 1 defined over K, and hence C is K-isomorphic to IP'1 . This shows that (c) implies (a) and completes the proof of the theorem. D
1.
:
--+
We have now assembled all of the tools that we need to prove the main theorem of this section. We state the theorem in full generality, but give the proof only for rational maps with trivial automorphism groups. The general case is proven similarly, but there are many additional technical complications. Theorem 4.92. Let ¢( z ) E
z ) be a rational map of degree d 2 2. Then the field of moduli of¢ is afield ofdefinition for ¢ in the following two situations:
R(
216
(a) (b)
4. Families of Dynamical Systems
¢(z) has even degree. ¢( z) is a polynomial.
Proof We prove the theorem under the assumption that Aut(¢) = 1. See [414] for
a proof in the general case. Without loss of generality, we may assume that K is the field of moduli of¢. Let Gal( R I K) PGL2 K) be the 1-cocycle associated to ¢ (Proposition 4.86) and let Cq, be the twist of I K associated to the 1-cocycle (Proposition 4.87). This means that there is a R -isomorphism
g
:
--+
lP'1 (
g
such that Using many of the results proven in this chapter, we have the following chain of equivalences: K is a field of definition for ¢ {=:::}
g is a 1-coboundary
{=:::}
Cq,(K) is not empty
{=:::}
{=:::}
'1/J
:
Cq, is K-isomorphic to lP' 1
(There is a divisor on Cq, (K) of) odd degree and defined over K
(Proposition 4.86), (Proposition 4.87), (Proposition 4.91 ) , (Proposition 4.91 ).
We are going to produce divisors and points on the curve Cq, using the map Cq, Cq, defined by the composition --+
- 1 ¢i is defined over K. To verify this, we i '1/J a ( a('l/J) a(i- 1 ¢i) = a(C 1 )a(¢)a(i) = a(i- 1 )¢9" a(i) a(i- 1 )g;/ ¢gaa(i) = a(i- 1 )(ia(C 1 )) - 1 ¢ia(i- 1 )a(i) C 1 ¢i '1/J. Hence '1/J Cq, Cq, is defined over K. We also note that ¢ and '1/J have the same degree, since i is an isomorphism. (a) Let D,p be the divisor of fixed points of '1/J, that is, the collection of fixed points of '1/J counted with appropriate multiplicities. In the language of algebraic geometry, D,p is the pullback by the diagonal map P 1------t ( P, P) , ofthe graph { (x, 'ljJ(x)) : x E Cq,} of'ljJ. A map of degree d has exactly d + 1 fixed points (counted with multiplicities), and if the map is defined over K, then the divisor of fixed points is defined over K,
We begin by checking that the map let E G al KI K) and compute
=
=
=
:
=
=
--+
i.e., it is fixed by the Galois group. Thus D,p is a divisor of degree d + 1 on Cq,
4.10. Fields of Definition and the Field of Moduli
217
and D,p is defined over K. By assumption, d is even, so D,p has odd degree. Hence by the chain of equivalences derived earlier, the field K is a field of definition for ¢. (b) The map ¢ lP'1 lP' 1 is a polynomial by assumption, so it has a totally ramified fixed point P. In other words, ¢( P) = P and the ramification index of¢ at P satisfies p ( ¢) d. The map i C
0 for n = tpk ,
exactly two t's and all k ::0: 0
Table 4.4: Values of aj,(n). 4.4. Let ¢(z) E K (z) be a rational function of degree d ::0: 2 and suppose that z = 0 is a fixed point of ¢ with multiplier A = 1. Write ¢(z) = z + z e 'lj;(z) with e ::0: 2 and '!j;(O) =!= 0, where e = ao (¢, 1) in the notation of Theorem 4.5. (a) Prove that ¢n (z) z + nz e'lj; (z) + O (z2e- 1 ) . =
(b) Assume further that K has characteristic p > 0 . Prove that
for all k ::0:
1.
4.5. Verify that the description of the values o f aj, (n) given in Table 4.4 i s correct. 4.6. Let ¢( z) be a rational function, let n ::0: 1, let p be a prime with p f n, and let P E (a) Prove that a ? (n , rjJP ) = a ? (n , ¢) + a ? ( np , ¢) . th (b) Deduce that the n dynatomic polynomial for cjJP factors as
(c) More generally, if gcd(n , k)
=
1P' 1 .
l, prove that
*n k ,¢ - Il (n* ,¢J. ) p( k fj)
·
jjk
4.7. * Let
a _ 1 zd-l + + a 2 z2 + a1z + ao d be a monic polynomial of degree d and let �. ¢ (z) be its nth dynatomic polynomial. Each root a of � , ¢ (z) has an associated multiplier A ¢ ( a ) = ( ¢n ) ' (a) . ¢(z) = zd
+
· · ·
(a) Prove that there is a (unique) monic polynomial whose nth power satisfies has a point of formal period n whose exact period m is strictly smaller than n if and only if there is a point of formal period m whose multiplier is a primi tive (n/m)1h root ofunity. 4.8. This exercise describes an analogue of Theorem 4.5 for automorphisms of projective space. Let 1> : IP'N _, IP' N be an automorphism defined over a field K, i.e., 1> E PGL N +l (K). We say that 1> is nondegenerate if the equation 1>( P ) = P has only finitely many solutions in IP'N (K). (a) Let A E GL N+ 1 (K) be an invertible matrix with coefficients in K representing the map ¢>. Prove that 1> is nondegenerate if and only if every eigenspace of A has dimen sion 1. (b) Assume that 1>n is nondegenerate, let rn c IP'N X IP'N be the graph of 1>n ' and denote the diagonal map by f:J. : IP'N _, IP' N x IP'N . Following Remark 4.4, we define the 0-cycle of n-periodic points of 1> to be the pullback
¢, n
= !:J. * (f¢) = L ap( cj>, n) ( P) PE'f' N
and the 0-cycle of formal n-periodic points to be
;, n = Let P E
L ll (�) kin
¢,n =
L af, (cj>, n)(P).
PEP N
IP'N be a point of primitive period m for ¢>. Prove that
a p ( n) =
{ ap(m) 0
ifmln, ifm f n,
{
ap(m) ( a*p n ) _
and
0
ifm = n, if m =1- n.
In particular, ;, n is an effective (i.e., positive) cycle. 4.9. Let F, G E K[X, Y] be homogeneous polynomials of degree d ::::: 2 with no common factors and let 1> = [F, G] : IP'1 _, IP'1 be the associated rational map. Define two sequences of polynomials inductively starting with Fo (X, Y) = X and Go (X, Y) = Y and then for n ::::: 0,
Fn + l = Fn (F(X, Y), G(X, Y))
and
Gn + l = Gn (F(X, Y), G(X, Y)) .
(a) Prove that Fn and Gn have no common factors.
227
Exercises (b) More precisely, prove that the resultant of Fn and Gn is given by Res(Fn, Gn) = Res(F, G) (2n - l )dn - l . (Hint. Use Exercise 2. 12.) (c) Prove that ¢n = [Fn, Gn]. (d) For all n , m � 0 , prove that
Fn+m(X, Y) Fn (Fm(X, Y), Gm(X, Y)) , Gn+m(X, Y) = Gn (Fm(X, Y), Gm(X, Y)) . =
4.10. With notation as in Exercise 4.9, we define the (generalized) (m, n )-periodpolynomial of 1> to be the polynomial
m,n(X, Y) n (Fm(X, Y), Gm(X, Y)) , where n(X, Y) = YFn - XGn is the usual n-period polynomial of ¢. =
(a) Prove that
m,n(X, Y) = Gm(X, Y)Fn+m(X, Y) - Fm(X, Y)Gn+m(X, Y). (b) Let P E lP' 1 ( K). Prove that
m,n(P) = 0 if and only if ¢m+n (P) ¢m (P) . Thus P is a root of m,n if and only if P is a preperiodic point with "tail" of length at =
most m and with period dividing n. (c) Prove that for all m, n � 1, the quotient
m,n m-l,n is a polynomial. 4.11. We continue with the notation from Exercises 4.9 and 4.10. Let �(X, Y) be the n1h dy natomic polynomial of ¢. Then for m, n � 1 we define the (generalized) (m, n)-dynatomic "polynomial" of¢ to be
� (Fm(X, Y), Gm(X, Y)) . m ' n (X, Y) (X, Y) ) ;', ( Fm-l(X, Y), Gm l Prove that ;, ,n (P) = 0 if and only if 1>m (P) has formal period n. Points satisfy ing ;, ,n (P) = 0 are called preperiodic points withformal preperiod ( m, n). ** Prove (or disprove) that ;, ,n (X, Y) is a polynomial. *
(a) (b)
=
Section 4.2. Quadratic Polynomials and Dynatomic Modular Curves 4.12. We continue with the notation from Exercise 4.7. Thus for any monic polynomial ¢(z), we define a monic polynomial on ,q,(x) by
4>n.q,()=O
4> n .q,()=O
and for min with m < n we set
�n,m = Res(Cn;m (x), 8m, q, (x)), where Ck (x) is the k1h cyclotomic polynomial. We now specialize to 1>c(z) write 8n(c, x) 8n, ¢c (x) and �n,m (c) to indicate the dependence on c. =
=
z2 + c and
228
Exercises
{a) Prove that Dn (c, x) E Z[c, x] and that � n ,m (c) explain the powers that appear in Table 4.3. (b) Prove that
E Z[c]. Then use Exercise 4.7(c) to
deg z ;', 8n (c, 2n x) is in Z[c, x]. {This eliminates many powers of 2 from the coefficients of Dn ( c, x).) Compute 8n ( c, x) for n 1 , 2 , 3, 4. ** Prove that there is a monic polynomial W n, m(x) E Z[x] such that �n,m(c) W n ,m (4c), Jn (c, x)
=
T
=
(c)
=
and deduce that the set
{c E Q : �n , m(c)
=
0 for some m
)(z) = c/>"'(z) c/>' (z) 2 c/>' (z) It measures the difference between 4> and the best approximation to 4> by linear fractional transformations. (a) If f(z) (az+ b)/(cz+d) is a linear fractional transformation, prove that (Sf)(z) = 0. (b) Let f be a linear fractional transformation. Prove that 4> and f o 4> have the same Schwarzian derivative. (c) Let c/>(z), 'lj;(z) E K(z) be nonconstant rational functions. Prove that
( )
=
(S(c/> o '¢)) (z)
=
(Sc/>) ('¢(z)) · ('¢' (z)) 2 + (S'Ij;)(z).
(d) In particular, iff is a linear fractional transformation, prove that and deduce that the quadratic differential form
232
Exercises
wq, (z)
=
(S¢)(z) (dz) 2
is invariant under the substitution ( ¢, z) f-+ ( ¢1, f - 1 z ) . Thus the map ¢ a natural map Md (quadratic differential forms on lP' 1 ). (e) Suppose that ¢(z) has a multiple zero or pole at z = a , say
f-+
wq, induces
---->
¢(z) a(z - a) m + =
with a =/= 0 and lml
· · ·
2: 2.
Prove that S¢ has a double pole at a. More precisely, prove that the Laurent series expansion of S¢ around a looks like
(S¢)(z)
=
1 - m2
-- (z - a ) - 2 + · · · . 2
(f) Prove that the map
(¢, P)
,___..
(S¢)(P),
is a morphism.
** An algebraic variety V is called unirational if there is a rational map lP'N ---+ V whose image is Zariski dense in V, and the variety V is called rational if there is a rational map lP'N V that is an isomorphism from an open subset of lP'N to an open subset of V. It is clear that every moduli space Md is unirational, since Ratd is an open subset of lP'2 d 1 and the map Ratd Md is surjective. We also know that M 2 is rational, since Theorem 4.56 says that M2 � A 2. For which values of d is Md a rational variety? In particular, is M3 rational? 4.25.
---+
+
---+
Section 4.5. Periodic Points, Multipliers, and the Multiplier Spectrum
This exercise asks you to prove the part of Theorem 4.50 that was left undone in the text. Prove that the map
4.26.
defines a function in Q[Ratd] . Prove that it is PGL2-invariant and deduce that it defines a function in Q[Md] . 4.27.
** What is the degree of the map
U3 ,N : M3 ----> A
k
for sufficiently large N? Similarly, what is the degree of 4 ,N on M4 away from the Lattes locus on which it is constant? u
Section 4.6. The Moduli Space
M2 of Dynamical Systems of Degree 2
4.28. Prove that ¢ E Rat2 is conjugate to a polynomial if and only polynomial maps in M2 � A2 trace out the line 2. x =
cr1
( ¢)
=
2. Thus the
4.29. Let ¢ E Rat 2 ( q be a rational map of degree 2 and suppose that one of its fixed points has a nonzero multiplier .-\.
233
Exercises (a) Prove that there are an f E
PGL2(1C) and a c E C such that
q/(z) � (z + c + �) . =
(4.66)
z + +
/
This generalizes the normal form given in Lemma 4.59. (b) Verify that the multiplier of ¢/ at the fixed point = oo is A00 ( ¢ ) A. (c) Let A 1 , A2, A3 be the multipliers of the fixed points of ¢ with A1 = A. Prove that the number in (4.66) satisfies c2 = 4 - A 1 (2 A2 A3).
c
=
4.30. Fix d 2: 2 and consider the subset of Ratd defined by
Bi Critd For d
=
ofRatd.
=
{¢ E
2 we have BiCrit2
=
Ratd
:
¢ has exactly two critical points}.
Rat2, but for larger d the set BiCritd is a proper subset
(a) Prove that BiCritd is an algebraic variety of dimension 5. (b) Prove that conjugation induces a natural action of PGL2 on morphism of varieties
(c) Suppose that ¢ E
BiCritd has critical points at 0 and
oo.
BiCritd,
Prove that ¢(
i.e., there is a
z) has the form (4.67)
(d) Let ¢ E BiCritd and apply a conjugation to move the critical points of ¢ to 0 and oo, so ¢ has the form (4.67) described in (c). Prove that the following quantities depend only on the conjugacy class of ¢:
(e) Let ¢, 'lj; E
BiCritd and suppose that w (¢) w ('lj;), II(¢) = u('lj;) , =
T (cp)
=
T ('lj;) .
Prove that ¢ and 'lj; are PGL2 conjugate (working over an algebraically closed field). (f) Prove that the three quantities (4 . 68) described in (d) satisfy the relation
+
UT wd - 1 (w l ) d+ 1 2 and no other relations. Conclude that M�icrit is isomorphic to A . (This generalizes The iCri orem 4.56, since M� t M 2.) =
=
(g) ** Describe the stable and semistable completions of M�icrit coming from geometric invariant theory. Section 4.7. Automorphisms and Twists 4.31. In Section 4.7 we defined two rational maps ¢ and 'lj; to be equivalent if there is an f E PGL2 (K) such that 'lj; = ¢! , and similarly to be K-equivalent if there is an f E PGL2(K) such that 'lj; = ¢! .
234
Exercises
(a) Prove that these definitions do indeed define equivalence relations on the set of rational maps. (b) More generally, suppose that a group G acts on a set X. Define a relation on X by setting y if there exists a E G such that y = ( ) Prove that this is equivalence relation. 4.32. Let ¢(z) E K(z) be a rational map. (a) Prove that Aut(¢) is a subgroup ofPGL2(K). (b) Let h E PGL2(K). Prove that Aut(¢h) = h -1 Aut(¢) h, so Aut(¢) and Aut(¢h) are conjugate subgroups ofPGL2(K). "'
x
a x .
a
an
4.33. Let ¢(z) E K(z), let f E Aut(¢), and let a be a critical point of ¢ (i.e., ¢'(a) = 0). Prove that f (a) is also a critical point of ¢. More generally, prove that a and f (a) have the same ramification index. 4.34. Describe all polynomials ¢(z) E K[z] whose automorphism group Aut(¢) is nontriv ial. (Hint. Iff E Aut(¢), what does f do to the totally ramified fixed point(s) of¢?) 4.35. Let ¢( z) E K ( z) be a rational map of degree d and write ¢( z) as a quotient of polyno mials d o ¢(z) abo + ab 1 z + .· .· ·· ++ bad zd · + 1z + dz Prove that f(z) = z -1 is in Aut(¢) if and only ifb; = ad - i for all i. 2 2z)/( -2z + 4.36. Let ¢(z) = ( z (a) Prove that Aut(¢) contains the maps z-1 z , =
1 ). { z, -,1 --, -1-, 1 --1 , 1 - z }
-
z
z
- z
z -
and that they form a group isomorphic to Ss . Prove that # Aut ( ¢) 6, so in fact Aut(¢) � Ss . (Hint. Find the fixed points of¢.) (b) Compute the values of a1 ( ¢), a2 ( ¢), and ( ¢). 4.37. Let ( E K be a primitive nth root of unity and let ¢(z) E K(z) be a nonconstant rational map. (a) Prove that Aut(¢) contains the map f(z) = (z if and only if there is a rational map 'lj;(z) E K(z) such that ¢(z) z'lj;(zn ). (b) Prove that Aut(¢) contains both of the maps f(z) (z and g(z) = 1/z if and only if there is a polynomial F(z) E K[z] such that the rational map 'lj;(z) E K(z) in (a) has the form 'lj;(z) F(z)/(zd F(z - 1 )), where d deg(F). Verify that the group generated by f and g is the dihedral group of order 2n. (c) Let ¢(z) = z d . Prove that Aut ( ¢) is a dihedral group of order 2d - 2. 2 4.38. We identify the set of rational functions Ratd of degree d with an open subset of !P' d + 1 as described in Section 4.3. (a) Let f E PGL2. Prove that the set {¢ E Ratd : qyf = ¢} is a (possibly empty) Zariski closed subset ofRatd. If f(z) i- z, prove that it is a proper subset ofRatd . (b) Let A C PGL2(K) be a nontrivial finite subgroup. Prove that the set a3
=
=
=
=
{ ¢ E Rat d ¢1 = ¢ for all f E A} is a proper Zariski closed subset ofRatd. :
235
Exercises
(c) Prove that up to conjugation, PGL2 ( K) has only finitely many distinct finite subgroups of any given order. (d) Prove that the set { ¢ E Ratd : Aut(¢) -1- 1} is a proper Zariski closed subset of Rat d. (Hint. Note that the order of Aut(¢) is bounded by Proposition 4.65.) (e) Prove that the set {¢ E Ratd : Aut(¢) #- 1} is PGL2-invariant and defines a proper Zariski closed subset of Md. (Hint. The groups Aut(¢) and Aut(¢1) are conjugate subgroups ofPGL2; see Remark 4.64 and Exercise 4.32.) 4.39.
Let a E C* and d � 1 and consider the rational function 1 d ¢(z) = a z +
( ) z-1
Prove that Aut(¢) = {f E PGL2(1C) : ¢1 = ¢} is trivial. 4.40. Let ¢( z) E K ( z) be a rational map of degree d � 2 and assume that Aut ( ¢) contains the element h(z) = -z oforder 2. (a) Suppose that d is even. Prove that at least one of the fixed points of ¢ ( z) is defined over K. (b) Suppose further that d = 2. Prove that there is an f E PGL2(K) such that h1 = h and such that ¢1 has the form ¢1 ( z) = az + b . z 1 (c) Write <Pa,b(z) = az + bz- • Prove that for all E K*, the maps <Pa ,bc2 and <Pa , b are PGL2(K)-equivalent. (d) In homogeneous form we have <Pa ( [X, Yl ) = [aX 2 + bY2 , XY]. Let .P�(a , b; X, Y) be the associated dynatomic polynomial. Prove that .P�(a, b; X, Y) E Z[a, b, X, Y]. (e) * If n � 4 is even, prove that .P� (a , b; X, Y) is reducible. More precisely, prove that there are nonconstant polynomials \lin , An E Z[a , b, X, Y] such that -
c
,
b
.P:(a, b; X, Y) = \lin(a , b; X, Y)An(a, b; X, Y). (Hint. Divide the set of points P of formal period n into two subsets depending on whether the involution h permutes the orbit O¢ (P) or sends it to a different orbit.) This result is due to Manes [286].
Let ¢ = az + bz- 1 E Rah (a) Prove that (171 ( ¢) , 172 ( ¢) ) depends only on a. (b) We saw in Exercise 4.40 that Aut(¢) contains { ±z }, so it has order at least 2. Find all values of a for which Aut(¢) is strictly larger than { ±z }. (c) As a varies, prove that the set of points
4.41.
describes the curve
C : 2x3 - 8xy + x2 y - 4y2 - x2 + 12x + 12y - 36 = 0
in .A.? � M 2 . (d) Prove that the curve C in (c) is singular at the point ( - 6, 12).
236
Exercises
(e) Move the singular point to the origin and perform a further change of variables to prove that the curve C is isomorphic to the curve
y2 = 4x3 + x4 . Thus the singularity of C is a cubic cusp. (f) The point ( -6, 12) is the unique singular point of the curve C, so the rational map ¢ satisfying (0"1 ( ¢) , 0"2 ( ¢) ) = ( -6, 12) should be special in some way. How is it special? (Hint. Compare with the answer to (a). See also Exercise 4.36.) Section 4.8. General Theory of Twists 4.42. Let X be an object defined over K, and for each twist phism iy : Y ----> X. Assume that Aut( X) is abelian. Prove that Twist (X/ K)
_____.
H1 (
G
Y of X, fix a K-isomor
l( K / K), Aut( X)) ,
a
is a well-defined one-to-one map of sets. (See Remark 4.78.) Section 4.1 0. Fields of Definition and the Field of Moduli 4.43. Let ¢ E K(z) be a rational function. We know from Proposition 4.84 that the field of moduli K.p is contained in every field of definition of ¢. Prove that K.p is equal to the intersection over all fields of definition of ¢. 4.44. Let Ca : x2 + y2 = a be the family of curves studied in Example 4. 76. (a) Let p be a prime number satisfying p = 1 (mod 4). Prove that Cp is Q-isomorphic to C1 . (b) Let p be a prime number satisfying p = 3 (mod 4). Prove that Cp(Q) = 0. Deduce that Cp is not Q-isomorphic to C1, and hence that [Cp]K represents a nontrivial element of Twist (Cl /Q). (c) Let p and q be distinct prime numbers that are congruent to 3 modulo 4. Prove that Cp and Cq are not Q-isomorphic. Deduce that Twist(C1/Q) is an infinite set. 4.45. Let g be the cocycle 9u
( )={ Z
z if 0" ( i) -1/z if O"(i)
= i, = -i,
described in Example 4.88, and let K be the associated fixed field in Q(w ) . (a) Prove that K = Q(u, v ) , where u = w - 1 /w and v = i ( w + 1/w). Deduce that C : u2 + v2 -4 is the twist of !P' 1 /Q associated to the cocycle g. (b) Let ¢(z) = i((z - 1)/(z + 1)) 3 be the rational map from Example 4.85 satisfying O"(f) = ¢g" . Our general theory says that K is a field of definition for ¢ if and only if C(K) =1- 0 , where C is the curve in (a). For example, C(Q ( H ) ) =1- 0, so Q ( H ) must b e a field of definition for ¢(z). Find an explicit linear fractional transformation f E PGL 2 (Q) such that ¢1 (z) E Q( H ) (z). =
Section 4.1 1 . Minimal Resultants and Minimal Models 4.46. ** This exercise raises some natural questions concerning global minimal models. (a) Is it true that every rational map ¢(z) E Q(z) of (odd) degree d 2: 2 has a global minimal model over Q?
237
Exercises
(b) Let K be a number field, let R be the ring of integers of K, let S be a finite set of primes, and let ¢(z) E K(z) be a rational map of degree d 2: 2 . We say that ¢ has a global S-minimal model if there is a linear fractional transformation f E PGL2 (K) and homogeneous polynomials F and G satisfying ¢f = [F, G] with the property that the coefficients of F and G are in the ring of S-integers Rs and ordp (Res(F, G))
=
ordp (9\q,)
for every prime p r:J. S .
Suppose that R s i s a principal ideal domain. Is i t true that every ¢ o f (odd) degree has a global S-minimal model? (c) Let K be a number field and ¢( z) E K ( z) a rational map of (odd) degree d 2: 2 . Suppose that the Weierstrass class iiq,; K is trivial. Is it then necessarily true that ¢ has a global minimal model? (This is true in an analogous situation for elliptic curves; see [410, VIII.8.2].) 4.47. ** Let K be a number field and let ¢ E K ( z) be a rational map of degree d 2: 2. Let S be a finite set of primes of K. Prove that
is a finite set.
{
1/1
E Twist(¢/ K) : 1/1 has good reduction at all p r:J. S }
4.48. ** Fix an embedding of the moduli space Md in projective space and let hM d denote the associated height function. Let K/Q be a number field, let d 2: 2 be an integer, and let B 2: 1 be a number. Prove that the set
contains only finitely many PGL2 (K)-conjugacy classes of rational maps. 4.49. * Let K be a number field, let R be the ring of integers of K, let S be a finite set of primes of R, and let d 2: 2 be an integer. Prove that there is a finite set of rational maps l3K,S,d C Ratd(K) such that if 4> E Ratd(K) is a rational map satisfying ( I ) ¢ has three or more critical points, (2) the critical points of ¢ remain distinct modulo all primes not in S, (3) the critical values of ¢ remain distinct modulo all primes not in S, where we recall that a critical value of¢ is the image of a critical point of ¢, then there are automorphisms f, g E P GL 2 ( Rs) such that
f o e/J o g E BK ,S,d · (Note that when we say that the critical points or values are distinct modulo a prime p, we really mean that they are distinct modulo \J3 for all primes \J3 lying above p in a suitable finite extension of K.)
Chapter 5
Dynamics over Local Fields : Bad Reduction In this chapter we return to the study of dynamical systems over complete local fields such as Qlp. We saw in Chapter 2 that if a rational map ¢(z) E Qlp(z) has good reduction, then its Julia set is empty, in which case considerable information about the dynamics of¢ on IP' 1 (Qlp) may be deduced by studying the dynamics of the reduction ¢ on IP' 1 (JFP). But if ¢( z) has bad reduction, then the situation is far more complicated. Indeed, since "interesting" unpredictable dynamics occurs only in the Julia set, we might say that the good reduction scenario studied in Chapter 2 is the uninteresting situation. This chapter is devoted to the interesting case! The field Qlp and its finite extensions have the agreeable property that they are complete, but they are not algebraically closed, so they are more analogous to lR than they are to
C.
This suggests that we work instead with an algebraic closure QP o f !Qp ,
but unfortunately we then lose the completeness property! Going one step further, we take the completion ofQp, and it turns out that this field, denoted by reP = the completion of the algebraic closure ofQlp, is both complete and algebraically closed. For proofs of the basic properties of Qlp and reP, see for example [8 1, 1 75, 1 83, 249, 382]. The fields re and reP share many common properties, but they also differ in crucial ways. In particular, the field of p-adic complex numbers reP is not locally compact! However, it is often essential to work in reP, rather than in a finite extension ofQlp, for example if we want to guarantee the existence of large numbers of periodic points. Table 5.1 compares some of the properties of the complete fields Qlp, QP, reP, IR, and C. The subject of p-adic and more general nonarchimedean dynamics is relatively new. After some early articles [50, 201, 278, 433, 438] in the 1980s, there was an explosion of interest that put the subject on a firm footing with a body of significant theorems and, just as importantly, an array of fascinating conjectures. In this brief 239
240
5. Dynamics over Local Fields: Bad Reduction
Nonarchimedean metric Algebraically closed Complete Locally compact Totally disconnected
.; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .;
Table 5.1: Comparison of complete fields.
chapter we can provide only a glimpse into this active area of current research, with many important topics omitted entirely in order to keep the chapter at a manageable length. For example, we do not touch on the important concept of local conjugacy, nor do we describe Rivera-Letelier's classification ofFatou domains [373, 375]. For the reader desiring further information, we mention the following articles on p-adic dynamics that are listed in the references: [4, 5, 13, 14, 22, 23, 26, 30, 29, 50, 63, 53, 54, 56, 57, 58, 59, 60, 62, 70, 71, 72, 73, 82, 84, 104, 1 16, 1 70, 168, 169, 1 85, 1 88, 1 89, 201 , 206, 208, 220, 222, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 262, 266, 267, 268, 271, 272, 273, 274, 278, 280, 282, 283, 320, 3 1 8, 3 19, 334, 337, 338, 339, 340, 344, 348, 352, 355, 372, 373, 374, 375, 376, 378, 389, 427, 433, 438, 449]. 5.1
Absolute Values and Comp letions
We recall from Section 2. 1 that a valued field is a pair ( K, I · I ) consisting of a field K and an absolute value I · I on K. In this section we briefly remind the reader of the construction and basic properties of the completion of a valued field. Definition. Let (K, I · I ) be a valued field. A sequence a1, a2 , a3, . . . E K is Cauchy if for every E > 0 there exists an N = N (E) such that for all
m, n
� N.
In other words, the sequence an is Cauchy if an - am � 0 as m, n � oo . It is clear that every convergent sequence is Cauchy. A valued field K is said to be complete if every Cauchy sequence in K converges.
{ }
i
i
Example 5.1. The real numbers lR and the complex numbers C are complete with
respect to their usual absolute values.
Theorem 5.2. Let ( K, I · I K ) be a valuedfield. Then ( k, I · I k), unique up to isomorphism of valuedfields,
ties:
(a) There is an inclusion K C k as valuedfields. (b) k is complete.
there exists a valuedfield with the following proper
241
5.1. Absolute Values and Completions
(c) K is a dense subset ofk. Further, k is the smallest valuedfield satisfYing (a) and (b) in the following sense: Let ( L, I · I L) be any complete valuedfield containing K. Then there is a unique inclusion k L ofvaluedfields that respects the inclusions of K into k and L. Thefield k is called the completion of K with respect to the absolute value I · I K · 00
In other words, the "nth term test" from elementary calculus becomes both necessary and sufficient in the nonarchimedean setting. A function ¢ : D(a, r) sented by a power series
Definition.
00
----t
K is holomorphic (or analytic) if it is repre
¢(z) = L ai (z - a) i E K[z - a]
i =O
(5.2)
that converges for all z E D(a, r). The order of¢ at a, denoted by orda (¢), is the smallest index i such that ai =1- 0. A meromorphic function on fJ (a, r) is a quotient ¢ = ¢1 /¢2 of functions ¢1 and ¢2 =/=- 0 that are holomorphic functions on D(r, a).1 A meromorphic func tion ¢ ¢1 /¢2 induces a well-defined map ¢ fJ ( a, r) lP'1 (K) , =
:
------7
The order of¢ at a is the difference We say that ¢ has a zero (respectively a pole) at z = a if orda ( ¢) > 0 (respectively iforda(¢) < 0). The next proposition describes some elementary properties of nonarchimedean holomorphic and meromorphic functions. Proposition 5.8. (a) Let ¢(z) be a holomorphicfunction on the closed disk D( a, r) and let b E D(a, r). Then ¢(z) is a holomorphicfunction on D ( b, r), i.e., ¢(z) is given by a convergent power series in K[z - b]. (b) Let ¢(z) be a nonzero holomorphicfunction on D(a, r). Then the zeros of ¢(z) in fJ (a, r) are isolated. This means that if¢( b) = 0, then there is a disk fJ ( b, E ) such that ¢(z) =/=- 0 for all z E D(b, E ) "- {b }. 1 More precisely, a meromorphic function is an equivalence class of pairs ( 0 such that
Proposition 5.16.
l ¢(z) - ¢(w) l
= l ¢' (a) l · \z - w\
for all z, w E D(a, t).
Proof Replacing ¢(z) with ¢(cz + a) - ¢(a) for some c E Cp satisfying \c\ = r, we may assume that ¢(z) E Cp[z] converges on D(O, 1) and that ¢(0) 0. Write ¢(z) = :Ln?: l an zn as usual (note that a0 = 0 since ¢(0) 0), and let =
=
s ( a)
=
s�up l ¢(z) l .
zED(O,l)
The maximum modulus principle (Theorem 5. 13(a)) says that s = \\¢\\ = sup \an \ · n ?:O
Let j 2: 1 be the smallest index such that s = \ a1 \. The definition of s clearly implies that ¢(.D(O, 1)) 0 there exists a 8 > 0 such that :
U JID1 ___.
U
D ) D (¢(P) , ) for every ¢ E . (b) is uniformly continuous on U if for every > 0 there exists a 8 > 0 such that ¢ (D (P, 8) n U) D ( ¢( P) , �:) for every ¢ E and every P E U. (c) is uniformly Lipschitz on U if there is a constant C such that ¢ ( ( P, 8)
c
c
E
c
p(¢(P), ¢(Q)) 5o C · p(P, Q)
for every ¢ E and every P, Q E U.
In the case that { ¢n} is the collection of iterates of a single function, we say simply that ¢ is equicontinuous, uniformly continuous, or uniformly Lipschitz.
=
It is important to understand that equicontinuity is weaker than uniform continu ity, because equicontinuity is relative to a particular point, while uniform continuity is uniform with respect to all points in In particular, uniform continuity is an open condition, whereas equicontinuity is not. Similarly, the uniform Lipschitz property
U.
255
5.4. The Nonarchimedean Julia and Fatou Sets
is an open condition, and indeed it is even stronger than uniform continuity. The following implications are easy consequences of the definitions: uniformly Lipschitz
=:::}
uniformly continuous
=:::}
equicontinuous at every point
As we will discover throughout this chapter, in a nonarchimedean setting it is often just as easy to prove that a family of maps is uniformly Lipschitz as it is to prove that it is equicontinuous. Definition. Assume first that K is algebraically closed. Then the Fatou set F( ¢) is the union of all open subsets ofJID1 ( K) on which ¢ is equicontinuous, i.e., F( ¢) is the largest open set on which ¢ is equicontinuous. The Julia set .:J ( ¢) is the complement of the Fatou set. In general, the Fatou set of¢ over K, which we denote by F( ¢, K), is the intersection of F(¢, with JID1 ( K). Similarly, the Julia set .:J ( ¢, K) is the complement ofF(¢, K) in JID1 ( K).
K)
Proposition 5.18. For every integer n 2:: 1,
and
We proved this over
p('1/J(z), '1/J( w )) :::; C1 p(z, w) for all '1/J E \jf and all z , w E D(a, r). Finally, let A ( z) be the linear fractional transformation
(5. 14)
The assumption that a and (3 are distinct implies that A is invertible. Then a very special case of Theorem 2. 14 (for the rational map A - 1 ) says that there is a constant C2 = C2 (A) = C2 (a, (3) > 0 such that
5. Dynamics over Local Fields: Bad Reduction
266
for all z, w E JP>1 (K). Further, by construction we have ¢ E z , w E D(a, r ) ,
�
if and only if A ¢ E
C1 p(z, w) ;::: p (A(¢(z)), A(¢(w)) ) ;::: c:; 1 p ( ¢(z), ¢(w) )
o
(5. 15) w,
so for any
from (5.14), from (5.15).
This completes the proof of the nonarchimedean version ofMontel theorem.
D
Remark 5.28. The proof of Theorem 5.27 is fairly straightforward. Later, in Sec tion 5.10.3.4, we describe a deeper p-adic version ofMontel's theorem on Berkovich space; see Theorem 5.80. In the classical setting, there are a number of important properties of the Julia set that follow more or less formally from Montel's theorem. We conclude this section with a few instances. Proposition 5.29. Let ¢ JP' 1 ( K) JP'1 ( K) be a rational map of degree d ;::: 2 and let U c JP' 1 ( K) be an open set such that U n .J ( ¢) # 0. In particular, we are :
-+
assuming that the Julia set of¢ is nonempty. (a) The set Un:::: o ¢n (U) omits at most one point of lP' 1 (K). (b) Suppose that the set in (a) does omit a point. Then ¢ is a polynomial function and the omitted point is the totally ramifiedfixedpoint. (In other words, there is a change ofvariables f E PGLz(K) such that ¢! ( z ) E K[z] and the omitted point has been moved to oo.)
D(a, r ) , which are both open and closed, so it suffices to prove the proposition under the assumption that U = D( a, r ) . If the union omits two or more points oflP' 1 (K), then Montel's theorem (Theorem 5.27) implies that ¢ is equicontinuous on U, contradicting the assumption that U contains a Julia point. This proves (a). If the union omits a:, then ¢ - l (a:) = {a:}, so a: is a totally ramified fixed point of ¢. Hence ¢ is a polynomial map by definition (page 1 7), and after a change of variables it becomes a polynomial in K [z] (Exercise 1 . 9(c)). D Let ¢ JP' 1 ( K) -+ JP' 1 ( K) be a rational map of degree d 2 2. Recall that a subset E of JP' 1 ( K) is said to be completely invariantfor ¢ if it is both forward and backward invariant, ¢- 1 (E) = E = ¢(E). Proposition 1 .24 says that the Fatou set F( ¢) and the Julia set .J( ¢) are completely invariant. (The proof in Chapter 1 is over C, but the proof works, mutatis mutandis, for any complete field.) In Chapter 1 we used the Riemann-Hurwitz formula to characterize all finite completely invariant sets (see Theorem 1 .6). More precisely, we showed that a finite completely invariant set E has at most two elements. Further, if #E = 1, then after a change of variables, ¢(z) E K[z] is a polynomial and E = { oo }, and if #E = 2, Proof The set U is covered by disks
:
5.6. A Nonarchimedean Montel Theorem
267
then again after a change of variables, ¢( z ) zd or ¢(z) = z- d and E = {0, oo }. We now show that except for these trivial cases, the Julia set is the smallest closed completely invariant subset of lP'1 ( K) . =
Proposition 5.30. Let ¢ : lP' 1 ( K) ----+ lP' 1 ( K) be a rational map of degree d 2 2, and let E � lP' 1 ( K) be a closed completely invariant subsetfor ¢ containing at least three points. Then E is an irifinite set and E :2 J ( ¢).
Proof Theorem 1 .6 tells us that a finite completely invariant subset contains at most
3
two points, so our assumption that #E 2 implies that E is infinite. Notice that the complete invariance of the closed set E implies the complete invariance of its com plement U, which is an open set. It follows that the union Un>o ¢n (U) omits at least two points, since it in fact omits the infinite set E. Montel's theorem (Theorem 5.27) tells us that U � :F( ¢). Hence E ;;2 J( ¢ ). D
Remark 5.3 1 . Proposition 5.30 tells us that if the Julia set J( ¢) is nonempty, then it
is the smallest closed completely invariant set containing at least two points. (Notice that the case of exactly two points is ruled out by the fact that if ¢ has a completely invariant subset containing exactly two points, then ¢ is conjugate to either z d or z- d , in which case its Julia set is empty.) Corollary 5.32. Let ¢ : lP' 1 ( K) assume that J ( ¢) -:/:- 0. (a) J(¢) has empty interior. (b) Let P E J(¢) and let
----+
lP' 1 ( K) be a rational map of degree d 2 2, and
O¢ (P)
= nU2:0 ¢-n (P)
be the backward orbit ofP. The Julia set J( ¢) is equal to the closure ofO¢ (P) in lP'1 (K).
(c) J( ¢) is a perfect set, i.e.,for every point P E J( ¢), the closure ofJ( ¢) "'- { P} contains P. (d) J ( ¢) is an uncountable set. Proof (a) Let 8J(¢) denote the boundary of the Julia set J(¢). Theorem 1 .24
tells us that :F( ¢) and 8:1 ( ¢) are completely invariant, so the same is true of their union 8J(¢) U :F(¢). This union is also closed, since its complement is the interior of J ( ¢). Proposition 5.20 says that the Fatou set :F( ¢) is always nonempty, and since it is open, it must contain infinitely many points. Hence the union 8:1 ( ¢) U :F( ¢) is an infinite, closed, completely invariant set, so Proposition 5.30 tells us that J(¢)
�
8J(¢) u :F(¢) .
=
But J(¢) and :F(¢) are disjoint by definition, which proves that J(¢) 8:1(¢), i.e., the Julia set has empty interior. (b) We know that J ( ¢) is completely invariant, so in particular 0¢ ( P) c J ( ¢) for any point P E J( ¢ ).
5. Dynamics over Local Fields: Bad Reduction
268
Next let U be any open set with U n .:J( ¢) -1- 0. Then Proposition 5.29(a) tells us that Un>o q;n (U) omits at most one point, and Proposition 5.29(a) says that if it does omit a point, that point is a totally ramified fixed point, hence is in the Fatou set. In particular, the possible omitted point cannot be P, since P E .:!(¢) by assumption. This proves that P E Un ->o q;n (U), or equivalently, that there is some 2:: 0 such that U n q;- n (P) -1- 0. This proves that every open set U that intersects .:J ( ¢) nontrivially also intersects the backward orbit o; ( P) nontrivially. Hence .:J ( ¢) is contained in the closure of 0¢ (P). ( c) Let Po E .:J ( ¢). We claim that the backward orbit o; (Po) must contain a non periodic point. To see this, suppose instead that 0¢ (Po) consists entirely of periodic points. Then ¢- 1 (Po) consists of a single point, so Po is a totally ramified periodic point and hence in the Fatou set, contrary to assumption. Therefore we can find a nonperiodic point P1 E 0¢ (Po). The point P1 is in .:J (¢), since .:J ( ¢) is completely invariant, so (b) tells us that n
Po E closure of 0¢ (PI ) .
On the other hand, Po i s not in 0¢ ( P1 ) , since otherwise P1 would be periodic. Hence Po E closure of ( .:!(¢) " {P} ) . (d) The Baire category theorem [387, §5. 1,5.2] implies that a nonempty perfect subset of IP'1 ( K) is uncountable. D 5. 7
Periodic Points and the Julia Set
Our goal in this section is to show that the Julia set .:J ( ¢) of a rational map ¢ is contained in the closure of the periodic points of ¢. We begin with an elementary lemma that is obvious in the classical setting by a compactness argument, but which requires a different proof over a non-locally compact field such as Cp . Lemma 5.33. Let ¢ 1 (z) and ¢2 (z) be power series that converge on suppose that ¢ 1 ( D(a, r )) n ¢z (D( a, r )) 0. Then
D(a, r ) , and
=
i_nf p ( ¢ 1 (z), ¢z (z) ) > 0.
z ED(a,r)
Proof Let
M1 = s_up l ¢ 1 (z) l
and
z E D (a,r)
M2
=
sup l ¢z(z) l .
z ED(a,r)
The maximum modulus principle (Theorem 5.13(a)) says that there are points z1 , zz E D(a, r ) such that ¢ 1 (zi ) M1 and ¢z (zz) Mz. In particular, M1 and M2 are finite, since ¢ 1 and ¢2 are power series that converge on D(a, r ).
=
=
269
5.7. Periodic Points and the Julia Set
Let M = max{M1 , M2 , 1}. Then for any z E D(a, r ) we have
On the other hand, the function (PI - ¢2 does not vanish on D( a, r ) by assumption, so Theorem 5.13(b) tells us that for all z E D(a, r ) . Hence
D
The next lemma is used in conjunction with Lemma 5.33 to move a varying set of pairs of points { a , ,8 } to the specific pair { 0, 1 }. Lemma 5.34. Let A, B C Cp be bounded sets that are at a positive distance from one another. In other words, there are constants �' > 0 such that
sup i al ::::; � ' aEA For each (a, ,8) E A
sup I,BI ::::; � ' (3EB x
and
o
inf ( a , ,8) = > 0. (5. 1 6) aEA , (3 EB p
o
B, define a linearfractional transformation
La,(3 (z) = ( ,8 - a)z + a. Then there is a constant C > 0, depending only on � and 8, such that
p(La ,(3 (z), La',(3' (z') ) ::::; C · m ax {p(a , a') , p (,8, ,8') , p (z, z') } for all a , a' E A, all ,8, ,8' E B, and all z , z' E JID1 (Cp). Remark 5.35. Although Lemma 5.34 appears somewhat technical, it is not saying
anything mysterious. The linear fractional transformation La,(3 is determined by the three conditions La ,(3 (0) = a ,
La,(3 (oo) = oo.
The lemma is asserting, roughly, that if we take two nearby ( a , ,8) values, then the associated transformations are close to one another, where we use the chordal sup norm
p ( L , L') =
sup {p (L(P) , L' (P) ) } PEIP'1 (ICp) to measure the closeness of two maps. Thus Lemma 5.34 is equivalent to the assertion that the map
( a , ,8 , z)
is Lipschitz.
�
La,(3 ( z) = (,8 - a)z + a
270
5. Dynamics over Local Fields: Bad Reduction
Proof ofLemma 5 .34. To ease notation, for x, y E Cp we write
lx, y l
= max{lxl, IYI } .
We also assume (without loss of generality) that Ll 2: 1 and 8 :::;; 1 . Then for any o:, o:' E A and /3, /3' E we have lo: - o:' l p(o:, o:' ) · lo:, 1 l · lo:', 1 1 :::;; Ll 2 p(o:, o:') , (5. 1 7) l/3 - /3'1 p(/3, /3') · 1/3, 1 1 · 1/3', 1 1 :::;; Ll2 p(/3, /3').
B = =
Let o:, o:' E A, let f3 , /3' E and let z, z ' E lP'1 ( Cp ). Directly from the definitions of L a ,(3 and the chordal metric, we have
B,
I ( (/3' - o:' )z' + o:') - ( (/3 - o: )z + o:) I max { 1 (/3 - o: )z + o:l, 1 } · max{ 1 (/3' - o:' )z' + o:' l , 1 } " (5. 1 8) Assuming for the moment that z -=1- oo and z ' -=1- oo, we multiply out the numerator and estimate it using the triangle inequality: p(La ,f3 (z), L a' ,!3' (z' ) ) =
1 ((/3' - o:')z' + o:' ) - ((/3 - o:)z + o:) l l/3'z' - f3z - o:' z' + o:z + o:' - o: l = l!3'(z' - z) + (/3' - f3)z - o:'(z' - z) + (o: - o:') z + (o:' - o:) l :S: max{l/3' - /3l · lzl, lo:' - o:l · lz, 1 1 , l z' - zl · lo:', /3' 1 } :::;; max{Ll2 p(/3, /3') · lzl, Ll2 p(o:, o:') · lz, 1 1 , l z' - zl · lo:', /3' 1 } from (5.17), :::;; max{Ll 2 p(/3, /3' ) · lzl, Ll2 p(o:, o:') · lz, 1 1 , Ll · lz' - z l } from (5. 16), :::;; max{ Ll2 p(/3, /3' ) · lzl, Ll 2 p(o:, o:') · lz, 1 1 , Llp(z, z' ) · lz, 1 l · l z', 1 1 } definition of p, :::;; Ll2 · max{p(/3, /3') , p(o:, o:') , p(z, z') } · lz, 1 l · lz', 1 1 . =
Substituting this into (5 . 1 8) and doing a little bit of algebra yields max{p(/3, /3' ) , p(o:, o:' ) , p(z, z ' ) } lz', 1 1 -' z '-.1-'-.., l < Ll 2 . -----;-:-,---1--' (5. 1 9) max{ l (/3 - o:)z + o:l, 1 } max{ I (/3' - o:' )z' + o:' l , 1 } · We are left to show that the righthand side is bounded in terms of 8 and Ll. By symmetry, it suffices to bound the first fraction. We consider two cases. First, if lzl :::;; Ll j8, then we have the trivial estimate -----,-�
lz, 1 1 < iz 1 1 < Ll _ max{ l (/3 - o:)z + o:l, 1 } - ' - 8
271
5.7. Periodic Points and the Julia Set
Second, suppose that lzl > fl/ 8 . Then the fact (5. 1 6) that 1/J - al � 8 implies that f (/J - a)z f > !:1
Hence
� lal,
1 (/J - a)z + al = f (/J - a)z f
so
� 8 lzl .
lz, l l = lz, 1 1 < � < !:1 max{ I (/J - a)z + al, l } max {8 lzl, l} - 8 - 8 ' Thus fl/ 8 serves as an upper bound in both cases, and substituting this bound into (5 . 1 9) yields the estimate
�
4
p(La,(3 (z) , La ' ,(3' (z' ) ) :::; 2 max{p(!), !)') , p(a, a' ) , p(z, z ' ) } .
This completes the proof of Lemma 5.34 with explicit dependence on 8 and !:1 in the case that z -1- oo and z ' -1- oo. The remaining cases are similar and are left to the �cr
0
We next prove a version ofMontel's theorem in which the two omitted points are allowed to move. Lemma 5.34 is the key technical tool that allows us to uniformly replace the two moving points with two particular points, thereby reducing the proof to our earlier result. Theorem 5.36. (Montel Theorem with Moving Targets, Hsia [208]) Let ¢ 1 , ¢2 be power series that converge on D( a, r ) , and suppose that
Further let be a collection ofrational, or more generally meromorphic, functions on D(a, r) such that for all ¢ E
and all z E D(a, r). Then satisfies a uniform Lipschitz inequality on D( a, r ) , so in particular is an equicontinuousfamily offunctions on D(a, r).
Proof The proof is very similar to the proof of Theorem 5 .26, but somewhat more
elaborate. First we note that since ¢ 1 , ¢2 : .D( a, r) K have disjoint images, there is at least one point a omitted by both of them. Making a linear change of variables z z - a , we may assume that ¢ 1 and ¢2 omit the value 0. Then Theorem 5 . 13(b) tells us that ---->
f--+
for all z E D(a, r). We are going to apply Lemma 5.34 to the disjoint bounded sets ¢1 ( D(a, r) ) and ¢2 ( D ( a , r ) ) . Thus for each point w E D (a , r ) , if we let
272
5. Dynamics over Local Fields: Bad Reduction
then Lemma 5.34 says that there is a constant C such that p(Lw (z), Lu(z') ) :::; C max{p( ¢1 ( w ) , ¢1 (u) ) , p (¢2 (w ) , ¢2 ( u) ) , p(z, z') } for all w , u E D(a, r) and all z, z' E lP'1 (Cp ). (5.20) (In the notation of Lemma 5.34, we have set a = ¢1 (w), (3 = ¢2 (w ) , a' = ¢1 ( u) , (3' ¢2 ( u).) We next use the fact that ¢1 and ¢2 themselves satisfy a Lipschitz condition. More precisely, Proposition 5.10 says that
=
for all w , u E D(a, r) and i Since w and u are bounded, this implies that there is a constant C'
2
for all w , u E D(a, r) and
p (tPi (w) , rPi (u) ) :::; C' p(w , u)
Substituting this into the inequality (5.20) yields
=
1 , 2.
1 such that
i = 1 , 2.
p(Lw (z) , Lu(z') ) :::; CC' max{p( w , u), p(z, z') } for all w , u E D(a, r) and all z, z' E 1P' 1 (Cp ) . (5.21)
We are now ready to prove Theorem 5.36. The idea is that we know that each ¢ E if> omits at least two values, but the omitted values may vary with ¢, so we use the linear transformations Lw to move the omitted values to two specific points. Thus for each ¢ E we define a new function and we consider the family of functions Each ¢ can be expressed as a rational function of ¢1 , ¢2 , and ¢, so it is a meromorphic function on D(a, r). Further, since we are given that ¢ (z) #- ¢1 (z) and ¢ (z) #- ¢2 (z) for all z E D( a, r ) , it follows that ¢
0 there is a polyno mial ¢( z) E Cp [z] of degree p + 2 that has no repelling periodic points of period smaller than yet ¢( z) does have repelling periodic points of higher periods. Example 5.41. Consider the polynomial map
n
n,
¢(z) =
zP - z
p
.
5.7. Periodic Points and the Julia Set
275
, then l aP I > a > 1 1 I ial, l ¢(a) l = l aP; a l =p · l aP - a l =p· l aP I > pi a l. Hence ¢n (a) --> so a E F(¢), since a is attracted to the attracting fixed point at infinity. We also observe that if a E D(O, 1) n Qlp = Zp, then Fermat's little theorem tells us that aP = a (mod p), so ¢(a) E Zp. Thus Zp is a completely invariant subset of ¢. Hsia [206, Example 4. 1 1] (see also [449]) explains how to identify the dynamics of ¢ on Zp with a shift map on p symbols, similar to the example studied in Section 5.5, from which one deduces the following facts: .J(¢,Qlp) = Zp. .J ( ¢, Qlp) contains all of the periodic points of ¢ (other than ) so in partic ular all of the periodic points of ¢ are defined over Qlp, and all except are It is clear that the Julia set of¢ is contained in D(O, ), since if I so oo,
•
oo
•
•
,
oo
repelling. .J( ¢, Qlp) .J ¢ Cp ), since Theorem 5.37 tells us that .J ¢ Cp ) is contained in the closure of the periodic points of ¢. Thus .J( ¢, Cp) is compact. (See also [73].)
= (
,
(,
3 + az2 + b with a, b E z;. p ¢(z) = pz We first consider the fixed points of ¢, which are the roots of the equation pz3 + az2 - z + b = 0. The assumption that a, b E z; implies that the roots satisfy l a 1 i = p and l a2 i (Look at the Newton polygon!) We also observe that paf and aai have a3 1 =p21, .while inorm a 1 - b has norm p, so a1 must have the form a1 = - p-a + c for some c with l ei = 1. Example 5.42. Let � 5 be a prime, and let
This allows us to compute
l ¢'(al ) l = j 3pai + 2aa1 - 1 1 = l al (3pal + 2a) - 1 1 = l al (-a +3pc) - 1 1 = i a 1 i = p. Thus a 1 is a repelling fixed point of¢. A similar, but more involved, calculation can be used to show that there are re pelling periodic points of higher orders. Alternatively, we can invoke Bezivin's The orem 5.40, which says that the existence of the single repelling fixed point a 1 implies that ¢ has infinitely many repelling points whose closure is .J ( ¢). In order to study the periodic points in the Fatou set, we observe that ¢ is non expanding on D(O, 1 ). To see this, note that ¢ maps the disk D(O, 1 ) to itself, so in particular 11¢11 :::; 1. Applying Proposition 5.10 yields l ¢(z) - ¢(w) l :::; i z - wl for all z, w E D(O, 1).
276
5. Dynamics over Local Fields: Bad Reduction
Hence 4> is uniformly Lipschitz on D(O, 1), so D(O, 1) c F( 4>) . Notice that the nth it erate of 4> has the following form (think about the reduction of 4>n modulo p, which is the same as the nth iterate of the reduction ¢(z) = iiz2 + b):
n (z) = Az3n +
·
· ·
+
Bz2 n + l + Cz2 n + Dz 2n - l +
coefficients in p'llp
i
c E z;
· ·
·
+ Ez + F .
coefficients in 'llp
Again using the Newton polygon, we see that the polynomial 4>n (z) - z has ex ::; 1. These roots are actly 2n roots (counted with multiplicity) satisfying points of period dividing n for 4> and they are in the Fatou set, since we showed that D(O, 1) c F(¢) . One can prove that this gives infinitely many distinct periodic points in D(O, 1). See [206, Example 4. 1 1].
o:
5.8
l o: l
Nonarchimedean Wandering Domains
We first recall a famous theorem from complex dynamics (see Theorem 1 .36). Theorem 5.43. (Sullivan's No Wandering Domains Theorem [426]) Let ¢( z) E C( z) be a rational function of degree d ;:=: 2 over the complex numbers and let U C F( 4>) be a connected component ofthe Fatou set of¢. Then U is prepe riodic in the sense that there are integers n > m > 0 such that
In other words, the connected component U does not wander, whence the name of the theorem.
The first obstacle to translating Sullivan's theorem to the nonarchimedean setting is the fact that Qp and Cp are totally disconnected. Thus with the classical definition of connectivity, there is no good way to break up the Fatou set into "connected" com ponents. Various alternatives have been studied, including disk connectivity, rigid analytic connectivity, and the use of Berkovich spaces. In this section we consider disk connectivity, which is the simplest of the three to describe. We state a theorem of Benedetto to the effect that a large class of rational maps in Qp ( z) have no wandering "disk domains" and illustrate the result by proving it under the somewhat stronger hypothesis that there are no critical points in the Julia set of ¢. We also note that Benedetto has shown that the statement is false over Cp , that is, there are rational maps over CP that do have wandering disk domains. Much work has been done on this problem, especially in a series of papers by Benedetto [54, 56, 57, 58, 59, 63, 62] and Rivera-Letelier [372, 373, 375, 376, 378], but there are still many open questions. The material that we cover in this section is due to Benedetto.
5.8. Nonarchimedean Wandering Domains 5.8.1
277
Disk Domains and Disk Components
The ordinary definition of connectivity is not useful for studying the totally discon nected spaces QP and Cp , nor is the notion of path connectedness helpful. We begin with an abstract notion of connectivity that uses a chain of "disks" in place of a path. Let X be a topological space and let V be a collection of open subsets of X. (In practice, V will be a base for the toplogy of X.) For convenience, we refer to the sets in V as disks. Let U c X be an open subset and let P E U . The disk component of U con taining P (relative to V) is the set of Q E U with the property that there is a finite sequence of disks D 1 , D2 , . . . , Dn C U such that Definition.
Di Di+ l =1- 0 n
for all l � i < n.
In other words, Q is connected to P by a path of disks, as illustrated in Figure 5.3, although as we shall see, in the nonarchimedean world, Figure 5.3 is somewhat mis leading. Note that we define disk components only for open subsets of X. It is easy to see that U breaks up into a disjoint union of disk components (Exercise 5.22).
Figure 5.3: A path of disks from P to Q. Example 5.44. Let X = C and let V be the usual collection of open disks in C. Then the disk components of an open set U c X are the same as the usual path connected components. This is clear, since if r is a path from P to Q, then r can be covered by open disks contained in U, and the compactness of r shows that it suffices to take a finite subcover. Thus the definition of disk components and the related notion of disk connectivity (see Exercise 5.23) are reasonable. For example, Sullivan's theorem may equally well be stated in terms of the disk components of the open set :F( ¢ ) . For the purposes of this section, we modify slightly our definition of disks in lP' 1 ( Cp ) . The resulting topology is the same, and indeed the disks contained within the unit disk D(O, 1) are the same, but the alternative definition is more convenient for working with disks that may contain the point at infinity. Definition.
given by
The standard collection ofclosed disks in lP' 1 (Cp ), denoted by Vclosed , is
278 Vclosed
=
{
5. Dynamics over Local Fields: Bad Reduction
all closed disks D(a, r) and the complement JP>1 (Cp) " D(a, r) of all open disks D(a, r).
Of course, despite the name, all of the disks in Vc1osed are both open and closed sets. Note that Dclosed includes all such disks, not only the disks of rational radius (cf. Remark 5.7). One can show (Exercise 5.24) that the disks of rational radius in Vc1osed are exactly the images of the unit disk D(O, 1) via elements ofPGL2 (Cp). Similarly, the standard collection of open disks in JP>1 (Cp), denoted by Dopen , is given by Dopen
disks D(a, r) and = { alltheopen complement JP>1 (Cp) " D(a, r) of all closed disks D(a, r).
In the nonarchimedean world, every disk component has a very simple form. More precisely, it is either a disk, the complement of a single point, or all of JP>1 ( Cp). Proposition 5.45. Let Dopen and Vc1osed be, respectively, the collections ofstandard open and closed disks in JP> 1 ( Cp) as defined above. (a) Let D1 , D2 E Dclosed· Then one of thefollowing is true:
(iii) D1 � D2 . (i) D 1 n D2 = 0. 1 (iv) D2 � D 1 . (ii) D 1 U D2 JP> (Cp). (b) Let U C JP>1 ( Cp) be an open set and let V be a disk component of U relative to Vclosed· Then V has one ofthefollowingforms:
=
(iii) V E Vclosed U Dopen · D2 = JP>1 (Cp), we are done. Otherwise, choose any point in the complement of D 1 U D2 and use a linear fractional transformation to move that point to oo. This reduces us to the case that neither D1 nor D2 contains oo, so they have the form D 1 D(a 1 , rl ) and D2 D(a2 , r2 ). If D 1 and D2 are disjoint, we are done, so we may assume that there is a point Proof (a) If D 1
U
=
=
and switching D 1 and D2 if necessary, we may also assume that r 1 ::; r2 . Let /3 E D 1 . Then so /3 E D2 . Hence D 1 � D2 . (b) If V = JP> 1 (Cp), we are done, so we assume that V -=1- JP> 1 (Cp) · Using a linear fractional transformation to move a point not in V to oo, we are reduced to the case that oo � V.
279
5.8. Nonarchimedean Wandering Domains
Let D 1 , . . . , Dn E 'Dc1osed be any path of disks contained in V. Each pair of adjacent disks (Di , DH 1 ) has nonempty intersection, so (a) tells us that one of them is contained within the other. Applying this reasoning to each pair, we see that the union U�=l Di is itself a closed disk, i.e., the union is in 'Dc1osed· This shows that every disk path in V actually consists of a single disk. Fix some point a E V and let R = sup {r 2: 0 : D(a, r) C V}. Note that R > 0, since a E V and V is open. If R = oo, then V done, so we assume that R < oo. We claim that D(a, R)
s;::
=
V s;:: D(a, R).
Cp and we are
(5.23)
The lefthand inclusion is clear from the definition of R. For the righthand inclusion, suppose that b E V and consider a disk path from a to b lying within V. From our previous remarks, this disk path consists of a single disk D( a, s). The definition of R and the fact that D (a, s) s;:: V tell us that s :s; R, and then the fact that b E D (a, s) tells us that b E D( a, R). This gives the other inclusion. But we get even more. If there is even a single b E V satisfying lb - ai = R, then s = R and D(a, R) D(b, s) s;:: V, so we find that V = D(a, R) E 'Dclosed · On the other hand, if l b - ai < R for every b E V, then V c D(a, R), so (5.23) tells us that V D( a, R) E 'Dopen· This completes the proof of Proposition 5.45. D =
=
5.8.2
Hyperbolic Maps over Nonarchimedean Fields
In this section we prove that the Julia set of a rational map ¢ contains no critical points if and only if¢ is strictly expanding on its Julia set. This result is used later to prove that such maps, which we call (p-adically) hyperbolic, satisfy a p-adic version of Sullivan's no wandering domains theorem. On first reading, the reader may wish to omit the somewhat technical proof of the main theorem in this section and proceed directly to the application of the theorem in proving Theorem 5.55 in Section 5.8.3. Theorem 5.46. (Benedetto [56]) Let K/Qp be a.finite extension ofp-adicjields and let ¢(z) E K(z) be a rationalfunction ofdegree d 2: 2. Proposition 5.20(c) tells us that F( ¢) =/= 0, so changing variables if necessary, we may assume that oo E F( ¢ ) . Then thefollowing are equivalent: (a) There are no critical points in .7( ¢ ) . (b) For everyfinite extension L / K there exists an integer m 2: 1 such that
for all a E .7(¢) n L. In other words, q;m is strictly expanding on .7 ( ¢)
n
L.
Definition. Let K/QP be a finite extension. A rational function ¢ E K(z) will be called p-adically hyperbolic if it satisfies the equivalent conditions of Theorem 5.46.
See Exercise 5.25 for the relationship with the classical definition ofhyperbolicity.
280
5. Dynamics over Local Fields: Bad Reduction
Remark 5.47. The classical analogue of Theorem 5.46 over C is much weaker. It
¢
says that some iterate of is strictly expanding on the Julia set if and only if the closure of the postcritical set is disjoint from the Julia set. (The postcritical set is the union of the forward orbits of the critical points.)
(¢), we can take L = K (a) and observe(a)thatis clear, since if a is a critical (¢m )'(a) m-iIT=O1 ¢'(¢i (a)) 0 (since ¢' (a) = 0). Thus the existence of a critical point in J ( ¢) implies that ( ¢m )' has a root in J ( ¢) for every so (b) cannot hold. The other implication is more difficult. Using the assumption that ¢' does not vanish on J ( ¢) , we can apply Proposition 5. 16(c) to every point a in J ( ¢) n L to find a disk D( a , r ) with the property that j ¢(z) - ¢(w)j j ¢'(a) j · l z - w l for all z , w E D(a, ra ) · These disks form an open covering of the compact set J ( ¢) n L, so we can find a finite subcover. Let E be the smallest radius of the disks in this finite subcover. We then consider a finite covering of J ( ¢) n L by disks of radius E, say ProofofTheorem 5.46. The implication (b)
point in J
=
=?
=
m,
a
=
The (nonarchimedean) triangle inequality implies that each disk in this covering is
contained in one of the disks of the previous covering, so we conclude that ¢ stretches by a constant factor on each In other words, there are constants such that for each 1 ::::: i ::::: n,
D( ai, E).
Si
z, w E D(ai, E). next step is to show that for any fixed a E J ( ¢) n L, the set of derivatives n 1 , 2, 3, . . . , (5.24) for all
The
=
is unbounded. We prove this by contradiction, using the following claim: Claim 5.48. Let J n L and suppose that there is an > 1 such that
a E (¢) R j(¢n )'(a)j :::; R for all n � 1. Then ¢n (lJ(a,E/R)) D(¢n (a),E) for all n 1. (5.25) ProofofClaim 5.48. We verify the claim by induction on n. The inclusion (5.25) is clearly true for n 0, since R > 1. Suppose the inclusion (5.25) is known for all 0 n N. Our choice of E ensures that for any f3 E J( ¢) n L, the map ¢ stretches C
:::::
�
=
1 on .J(¢) L, which will complete the proof of the theorem. Let
2N
4M
n
To verify this claim, let a E :1 ( ¢) n L and let a0, a 1 , . . . and i0, i 1 , . . . be the orbit and itinerary of a as described earlier. We use the chain rule to compute
N -mio -mi1 - · · -mik first be M. M > N-m· -m· - ··· -m·'l.k -> N- (k+ 1)M so k + > N/M. We have thus found integers k and satisfying •
We continue this process until the exponent comes smaller than Notice that this implies that to
2
21
'
r
5.8. Nonarchimedean Wandering Domains
283
(¢N) ' (o:) = 1/J�0 (o:o ) 7/( (o:I )'lj;�2 (o:2) · · · 1/J�k (o:k ) · (¢r ) ' (o:k+ l )
with > N/M - 2 and r < M.
k
Hence
I (¢N) ' (o:) l = I1/J�o (o:o ) 1/J�l (o:I ) 'lj;�2 (o:2 ) · · · 1/J�k (o:k ) l · l (¢r ) ' (o:k+I ) I 2k 2: 2 / M - 2 . /-LM since k > N/M - 2 and M, N �------�--� � 2 J.'r from (5.27) 2 from (5.26) r
1
from the choice (5.28) of N.
This shows that I ( ¢N) ' ( o:) I > 1 for all N > which completes the proof of Theorem 5.46. 5.8.3
M2 log2 (J-L- l ) and all o: E .7( ¢)
n L,
D
Wandering Disk Domains
If U is a disk component of the Fatou set F(¢) of a rational map ¢ E Cp(z), then ¢(U) may not be a full disk component of F(¢). This situation, which does not occur in the complex case, prompts the following definition. Definition.
field, and let
Let ¢( z) E Cp ( z) be a rational map defined over a nonarchimedean 'DC
(¢)
=
(
{ disk components of the Fatou set F ¢) }
be the collection of disk components of the Fatou set of ¢. Then ¢ induces a map of the set 'DC ( ¢) to itself according to the rule
(¢), U (disk component ofF(¢) containing ¢( U)). (5.29) We say that U E 'DC ( ¢) is periodic, preperiodic, or wandering according to the 'DC
(¢)
------>
'DC
1-------t
behavior of U under iteration of the map (5.29).
Example 5.49. Let p be an odd prime and let
¢(z) =
z2 - z p
be the function that we studied in Section 5.5. We proved (Proposition 5.22) that the Julia set of¢ is contained in the union of two open disks,
D(O, 1) U D(1, 1), and that :T (¢) contains the repelling fixed points 0 and 1. .7(¢)
c
For o: E Cp, let B ( o: ) denote the disk component of o: in F(¢). We claim that B ( -1) = D ( - 1, 1). To see this, we observe that B ( - 1 ) cannot contain any larger disk, since it does not contain 0. On the other hand, D(-1, 1 ) is in F(¢), since it is disjoint from D(O, 1) U D ( 1, 1). Hence B ( - 1) = D ( -1, 1).
284
5. Dynamics over Local Fields: Bad Reduction
Now consider the image point ¢( - 1) = 2p- 1 and the image of the associated disk component ¢ (B ( - 1 ) )
=
¢ ( D( - 1 , 1)) = D(2p- 1 , p).
The disk D(2p-I , p) is contained in :F( ¢) , but it is certainly not the largest disk around 2p- 1 contained in :F(¢) . Indeed, 2p-l E JP 1 (Cp) " D(O, 1)
c
:F(¢).
It is not hard to check that JP1 (Cp) " D(O, 1) is the full disk component of :F(¢) containing ¢ (B( - 1) ) . (No Wandering Disk Domains Conjecture) Let K/Qp be a finite extension and let ¢( z) E K ( z) be a rational map of degree d :::: 2. Then the Fatou set :F( ¢) has no wandering disk components. Conjecture 5.50.
Benedetto proves Conjecture 5.50 for a large class of rational maps. In order to state his result, we give four definitions (some of which we already know): Definition. Let ¢ E Cp ( z) be a rational map of degree d :::: 2 and let P E JP1 ( Cp). We say that P is:
(i) (ii) (iii) (iv)
Julia if P E .J(¢). critical if e p ( ¢) :::: 2. wildly critical if ep(¢) 0 (mod p). recurrent if there is a sequence of integers i.e., if P is in the closure of the set { ¢n ( P) =
n: in----+:::: oo1}.such that c/Jni (P) ----+ P,
Theorem 5.51. (Benedetto [54]) Let KjQp be a finite extension of p-adic fields and let ¢( z) E K ( z) be a rational map ofdegree d :::: 2. Assume that ¢ has no wildly critical recurrent Julia points (defined over Then the Fatou set :F( ¢) has no wandering disk components.
K).
We make three short observations concerning Theorem 5.5 1 . Remark 5.52. If p is odd, then Theorem 5.51 is true for "almost all" rational maps in Cp(z). This is true because if ¢ ( z) has a wild critical point P, then in particular it has a point whose ramification index satisfies e p ( ¢) :::: p It is not hard to show that all of the critical points of a "generic" rational map of degree d have ramification index equal to 2. (See Exercise 4.23 for a more precise statement.) Remark 5.53. For a fixed degree d, Theorem 5.51 is true for all primes p > d, since p > d rules out the existence of wild critical points. Remark 5.54. If a recurrent critical point P is in the Fatou set :F( ¢) , then one can show that P is in fact periodic (Exercise 5.26). On the other hand, if P is critical, then locally around P the map ¢ looks like ¢ (z) = ¢(P) + a(z - z(P))k + for some a E CP and some :::: 2. Thus if Q is sufficiently close to P, then
:::: 3.
k
· ·
·
5.8. Nonarchimedean Wandering Domains
285
p ( ¢(Q) , ¢(P) ) = p(P, Q) k , so ¢ is highly contractive near P. And if P is also recurrent, then ¢n (P) gets very close to P infinitely often, so it receives the highly contractive effect of ¢ infinitely often. This should cause points near P to remain near P, and thus force P into the Fatou set. The critical recurrent condition and the Julia condition are thus in opposition to one another, which means that nonperiodic recurrent critical points should be quite rare. On the other hand, Rivera-Letelier [378] has shown that there are maps having wildly critical recurrent points (which are then necessarily in the Julia set) in lP' 1 (Qp). It is not known whether Rivera-Letelier's examples have wandering disk domains. In the other direction, it is known that there are rational maps defined over Cp that have wandering disk domains [63, 59, 62, 378]. In these examples the critical points are in the Fatou set, so Theorem 5.55 implies that the maps cannot be defined over a finite extension of QP . We now use Theorem 5.46 and a simple compactness argument to prove that the Fatou sets of p-adically hyperbolic maps over finite extensions of Qp have no wandering disk domains. This is not as strong as Theorem 5.5 1, but still covers an important class of maps. The proof of Theorem 5.51 uses similar ideas, but is more complicated; see [54]. Theorem 5.55. (Benedetto [56]) Let K/QP be a.finite extension ofp-adicfields, let ¢( z ) E K ( z ) be a rational map ofdegree d 2: 2, and assume that the Julia set .:J ( ¢) contains no critical points of¢, i.e., assume that ¢ is p-adically hyperbolic. Then the Fatou set F( ¢) has no wandering disk components. Proof Proposition 5.20 assures us that F( ¢) is nonempty, and it is open, so it con
tains an algebraic point. (Note that Qp is dense in Cp.) Replacing K by a finite extension and changing coordinates, we may assume that oo E F( ¢), and indeed we may even assume that .:J(¢) c D(O, 1). Equivalently, we may assume that the disk component of oo contains lP'1 (Cp) ' D(O, 1 ) . We suppose that U c F (¢ ) is a wandering disk component of F(¢) and de rive a contradiction. Replacing U with the disk component containing ¢n (U) for a sufficiently large n, we may assume that the orbit of U does not include the disk component at oo. In particular, ¢n (U) c D(O, 1) for all n 2: 0. Again taking a finite extension of K if necessary, we can find a point a0 E K and a radius r0 such that D(ao, ro) C U. At this stage we fix the field K and we use Theorem 5.46 to find an integer m such that I ( ¢m )'I > 1 on .:J ( ¢) n K. Replacing ¢ by ¢m , it suffices to consider the case that W I > 1 on .:J(¢) n K. For each n 2: 0, the image ¢n (iJ(a0 , r0 )) is a (closed) disk centered at the point an = ¢n (ao), say In particular, ¢ (D(an , rn ) ) = D(an+l, rn + d, so applying Proposition 5 . 16(b) yields
286
5. Dynamics over Local Fields: Bad Reduction
(5.30)
We also know that the disks D (O:n , rn ) are disjoint, since D (O:n , rn ) is contained in ¢n (U), and further, each disk D(o:n , rn ) contains a point of D(O, 1)nK. It follows that the radii must satisfy limoo rn = 0, (5.3 1) n---+ since the set D(O, 1) n K has finite volume, so can contain only finitely many nonempty disjoint disks of any fixed radius. It follows from (5.30) and (5.31) that there are infinitely many with the property that
n
i.e., since rn of points
----t
0, we must have rn+ 1 < rn infinitely often. Consider the infinite set
{ o:n : l ¢' (o:n ) l < 1, n = 1,2,3, . . } . It is contained in the compact set D(O, 1) n K, so it contains an accumulation point f3 E K. The continuity of ¢' implies that l ¢'(!3) 1 :::; 1, which shows that f3 E F(¢), since we used Theorem 5.46 to ensure that I ¢'1 > 1 on :J ( ¢). .
Let V be the disk component of F(¢) containing {3. Then by construction V contains infinitely many of the iterates O:n = ¢n ( o:0). Since the radii of ¢n ( U) and ¢n (V) shrink to 0 as oo, it follows that some iterate ¢n (U) is contained in V and that some (nontrivial) iterate of ¢n (V) is contained in V. Therefore U is not wandering, contradicting our original assumption. D
n
5.8.4
----t
Wandering Disk Domains Exist in CP
We have proven that p-adically hyperbolic maps defined over Qp have no wandering disk domains. More generally, Theorem 5.51 shows that rational maps ¢( z) E Qp ( z) with wandering disk domains are very rare, if they exist at all. And of course, Sul livan's theorem 5.43 says that rational maps ¢(z) E C (z) defined over the complex numbers never have wandering domains. It is thus somewhat surprising to discover that there are very simple polynomial maps defined over Cv that have wandering disk domains. Theorem 5.56.
(Benedetto [59]) For c E Cv, let ¢c(z) be the polynomial cPc (z)
=
(1 - c)zP+ l + czP .
Then there exists a value a E Cv such that: (1) :J(c/Ja) -::/- 0, (2) F( cPa) has a wandering disk domain, (3) F( cPa) contains every critical point of cPa·
Proof See [59] for a proof of this specific theorem, and see [63, 62, 373, 378, 380]
for generalizations and related results.
D
5.9. Green Functions and Local Heights
5.9
287
Green Functions and Local Heights
The canonical height h¢ associated to a morphism ¢ JID1 --+ JID 1 is defined as the limit of d - n h ( ¢n ( P)) as n --+ oo. The utility of h1; lies in the two formulas :
( ) + 0( 1)
h¢(P) = h P
and
( ))
h¢ (¢ P = dh¢(P),
where the first says that h1; contains arithmetic information and the second says that h1; transforms canonically. It is tempting to try a similar construction locally and define (say) 1 (v-adic local height of a) = lim (5.32) log max{ l ¢n (a ) I 1 } . -+oo n d n It is clear that if the limit (5.32) exists, then it transforms canonically, and indeed if ¢(z) is a polynomial, then the limit does exist and everything works quite well (see Exercise 3.24). Unfortunately, for general rational maps the limit (5.32) may not exist. Rather than working directly with a rational map ¢ JID1 JID1 , it turns out to be easier to develop a theory oflocal heights by first lifting ¢ to a map A2 --+ A2 and then constructing a real-valued function g on A2 that satisfies the canonical trans formation formula g ( = dg In this section we construct the Green function g, prove some of its basic properties, and then use g to construct local canonical height functions on JID1 as described in Theorem 3.27. A point in projective space y] E JID 1 is given by homogeneous coordinates, so We make explicit the natural projec it is really an equivalence class of pairs tion map v
:
(x, y))
(x, y).
[x,
that sends a point (x, y) we write
E
--+
(x, y).
,
:
(x, y) � [x, y],
A2 to its equivalence class [x, y] A; = A2 {0, 0}
E
JID 1 . To ease notation,
"-
for the affine plane with the origin removed. Let ¢ JID1 JID 1 be a rational map of degree d. Then ¢ can be written as usual in the form ¢ = with homogeneous polynomials and of degree d having no common factors. The polynomials and then define a map :
--+
[F, G]
F G F G (x,y) = (F(x,y),G(x,y)),
that fits into the commutative diagram A 2 .p A 2 &* ------+ &* JID l � JID l
We call a lift of¢. By homogeneity, if = is one lift of¢, then every other lift of ¢ has the form for some constant c E K*.
c [cF, cG] [F, G] =
5. Dynamics over Local Fields: Bad Reduction
288
Definition. Let K be a field and let v be an absolute value on K. We denote the absolute value (or sup norm) ofa point E A2 ( K) by
( x, y)
Similarly, the absolute value (or sup norm) of one or more polynomials is the maxi mum of the absolute values of their coefficients. (We have already made use of this convention in the proof of Theorem 3 . 1 1 .) We begin by recalling how a map of a point.
:
A2(K)
-t
A2(K) affects the v-adic norm
Proposition 5.57. Let K be afield with an absolute value v. Let
be given by homogeneous polynomials F, G E ofdegree 2 1, and assume that F and G have no common factors in K (a) There are constants c1 , c2 > 0, depending only on and v, such that
[x, K[y]. x, y]
d
I (x,y) l v Cl - I (x,y) I v - C2 forall (x,y) E A; (K). (5.33) (b) If v is nonarchimedean and satisfies I I v = 1, then (a) is true with the explicit constants (x, y) l l v l l G) l IRes (F v -< l l (x, Y)ll � - for all (x,y) E A; (K). (5.34)
as a sum of local canonical height functions �c/>,v• but we deferred the proof. The intuition in Section 3.5 was that the local canonical height should measure
�¢,v(P) = - log( v-adic distance from P to oo ) . More generally, it is convenient to define a local canonical height that measures the v-adic distance from P to a collection of points. In the following theorem we de scribe a set of points, possibly with multiplicities greater than 1, by specifying the homogeneous polynomial E E K[x , y] at which they vanish. In slightly fancier ter minology, we are identifying positive divisors in D iv(IP'1 ) with homogeneous poly nomials in K[x, y] . IP'1 --+ IP'1 be a rationalfunction of degree d 2: 2 defined over K. Fix a lift = ( F, G) of¢ and let Qcp be the associated Green function. Theorem 5.60. Let K be afield with an absolute value v and let ¢
:
5. Dynamics over Local Fields: Bad Reduction
292
(a) For any homogeneous polynomial E ( x, y) E K [x, y] ofdegree e 2': 1 we define
�q,, e ([x, yl ) eQq,(x, y) - logiE(x, y) l v for [x, y] E lP' 1 (K) with E(x, y ) -1- 0.
=
Then �q,,E is a well-definedfunction on lP'1, i.e., the value of�q,, e (P) does not depend on the choice ofhomogeneous coordinates [x, y]for P. Thefunction �¢,E is called a local canonical height associated to ¢ and E. In the special case that E(x, y ) y, we drop E from the notation and refer simply to a local canonical height associated to ¢. (b) For all P E lP'1 (K) with E(P) -1- 0 and E(if.>(P)) -1- 0 we have
=
A
A
I (E o if.>)(P) d I
. v (Note that the homoegeneity of E and if.> ensures that the ratio ( E o if.>)/ Ed is a >.. q, ,E ( ¢(P)) d>..q, , e (P) - log
=
well-definedfunction on lP'1.) (c) Thefunction
E(P)
E(P) (5.42) >.. q, ,e(P) + log I liP I v , II � which a priori is defined only at points P satisfYing E(P) -1- 0, extends to a P
A
f-------7
bounded continuous function on all oflP'1 (K). (d) Given the particular choice of lift if.>, the function unique real-valuedfunction
�q,,E defined in (a) is the
satisfYing (b) and (c). If cif.> is a different lift, then with the obvious notation,
In particular, any two local canonical heights differ by a constant.
Qq,(cx, cy) Qq,(x, y) + log l clv, while the ce E(x, y), so the difference
Proof The Green function satisfies
polynomial E satisfies E(cx, cy)
=
=
eQq,(x, y) - log I E(x, y) lv defining �¢,E does not change ifwe replace (x, y) by (ex, cy). This proves (a). (b) We compute
�q, , e (¢(P)) = eQq, (if.>(x, y)) - logiE( if.> (x, y)) l v edQq,(x, y) - log i E(if.>(x, y)) l v from Proposition 5.58(b), = d�q,, e (P) + d log I E(x, y) lv - log i E(if.>(x, y)) l v ·
=
5.9. Green Functions and Local Heights
293
( c ) Directly from the definition of �.E we see that the righthand side of (5A2) is equal to (5,43) e(Qq,(P) - log II P II v) · The boundedness of(5A3) is exactly (5.37) in Proposition 5.58(b). Further, we know from Proposition 5.58(e) that Qq,(x, y) is a continuous function on A;(K), and it is clear that logl l (x, y) ll v is also a continuous function on A; (K), so the differ ence (5,43) is a continuous function on A;(K). Further, the difference is invariant under ( x, y) (ex, cy) , so it descends to a continuous function on lP' 1 ( K). Suppose that � and �� both satisfy (b) and (c), and let ��� = � - �1 • Writing
(d)
r--+
(
Iv �" (P) = �(P) + log I E(P) P II II �
Iv ) ) (�'(P) + log I E(P) P II II � ' _
we see from (c) that ��� extends to a continuous bounded function on all of lP'1 ( K). Let C be an upper bound for I �"I · From (b) we find that ��� ( ¢(P)) = d�" (P) for all P E lP' 1 (K) with E(P) =/- 0 and E ( ¢(P)) =/- 0, and iterating this relation yields �11 ( ¢n ( P) ) = dn �11 (P ) provided E ( ¢i ( P)) -1- 0 for all 0 ::; i ::; Hence (5,44) for all points P E lP' 1 ( K) satisfying E ( ¢i ( P)) -1- 0 for all 0 ::; i ::; n. But each equation E ( ¢i ( P)) = 0 eliminates only finitely many points, so the inequality (5,44) is true for all but finitely many points oflP' 1 (K). Then the continuity of ��� tells us that I � �� ( P) I ::; Cd-n is true for all P E lP'1 ( K). Since is arbitrary, this proves that ��� ( P) = 0, so � = �'. Finally, the effect of replacing by c follows immediately from the definition of �,E in terms of the Green function Qq, and from the corresponding transformation D formula for Qq, given in Proposition 5.58(d). n.
n
Finally, as promised in Section 3.5, we prove that the global canonical height is equal to the sum of the local canonical heights. Theorem 5.61. Let K be a number field, let ¢ lP' 1 ----+ lP' 1 be a rational func tion of degree d 2: 2 defined over K, and fix a lift = (F, G) of ¢. Choose a :
homogeneous polynomial E(x, y) E K[x, y], andfor each absolute value v E MK, let �,E,v be the associated local canonical height described in Theorem 5.60, where we now include v in the notation so as to distinguish between different absolute values. Then the (global) canonical height has a decomposition as a sum of local canonical heights, for all P E
lP'1 (K) with E(P) -1- 0.
294
5. Dynamics over Local Fields: Bad Reduction
Proof We use the definition of �c/>,E,v in terms of the associated Green function Q�, v
from Theorem 5.60(a) to compute
�
�
( de de E L nv �cf> ,E , v (P) = de E L nv ( g E)Q�, v (P) - logiE(P ) IJ v EMK vEMK
Theorem 5.59 says that the first sum is equal to h¢(P), while the product formula (Proposition 3.3) tells us that the second sum is 0. (Note that this is where we use the D assumption that E(P) -=1- 0.) Remark 5.62. If v E M� is nonarchimedean and ¢ has good reduction at v, then the
Green function and the local canonical height are given by the simple formulas
(
)
max { lxl v , I Y i v } Q� ,v (x, y) = log max { lxl v , I Y i v } and >.., c/>, E,v ( [x, y] ) = log · I E(x, y) lv Thus it is only for maps with bad reduction that the Green and local canonical height functions are interesting. This should not come as a surprise to the reader, since bad reduction is the situation in which dynamics itself becomes truly interesting. Of course, this is said with the understanding that every rational map over C has "bad reduction," so the dynamics of holomorphic maps on ( are always interesting and complicated. Remark 5.63. For additional material on dynamical Green functions and dynamical local heights, see [21 , 88, 234, 233].
lP'1 q
5.10
Dynamics on Berkovich Space
We have seen that Cp is a natural space over which to study nonarchimedean dy namics, since it is both complete and algebraically closed. However, the field Cp has various unpleasant properties: Cp is totally disconnected. Cp is not locally compact, so the unit disk { z E Cp : l zl ::; 1 } and projective line ( Cp ) are not compact. The value group IC;I = l� consists of the rational powers of p, so it is not discrete in IR>0, yet neither is it all of!R> O · This list suggests that it might be better to work in some larger space. There is a general construction, due to Berkovich [64, 67], that solves these problems for Cp and other more complicated spaces. The study of dynamics on Berkovich spaces started during the 1990s and is an area of much current research. In this section we briefly describe the Berkovich disk and associated affine and projective lines and •
•
•
lP'1
295
5.10. Dynamics on Berkovich Space
discuss some very basic dynamical results. In a final subsection we state without proof some recent results. For further reading, see [26, 5 1 ] for an introduction to dynamics on Berkovich space and see [373, 375, 376, 379, 377, 380, 381] for Rivera Letelier's fundamental work in this area. 5.10.1
The Berkovich Disk over CP
The unit Berkovich disk [JB is a compact connected metric space that contains the totally disconnected non-locally compact unit disk in Cw We describe two construc tions of [JB, the first an explicit description as the union of four types of points and the second as a set ofbounded seminorms on Cp[z]. 5.10.1.1 The Four Types of Berkovich Points
The most concrete description of [JB is as the union of the following four sets of points: Each point a in the standard unit disk D(O, 1) of Cp is associated to a point of the Berkovich disk, which we denote by
Type-I Berkovich Points.
�a, o
Each closed disk D(a, r ) contained in D(O, 1) with ra
Type-11 Berkovich Points.
dius r E I C; I denote by
=
E D- s .
pifl is associated to a point of the Berkovich disk, which we �a, r
E D- s ·
Type-III Berkovich Points. Similarly, each closed disk D( a, r ) C D(O, 1) with positive radius r ¢. I C; I = pQ is associated to a point of the Berkovich disk, which we naturally also denote by �a,r·
These are the trickiest points in [JB. They are associ ated to nested sequences of closed disks
Type-IV Berkovich Points.
with the property that
n D(an , Tn ) = 0.
n2: 1
We denote these points by �a ,r, where, as the notation suggests, the vectors a and r are a = (a 1 , a2 , . . . ) and r = (r1 , r2 , . . . ) . Remark 5.64. Note that Berkovich points �a, r of Types II and III are disks D(a, r ) ,
so different values of a may yield the same Berkovich point. Indeed, we have if and only if
r = s
and I a - bl :::; r,
5. Dynamics over Local Fields: Bad Reduction
296
since these are the conditions for the disks D( a, r) and D ( b, s ) to coincide. Similarly, two Berkovich points of type IV are the same if their sequences of disks can be suitably intertwined. See Exercise 5.40 for details. Remark 5.65. A point �a, r of Type-I, II, or III corresponds to a disk (possibly of radius 0), so we define the radius of �a, r to be r. The radii ro, r 1 , r2 , . . . of a Type IV point �a,r are nonincreasing, so the limit r = limi_,oo ri exists and is called the
radius of�a,r·
We claim that the radius of a Type-IV point is strictly positive. To see this, sup pose that �a,r has radius 0. Then the sequence of points a 1 , a2 , . . . is a Cauchy se quence in Cp, so it converges to a point a E Cp. Let
i5i = jnf ,ri zED(ai )
lz - al
be the distance from a to the ith disk. Notice that 0 � n Zn E CCp [z] : J�moo l cn iRn n 2:0
=
0
}·
298
5. Dynamics over Local Fields: Bad Reduction
The ring 1l'R is a Banach algebra over Cp using the Gauss norm, which the maximum modulus principle tells us is the same as the sup norm. The Berkovich disk D� is often defined to be the set of bounded seminorms on the ring 1l'R - One can show, using the Weierstrass preparation theorem, that this leads to the same set of points and the same topology as taking the I · llwbounded semi norms on the polynomial ring Cp [z]; see [26]. The ring 1l'R is an example of a Tate algebra. Applying the same construction to more general Tate algebras allows one to construct Berkovich spaces for the associated rigid analytic spaces. 5.10.1.3 Visualizing the Berkovich Disk
In order to visualize the Berkovich disk, we place the Gauss point �0, 1 at the top of the page and observe that there is a line segment running from any point � -=1- �0, 1 up to the Gauss point. If� = �a,r is of Type-I, II, or III, then this line segment is the set of points La,r = { �a , t : r S t S 1}. Notice that any two line segments La,r and Lb,s merge with one another at the point �a, t = �b,t determined by t = max{r, s, l b - a l } . This is the smallest allowable value of t for which the disks D(a, t) and D(b, t) acquire a common point, hence for which they coincide. Thus one can imagine the various line segments continually merging as they run upward toward the Gauss point at the top of the tree. The line segment running up from a Type-IV point �a,r is slightly more compli cated. It is the set of points 00
00
i= 1
i=1
(5.45)
(Note that ro = 1 by definition.) The Type-IV point �a,r is included in the Berkovich disk precisely to provide an endpoint for the union of line segments U La; ,r; . Now imagine starting at the Gauss point and moving downward through the tree. We claim that at any instant, there are three possible scenarios: 1 . If you have reached a point �a,o of Type I or a point �a,r of Type IV, then you have hit the end of a segment and cannot proceed further. 2. If you have reached a point �a,r of Type II, then r > 0 is in the value group of Cp and there are countably infinitely many branches along which you can move down the tree. 3. If you have reached a point �a,r of Type III, then r > 0 is not in the value group of Cp and there is only one direction to move down the tree.
5.10. Dynamics on Berkovich Space
299
The picture for points of Type I is clear. In order to understand Types II and III, suppose that we fix a point of Type II or III. Each b E D(a, r) gives a line segment that runs up from the Type-I point and through the point Two such line segments and if and only if merge before reaching l b - b'l < r, so it is really each open disk D(b, r) inside the closed disk D( a, r) that gives a line segment running up to If is of Type III, then D(b, r) = D( a, r) for any b E D( a, r ) , so there is only one segment running downward from The situation is much more interesting, and complicated, if is of Type II. In this case D( a, r) is covered by a countable union of open disks D(b, r ) , so there is a countable set of branches downward from A convenient, although noncanon ical, way to describe the branches is as follows. Let lfJ = { z E Cp : I z I < 1} denote the maximal ideal in the ring of integers of Cp and fix some c E Cp with l ei = r. Then the open disks of radius r are in one-to-one correspondence with the residue field lFP via the map
�a,r Lb,o Lb' ,o �a,r · �a,r ·
Lb,o
�a,r
�b,o
�a,r
�a,r
�a,r ·
{ disks D(b, r) inside D(a, r) }
-+
JFP ,
�r,a ·
D(b, r)
f-------7
(b/c) mod \l}.
The surjectivity of this map is clear, and the injectivity follows from the fact that D(b, r) = D(b' , r) if and only if l b - b'l < r = l eiIn order to fit the Type-IV points into the picture, let
be any sequence of nested disks and consider what happens as we move down the line segments The fact that the disks D(ai , ri ) are nested implies that the line segments extend one another downward as i increases. If the intersec tion n D (a i , ri ) is nonempty, then it is equal to D (a, r) for some a E Cv and some r 2:: 0, so the intersection corresponds to a point of Type I, II, or III forms a closed line segment. and the union together with the endpoint However, if the intersection n D ( ai , ri ) is empty, then there is no actual disk D(a, r) sitting at the bottom of the union of the line segments Thus as already noted, the points of Type IV exist precisely to remedy this situation and to ensure that every downward path has a termination point. Further, this explains why we defined by (5.45) to be the line segment running from to (See Exercise 5 .41 .) The Berkovich disk [JB is "illustrated" in Figure 5.4. Of course, Figure 5.4 is at best a pale imitation of the true glory of the Berkovich disk, since in a complete picture of [JB, every line segment contains a countable number of Type- II points, off of each of which there is a countable number of downward branches. Unfortunately, despite advances in modem technology, there are still no computer packages capable of fully rendering a (countably) infinitely branched broomstick.
LaLa,,,,rrii · U La,,ri
�a,r �a,r
La,r
5.10.1.4 The Gel'fond Topology on the Berkovich Disk
The next step is to put a topology on the Berkovich disk.
ULa,,r, · �a,r �0, 1 .
300
5. Dynamics over Local Fields: Bad Reduction
�0, 1 � Gauss point I
I Type-I point I
-
�0 , 0
�a ,O
�b,O
Figure 5.4: The Berkovich disk D 8 . Definition. The Gel'fond topology on D 8 is the weakest topology such that for every f E Cv [ z] and every B > 0, the following sets are open:
U(f, B) = {x E D8 : lfl x < B}
V(f, B) = {x E D8 : lfl x > B}.
and
Theorem 5.68. (Berkovich) The Berkovich disk fJB with the Gel 'fond topology is a compact path-connected Hausdorffspace.
Proof See [64, Theorem 1 .2.1] or [29, Appendix D) for the proof that [JB is compact
and Hausdorff and [64, Corollary 3.2.3] for the proof that it is path connected.
D
Basic open sets for the Gel ' fond topology on JJ!3 , viewed as a tree, can be de
scribed by taking (and deleting) branches of the tree as described in the following definition. The closed branch of [JB rooted at (a, r), denoted by (r.. a , r , consists of all points �b,s such that �a , r is on the line segment Lb,s running from �b,s to �0,1, together with whatever Type-IV points are needed to finish off the bottom of any open line segments. Thus
Definition.
{
lr-.. a , r = �b,s s :
:S r and l b - a l :S r }
U
{ appropriate Type-IV points}.
The open branch rooted at (a, r), denoted by �r-..: , r is obtained by starting with the closed branch lr-.. a ,r • removing the point �a , r. and then taking the connected com ponent that contains �a,O · If �a,r is of Type III, this is simply the closed branch at �a , r with the single point �a,r removed; but if �a,r is of Type II, then there are countably many branches at �a , r and we select the one that includes the point �a ,O · It is not hard to see that �r-..: , r = �b,s : s < r and l b - a l
{
1 and r < 1} U {�oo , o}.
The open and closed branches in A\:0 are defined using the natural identification �
Ao Ao /1\0, 1 f-------> /1\ oo •
For example, a basic open neighborhood of the Gauss point �o, 1 is obtained by re moving from JP'8 a finite number of closed branches, some of which may be in the vertical branch A\"oo at infinity.
303
5.10. Dynamics on Berkovich Space
Extra branch "at infinity"
Figure 5.5: The Berkovich projective line
JP'8 .
Remark 5.70. There is a natural embedding of A.S into lP'8 . However, the inversion map f ( z ) = 1/z used to glue together the two pieces of can cause notational
8 lP' confusion regarding the "radius" of points of Types II and III. For example, the point �p-2 ,p-3 in the branch fr..:O of lP'8 would be denoted by �p-1 ,p when viewed as a point in A8 . Thus it might be wiser to denote the points in fr.. using some :O
alternative notation, for example ta, r, but we will not do so. Remark 5.71. A useful alternative construction oflP,s mimics the construction of the scheme lP'i as the set of homogeneous prime ideals. It starts with the set ofbounded seminorms on the two-variable polynomial ring Cp [x , y] that extend the usual ab solute value on Cp and that do not vanish on the maximal ideal (x, y). Two semi norms I 1 1 and I · are considered equivalent if there is a constant c > 0 such that for all homogeneous f E Cp[x, y]. Then is the set of equivalence classes of such seminorms. For details of this construction, see [66]. Let P E lP' 1 (Cp). To create a seminorm from P, choose homogeneous coordi nates P = [a, b] and set I f I = lf(a, b) 1. Notice that a different choice of homoge neous coordinates for P gives an equivalent seminorm. This embeds lP' 1 ( Cp) into
12
lP'8
5.10.2.3
Properties of
A8 and JP'8
JP'8 .
As repayment for the effort required to construct them, Berkovich spaces have many nice properties that Cp lacks.
304
5. Dynamics over Local Fields: Bad Reduction
(a) The Berkovich disks D� are compact, Hausdorff, and uniquely path connected. (b) The Berkovich affine line A.6 is locally compact, Hausdorff, and uniquely path connected. (c) The Berkovich projective line JID6 is compact, Hausdorff, and uniquely path con nected.
Theorem 5.72.
5
D
Proof See [26] and [64].
Remark 5.73. As noted earlier, the Berkovich affine line A6 contains a copy of A 1 (Cp) = Cp, since each a E Cp gives an associated Type-1 point in .D�
�a,o
provided R ;::: l aJ. Similarly, the Berkovich projective line JID6 contains a copy of the classical projective line JID1 ( Cv) via the map a f-----+
E
{ �a�a,,oO �oo,O
D (O , 1) if lal :::; 1,
E /r..:O E
/r..:O
if 1 < Ia I < oo , if a = 00.
One can show that the restriction of the Gel'fond topology on A6 and JID6 to A 1 (Cp) and JID1 ( Cv), respectively, gives the topology induced by the usual metric on A 1 ( Cp) and the chordal metric on JID1 (Cv) · See Exercise 5.44. This explains why the Gel'fond topology is the "right" topology to use on Berkovich spaces. Remark 5.74. We have constructed A6 and JID6 purely as topological spaces. It is more difficult, but very important, to refine this construction and make A6 and JID6 into ringed spaces with structure sheaves built up naturally from rings of functions. There are two approaches, both due to Berkovich. The first takes unions of open Berkovich disks, which have a natural structure as analytic spaces, and glues them along open annuli; see [26, 64]. The second glues affinoids (which are closed) using nets; see [65]. This second construction is less intuitive, but it allows one to functo rially attach a Berkovich analytic space to any reasonable rigid analytic space. 5.1 0.3
Dynamics on Berkovich Space
Having constructed the Berkovich spaces .06, A 6, and JID6, we are finally ready to study iteration of maps on these spaces. 5.10.3.1
Polynomial and Rational Maps on Berkovich Space
Let ¢(z) E Cp[z] be a polynomial. There is a natural way to extend the map ¢ : A 1 (Cp) A 1 (Cp) to a map on Berkovich affine space ¢ : A6 � A.6. In terms of seminorms, the action of ¢ is simply given by composition, �
lfi 1 .
Note that tlle assumption tllat ¢ does not vanish on D(a, r) is equivalent to the in equality 1¢( a) I > s, so the indicated points are in D8. The description of ¢(�a, r ) when ¢ has zeros and/or poles on D(a, r) is more complicated. An explicit description in terms of open annuli is given by Rivera Letelier [373, 375, 376]. (See also [26, Section 2].) A succinct, but less explicit, way to specify the induced map ¢ JID8 JID8 is to use the construction of JID8 as a space of homogeneous seminorms as described in Remark 5. 7 1 . Then for a given seminorm � E JID8, the seminorm ¢( � ) is determined by writing ¢ = [F, G] using homogeneous polynomials F and G and setting :
---+
if(x, y ) l ¢ (� ) = if(F(x, y), G(x, y)) i
for all homogeneous f E Cp[x, y].
306 5.10.3.2
5. Dynamics over Local Fields: Bad Reduction The Julia and Fatou Sets in Berkovich Space
A natural way to put a metric on the Berkovich spaces [JB, A 8, and !P'8 is to use the underlying tree structure and measure distances along line segments. Unfor tunately, this path-length metric does not give the Gel'fond topology, and as we have observed, it is the Gel'fond topology that extends the natural metric topolo gies on D (O , 1), A 1 (Cp), and IP' 1 (Cp) . (See Exercises 5.42 and 5.44.) It is possible to define a metric that does yield the Gel'fond topology, but the definition of the "Gel'fond" metric is quite indirect. See [26, Corollary 1 .3]. So rather than using a metric, we instead characterize the Fatou and Julia sets in Berkovich space using an abstract topological version of equicontinuity. Let X and Y be topological spaces and let be a collection of con tinuous maps X Y. The set is (topologically) equicontinuous at x if for every point y E Y and every neighborhood V c Y of y there are neighborhoods U c X of x and W c Y of y such that for every rjJ E , the following implication is true: Definition.
-+
rjJ(U) n W -1- 0
==?
rjJ(U) c V.
Intuition: is equicontinuous at x if for each y E Y, whenever rjJ E sends some point close to x to a point that is close to y, then rjJ sends every point close to x to a point that is close to y. One can show that if Y is a compact metric space, then topological equicon tinuity agrees with the usual metric definition of equicontinuity. (See [26, Proposi tion 7 . 1 7].) We say that is (topologically) equicontinuous on X if it is topologically equicontinuous at every point of X.
Let rjJ(z) E Cp(z) be a rational map. The (Berkovich) Fatou set of rjJ is the largest open subset of lP'8 on which rjJ is equicontinuous, or more precisely, on which the set of iterates { rjln } is equicontinuous. The (Berkovich) Julia set ofrjJ is the complement of the Berkovich Fatou set. We denote these sets by :F8 ( ¢ ) and .:J8( rjJ ), respectively. Definition.
Remark 5.76. Recall that the classical points in !P'8, i.e., the points of Type-1, form
a copy of IP' 1 ( Cp) sitting inside !P'8. As noted earlier in Remark 5. 73, the restric tion of the Gel'fond topology on !P'8 to the classical points gives the same topol ogy on IP'1 (Cp) as that induced by the chordal metric. Using this one can show that equicontinuity at a classical point of!P'8 using the Gel'fond topology is equivalent to equicontinuity using the chordal metric. Hence the classical Fatou and Julia sets sit within their Berkovich counterparts: and
rjJ(z) E Cp(z) be a rational map of degree at least 2. Various authors have shown that there is a unique probability measure 1-l¢ on !P'8 satisfying
Remark 5.77. Let
¢ * 1-l¢ = d . 1-l¢
and
5.10. Dynamics on Berkovich Space
307
(Recall that a probability measure is a nonnegative measure of total mass 1 .) We call f-l the canonical measure associated to ¢, since the property ¢* f-l = d f-l resembles the analogous property h4> ( ¢( P)) = d · h4> (P) of the canonical height. The reader should be aware that other common names for f-l in the literature in clude Brolin measure, Lyubich measure, and invariant measure. For the construction and applications of fJ, see Baker and Rumely [26, Theorem 7.14], [29], Chambert Loir [98], Thuillier [434], and Favre and Rivera-Letelier [168], as well as [360]. ·
Theorem 5.78. Let rjJ( z ) E Cp(z) be a rational map of degree at least 2. The sup port of the canonical measure f-l is equal to the Julia set .J8 ( ¢) . In particular, the Berkovich Julia set .J8 ( ¢) is not empty.
Proof This theorem is an amalgamation of results due to Baker, Rumely, and Rivera
Letelier. We refer the reader to [26, Section 7.5] for the construction of the canonical measure and to [26, Theorem 7. 1 8], [27, Theorems 8.9 and A.7], and [38 1 ] for the proof that f-l is supported exactly on .J8 ( ¢). The last part of the theorem is then clear, since the empty set cannot provide the support for a nontrivial measure. D
Example 5.79. Let ¢( z ) E Cp(z) be a rational map of degree at least 2 and sup
pose that ¢ has good reduction. We know (Theorem 2.17) that the classical Julia set .J ( ¢) c lP'1 ( Cp) is empty. Using the construction of the canonical measure, it is not hard to show [26, Example 7.2] that for a map of good reduction, the canonical measure is entirely supported at the Gauss point, i.e., f.l(U) = 1 if �o, 1 E U
and
f.l (U) = 0 if �o, 1 � U.
Thus .18(¢) = {�0, 1 }, so the nonempty Julia set guaranteed by Theorem 5.78 is not very interesting, since it consists of a single point. Hence even in Berkovich space, the most interesting dynamical behavior occurs for maps of bad reduction. On the other hand, if the conjugates ¢! of ¢ have bad reduction for every f E PGL2 (Cp), then .J(¢) is a perfect set, and hence uncountable. (See Theorem 5.82.) 5.10.3.3 The Map ¢ ( z )
=
z 2 on Berkovich Space
To conclude our brief foray into Berkovich space, we illustrate Berkovich dynamics by studying the simplest possible map, namely ¢( z ) z 2 • For any a E Cp, we expand ¢( z ) - ¢( a ) = z 2 - a2 = 2a ( z - a ) + ( z - a ) 2 . =
Assuming henceforth that p 2: and using our convention that all points in !P'8 have radius satisfying r ::; 1, we find that
3
r/J (�a,r) = �(a),s
with s = max{ l2a l r, r 2 } = r · max{ lal, r }.
(5.46)
This explicitly gives the action of ¢ on points of Types I, II, and III in fJB, and the action of ¢ on Type-IV points is given by the appropriate limit.
308
5. Dynamics over Local Fields: Bad Reduction
The formula (5.46) allows us to compute many orbits O (�a ,r ) . For example, suppose that ia l < 1 and r < 1. Then I n such that Then q;m(�a,r) q;n(�a ,r ), so �a , r is preperiodic. We conclude this section by using the definition of topological equicontinuity to directly demonstrate that the Gauss point �0, 1 is in the Julia set of¢. The intuition is that any neighborhood of �0, 1 contains points �o,r on the line segment connecting �o, 1 to �0,0. If < 1, then the iterates q;n (�o,r) approach �o,o, but the Gauss point �o, 1 is fixed. Hence q;n(�0, 1 ) does not remain close to q;n(�o,r ). We now make this argument ngorous. We suppose that x = �0, 1 E :F8 ( ¢) and derive a contradiction. Let y �0,0, and let V /1-..� ,1 ; 2 be our chosen neighborhood of �0,0. The definition of equicontinuity says that there are neighborhoods �0, 1 E U and �o,o E such that for all n ;::: 0,
=
r
=
=
W
The implication (5.47) remains true if we replace U and hoods, so we may assume that U
=
JP'B "-
k
U /1-..ai ,ri
with 0
IP'1 (K), and suppose that o: , (3 E IP'1 ( K) do not lie in rjJ (D( a, r) ) for all rjJ E . Our proof ofthe nonar chimedean Monte! theorem (Theorem 5.27) shows that there is a constant C C( o:, (3, a, r) such that =
p(rj;(z), rf;(w)) S Cp(z, w) Find an explicit value o f C i n terms o f
o: ,
E and all z, w E D(a, r). (3 , a, and r. for all rjJ
316
Exercises
Section 5.7. Periodic Points and the Julia Set 5.19. As in the statement of Lemma 5.34, let A, B C Cp be bounded sets for which there are constants 0 < 8 ::; 1 ::; � such that
sup I ::; �, sup 1 13 1 ::; �, and aEA,inf(3EB p(o:, !3) = 8 > 0. aEA Ia: (3EB For each ( o:, 13) E A x B, let La , f3 (z) (13 - o:)z + o:. During the proof of Lemma 5.34 we =
showed that
p ( La , f3(z), La',f3' (z ' ) ) ::;
�24
max {p (o: , o:'), p(l3, 13'), p(z, z') } for all o: , o:' E A, all 13 , 13' E B , and all z , z' E IP 1 (Cp)· Improve this result b y reducing the exponent o f � and/or 8 in the constant. Try to find the •
best-possible exponents. 5.20. Let A, B , C c IP1 (Cp) be three sets that are at a positive distance from each other, i.e., there is a constant 8 > 0 such that
aEA,inf(3EB p( o:, 13) 2: 8, aEA,inf-yEC p(o:, 'Y) 2: 8, For any triple of points (o:, 13, 'Y) E A x B x C, define
inf
(3EB, -yEC
p(l3, 'Y ) 2: 8.
to be the unique linear fractional transformation satisfying
La,(3, -y (0)
= 0: ,
Prove that the map
( o: , /3 , ')' , z)
f-->
La,(3, -y (z),
is Lipschitz and find an explicit Lipschitz constant depending only on 8. (This exercise gener alizes Lemma 5.34, which is the case that C = { oo} is the single point at infinity.) 5.21. This exercise can be used in place of Lemma 5.34 to prove Mantel 's theorem with moving targets (Theorem 5.36). (a) Let P, Q E IP 1 (K) be distinct points. Prove that there is a linear fractional transformation A = AP, Q E PGL2 (K) satisfying
A(O)
=
P,
A(oo)
=
Q,
IRes( A) I
=
p(P, Q).
(Hint. If A = ( � S) is normalized to satisfy max { l a l , l b l , l ei , ldl } = 1, then Exer cise 2.8 says that the resultant of A is equal to the determinant ad - be.) (b) Let A, B C Cp be bounded sets that are at a positive distance from each other. Prove that the map
(P, Q , R) i s Lipschitz.
f-->
A P, Q ( R) ,
317
Exercises Section 5.8. Nonarchimedean Wandering Domains
5.22. Let X be a topological space and let D be a collection of open subsets of X that form a base for the toplogy of X. We use D to define disk components as described on page 277. (Note that the sets in D need not be actual disks, since the space X is merely assumed to be a topological space.) (a) Let U c X be an open set, let P1 , P2 E U be points, and let V1 and V2 be the disk components of U containing H and P2 , respectively. Prove that either V1 = V2 or vl n v2 = 0. (b) Prove that U is a disjoint union of disk components. (c) Prove that the disk components of U are open. 5.23. Let X be a topological space and let D be a collection of open subsets (disks) of X that form a base for the topology of X. An open set U C X is defined to be disk-connected (relative to D) if for every pair of points P, Q E U there is a finite sequence of disks D 1 , D 2 , . . . , D n C U such that for all l ::; i
IP' be a morphism of degree d ::;,: 2, fix a lift of ¢, and for each homogeneous polyno mial E E K[x, y], let 5..q, , E be the associated local canonical height function. (See Theo rem 5.60.) (a) If E1 , E2 E K[x, y] are homogeneous polynomials, prove that
)..¢ , E 1 E2 ( P) = )..¢ ,E1 ( P) + 5..q, ,E2 ( P) at all points such that E1 ( P) =I 0 and E2 ( P) =I 0. (b) Let D = n1 (Q1 ) + n2 (Q 2 ) + · · · + nr(Qr) be a divisor on IP'1 , i.e., D is a formal sum of points with n1 , . . . , nr E Z. Use (a) to associate to D a local height function (c) Prove that there is a rational function fv E K(z) with the property that
at all points where 5.. q, ,v (¢(P)) and 5.. q, ,v( P) are defined. (d) Let g(z) E K(z) be a rational function and let D9 be the divisor ofg, which by definition is the formal sum
Exercises
321 D9
= L ordQ(g)( Q ).
Q EII'l(K) (Here ordQ (g) is the order of vanishing of g at the point Q ; see Example 2.2.) Prove that
the function
P
>----+
5-..q, ,v9
(P) + loglg(P) l v
extends to a bounded continuous function on all of lP'1 (K) . Section 5 . 1 0. Berkovich Space and Dynamics 5.33. Let ¢(z) in z - a, say
E
Cp[z] be a polynomial and let a
Prove that
E
D(O, 1). Write ¢(z)
as a polynomial
¢(D(a, r)) = D (¢(a), s) ,
where the radius s of the image disk is given by
s = r · max{ lcl l, lc2 l r, lc3 l r2 , . . . , l cd l rd } . (Hint. We already proved that ¢ (D (a, r)) is a disk; see Proposition 5. 16. Now use the maxi mum modulus principle to find the radius.) 5.34. This exercise develops some elementary properties of bounded seminorms as defined on page 296. (a) Let I · I be a nonconstant seminorm. Prove that I 0 I = 0 and I l l = 1. (b) Suppose that I · I is an I · ll wbounded seminorm. Let f(z) = c be a constant polyno mial. Prove that If I = l e i is the usual absolute value on Cp . (c) Suppose that we replace property (3) of bounded seminorm with the usual triangle in equality If + gl ::; If I + lg l. Prove that I · I satisfies (3). (d) Suppose that we replace property (4) of bounded seminorm with the weaker statement that there is a constant K such that If I ::; K I I f II R for all f E Cp [z ] . Prove that (4) is true, i.e., we can take K = 1. 5.35. Prove that the seminorms associated to points of Types II, III, and IV are actually norms. (Hint. For Type IV, use the fact that the limiting radius is positive. See Remark 5.65.)
5.36. Let d 2: 2. Prove that the map ¢( z) = z d has no Type-IV fixed points in A 8. Does cp(z) have any Type-IV periodic or preperiodic points? 5.37. Let D(a, r) be a disk with 0 �
D(a, r) and let f(z) = 1/ z . Prove that f (D(a, r)) = {z- 1 : l z - a l :S: r} = D (a - 1 , r/lal 2 ) .
5.38. Let
d ¢(z) = L cd E Cp[z] i=O
be a polynomial with lc; l ::; 1 and l ed I
The polynomial ¢ induces a map ¢ : D8 --+ D8 on the Berkovich disk. (a) Prove that the Gauss point �o, 1 is fixed by ¢.
= 1.
(5.49)
Exercises
322
(b) Let a E Cp be a fixed point of c/> : Cp --+ Cp. Prove that �a , o is fixed by ¢. (c) Let a E Cp be a neutral fixed point, i.e., ct>' (a) = 1, and let 0 � r � 1. Prove that �a,r is fixed by ¢. (In other words, if a is a neutral fixed point, then the entire line segment La,o C D 8 is fixed by ¢.) (d) Prove that every Berkovich fixed point of c/> of Type I, II, or III is one of the three types listed in (a), (b), (c). (Hint. Use the explicit description of c/>(�a,r) given in Exercise 5.33.) (e) Can c/> have fixed points of Type IV?
l l
5.39. Let c/>( z ) = 2:: c;zi E Cp[z] be a polynomial as in (5.49). Prove directly that the Julia set of c1> is .J8 (c/>) = { 6,0 } . (This is a special case of the result described in Example 5.79, since the conditions on c/> imply that it has good reduction.) 5.40. For any nested sequence of closed disks
regardless of whether the intersection is empty, we define a seminorm in the usual way,
(a) Let r = lim r; . If the intersection n D (a; , r;) is nonempty and a is a point in the intersection, prove that lfl a,r = lfla ,r · Thus every point in D 8 , regardless of whether it is of Type I, II, III, or IV, can be represented by a nested sequence of closed disks. (b) Two nested sequences of closed disks D (O, 1) D (O, l)
::::l D (a1 , r l ) ::::J
D (b 1 , s1 )
::::l
::::J
D (a2 , r2 ) ::::l . . , D (b2 , s2 ) ::::J . . . , •
are defined to be equivalent if the following two conditions are true: • For every i 2 1 there exists a j 2 1 such that D (bj , Sj ) c D (a; , r;). • For every i 2 1 there exists aj 2 1 such that D (aj , rj ) C D (b; , s;). Prove that two sequences of disks are equivalent if and only if their seminorms I · l a,r and I · l b,s are equal. 5.41. Let �a,r E D 8 be a Type-IV point. We defined the "line segment" running from �a, r to the Gauss point �o,1 to be the set
00
L a, r
=
{�a,r } U U {�ai , t : r; � t � r; - d . i= l
(Note that r0
=
1 by definition.) Let r =
lim;___, 00 r;. Prove that the map
h : [r, 1]
h(t)
---->
L a,r ,
=
{ �
�a,r f t = r, �ai ,t Ifr; � t � Ti- l ,
is a homeomorphism, where [r, 1] C R has the usual topology and D 8 has the Gel'fond topology. Thus La,r is indeed a line segment.
Exercises
323
5.42. There is a natural metric on the Berkovich disk D8 coming from the tree structure. We first identify the interval [0, 1] with each of the line segments
La,o
=
{ �a,r
: 0 .,:; r .,:; 1} E D , -B
so each La,o has length 1. We then define the distance between two points �a, r and �b,s to be the length of the shortest path in the tree connecting them. Denote this distance by ��:(�a,r, �b,s). (For Type-IV points, take the limit.) (a) If �a,r and �b,s both lie on some line segment Lc,o, prove that ��:(�a,r, �b,s) = lr - s l is simply the distance between them on that line segment. (b) In general, prove that
��:(�a,r, �b,s)
=
max { lr - s l , 2lb - al
- r - }. s
(Hint. How far above �a,r and �b,s do the line segments La,r and Lb,s merge? Go up one
line segment and then down the other.) (c) Prove that the set
{ � E D8 : ��:(�, �o, ! ) < �}
contains no Type-I points. Prove that every neighborhood of �0, 1 in the Gel'fond topol ogy contains infinitely many Type-I points. Deduce that the path metric 11: does not define the Gel'fond topology, and indeed that ��: is not even continuous in the Gel'fond topol ogy! In Baker and Rumely's terminology, the set D8 with the metric ��: is called the small model. There is also a big model, in which the edges are reparameterized so that if D(a, r) c D(b, s ) , then the distance from �a,r to �b,s is J log(r / s ) J . In particular, points of Type I are at infinite distance from each other and from all points of Types II, III, and IV. It is the big model that is best adapted for doing potential theory on Berkovich space; see [26] for details. 5.43. For points of Type I, II, and III in D8, the Hsia kernel is defined to be
8 (,;a, r , ,;b, s )
=
max(r, Ia - bl) . s,
(For points of Type IV, one takes the appropriate limit.) The Hsia kernel is used in studying potential theory on D8; see [26, 209]. (a) Prove that 8(�a,r, �a,r ) = r. (b) Prove that 8(�a,o, �b,o) = Ia - bl. Thus 8 extends the usual norm on ICp, where we identify the unit disk in ICp with the Type-I points in D8. (c) Let �c,t be the point where the line segments La,r and Lb,s first meet. Prove that o(�a,r, �b,s) = t. (d) Prove that
8(�, (') .,:; max{o(�, (), o(( , (' ) } 5.44. There are natural inclusions
D(O, 1 ) -
'---->
D , -B
In each case, prove that the restriction of the Gel'fond topology on the Berkovich space yields the usual metric topology on the smaller space, where we use the usual p-adic met ric on D(O, 1) and A 1 (ICp) and the p-adic chordal metric on lP'1 (ICp) .
Exercises
324 5.45. Let ¢ be a rational map of degree at least 2, let P E hood of P with the property that
¢( U ) �
U
and
n¢
n ;:> O
n
lP'5, and let U c lP'5 be a neighbor
( U)
= { P} .
Prove that P is a Type-I point and that, considered as a point in 1P'1 (Cp), the point P is an attracting fixed point for ¢.
Chapter 6
Dynamics Associated to Algebraic Groups In the forest of untamed rational maps live a select few whose additional structure allows them to be more easily domesticated. They are the power maps, Chebyshev polynomials, and Lattes maps, whose complex dynamics were briefly discussed in Section 1 .6. The underlying structure that they possess comes from an algebraic group, namely the multiplicative group for the power maps and Chebyshev poly nomials and elliptic curves for the Lattes maps. Although such maps are special in many ways, they yet provide important examples, testing grounds, and boundary conditions for general results in dynamics. In this chapter we investigate some of the algebraic and arithmetic properties of the rational maps associated to algebraic groups. 6.1
Power Map s and the Multiplicative Group
The simplest rational maps are the power maps given by monic monomials,
where the integer d may be positive or negative. These maps obviously commute with one another under composition,
A more intrinsic description of the maps associated to the monic monomials Md is that they are endomorphisms of the multiplicative group,
More precisely, there is an isomorphism of rings, 325
326
6. Dynamics Associated to Algebraic Groups
The fact that the Md are endomorphisms of the multiplicative group Gm makes it quite easy to describe their preperiodic points. IP'1 be the power Proposition 6.1. Let d E Z with l d l .2': 2 and let Md : IP'1 map Md(z) z d. Then -+
=
PrePer(Md)
=
(Gm)tors
=
{ ( E Gm : C
=
lfor some n .2':
where recall that J.Ln denotes the group ofn1h roots ofunity.
1}
=
U J.Ln ,
Proof We proved this long ago for any abelian group G and homomorphism z zd with d .2': 2; see Proposition 0.3. The proof for d :::; -2 is similar and left to the reader. D t--+
The iterates of Md are given by
so the periodic points of Md are also easy to characterize, l d l .2': 2 and let ( E Per�* (Md) be a point of exact pe 2. Then the multiplier ofMd at ( is given by
Proposition 6.2. Let
riod
n
:2:
Proof Using M:J:(z)
=
z dn , we can directly compute
In particular, if we are working over C, then every periodic point of Md in C * is repelling. On the other hand, over a p-adic field with p f d, the multiplier dn is a unit, so the periodic points are indifferent. And if p I d, then all of the periodic points are attracting. Of course, there are also the two superattracting fixed points 0 and oo . It is not hard to find all rational maps that commute with the power maps. In particular, we can compute the automorphism group Aut(Md). Proposition 6.3. Let K be afield and let Md(z) z d be the d1h-power map for some l d l .2': 2. Further, if K hasfinite characteristic p, assume that p f d. (a) The set ofrational maps that commute with Md(z) is given by =
{-+ P+Q E,
E P >-+ -P E
'
are morphisms, i.e., are given by everywhere defined rationalfunctions. (c) The subset E ( K) consisting ofpoints ofE that are defined over K is a subgroup ofE(K). D
Proof See [410, III §§2,3].
It is not hard to give explicit formulas for the group law on an elliptic curve, as in the following algorithm. Proposition 6.17.
(Elliptic Curve Group Law Algorithm) Let E be an elliptic curve
given by a Weierstrass equation
y2 = x3 + ax + b, and let P1 = (x 1 , Yl ) and Pz = (xz, yz) be points on E. IfX 1 = xz and Yl = -yz, then P1 + Pz = 0. Otherwise, define quantities A = XYzz -- XY11 ' xz - xl i l + 2b ' ifx1 = xz. A = 3x2yl+ a ' V = --xr--"---+-2axy--l Then y = AX + is the line through P1 and P2 , or tangent to E if P1 = P2 , and the sum of P1 and Pz is given by E
:
v
P = (x, y) is 2 - 8bx + a2 x([2]P) = x4 -4x2ax 3 + 4ax + 4b
As a special case, the duplication formula/or
Proof See [410, III.2.3] 6.3.3
Divisors and Divisor Classes
Definition.
A divisor on E is a formal sum of points D
= LEE np(P), P
D
339
6.3. A Primer on Elliptic Curves
with np E Z and all but finitely many np = 0. The set of divisors under addition forms the divisor group Div( E). The degree of a divisor d is deg(D) =
L np.
PEE
The degree map deg : Div(E) Z is a group homomorphism. There is a natural summation map from Div(E) to E defined by ----+
sum : Div(E) ----> E,
L np(P)
PE E
f------t
L [np] (P).
PE E
(N.B. The two summation signs mean very different things. The first is a formal sum of points in Div(E). The second is a sum using the complicated addition law on E.) The zeros and poles of a rational function f on E define a divisor div(f) =
L ordp(f)(P), PEE
where ordp(f) is the order of zero of f at P if f(P) = 0, and ordp(f) is negative the order of the pole off at P if f(P) = oo. A divisor of the form div(f) is called a principal divisor. The principal divisors form a subgroup ofDiv(E), and the quotient group is the Picardgroup Pic( E). Within Pic( E) is the important subgroup Pic0 (E) generated by divisors of degree 0. The next proposition describes the basic properties of divisors on E. Proposition 6.18. Let E be an elliptic curve.
(a) Every principal divisor on E has degree 0. (b) A divisor D E Div(E) is principal if and only if both deg(D) sum(D) = 0.
0 and
(c) The summation map induces a group isomorphism
sum : Pic0 (E) ----> E. Proof See [410, 111.3.4 and 111.3.5]. 6.3.4
D
Isogenies, Endomorphisms, and Automorphisms
Definition. An isogeny between two elliptic curves E1 and E2 is a surjective morphism 'ljJ : E1 E2 satisfying 'lj;( 0) = 0. (Note that any nonconstant mor phism E1 E2 is automatically finite and surjective.) The curves E1 and E2 are said to be isogenous if there is an isogeny between them. Remark 6. 19. Every nonconstant morphism 'ljJ E1 E2 is the composition of an isogeny and a translation (cf. [410, III.4.7]). To see this, let ¢(P) = 'lj;(P) - 'lj;(O). Then the map ¢ : E E is a morphism, and ¢( = 0, so ¢ is an isogeny. Hence ----+
----+
:
----+
----+
0)
'lj;(P) = ¢(P) + 'lj;(O) is the composition of an isogeny and a translation.
6. Dynamics Associated to Algebraic Groups
340
Remark 6.20. We observe that an isogeny is unramified at all points. This fol
lows from the general Riemann-Hurwitz formula (Theorem 1 .5) applied to the map
7/J : E1 E2 , 2g(E1 ) - 2 = (deg 7j;)(2g (E2 ) - 2) + 2::: (ep('lj;) - 1 ) . The elliptic curves E1 and E2 both have genus 1, and the ramification indices sat isfy ep('lj;) :2: 1, so it follows that every ep('lj;) is equal to 1, so 7/J is unramified. Theorem 6.21 . An isogeny 7/J : E1 E2 is a homomorphism ofgroups, i.e., 7/J(P + Q) = 7/J(P) + 7/J( Q) for all P, Q E E1 (K). -+
P E E1
-+
0
Proof See [410, 111.4.8]
7/J : EE1 . E2 Q Q,7/J 2 7/J deg(7/J) 7/J. Theorem 6.22. Let 7/J : E1 -+ E2 be an isogeny ofdegree d. Then there is a unique isogeny � : E2 E1 , called the dual isogeny of 7/J, with the property that �(7/J(P)) = [d]P and 7/J(�(Q)) = [d]Q for all P E E1 and Q E E2 .
The degree of an isogeny is the number of points in the inverse image for any point E This number is independent of the point since, as noted earlier, is an unramified map. It is clear that if > 1, then is not invertible, since it is not one-to-one. However, there does exist a dual isogeny that provides a kind of"inverse" for -+
'lj;- 1 (Q)
-+
0
Proof See [410, III §6].
End( E),E
E,
Definition. Let be an elliptic curve. The endomorphism ring of which is de noted by is the set of isogenies from to itself with addition and multipli cation given by the rules
E
E d(E) End(E).
(In order to make n into a ring, we also include the constant map that sends every point to 0.) The automorphism group of denoted by is the set of endomorphisms that have inverses. Equivalently, is the group of units in the ring
E,Aut(E) = End( Aut(E)*E), Every integer m gives a multiplication-by-m morphism in End( E). For m this is defined in the natural way as m terms
>
0
[m] : E -+ E, [m](P) = P + P + · · · + P . For m < 0 we set [m](P) = -[ -m](P), and of course [O](P) = 0. This gives an embedding of Z into End( E), and for most elliptic curves (in characteristic 0), there are no other endomorphisms.
341
6.3. A Primer on Elliptic Curves
E d( )
An elliptic curve E is said to have complex multiplication if n E is strictly larger than Z. The phrase "complex multiplication" is often abbreviated by CM. Example 6.23. The elliptic curve E y2 x3 + x has CM, since the endomorphism Definition.
:
'ljJ : E
-----+
=
'1/J(x, y) ( - x , iy ),
E,
=
i s not in Z . An easy way to verify this assertion i s to note that
'ljJ2 (x , y ) (x, - y ) = -(x , y ),
=
so 'ljJ2 [ - 1]. This gives an embedding of the Gaussian integers Z[i] into via the association m + ni [m] + [n] '1/J, and in fact it is not hard to show that is isomorphic to Z[i]. Example 6.24. More generally, there are two special families of elliptic curves that have CM, namely those with a 0 and those with b 0. These are the curves
= End(E)
f---)
End(E)
o
= = j(E�) = 1728, End(E�) = Z[i], Aut(E�) = J-L4 , j(E� ) = 0, End(E� ) = Z[p], Aut(E� ) = J-L6 .
E� : y2 = x3 + ax, E� y2 = x3 + b, Here p = ( -1 +A)I2 denotes a cube root ofunity and of unity. :
1-Ln
is the group ofn1h roots
Of course, all of the E� are isomorphic over an algebraically closed field, since they have the same j-invariant, and similarly for all of the E�. However, the curves in each family may not be isomorphic over a field K that is not algebraically closed. This is an example of the phenomenon of twisting as described in Section 4.8 (see also [410, X §5]). Proposition 6.25. Let E I K be an elliptic curve. Then the endomorphism ring of
E (a) End(E) = Z. (b) End( E) is an order in a quadratic imaginaryfield F. This means that End(E) is a subring ofF and satisfies End( E) Q = F. In particular, End( E) is a subring offinite index in the ring of integers of F. (c) End( E) is a maximal order in a quaternion algebra. (This case can occur only ifE is defined over afinitefield.)
is one of thefollowing three kinds ofrings:
0
0
Proof See [410, III §9].
The automorphisms of an elliptic curve are very easy to describe. Proposition 6.26. Let K be afield whose characteristic is not equal to or and let EI K be an elliptic curve. Then
2 3
{ Aut(E) =
J-L2 J-L4
J-L6
ifj (E) =1- 0 and j (E) =1- 1728, ifj(E) = 1728,
ifj( E)
=
0.
342
6. Dynamics Associated to Algebraic Groups D
Proof See [410, III § 1 0].
Aut(E)
Remark 6.27. It is easy to make the description of in Proposition 6.26 com pletely explicit. Assuming that E is given by a Weierstrass equation (6. 1 1) as usual,
for an appropriate choice of n there is an isomorphism [ · ] : J.Ln
_____. Aut(E), (6. 13) Here we take n 4 if j ( E ) = 1728, we take n 6 if j (E) 0, and we take n 2 otherwise. Of course, for n 2 and n 4 the formula simplifies somewhat to (x, � ) and (x, �- 1 ), respectively. =
=
y
=
=
y
=
=
Minimal Equations and Reduction Modulo p
6.3.5
Let K be a local field with ring of integers R, maximal ideal p, and residue field Rip . As in Section 2.3, we write x for the reduction of x modulo p .
k
=
Let EI K be an elliptic curve defined over a local field K. A minimal Weierstrass equation for E is a Weierstrass equation whose discriminant Ll(E) has
Definition.
minimal valuation subject to the condition that the coefficients of the Weierstrass equation are all in R.
k
2 3,
Example 6.28. If does not have characteristic or then a Weierstrass equation (6. 14)
for E is minimal if and only if a,
bER
and
min{ 3 ordp ( ) 2 ordp (b) } < 12. a ,
k 2 3,
2 3,
In general, if the residue field does not have characteristic or then any Weier strass equation (6. 14) can be transformed into a minimal equation by a substitution of the form (x, y) (u2 x, u3 y ) for an appropriate u E K*. If has characteristic or then a minimal Weierstrass equation may require the general form (6.12). There is an algorithm of Tate [412, IV §9] that transforms a given Weierstrass equation into a minimal one. �--'
k
Definition. Fix a minimal Weierstrass equation for EI K. Then we can reduce the coefficients of E to obtain a (possibly singular) curve I We say that E has good reduction if is nonsingular, which is equivalent to the condition that Ll(E) E R* . In any case, we obtain a reduction modulo p map on points,
E k.
E
E(K)
_____.
E(k),
P f-------+ P.
Proposition 6.29. If E has good reduction, then the reduction modulo p map
E(K) --+
E(k) is a homomorphism.
Proof See [410, VII.2. 1].
D
343
6.3. A Primer on Elliptic Curves
Remark 6.30. For elliptic curves defined over a number field K, we say that E has good reduction at a prime p of K if it has a Weierstrass equation whose coefficients
are p-adic integers and whose discriminant is a p-adic unit. Note that one is allowed to use different Weierstrass equations for different primes. If there is a single Weier strass equation that is simultaneously minimal for all primes, then we say that E K has a global minimal Weierstrass equation. Global minimal equations exist for el liptic curves over Q, and more generally for elliptic curves over any number field of class number 1, but in general the existence of global minimal equations is somewhat subtle; see [410, VIII §8] and [48]. We discussed a related notion of global minimal models of rational maps in Section 4.1 1 .
I
6.3.6
Torsion Points and Reduction Modulo p
The kernels of endomorphisms help to determine the arithmetic properties of elliptic curves. Definition. Let E be an elliptic curve. For any endomorphism 'ljJ E End(E) we write E['lj;] = Ker('lj;) = { P E E : 'lj;(P) = 0 } . Of particular importance is the kernel of the multiplication-by-m map, E[m] = { P E E : [m]P = 0 } . The group E[m] is called the m-torsion subgroup ofE . The union of all E[m] is the torsion subgroup of E,
Etors =
I
U E[m].
m� l
Theorem 6.31. Let E K be an elliptic curve and assume that either K has char acteristic 0 or else that K has characteristic p > 0 and p f m. Then as an abstract group, E[m] = ZlmZ x ZlmZ. In other words, E [m] is the product oftwo cyclic groups oforder m. Proof See [410, III.6.4]. D
The next result gives conditions that ensure that the reduction modulo p map respects the m-torsion points. It may be compared with Theorem 2.2 1 , which tells us what reduction modulo p does to periodic points of a good-reduction rational map. Theorem 6.32. Let K be a local field whose residue field has characteristic p,
let E K be an elliptic curve with good reduction, and let m ;::: 1 be an integer with p f m. Let E(K) [m] denote the subgroup of E[m] consisting ofpoints defined over K, i.e., E(K) [m] = E[m] n E(K). Then the reduction map
I
E(K) [m] --+
E(k)
is injective. In other words, distinct m-torsion points have distinct reductions mod ulo p.
344
6. Dynamics Associated to Algebraic Groups
D
Proof See [410, VII.3. 1].
EIK K,
K.
K. E[m]
Let be an elliptic curve defined over the field Then the points in are algebraic over so their coordinates generate algebraic extensions of An im mediate corollary of the preceding theorem limits the possible ramification of these extensions. Corollary 6.33. Let be a local field whose residue field has characteristic p, let be an elliptic curve with good reduction, and let m 2: 1 be an integer with p f m. Then the field obtained by acijoining to the coordinates of the m-torsion points of is unramified over
K
EI K
K EK (E[ml) K. ProofSketch. Let K' K(E[ml), let p ' be the maximal ideal of the ring of integers of K' , and let k' be the residue field. Suppose that CY E Gal( K' I K) is in the inertia group. Then CY fixes everything modulo p', so in particular, for all P E E[m]. (6. 1 5) CY(P) P ( mod p') But from Theorem 6.32, the reduction map E[m] ----+ E(k') is injective, so (6. 1 5) implies that CY(P) = P for all P E E[m]. The points in E[m] generate K' IK, CY fixes K' . Hence Gal(K' I K) has trivial inertia group, so K' IK is unramified. (For further details, see [41 0, VII.4. 1].) D Remark 6.34. The coordinates of the points in E[m] are algebraic over K, so the absolute Galois group Gx Ga!(KIK) acts on E[m] compatibly with the group =
=
so
=
structure. In this way we obtain a representation p :
G x --+ Aut(E[m]) 1
E
The quotient of by a finite group of automorphisms gives a map from These quotient maps play an important role in dynamics.
Aut(E). E EThen If theIID1 quotient is given
Proposition 6.37. Let r be a nontrivial subgroup of curve is isomorphic to IID 1 and the projection map 1r explicitly by
EIf
E to IID1 .
:
---+
�
r = M2 (j (E) if { ( y) = x2 if r = f-L4 (j(E) = 1728
347
6.3. A Primer on Elliptic Curves
x
1f
X
'
arbitrary), only), only), only).
y
if r = M3 (j(E) = 0 x3 if r = f-L6 (j(E) = 0 Proof By definition, the quotient curve E /f is the curve whose function field is the subfield of K(E) = K(x, y) fixed by r . Using the explicit description (6. 13) of the action of Aut(E) on the coordinates of E, it is easy to find this subfield. For example, iff = JL2 , it consists of those functions that are invariant under (x, y ) (x, -y ), so the fixed field K(Et is K(x, y2 ) = K(x). As a second example, if r = JL6 , then we need functions invariant under (x, y) (px, -y ), where p is a primitive cube root of 1 . This fixed field is K ( El = K (x3, y2 ) = K (x3), since in this case the elliptic curve is given by an equation of the form y2 = x3 + b. The other cases are 0 similar. For later use, we prove that the isomorphism class of an elliptic curve E is deter mined by the critical values of any double cover E JID1 . Lemma 6.38. Let E be an elliptic curve defined over a field of characteristic not equal to 2 and let 1r : E ---+ JID 1 be a rational map of degree 2. Then 1r has exactly four critical values and they determine the isomorphism class ofE. Proof The Riemann-Hurwitz formula (Theorem 1 .5) for the map 1r : E JID 1 says that 2g (E) - 2 = (deg 7r)(2g(JID1 ) - 2) + L (ep(¢) - 1). The map 1r has degree 2, the elliptic curve E has genus 1, and JID 1 has genus 0, so we find that L (ep (¢) - 1) = 4. The ramification indices satisfy 1 e p ( ¢) deg 1r = 2, so we conclude that there are exactly four critical points, i.e., four points with ep ( ¢) = 2 and all other points satisfy e p ( ¢) 1. Further, these four critical points must have distinct images in JID1 , since for any point E JID1 we have L ep (¢) deg(1r). Let fi, t2 , t3 , t4 E JID1 be the four critical values of1r, i.e., the images ofthe critical points, and let f E PGL2 be the unique linear fractional transformation satisfying f---*
f---*
---+
---+
PEE
PEE
:S
=
t
PE7r- 1 (t)
Explicitly,
:S
=
348
6. Dynamics Associated to Algebraic Groups
(If any of t 1 , t2 , t3 equals oo, take the appropriate limit.) The quantity
is called the cross-ratio oftr, t2 , t3 , t4; cf. Section 2.7, page 7 1 . We let x f so x is a rational function of degree 2 on E with critical values 0, 1, oo, and "' To ease notation, we let =
o 1r, ·
Taking 0 to be the identity element for the group law on E, we see that
div (x)
=
2 (To ) - 2 ( 0 ) ,
div(x - 1)
=
div (x- K, ) = 2 (T,. ) - 2 ( 0 ) ,
2 (TI ) - 2 ( 0 ) ,
so To , T1 , T,. are in E[2] , i.e., they are points of order 2. The sum of the three non trivial 2-torsion points on any elliptic curve is equal to 0, so Proposition 6. 1 8 tells us that there is a rational function y on E with divisor
div (y) = (To ) + (TI ) + (T,.) - 3 ( 0 ) . After multiplying y by an appropriate constant, it follows that x and y are Weierstrass coordinates for E (cf. [198, IV.4.6] or [410, 111.3. 1 ]). More precisely, the rational functions x and y map E isomorphically to the curve with Weierstrass equation E : y2 = X (X 1) (X ) -
0
- K,
It is easy to compute the j-invariant of E in terms of"' (cf. [410, III. l . 7(b)]). We find that 2 " s ("' - "' + 1) 3 J (E) = 2 /'1,2 ( /'1, - 1) 2 so in particular j (E) is uniquely determined by t 1 , t2 , t3 , t4. Then we apply Propo sition 6. 1 5 (or [410, III.l .4(b )]), which says that the isomorphism class of E is de D termined by its j-invariant. This completes the proof of Lemma 6.38. '
6.3.9
Complex Multiplication
Let E/ K be an elliptic curve with complex multiplication defined over a number field. As described in Proposition 6.25, the endomorphism ring of E is isomorphic to a subring of the ring of integers of a quadratic imaginary field. We briefly recall an analytic proof of this important fact and then discuss the relationship between com plex multiplication and the ideal class group of the associated quadratic imaginary field. This material is used only in Section 6.6, so may be omitted at first reading. Proposition 6.39. Let E /C be an elliptic curve with complex multiplication, i.e., the endomorphism ring End( E) is strictly larger than Z. Choose a lattice L C C such that E(C) � Cj L, let
349
6.3. A Primer on Elliptic Curves
R = {a E C aL E,:
7/J : E d 2:E,2
lP'1 lP'1 of degree rational map is called a Lattes map if there are an elliptic curve a morphism and a finite separable' covering lP' 1 such that the following diagram is commutative:
Definition.
�
�
�
(6.22)
lP' ' � lP'' . be an elliptic curve. Then the classical Example 6.41 . Let E = lP'1 yield formula for (Proposition 6. 1 7) and the isomorphism the Lattes map
2 x3 + ax + b : y x(2P) x : E/ { ±1} 2 - 8bx + a2 ¢>(x) = x(2P) = x4 -4x2ax 3 + 4ax + 4b Here 7/J is the duplication map 7/J ( P) [2]P, and the projection 1r is given by 1r(P) 1r(x, y) = x. Example 6.42. Let E be the elliptic curve E : y2 x3 + ax with j (E) 1728 and again let 7/J(P) = [2]P be the doubling map. If we take 1r(x, y) = x, then we are in the b 0 case of Example 6.4 1 , and we obtain the Lattes map ¢>(x) = x(2P) = 4(xX (2X-2 +a)a2) However, for this curve we may instead take (x, y) = x 2 . This gives a new Lattes map ¢ 1 . We find a formula for ¢ 1 using the relation (x - a)2 ) 2 (x - a)4 2 · q;,(x) (JX) 2 ( 4fo(x + a) 16x(x + a) Note that the map 1r(x, y) = x2 corresponds to taking the quotient of E by its automorphism group Aut(E) � 114 via the association described in Remark 6.27. Example 6.43. In a similar manner, the doubling map on the elliptic curve E : y2 = x 3 + 1 with j(E) 0 and Aut(E) 116 gives various Lattes maps corresponding to taking the quotient of E by the different subgroups of 116 . Explicitly, the Lattes maps corresponding, respectively, to 112 , 113 , and 116 are �
=
=
=
=
=
'
1r
=
=
=
=
We leave the verification of these formulas to the reader; see Exercise 6. 12. 1 The assumption that 1r i s separable i s relevant only when one i s working over a field o f characteris tic p, in which case it is equivalent to the assumption that 1r does not factor through the p-power Frobenius map.
6. Dynamics Associated to Algebraic Groups
352
We begin with an elementary, but useful, characterization of the preperiodic points of a Lattt��s map (cf. Proposition 1 .42). Proposition 6.44. Let ¢ be a Lattes map associated to an elliptic curve E. Then
PrePer(¢) = ;r (Etors ) · Proof Let ( E lP' 1 and let P E E be any point satisfying ;r(P) = (. We consider the orbits of ( and P. Thus ;r (O,p (P)) = 1r ({�n (P) : n � 0 }) = {;r�n (P) : n � 0} = {¢n ;r(P) : n � 0} = {¢n (() : n � 0 } = Oq, ((). The map 1r is finite, so this shows that O,p( P) is finite if and only if Oq, ( ( ) is finite. Hence
PrePer(¢) = ;r(PrePer(�)), and it i s left to prove that PrePer(�) = E1ors · We observe that the map � : E E has the form �(P) = �o (P) + T for some �o E End(E) and some point T E E. (See [410, III.4.7].) We are going to prove Proposition 6.44 in the case that � = �0 E End(E), i.e., assuming that = 0. �
T For the general case, which requires knowing that the point T is a point of finite or
der, see Exercise 6. 14. Suppose first that P E E1ors . say [n]P = 0 for some n � 1. Consider the images of the iterates �, �2 , �3 , . . . in the quotient ring End (E)/ n End (E). It follows from the description of End( E) in Proposition 6.25 that this quotient ring is finite, so we can find iterates i > j � 1 such that
�i �j (mod n End(E)). In other words, there i s an endomorphism E End(E) such that �i = �j + f3n. Evaluating both sides at P and using the fact that [n]P = 0 allows us to conclude that �i (P) = �j (P). Hence P E PrePer( �), which proves that E1ors C PrePer(� ). Next suppose that P E PrePer(�), say �i (P) = �J (P) for some i > j. We ----i i rewrite this as ( � - �j ) ( P) = 0 and apply the dual isogeny � - �j described in =
f3
Theorem 6.22 to obtain
[deg(�i - �j )](P) = 0. We know that �i -1- �j ' since deg(�) = deg(¢) � 2 and i > j, so �i - �j has positive degree. This proves that P E E1ors . which gives the other inclu sion PrePer(�) C Etors · 0 Many dynamical properties of a rational map can be analyzed by studying the behavior of the critical points under iteration of the map. This is certainly true for Lattes maps, whose postcritical orbits have a simple characterization, which we give after setting some notation.
353
6.4. General Properties of Lattt�s Maps
Let ¢ C1 C2 be a nonconstant rational map between smooth projective curves. The set of critical points (also called ramification points) of ¢ is denoted by Definition.
:
CritPt,p
=
----+
{P E C1 : ¢ is ramified at P}
{P E C1 ep(¢) 2 2}.
=
:
The set ofcritical values of ¢ is the image of the set of critical points and is denoted by CritVal¢
=
¢(CritPt¢) ·
If ¢ C C is a map from a curve to itself, the postcritical set is the full forward orbit of the critical values and is denoted by :
----+
00
PostCrit¢
00
U=O ¢n (CritVa1¢) U=l CritValn .
=
=
n
n
(See Exercise 6. 1 5.) Proposition 6.45. Let ¢ diagram (6.22). Then
:
IP'1
----+
IP'1 be a Lattes map thatfits into a commutative
CritVal7r
=
PostCrit .
In particular, a Lattes map is postcritically finite. Proof The key to the proof of this proposition is the fact that the map 'lj!
:
E E ----+
is unramified, i.e., it has no critical points, see Remark 6.20. (In the language of modem algebraic geometry, the map 'lj! is etale.) More precisely, the map 'lj! is the composition of an endomorphism of and a translation (Remark 6.19), both of which are unramified. For any n 2 1 we compute
E
CritVal7r
CritVal1r,p n CritVal¢n7r = CritValn U ¢n ( CritVal7r) :2 CritValn .
= =
because 'lj! is unramified, from the commutativity of (6.22), from the definition of critical value,
This holds for all n 2 1, which gives the inclusion CritVal7r :2
00
U=O ¢n (CritVa1¢)
=
PostCrit¢ .
n
In order to prove the opposite inclusion, suppose that there exists a point Po E satisfying and 1r(P0 ) � PostCrit¢ . (6.23) Po E CritPt7r Consider any point Q E 'lj! - I (Po). Then Q is a critical point of 1r'lj!, since 'lj! is unramified and 1r is ramified at 'lj! ( Q) by assumption. But 1r'lj! ¢1r, so we see that Q is a critical point for ¢1r.
E
=
6. Dynamics Associated to Algebraic Groups
354
On the other hand, ¢ (1r( Q ) )
=
1r(Po) tJ_ CritVal
=
cj>(CritPt¢) ,
so 1r(Q ) i s not a critical point for ¢. It follows that Q i s a critical point of 1r. Further, we claim that no iterate of 4> is ramified at 1r ( Q). To see this, we use the given fact that 1r(P0) is not in the postcritical set of 4> to compute for all n � 1, 1r(Po) tJ_ PostCrit ===} 1r(P0) tJ_ c/>n (CritPt¢) for all n � 1, ===} 1r(1jJ(Q)) tJ_ c/>n (CritPt¢) for all n � 1, ===} ¢(1r(Q)) tf_ c/>n (CritPt¢) for all n � 0, ===} 1r( Q ) tJ_ c/>n (CritPt¢) ===} 1r( Q) tJ_ PostCrit . To recapitulate, we have now proven that every Q E 1/J- 1 (Po) satisfies Q E CritPt1r
1r( Q) tJ_ PostCrit,p .
and
In other words, every point Q E 1/J-1 (Po) satisfies the same two conditions (6.23) that are satisfied by P0. Hence by induction we find that if there is any point Po satisfying (6.23), then the full backward orbit of 1/J is contained in the set of critical points of 1r, i.e., CritPt71'
:::>
00
U 'lj;- n (P0 ) .
n= 1
But 1/J is unramified and has degree at least 2 (note that deg 1/J
=
deg 4>), so
This is a contradiction, since has only finitely many critical points, so we conclude that there are no points Po satisfying (6.23). Hence 1r
P0 E CritPt1r ===} 1r(P0) E PostCrit , which gives the other inclusion CritVal1r � PostCrit.
D
As an application of Proposition 6.45, we show that Lattes maps associated to distinct elliptic curves are not conjugate to one another. Theorem 6.46. Let K be an algebraically closedfield of characteristic not equal to 2 and let 4> and 4>' be Lattes maps defined over K that are associated, respec tively, to elliptic curves E and E' . Assume further that the projection maps and 1r1 associated to 4> and 4>' both have degree 2. If 4> and 4>' are PG L2 ( K)-conjugate to one another, then E and E' are isomorphic. 7r
355
6.5. Flexible Lattes Maps
Proof Let f E
PGL2 (K)
be a linear fractional transformation conjugating ¢/ to ¢. Then we have a commutative diagram 7r'
,____
E'
1 ,p'
We let 1r11 = f o 1r1• Note that since f is an isomorphism, the map 7f11 still has degree 2. This yields the simplified commutative diagram
E E'.
showing that ¢ is a Lattes map associated to both elliptic curves and Applying Proposition 6.45, first to and then to we find that CritVal1r
E
=
PostCritq,
=
E' ,
(6.24)
CritVal1r" .
1r" :EEand the exact ' ---+ E'lP'1 arehaveisomorphic.
: E ---+
In other words, the degree-2 maps 1r lP'1 and same set of critical values. Then Lemma 6.38 tells us that
6.5
D
Flexible Lattes Maps
A Lattes map is a rational map that is obtained by projecting an elliptic curve endomorphism down to lP'1 . For any integer m 2: 2, every elliptic curve has a multiplication-by-m map and a projection � lP' 1 , so every elliptic curve has a corresponding Lattes map. As varies, this collection of Lattes maps varies continuously, which prompts the following definition. Definition. A flexible Lattes map is a Lattes map ¢ lP'1 lP'1 that fits into a Lattes commutative diagram (6.22) in which the map 'ljJ has the form
EE ---+ Ej { ±1} : E: ---+ E 'lj;(P) [m](P) + T for some m E Z and some T E E and such that the projection map 1r : E ---+ lP'1 satisfies and deg(1r) 2 1r(P) 1r(- P) for all P E E. Remark 6.47. The condition that 1r be even, i.e., that it satisfy 1r(- P) 1r(P), is included for convenience. In general, if 1r E ---+ lP'1 is any map of degree 2, then there exists a point Po E E such that 1r( -(P + Po)) 1r(P + P0 ) for all P E E. Thus 1r becomes an even function if we use Po as the identity element for the group law on E. See Exercise 6.1 6. ---+
=
=
=
=
:
=
356
6. Dynamics Associated to Algebraic Groups
Remark 6.48. We show in this section that the Lattes maps of a given degree have
identical multiplier spectra. This is one reason that these Lattes maps are called "flex ible," since they vary in continuous families whose periodic points have identical sets of multipliers. We saw in Section 4.5 that symmetric polynomials in the multipliers give rational functions on the moduli space Md of rational maps modulo PGL2 conjugation. Flexible families of rational maps thus cannot be distinguished from one another in Md solely through the values of their multipliers. Example 6.49. We saw in Example 6.41 that the Lattes function associated to the duplication map 7/J(P) = [2] (P) on the elliptic curve E : y2 = x3 + ax + b is given by the formula
2 - 8bx + a2 4 = x(2P) = x - 2ax 4x3 + 4ax + 4b It is clear that if a and b vary continuously, subject to 4a3 + 27b2 -1=- 0, then the Lattes maps c/Ja,b vary continuously in the space of rational maps of degree 4. ¢a ' b(x)
More precisely, the set of maps c/Ja,b is a two-dimensional algebraic family of points in the space Rat4, given explicitly by
(a, b)
f------+
[1, 0, - 2a, - 8b, a2 , 0, 4, 0, 4a, 4b].
lfwe conjugate by fu (x) = ux , the Lattes map c/Ja,b transforms into Thus assuming (say) that ab -1=- 0, we can take u = b/a to transform c/Ja,b into .+,fb/a
'f'a,b
= '.+.f'c,c
'th C = a3 /b2 .
Wl
In other words, the two-dimensional family ofLattes maps { c/Ja,b} in Rat4 becomes the one-dimensional family of dynamical systems (6.25) Of course, it is not clear a priori that the map (6.25) is nonconstant. But if the Lattes maps c/Jc,c and ¢c' ,c' are PGL2 -conjugate, then Theorem 6.46 tells us that their associated elliptic curves E and E' are isomorphic. The j-invariant of the elliptic curve Ec : y2 = x3 + ex + c is 8. 3 c '
J
( Ec ) = 2
3
4c + 27 ' so we see that j ( Ec) = j ( Ec' ) if and only if c = c'. This proves that the map (6.25)
is injective, so these flexible Lattes maps do indeed form a one-parameter family of nonconjugate rational maps with identical multiplier spectra, i.e., they are a nontrivial isospectral family.
357
6.5. Flexible Lattes Maps
y2 = x3
ax2
+ bx has the 2-torsion point T = (0, 0). To compute the Lattes function ¢ lP'1 -+ lP'1 associated to the translated duplication map 1/J(P) = [2] (P) + T, we first use the classical duplication formula to compute
Example 6.50. The elliptic curve E
:
+
:
2P
- ( x4 - 2bx4y22 _
+
)
b2 x6 + 2ax5 + 5bx4 - 5b2 x2 - 2ab2 x - b3 . ' 8y3
Then the addition formula and some algebra yield + ax2 + bx) . ¢(x) = x ( 2 P + T) = 4b(x3 x4 - 2bx2 + b2
As in the previous example, these Lattes maps form a one-dimensional family in M4 . We begin with a few elementary, but useful, properties of flexible Lattes maps. Proposition 6.51. Let ¢ : lP'1
-+ lP'1 be a flexible Lattes map whose associated map 1/J : E -+ E has theform 1/J(P) = [m]P + T. (a) The map ¢ has degree m2 . (b) The point T satisfies [2]T = 0. (c) Fix a Weierstrass equation (6. 1 1) for E. Then there is a linearfractional trans formation f E PGL 2 such that 1r = f o x. Hence ¢! fits into a commutative diagram (6.26) ]p>l
�
lP' l .
Proof (a) The commutativity of the diagram (6.22) tells us that
deg( ¢) deg( 1r ) = deg( 1r ) deg( 1jJ ). The map 1/J has degree m2 , since multiplication-by-m has degree m2 and trans1ation by-T has degree 1. Therefore deg(¢) = m2 .
(b) We are given that the map 1r E -+ lP'1 has degree 2 and satisfies 1r ( P) = n ( -P). It follows that n (P) = n (Q) if and only if P = Q or P = -Q. We use the :
commutativity of (6.22) to compute
n ( - [m]P - T) = n ( [m] P + T) = n (?jJ (P)) = ¢( n (P)) = ¢( n ( -P)) = n ( ?jJ ( -P)) = n ( - [m]P + T) .
Hence for every P E E we have either
- [m]P - T = - [m]P + T or - [m]P - T = -( - [m]P + T) = [m]P - T. Simplifying these expressions, we find that every point P E [2]T = 0
or
[2m]P = 0.
E satisfies either
358
6. Dynamics Associated to Algebraic Groups
P E
[2m]P = 1r(-P) = 1r(P). K(E); f(z) K(z) 1r = f(x).
But there are only finitely many points E satisfying 0; hence we must have 0. ( c) The map is a rational function on E satisfying It follows that is in the subfield of the function field see [41 0, 111.2.3 . 1 ] . In other words, there is a rational function E such that Equivalently, factors as the map
1r
[2]T1r=
1r : E lP'1
K(x)
___,
In particular,
2 = deg(1r) = deg(f o x) = deg(f) deg(x) = 2 deg(f), so we see that deg(f) = 1. Hence f is a linear fractional transformation, which proves the first part of(c). Finally, we compute q/ X = f - 1 0 deg(Gm)· We also note that all ofthe x( Q) are integral over R, since they are roots of Gc(X) and £ E R*. lt follows that H(X) E R[X]. Further, Proposition 2. 1 3(b) (see also Exercise 2.6) tells us that the resultant of H(X) and Gm(X) is Res(H (X), Gm(X ) )
=
±m2 deg H II H(() Gm (() =O
=
±m2deg H II Fm(( /2-1 Gm (() =O
=
Res (Fm(X), Gm(X)) £2-1 .
Hence in order to show that ¢ has good reduction, it suffices to prove that Res (H(X), Gm(X) ) E R* .
Let K' K (E[m£]) be the field extension obtained by adjoining the coordi nates of the points of order m£ to K, let R' be the ring of integers of K', let p' be the maximal ideal in R' , and let R' jp' be the residue field of R' . The ex tension K' / K is unramified, because we have assumed that E has good reduction and m.e is a unit in R; see Corollary 6.33 . Note that H (X ) and Gm (X) factor completely in K', and in fact their roots are in R'. This is clear for Gm (X), since its roots are the x-coordinates of the points in E[m] and its leading coefficient is m2 , which is a unit in R. We now analyze H (X) more closely. =
k'
=
Claim 6.56. The roots ofH ( X ) are given by
{roots of H(X)}
=
x( E[m£] " E[ml) C R' .
Proof ofClaim. The roots of H (X) are the solutions to
Fm(X) Gm(X)
=
X (Q)
for some Q E E[£].
Writing a root of H (X) as x(P) for some P E E, this means that Fm ( P) ) x([m]P) = Gm (x ( P) (x )
=
x( Q ) ,
6. Dynamics Associated to Algebraic Groups
364
and hence [m]P ±Q. But Q E E[£], so P E E[m£]. This shows that the roots of H(X) are contained in x (E[m£]). Further, if P E E[m], then Gm (x(P)) 0 and Fm (x(P)) =f- 0, so =
=
H (x(P))
=
£2 I Fm (x(P)) - =/- 0.
This gives the inclusion {roots of H(X)}
C
x (E[m£] " E[ml).
The other inclusion is clear from the definition of H(X), since
E[m£] "- E[m] ===? [m]P E E[£] "- {0}. Thus x(P) is a root of Fm (X) - x([m]P)Gm (X), which is one of the factors in the PE
product defining H(X). Finally, we note that K' contains the x-coordinates of the points in E[m£] by construction. Further, these x-coordinates are the roots of the polynomial Fme (X) E R[X] whose leading coefficient is m2£2 E R*, so the roots are integral over R, hence are in R' . D We now resume the proof of Proposition 6.55. We assume that the resultant
Res ( H (X), Gm (X)) is not a unit in R and derive a contradiction. This assump tion means that H(X) and Gm (X) have a common root modulo p ' , so we can find X I , x2 E R' such that
and
x1 = x2 (mod p ' ) .
From our description of the roots of H(X) and Gm (X), this means that we can find points PI E E[m£] " E[m] and P2 E E[m] " {0} satisfying PI = P2 (mod p ' ). (In principle, we might get PI = -P2, but if that happens, then just replace P2 by -P2 .) Since clearly P1 =f- P2, this proves that the reduction modulo p ' map
E ( K' ) �
E(k')
is not injective on E[m£]. This is a contradiction, since Theorem 6.32 tells us that D the prime-to-p torsion injects on elliptic curves having good reduction. 6.6
Rigid Lattes Maps
In general, a Lattes map ¢ JIDI ----+ lP'I is defined via the commutativity of a diagram :
E � E (6.36)
6.6. Rigid Lattes Maps
365
where 7/J is a morphism of degree 2 and is a finite separable map. Ev ery morphism of an elliptic curve to itself is the composition of an endomor phism and a translation (Remark 6. 19), so 7/J has the form 7/J( P) a(P) + T for some a E End(E) and some T E E. However, it turns out that the commutativity of (6.36) puts additional constraints on ¢, 7/J, and More precisely, it forces the ex istence of a similar diagram in which 1r has a special form. We state this important result and refer the reader to [300] for the analytic proof.
d>
1r
=
'Tf.
Theorem 6.57. Let K be afield ofcharacteristic 0 and let ¢ be a Lattes map defined over K. Then there exists a commutative diagram of the form (6.36) such that the map 1f has the form
E -------+ E/f � lP' 1 for some nontrivialfinite subgroup C Aut(E). 1r :
r
C, see [300, Theorem 3.1]. The general case for character istic-a fields follows by the Lefschetz principle, cf. [410, VI §6].
Proof For a proof over
D
Definition. Let ¢ be a Lattes map. A reduced Lattes diagramfor ¢ is a commutative diagram of the form
E
E (6.3 7)
lP' 1 � lP'1 � E;r Theorem 6.57 says that every Lattes map fits into a reduced Lattes diagram.
E
;r
�
Corollary 6.58. Let ¢ be a Lattes map given by a reduced diagram (6.37). Then the point 7/J( 0) isfixed by every element off, so in particular, 7/J ( 0) E Etors· Iffurther j (E) =1- 0 and j (E) =1- 1728, then
f
=
J.L2 ,
deg 1r = 2,
Proof We defer the proof that
and
7/J ( 0) E E[2].
7/J( 0) is fixed by every � E f until Proposi
tion 6.77(b), where we prove it in a much more general setting. (Cf. the proof for flexible Lattes maps in Proposition 6.5 l (b).) To see that 7/J ( O) is a torsion point, let � E be a nontrivial element of f. Then �(7/J(O)) = 7/J(O), so applying Theo rem 6.22 to the isogeny � - 1, we find that
r
[deg(� - 1) ] (7/J(O)) = (f=l_) o (� - 1) (7/J(O))
=
0.
For the final statement of the corollary, we note that if j(E) is not equal to 0 or 1728, then Proposition 6.26 tells us that Aut(E) J.L2 • Hence r J.L 2 and deg 1r = 2. Further, since 7/J( 0) is fixed by every element of we D have [-1 ]7/J (O) = 7/J (O), so [2] 7/J(O) = 0. =
r,
=
366
6. Dynamics Associated to Algebraic Groups
Remark 6.59. The proof of Theorem 6.57 in [300] actually shows something a bit stronger. Suppose that ¢ is a Lattt'�s map fitting into the commutative diagram (6.36).
It need not be true that the map 1r : E IP'1 is of the form E ----> E ;r, i.e., the given diagram need not be reduced, and indeed the map 1r may have arbitrarily large degree. However, what is true is that there are an elliptic curve E' , an isogeny E E' , and a finite subgroup r' c Aut(E' ) such that 1r factors as E E' E'/ f � IP'1 . ---->
---->
�
�
Further, this factorization is essentially unique. See [300, Remark 3.3]. Remark 6.60. The proof of Theorem 6.57 is analytic and does not readily generalize to characteristic p. A full description ofLattes maps in characteristic p is still lacking. Aside from the curves having j-invariant 0 or 1728, every Lattes map has r = IL2 = Aut(E) and deg 1r = 2, so after a change of coordinates, the projec tion 1r : E ----> IP'1 is 1r(x, y) = x. For simplicity, we will concentrate on this situation, although we note that the two special cases with Aut( E) = IL4 and Aut( E) = 1L6 have attracted much attention over the years for their interesting geometric, dynami cal, and arithmetic properties. Our next task is to describe the periodic points of (rigid) Lattes maps and to compute their multipliers. Proposition 6.61. Let ¢ IP' 1 IP'1 be a Lattes map and fix a reduced Lattes diagram (6.3 7) for ¢. We assume that j (E) =j:. 0 and j (E) =j:. 1728. We further assume that 'ljJ is an isogeny, i.e., with our usual notation 7/J(P) [a](P) + T, we :
---->
=
are assuming that = 0. (See Exercise 6 .24 for the other cases.) (a) The set affixedpoints of¢ is given by
T
Fix(¢) = 1r(E[a + 1] U E[a - 1]) .
(6.38)
(b) The intersection satisfies
E[a + 1] n E[a - 1] c E[2]. If deg( a - 1) is odd, then the intersection is 0. (c) Let 1r(P) E Fix(¢). The multiplier of¢ at 1r(P) is
ifP E E[a - 1] and P rf. E[a + 1] , ifP E E[a 1] and P rf. E[a - 1], ifP E E[a + 1] n E[a - 1].
+
(6.39)
Proof (a) We have 1r(P) E Fix(¢) if and only if
1r(P) = ¢ (1r(P)) = 1r('¢ (P)) . Our assumption on j(E) means that r = Aut( E) = /.L2 , so 1r(P) is fixed by ¢ if and only if 'lj;(P) = ±P. Since we are also assuming that 7/J(P) = [a](P), this is the desired result.
367
6.6. Rigid Lattes Maps
(b) Let P E E[o: - 1] n E[o: + 1]. Adding [o: - 1] (P) 0 to [o: + 1] (P) 0 yields [2]P = 0, so P E E[2]. To ease notation, let m deg(o: - 1). Then using Theorem 6.22, we find that =
=
=
[m] (P)
=
[c;=-y] o [o: - 1] (P) = [c;=-y] (O) = 0,
so P E E[m] . Hence P E E[2] n E[m], so ifm is odd, then P 0. ( c ) The proof is identical to the proof of Proposition 6.52. The only difference is that 'ljJ = [a] may no longer be multiplication by an integer, but we still have the key formula =
'lj; * (w) = [o:] * (w) = o:w
giving the effect of 'ljJ on the invariant differential of E. Using this relation in place offormula (6.29) used in proving Proposition 6.52 and tracing through the argument yields the desired result. D w
We recall from Section 4.5 that O" n) ( ¢) denotes the i1h symmetric polynomial of the multipliers of the points in Pern ( ¢), taken with appropriate multiplicities. For d � 2 and each N � 1, we write
i
(6.40)
i
for the map defined using all of the functions O" n) with 1 :::; n :::; N. McMullen's Theorem 4.53 says that for sufficiently large N, the map ud, N is finite-to-one away from the locus of the flexible Lattes maps. As noted by McMullen in his paper and stated in Theorem 4.54, rigid Lattes maps can be used to prove that ud, N has large degree. For the convenience of the reader, we restate the theorem before giving the proof. Theorem 6.62. Define the degree of d,N to be the number ofpoints in O"" ;t, }v (P)for a generic point P in the image ud, N (Md)· One can show that the degree of ud , N stabilizes as N oo. We write deg(ud)/or this value. Then for every E > 0 there is a constant c€ such that 0"
____,
for all d. In particular, the multiplier spectrum ofa rationalfunction ¢ E Ratd determines the conjugacy class of¢ only up to 0< ( d � - < ) possibilities. Proof We prove the theorem in the case that d is squarefree and leave the general
case for the reader. Let F = Q( R ) , let RF be the ring of integers of F, and let a1 , . . . , ah be fractional ideals of F representing the distinct ideal classes of RF. Consider the elliptic curves E1 , . . . , Eh whose complex points are given by 1 :::; i :::; h.
Each Ei has End ( Ri )
9:!
RF (Proposition 6.40), and we normalize an isomorphism
368
6. Dynamics Associated to Algebraic Groups
as described in Proposition 6.36. We fix a Weierstrass equation for each Ei and we define a Lattes map c/Ji by c/Ji 0 X = X o [ yCd ] . Then deg( c/Ji ) = d from Proposition 6.39, and Proposition 6.61 tells us that the multipliers of c/Ji are given by (6.39). In particular, they are the same for every c/Ji , i.e., the set of maps { ¢ 1 , . . . , ¢h} is isospectral, so we see that Next we observe that ¢ 1 , . . . , ¢h give distinct points in M d , because Proposi tion 6.40 says that E1 , . . . , Eh are pairwise nonisomorphic, and then Theorem 6.46 tells us that ¢ 1 , . . . , ¢h are pairwise nonconjugate. This proves that d , N is generi cally at least h-to-1, where h is the class number of the ring of integers of Q ( R ) . (Note that the c/Ji are not flexible Lattes maps, and McMullen's Theorem 4.53 tells us that d , N is finite-to-one away from the flexible Lattes locus.) To complete the proof we need an estimate for this class number. Such an esti mate is given by the Brauer-Siegel theorem [258, Chapter XVI], which for quadratic imaginary fields says that a
rr
log (class number of Q ( R ) ) log d d squarefree li d--+oo m
1 2
(Note that this is where we use the assumption that d is squarefree, since it implies that the discriminant of Q ( R ) is equal to either d or 4d.) In particular, the class number is larger than d 1 1 2 -E for all sufficiently large squarefree d, which completes the proof of Theorem 6.62 for squarefree d. In the general case, there are two ways to proceed. The first, which is sketched in Exercise 6.25, is to find a quadratic imaginary field F whose discriminant is O(d1 -E) and whose ring of integers contains an element of norm d. The second is to write d = ab2 with a squarefree and use elliptic curves whose endomorphism rings are isomor phic to the order Rb = Z + bRF in the field F = Q ( Fa) . The class number of Rb is equal to hFb times a small correction factor; see [399, Exercise 4. 12]. D 6.7
Uniform Bounds for Lattes Maps
A fundamental conjecture in arithmetic dynamics asserts that there is a constant C = C(d, D) such that for all number fields K/Q of degree D and all rational maps cp(z) E K(z) of degree d the number of K-rational preperiodic points of ¢ satisfies # PrePer(¢, 1P' 1 (K)) ::::; C(d, D). (See Conjecture 3.15 on page 96.) Aside from monomials and Chebyshev polynomi als, the only nontrivial family of rational maps for which Conjecture 3.15 is known
� 2,
369
6.7. Uniform Bounds for Latt(�s Maps
is the collection of Lattes maps. The proof uses the following deep theorem, whose demonstration is unfortunately far beyond the scope of this book. Theorem 6.63. (Mazur-Kamienny-Merel) For all integers D 2:: 1 there is a con stant B(D) such thatfor all numberfields KIQ ofdegree at most D and all elliptic curves EI K we have
# E(K) tors ::; B(D).
Discussion. This deep result was first proven by Mazur [292] for K Q, then by Kamienny [225] for [K : Q] 2, and then was extended to various specific larger degrees before the proof was completed for all degrees by Merel [297]. The proof uses the theory of modular curves and Jacobians, which do have counterparts in arithmetic dynamics (cf. Sections 4.2-4.6). However, the proof also relies in a fundamental way on the fact that E is a group, and hence that there exist a large number of commuting maps E E. This is in marked contrast to the situation for a general rational map rjJ lP' 1 lP'1 , for which only the iterates of rjJ commute with ¢. The inclusion c End(E) leads to the existence ofHecke correspondences on elliptic modular curves, and these correspondences provide an essential tool in the proof of Theorem 6.63. Unfortunately, there does not appear to be an analogous theory of correspondences for the dynamical modular curves and varieties attached to non-Lattes maps on lP' 1 . D =
=
___,
:
Z
___,
Corollary 6.64. For all integers curves E I K we have
#
n
> 1, all number fields
KIQ,
( U E(L)tors) ::; B (n[K : Ql ) 3 ,
and all elliptic
(6.41)
[L:K] :S: n
where B(D) is the constant appearing in Theorem 6.63. Proof To ease notation, we let D
=
in (6.41) satisfies
[K : Q]. Every field L appearing in the union
[L : Q] [L : K] [K : Q] ::; nD, so Theorem 6.63 tells us that # E(L) tors ::; B(nD). In particular, E(L) contains no points of order strictly larger than B(nD). This is true for every such L, so we =
conclude that
U
Then using # E[b]
#
=
E[b].
#E[b] =
L
I :S:b:S:B(nD)
b2 yields
( U E(L)tors) ::; [L:K] :S:n
u
E(L) tors C
[L:K] :S:n
L l :S:b:S:B(nD)
l :S: b:S: B (nD)
b2 ::; B(nD) 3 .
D
6. Dynamics Associated to Algebraic Groups
370
We now use Theorem 6.63 to prove uniform boundedness of preperiodic points for Lattes maps. This bound is in fact independent of the degree of the Lattes map ¢, which may be surprising at first glance. However, it is easily explained by the fact that Lattes maps associated to the same elliptic curve all commute with one another, so they have identical sets of preperiodic points. Theorem 6.65. Let D 2: 1 be an integer. There is a constant C(D) such thatfor all numberfields K/Q ofdegree D and all Lattes maps ¢ lP'1 -+ lP' 1 defined over K we :
have
# PrePer(¢, 1P'1 (K)) :::; C(D).
Proof Without loss of generality we fix a reduced Lattes diagram (6.37) for ¢. Then Proposition 6.26 says that the projection map 1r : E -+ lP'1 has degree at most 6, and indeed if j (E) -1- 0 and j (E) -1- 1 728, then deg( 1r ) = 2. Proposition 6.44 tells us that
PrePer(¢, 1P'1) n(Etors ), =
so the fact that deg( ) 1r
:::;
6 yields
PrePer(¢, lP'1 (K))
C
U n(E(L)tors) ·
(6.42)
[L:K] ::0:6
Corollary 6.64 says that the set on the righthand side of (6.42) has size bounded solely in terms of D, hence the same is true of # PrePer( ¢, lP' 1 ( K)). D Example 6.66. The rational map
<Pa,b(x)
=
x4 - 2ax2 - 8bx + a2 4x3 4ax + 4b
+ is the Lattes map associated to multiplication-by-2 on the elliptic curve Ea,b : y 2 x3 + ax + b. =
The j-invariant and discriminant of Ea,b are given by the usual formulas
3 j(Ea) 1 728 4a3 4a + 27b2 =
and
Theorem 6.65 tells us that # PrePer(<Pa,b, lP' 1 (K)) is bounded solely in terms of the degree [K Q]. In general, the best known bounds are exponential in but if j ( Ea,b) is an algebraic integer, then much stronger bounds can be proven as in the following result. Theorem 6.67. Let K be a number field of degree D 2: 2, let E / K be an elliptic
d
=
:
d,
curve whose j-invariant is an algebraic integer, and let ¢ be a Lattes map associated to E. Then there is an absolute constant c such that
371
6.7. Uniform Bounds for Lattes Maps
Proof The assumption that the elliptic curve E has integral j-invariant means that it has everywhere potential good reduction. Replacing K by an extension of bounded degree, we may assume that E has everywhere good reduction. (In fact, it suffices to go to the field K (E[3l), a field of degree at most 48 over K.) Then a result of Hindry-Silverman [204] implies a bound slightly stronger than
# E(K) tors �
221 D log D.
Finally, we note that as in the proof of Corollary 6.64, a bound for E(K) tors of the form # E(K) tors � B ([K : Ql) for all number fields K implies a bound of the form
#
([L:UK)S:nE(L)tors) � B (n[K : QJ(
Hence as in the proof of Theorem 6.65 we have
PrePer(¢, lP' 1 (K)) C for an absolute constant c.
7r(E(L)tors) � c(D log D) 3 U [L:K)S:6
D
Theorem 6.65 proves uniformity for rational preperiodic points of Lattes maps. In the other direction, recall that we proved (Theorem 3.43) that the orbits of rational wandering points contain only finitely many integers except in a few precisely spec ified situations. In particular, Lattes orbits contain only finitely many integer points, since Lattes maps are not polynomial maps. Using deep results from the theory of elliptic curves, it is possible to obtain strong uniformity estimates for the number of integer points lying in Lattes orbits. For simplicity we state results over 0 are pairwise relatively prime integers satisfying A + B = C, then C «€ Tip( A BC pl+E.
373
6.7. Uniform Bounds for Latti�s Maps
(b) If the ABC conjecture is true, then the number of integer points in bounded by an absolute constant independent of E and (.
0¢ ( ( ) is
Proof Write the given Weierstrass equation for E as
with a, b E Z. We begin by showing that the minimality assumption (6.44) implies that there are no primes p with p2 l a and p3 l b. The rational function ¢( x) = F ( x) / G ( x) is associated to the multiplication-by-m map, so it is given by polynomials
F(a, b; x), G(a, b; x) E Z[a, b, x] that are weighted homogeneous in the sense that
For example, if m
=
2, then
2 - 8bx + a2 4 cp(x) = x -4x32ax+ 4ax + 4b Hence ifp2 l a and p3 l b then conjugating cp(x) by f (x) = px yields b ; px) = F(p- 2 a , p-3b; x) . cpf (x) p _1 F(a, G (a, b; px) G(p- 2 a, p-3b; x) The assumption that p2 l a and p3 l b implies that these polynomials have integer coef ,
=
ficients, and then homogeneity yields
Res(q/) Res(F(p- 2 a, p-3 b ; x), G(p- 2 a, p-3 b; x) ) m m = p- 2 ( 2 -l ) Res(F(a, b ; x), G (a, b ; x) ) p_ m2 ( m2 - 1 ) Res(¢). =
=
This contradicts (6.44), so we have proven that there are no primes p satisfying p2 l a
and p3 l b.
We would like to apply Theorem 6.69 to the rank-1 subgroup generated by a point P = ( (, TJ) of E lying above (. Unfortunately, although ( E Q, there is no reason that rJ need be rational. So it is necessary to move to a twist of E. We are given a point ( E Q and we choose a point P ( ( , TJ) E E lying above ( . We do not assume that TJ = J (3 + a( + b is rational. We write =
then we factor and consider the elliptic curve
with u1 squarefree,
6. Dynamics Associated to Algebraic Groups
374
(6.45)
(In the terminology of Section 4.7, E' is a twist of E; cf. Example 4.7 1 .) Notice that the point P{ (u 1 x 1 , uiv1 ) E E' (IQ!) is a point of E' having integer coordinates. We claim that the Weierstrass equation (6.45) for E' is quasiminimal. To prove this claim, let p be any prime. We showed earlier that either =
ordp(b) < 3.
or
Since u 1 is squarefree by construction, it follows that either or which shows that the Weierstrass equation (6.45) is quasiminimal. This means that we can apply Theorem 6.69 to E' and the rank-1 subgroup gen erated by P' to conclude that '
#{ n 2: 1 : [n]P' E E' (Z) } :S C"(E )+l .
(6.46)
Further, if the ABC conjecture is true, then the upper bound may be replaced by C. The two elliptic curves E and E' are isomorphic, although the isomorphism is defined only over 1Q! ( JUl ) . This isomorphism, which we denote by F, is given explicitly by E y2 :
+
x3 ax + b (x, y)
=
In particular, j (E) j (E'), so v(E) v(E'). In order to relate integers in Oq,( () to integer points in E' (IQ!), we write =
=
P�
and
=
[n]P{
=
(x� , y�) .
Since the isomorphism F respects multiplication by n, we have
In particular, since P{ E E' (IQ!), it follows that [n]P{ P� E E'(IQ!), so F maps the multiples of P1 , which, note, are not in E(IQ!), to points in E'(IQ!). Further, if Xn E Z, then it is clear from the definition of F that u 1 xn E Z, and hence y� is also in Z, since we just showed that y� is in 1Q! and the equation of E' shows that y� is the square root of an integer. To summarize, we have proven that =
x�
Xn E Z
===?
P�
=
=
F(Pn ) E E' (Z).
(6.47)
375
6.8. Affine Morphisms and Commuting Families
By construction we have q} (() = x([mk]P)
=
# (0¢(P) n z) = #{k :S #{ :S #{ n
2': 0 : Xmk E Z} k 2': 0 : P:r,k E E'(Z) } 2': 0 : P� E E'(Z) }
Xm k , and hence
from (6.47),
v(E' ) + l
from (6.46), since E and E' are isomorphic. = c Further, if the ABC conjecture is true, then we may replace the upper bound by C. :S C
v(E) + l
D Remark 6.7 1. We note that something like the affine minimality of¢ is necessary in
the statement of Theorem 6.70. Indeed, without some kind ofminimality condition, we saw in Proposition 3.46 that we can make # (0¢(() n z) arbitrarily large by replacing ¢ ( z ) with Bcp(B- 1 z). This conjugation has the effect of multiplying every point in the orbit by B, hence allows us to clear an arbitrary number of denominators. Remark 6.72. Continuing with notation from the statement of Theorem 6.70, we note that there is a cutoff value k0 such that cpk (() E 7!. for 0 :S k :S ko and cpk (() � 7!. for k > ko . This reflects the more general fact that if x ( [n] P) E 7!. and if r I n, then x ( [r] P) E 7!. ([410, Exercise 9. 12]). Note that no such cutoff statement holds for general rational maps that are not non-Lattes maps. 6.8
Affine Morphisms, Algebraic Groups, and Commuting Families of Rational Maps
Power maps and Chebyshev maps are attached to endomorphisms of the multiplica tive group Gm and its quotient Gm/{z = z-1 }, and similarly Lattes maps are attached to maps of quotients of elliptic curves. In this section we put these con structions into a general context and state a classical theorem on commutativity of one-variable rational maps. Definition. Let G be a commutative algebraic group. An affine morphism of G is the composition of a finite endomorphism of degree at least 2 and a translation. Remark 6.73. The reason for this terminology is as follows. Let G /C be a connected commutative algebraic group of dimension g. Then its universal cover is C9 and every affine morphism 'ljJ G G lifts to an affine map lf29 lf29, i.e., there are a matrix and vector a such that the following diagram commutes:
A
:
---+
---+
z
1
G
>->
Az + a
1
G
376
6. Dynamics Associated to Algebraic Groups
Example 6.74. Every affine morphism of the multiplicative group Gm has the form 'lj;(z) = azd for some nonzero a and some d E Z. More generally, for any commu tative group G, any a E G, and any d E Z there is an affine morphism '1/J(z) = azd .
Notice that it is easy to compute the iterates of this map,
Proposition 6.75. Let 'ljJ : G ---T G be an affine morphism of an algebraic group G, so 'ljJ has the form 'lj; (z) = a · a(z) for some a E End( G) and some a E G. (a) The endomorphism a and translation a are uniquely determined by 7/J. (b) Let a and a be as in (a). Then the iterates of'lj; have theform
Proof The definition of affine morphism tells us that there are an element a E G and an endomorphism a of G such that the map 'ljJ has the form 'lj;(z) = aa(z). Evaluating at the identity element e E G yields 'lj; (e) = aa(e) = a, so a is uniquely determined by 'lj;. Then a(z) = a- 1 7/J(z) is also uniquely determined by 7/J. This proves (a). The proof of (b) is an easy induction, using the commutativity of G and the fact that a is a homomorphism. 0 Definition. A self-morphism of an algebraic variety ¢ : V ---T V is dynamically affine if it is a finite quotient of an affine morphism. What we mean by this is that there are a connected commutative algebraic group G, an affine morphism 'ljJ : G ---7 G, a finite subgroup r c Aut (G), and a morphism G;r ---7 V that identifies G;r with a Zariski dense open subset of V (possibly all of V) such that the following
diagram is commutative:
G
1
Gf r
11
v
-----7
7/J
G
-----7
1
q,
-----7
Gfr
(6.48)
11
v
Example 6.76. Examples of dynamically affine rational maps ¢ : lP' 1 ---7 lP' 1 include the power maps ¢( z) = zn with G = Gm and r = { 1}, the Chebyshev polyno mials Tn (z) with G = Gm and r = {z, z- 1 }, and Lattes maps with G an elliptic curve E and r a nontrivial subgroup of Aut( E). Proposition 6.77. Let ¢ : V ---7 V be a dynamically affine map and let 'ljJ : G ---7 G and r C Aut( G) be the associated quantities fitting into the commutative dia gram (6.48). (a) For every � E f there exists a unique ( E f with theproperty that 7/J o � = ( o 'lj;. (b) Write 7/J(z) = a · a(z) with a E G and a E End( G) as in Proposition 6.75. Then �(a) = afar every � E r .
377
6.8. Affine Morphisms and Commuting Families
(c) Assume that #f 2 2 and that G is simple. (An algebraic group is simple if its only connected algebraic subgroups are {1} and G. ) Then a E Gtors. i.e., the translation used to define 'lj; is translation by a point of.finite order. Proof (a) The uniqueness is clear, since if 'lj; o � =
6 o 'lj; = 6 o 'lj;, then 6 = 6
because the finite map 'lj; G ----+ G is surjective. We now prove the existence. The commutativity of (6.48) tells us that for all z E G and all � E r, :
(n o 'lj; o O (z) = (¢ o o �)(z) = (¢ o n) (z) = ( n o 'lj;) (z) . 1r
Thus ('lj; o �) (z) and 'lj;(z) have the same image for the projection map so there is an automorphism e E f satisfying
1r :
G ----+ G
/f,
'1/J (�(z)) = ( ( '1/J (z)) .
We claim that the automorphism e, which a priori might depend on both � and z, is in fact independent of z. To see this we fix � and write 'lj; (� ( z)) = �� ('lj; ( z ) ) to indicate the possible dependence of e on z. In this way we obtain a map (of sets) G
______.
r,
Since r is finite, there exists some (' E r such that �� = (' for a Zariski dense subset of z E G. (Note that a variety cannot be a finite union of Zariski closed proper subsets.) It follows that 'lj; o � is equal to (' o 'lj; on a Zariski dense subset of G, and hence they are equal on all of G. (b) From (a) we see that there is a permutation off defined by the rule T
:
r
______.
r,
Evaluating both sides of 'lj; o � = T(�) o 'lj; at the identity element 1 E the fact that �(1) = 1 and 'lj;(1) = a · o:(1) = a, we find that
G
and using
a = 'I/J (�(1)) = T(0 ( '1/J (1)) = T(0(a).
But T is a permutation of r' so as � runs over r' so does T (0. Hence a is fixed by every element off. ( c ) From (b) and the assumption that #f 2 2, there exists a nontrivial � E r with �(a) = a. It follows that a is in the kernel of the endomorphism G
______.
G,
The kernel is not all of G, since � is not the identity map, so the simplicity of G tells D us that the kernel is a finite subgroup of G. Hence a has finite order. Remark 6.78. In this book we are primarily interested in dynamically affine maps of lP'1 , but higher-dimensional analogues, especially of Lattes maps, have also been studied. See for example [68, 1 34, 145, 439].
6. Dynamics Associated to Algebraic Groups
378
The commutativity of (6.48) implies that deg(¢) deg(?j!). It follows that all dynamically affine maps for the additive group Ga have degree 1 , since every affine morphism of Ga has the form ?j!(z) az + b. Hence nonlinear dynamically affine maps on lP' 1 are attached to either the multiplicative group Gm or to an elliptic curve, since these are the only other algebraic groups of dimension 1 . We note that over a field of characteristic 0, the endomorphism ring End( G) of a one-dimensional algebraic group G is commutative. 3 More precisely, the mul tiplicative group has endomorphism ring End(Gm) Z, and the endomorphism ring End(E) of an elliptic curve E is either Z or an order in a quadratic imaginary field. The commutativity of End( G) means that dynamically affine maps commute with many other maps. An appropriately formulated converse of this statement is a classical theorem of Ritt. =
=
=
Theorem 6.79. (Ritt and Eremenko) Let ¢, 7jJ E q z) be rational maps ofdegree at least 2 with the property that ¢ o 7jJ = 7jJ o ¢. Then one ofthefollowing two conditions is true: (a) There are integers m, n 2 1 such that cpn ?j!m . (b) Both ¢ and 7jJ are dynamically affine maps, hence they are either power maps, Chebyshev polynomials, or Lattes maps. In all cases, the commuting maps ¢ and 7jJ satisfy =
F(¢)
=
F(?j!),
.J(¢)
=
.J(?j!) ,
and
PrePer( ¢)
=
PrePer( 7jJ).
Proof The first part of the theorem, in somewhat different language, is due to
Ritt [371]. See Eremenko's paper [152] for a proof of both parts of the theorem and some additional geometric dynamical properties shared by commuting ¢ and ?j!. A higher-dimensional analogue is discussed in [135]. We remark that the equality PrePer( ¢) PrePer( 7jJ) is a formal consequence of the commutativity of ¢ and 7jJ and the fact that the preperiodic points of a nonlinear rational map are isolated; see D Exercise 1 . 15. =
Although we do not give a proof of Ritt's theorem, we conclude this section by proving the easier statement that only polynomial maps can commute with polyno mial maps. This result was used in our description of the rational maps commuting with the Chebyshev polynomials (Theorem 6.9). Theorem 6.80. Let K be afield, let cp(z) E K[z] be a polynomial ofdegree d 2 2, and let ?j! (z) E K(z) be a nonconstant rational map. We assume that both ¢ and 7jJ are separable, i.e., neither ofthe derivatives ¢' ( z) and ?j!' ( z) is identically 0. Suppose further that ¢ and 7jJ commute under composition, ¢ o 7jJ = 7jJ o ¢. Then one of the following is true: (a) ?j!(z) E K [z], i.e., 7jJ is also a polynomial. 3Even in characteristic p, most elliptic curves have commutative endomorphism ring. However, there are a finite number of elliptic curves whose endomorphism ring is a maximal order in a quatemion algebra. These supersingular curves are all defined over IFp2 . See [410, V §3].
6.8. Affine Morphisms and Commuting Families
379
(b) After simultaneous conjugation by an affine map f ( z) z + (3, the polynomial ¢(z) has the form ¢(z) = a zd and the rational map '1/J(z) has the form '1/J(z) = bzr for some r < 0. Proof The proof is an application of ramification theory and the Riemann-Hurwitz formula (Theorem 1 . 1 ). By assumption, the map ¢ is a polynomial, so oo is a totally ramified fixed point of¢. Suppose that '1/J(z) is not a polynomial. This means that we can find a point a E '1/J- 1 ( oo) with a =/=- oo. We use the commutativity of ¢ and 'ljJ to compute ea ( c/Jn o '1/J)
-1
= ea ( 'l/J ) nII e qhj;(o: ) (¢) = ea ('l/J)e oo ( c/Jt = ea ( 'l/J) dn i=O
(6.49)
Hence
IT e¢i (a) (¢) =
ea ( 'l/J ) > e E Ip, � IP' 1 . Prove that we can take 1r( x, y) = x and that the Lattt:s map 2 corresponding to 1/J is
(b)
z(z3 - 8b) EI p,3 � IP' 1 . Prove that we can take ( x, y) = y and that the Lattes map corresponding to 1/J is z4 + 18bz2 - 27b2 '+'2 ( z ) = 8z3 Let E --> Elp,6 � IP' 1 . Prove that we can take 1r(x, y) = x3 and that the Lattes map 1r :
1r
A-
(c)
1r :
corresponding to 1/J is
z(z - 8b) 3 64(z + b) 3 Compute the conjugate ¢3 ( z - b) + b of ¢3 ( z ) , compare it to ¢2 ( z), and explain. 6.13. Let ¢ be a Lattes map. Prove that there does not exist a linear fractional transforma tion f E PGLz such that the conjugate ¢1 is a polynomial. (Cf. Exercise 6.8.) ¢3 (z)
=
Exercises
382
6.14. Complete the proof of Proposition 6.44 in the general case that 'lj;(P) = a ( P) + T with a E End(E) and T E E not necessarily equal to 0. However, you may assume that T E Etors , i.e., T is a point of finite order. 6.15. Let ¢ : C ----> C be a nonconstant rational map from a smooth curve to itself. Recall that CritVal denotes the set of critical values of¢. Prove that 00
00
n=O
n =l
Section 6.5. Elliptic Curves and Flexible Lattes Maps 6.16. Let E be an elliptic curve and let 1r : E ----. IP' 1 be a map of degree 2. (a) Let R be any point on E. Show that we can define a new group law (call it *) on E by the rule P * Q = P + Q - R.
Show that R is the identity element for the group (E, * ). (b) Prove that there exists a point Po E E such that 1r ( - ( P + Po) ) = 1r ( P + Po) for all P E E. (c) Conclude that after choosing a new identity element for E, the map 1r is even, i.e., satisfies 1r(P) = 1r(-P) for all P E E. 6.17. Fix an elliptic curve E and and a degree-2 map 1r : E ----. lP'1 satisfying 1r( P) = 1r(-P) . For any integer m and any point T E E[2], let rPm,T : lP' 1 ----> IP' 1 be the flexible Lattes map associated to the map 'lj;(P) = [m] (P) + T as in the commutative diagram (6.22). (a) Prove that rPm,r o r/Jm' ,T' = rPmm' ,mT'+T · In particular, the maps rPm,o commute under composition. (b) Prove that ¢;', , T is either c/Jmn ,r or c/Jmn , o . More precisely, if m is odd and n is even, prove that ¢;', ,T = r/Jmn , o , and prove in all other cases that ¢;', ,r = rPmn ,r. (Of course, if T = 0, the cases are all the same.) (c) It follows from (a) that the collection of maps { rPm,T : m 2: 1 , T E E[2J } is closed under composition. Prove that r/J 1 ,o is the identity element and that the associative law holds. Thus this set of flexible Lattes maps for E is a noncommutative monoid. 6.18. Let ¢ : lP' 1 ----> lP'1 be a flexible Lattes map associated to 'lj;(P) = [m]P + T, where the point T E E[2] is not necessarily equal to 0. (a) Prove that the set of fixed points of ¢ is given by
Fix(¢) = x ( [m - lr 1 (T)) U x ( [m + lr 1 (T)) . (b) Compute the multiplier of ¢ at each point in Fix(¢). (c) Use the results from (a) and (b) and the formula for the composition of Lattes maps in Exercise 6. 1 7 to describe the periodic points of ¢ and to compute their multipliers. (Hint. Mimic the proof of Proposition 6.52, which dealt with the case T = 0.) 6.19. Proposition 6.52 describes the multipliers of a flexible Lattes map. Using these values, verify directly that the formula
from Theorem 1 . 1 4 is true for flexible Lattes maps.
Exercises
383
6.20. Complete the proof of Proposition 6.52(b) by computing the multiplier A¢ ( oo ) at the fixed point oo = x(O). (Hint. Move 0 to (0, 0) using the change of variables z = xj y and w = 1/y . Then write the invariant differential in terms of z and w and mimic the proof in the text.) 6.21. Let K be an algebraically closed field and let ¢ and ¢' be flexible Lattt:s maps defined over K that are associated, respectively, to elliptic curves E and E'. Suppose that ¢ and ¢' are PGL2(K)-conjugate to one another. We proved (Theorem 6.46) that if the characteristic of K is not equal to 2, then E and E' are isomorphic. What can be said in the case that K has characteristic 2? (Note that in characteristic 2 it is necessary to use a generalized Weierstrass equation (6. 1 2) to define E.) 6.22. We proved Proposition 6.55 in the case that '1/J( P) = [m] ( P). (a) Prove Proposition 6.55 for general flexible Latt($ maps, i.e., Lattes maps associated to maps of the form 'lj;(P) = [m](P) + T with T E E[2]. (b) Formulate and prove a version of Proposition 6.55 for rigid Lattes maps. (c) ** To what extent is the converse of Proposition 6.55 true? More precisely, if ¢ is a Lattes map fitting into a reduced Lattes diagram (6.37) and if rpf has bad reduction for every f E PGL2 (K), does the elliptic curve E necessarily also have bad reduction? 6.23. Let E be an elliptic curve given by a Weierstrass equation
E : y2 = x3 + ax + b. Let m � 1 be an integer and write x ([m] P) as a quotient of polynomials _
x ( [m]P) -
Fm (x(P)) . Gm x(P) )
(6.52)
( (a) Prove that Fm and Gm can be taken to be polynomials in x, a, and b. More precisely, prove that there are polynomials Fm , Gm E Z[a, b, x] satisfying (6.52) and that they are uniquely determined by the requirement that Fm be monic in the variable x. (b) Prove that deg(Fm) = m2 and deg(Gm) = m2 - 1 and that their leading terms are Fm(x) = xm2 + and Gm(x) = m2x m2- 1 + (c) If m is odd, prove that there is a polynomial '1/Jm (x) E Z[a, b, x] such that Gm(x) = 'I/Jm(x)2• Similarly, ifm is even, prove that there is a polynomial '1/Jm (x, y) E Z[a, b, x, y] such that Gm (x) = '1/Jm (x, y ? , where in the computation we replace y2 by x3 + ax + b. The polynomial '1/Jm is called the m'h division polynomial for E, since its roots are the nontrivial points of order m. (d) Prove that Fm and Gm satisfy · · ·
· · ·
.
Fm(t2 a, t 3 b; tx) = tm2 Fm(a, b; x) and Gm ( t2 a, t3 b; tx ) = tm2 -1 Gm ( a, b; x ) . Thus Fm and Gm are homogeneous if x, a, and b are respectively assigned weights 2, 4,
and 6. (e) Let !::,. ( E) = -16(4a3 + 27b2). Prove that the resultant of Fm and Gm with respect to the variable x is given by
384
Exercises
Section 6.6. Elliptic Curves and Rigid Lattes Maps 6.24. This exercise extends Proposition 6.61 . Let ¢ : 1P'1 --+ 1P'1 be a Lattes map and fix a reduced Lattes diagram (6.37) for ¢. Write 'lj;(P) = [o:] (P) + T as usual, where we use the standard normalization described in Proposition 6.36 to identify End( E) with a subring
of !C. (a) Prove that the fixed points of ¢ are given by Fix(¢) =
u { 1r(P) : [o: - �] (P) = -T } .
(6.53)
� EI'
(b) Let 1r(P) E Fix(¢). Prove that P is a critical point for 1r if and only if P is fixed by a nontrivial element of�- More generally, prove that the ramification index is given by ep ( 1r ) = {� E f : [�]P = P } .
I n particular, i f P is not a critical point, then there i s a unique � E f that fixes P. (c) Assume that T = 0. Let 1r (P) E Fix(¢) and choose some automorphism � E f such that 'lj;(P) = [WP). Compute the multiplier of ¢ at 1r(P) as in the following table (we have given you the first four values):
A7r(PJ (¢) =
C 1 o: if ep (1r) = 1, o:2 if f = {t2 and ep ( 1r ) o:3 if f = JL3 and ep (1r) e o:2 iff = /.L4 and ep ( 7r) if f = {t4 and ep ( 1r) iff = J-L6 and ep ( 1r ) if f = J-L6 and ep ( 1r)
= 2, = 3, = 2, = 4, = 2, = 3.
6.25. In the text we proved Theorem 6.62 under the assumption that d is squarefree. This
exercise sketches an argument to eliminate the squarefree hypothesis. We set the notation S(b) for the squarefree part of the integer b 2 1 . (a) For each integer d 2 2 , let Dd 2 1 be an integer with the property that d is a norm from the ring Z [ J- Dd ] down to Z. In other words, there are integers u and v such that
u2 + Ddv2 = d.
Then with notation as in the statement of Theorem 6.62, prove that for every E > 0 there is a constant C, such that for all d. (Hint. Use elliptic curves with CM by the ring Z [ J- Dd ] and Lattes maps associated
to the endomorphism [u + vv- Dd ] and follow the proof of Theorem 6.62.) (b) Prove that for every E > 0 there is a constant C� > 0 such that max S ( d - u2 ) 2 C�dl -
- 1 = (X0e , G- l , G2 , . . . , G N ) be the lifts of ¢ and ¢ 1 , respectively. The fact that ¢ and ¢ 1 are inverses of one ci>
=
-
-
another implies that there is a homogeneous polynomial f of degree property that
=
-
1 with the
xge - 1 . Thus ( xge ' xge - 1 xl , xge - 1 xl , . . . ' xge - 1 XN ) ,
But the first coordinate of the composition is xge , so we see that f
( cp - 1 0 ci> )(Xo , . . . ' XN )
de
=
or equivalently,
(7.3) G- 1 (X0d , F1 , . . . , FN ) X0de - l X1 for all l S: j S: N. Now let P = [0, x 1 , . . . , x N ] E H0 " Z(¢), so ¢(P) [0, F\ (P), . . . , PN (P)] -
=
with at least one Pi (P) =I- 0 . From (7.3) we see that
=
Hence so ¢ - 1 is not defined at ¢(P). Therefore ¢(P) E Z( ¢- 1 ) . Lemma 7.8. Let ¢ let H0 = {Xo = 0 }
:
=
0
AN ----+ AN and '1/J A N AN be affine morphisms, and lP'N "- A_N be the usual hyperplane at infinity. Then :
----+
deg( 'lj; o ¢) < deg( 'I/J) deg(¢) ifand only if ¢( Ho "- Z ( ¢)) c Z('ljJ ).
7.1. Dynamics of Rational Maps on Projective Space
393
Proof Let d deg( ¢ ), let e deg( 7f} ) , and let and \f! be lifts of ¢ and if;, respectively. We write explicitly as =
=
The composition \f! o has the form where EI , . . . , EN are homogeneous polynomials of degree de. The degree of 7f} o ¢ will be strictly less than de if and only if there is some cancellation in the coordinate polynomials of \f! . Since the first coordinate is xge, this shows that o
X0 divides Ej for every 1 ::; j ::; N.
{::::::}
deg( 7f} ¢) < deg( 7f}) deg( ¢) o
Suppose now that Xa iEj for every j and let P = [0, XI , . . . , x N ] E H0 Since ¢ is defined at P, some coordinate of
Eventually we will apply the following height estimate to a regular affine auto morphism ¢ and its inverse ¢ - l , but it is no harder to prove the result for any pair ofjointly regular maps, and working in a general setting helps clarify the underlying structure of the proof. Theorem 7.15. Let ¢ 1 the property that
: AN ---> A N and ¢2 A_N ---> AN be affine morphisms with :
Z(¢ 1 ) n Z(¢2) (We say that rPI and ¢2 are jointly regular.) Let
=
0.
7. Dynamics in Dimension Greater Than One
398
and There is a constant C = C((fJI , ¢2) such thatfor all P E AN (Q),
1 1 ¢1 (P)) + h ( ¢2 (P) ) 2: h (P) - C. d1 h ( d2
(7.4)
Remark 7 .16. We recall that the upper bound
h(1/J(P) ) ::; (deg 'lj;) h(P) + 0(1)
(7.5)
is valid even for rational maps 1j; lP'N --. lP'N (see Theorem 3. 1 1 ), since the proof of (7.5) uses only the triangle inequality. Thus Theorem 7.15 may be viewed as providing a nontrivial lower bound complementary to the elementary upper bound :
Proofof Theorem 7.15. Write the rational functions lP'N --. lP'N induced by ¢1 and ¢2 as
and where the Pi are homogeneous polynomials of degree d1 and the Gi are homoge neous polynomials of degree d2 . The loci of indeterminacy of ¢1 and ¢2 are given by Z(¢1) = {Xo = F1 = = PN = 0}, Z(¢2) = { Xo = G1 = = GN = 0}. We define a rational map 1j; lP'2N --> lP'2N of degree d1 d2 by .
.
·
.
.
·
:
The locus of indeterminacy of 1j; is the set since by assumption Z ( ¢1 ) and Z ( ¢2) are disjoint. Hence 1j; is a morphism, so we can apply the fundamental height estimate for morphisms (Theorem 3. 1 1) to deduce that (7.6) for all P E lP'2N (Q). The following lemma will give us an upper bound for the height of 1j; ( P) . Lemma 7.17. Let u, a1 , . . . , a N , b1 , . . . , b N E Q with u -/=- 0. Then
7.1. Dynamics of Rational Maps on Projective Space
399
adu and /3i = bi/u for 1 :S i :S N. Then for any absolute value v we have the trivial estimate
Proof Let a i
=
l l , i aN iv, I/3I Iv, , if3Niv}
max{ 1, a 1 v ,
· · ·
···
:S max{ 1, la 1 l v , . . . , iaNiv} · max{ 1 , I/3I Iv,
· · ·
, if3Niv }
·
Raising to an appropriate power, multiplying over all absolute values, and taking logarithms yields
This is the desired result, since the height does not depend on the choice of homoge neous coordinates of a point. 0 We apply Lemma 7.17 to the point with P E AN (Q), which ensures that X0 (P) -1- 0. The lemma tells us that h ('1/J (P) ) ::; h ( [X0 (P)d1d2 , FHP)d2 , . . . , PN (P)d2 ] ) + h ( [Xo (P)d t d2 ' G l (P)d t ' . . . ' G N (P)d t ] ) = d2h ( [Xo(P)d 1 , F1 (P), . . . , FN (P)] ) + d1h ( [Xo (P)d2 , G 1 (P), . . . , G N (P)] ) = d2h ((!>I (P)) + d1h (¢2 (P)) .
We combine this with (7.6) to obtain Dividing both sides by d1d2 completes the proof of Theorem 7.15.
0
For regular affine automorphisms, it is conjectured that the height inequality (7 .4) in Theorem 7. 1 5 may be replaced by a stronger estimate. Conjecture 7.18. Let ¢ : A N ----+ AN be a regular affine automorphism. Then there is a constant C = C(¢) such thatfor all P E A N (Q),
1 1 h (¢(P)) + h ( ¢- l (P)) � d2 dl
(
1+
)
1 h(P) � C. dl d2
(7.7)
Kawaguchi [230] proves Conjecture 7.1 8 in dimension 2, i.e., for regular affine automorphisms ¢ A2 ----+ A2 ; see also [413]. However, for general jointly regular affine morphisms, it is easy to see that (7.4) cannot be improved; see Exercise 7.8. Kawaguchi also constructs canonical heights for maps that satisfy (7.7); see [230] and Exercises 7.17-7.22. :
7. Dynamics in Dimension Greater Than One
400 7.1.5
Boundedness of Periodic Points for Regular
Automorphisms of AN
Theorem 7. 1 5 applied to a regular affine automorphism ¢ and its inverse implies that at least one of ¢(P) and ¢ - 1 (P) has reasonably large height. This suffices to prove that the periodic points of¢ form a set of bounded height, a result first demonstrated by Marcello [287, 288] (see also [ 1 3 1 , 4 1 8]) using a height bound slightly weaker than the one in Theorem 7 . 1 5 . Theorem 7.19. (Marcello) Let ¢ : AN ____, AN be a regular affine automorphism of degree at least 2 defined over Q. Then Per(¢) is a set of bounded height in A N ( Q). In particular,
Per(¢) n AN (K) isfinitefor all number fields K. Proof Let
and Applying Theorem 7. 1 5 with ¢ 1 = ¢ and ¢2
=
¢ - 1 yields the basic inequality (7.8)
where C is a constant depending on ¢, but not on P E AN (Q). We prove the theorem initially under the assumption that d1 d2 function c 1 ( 1 f(P) = d h (P) - ad h ¢ - 1 (P) ) - a 2 1 where the real number a > will be specified later. Then f satisfies
1'
1
(
>
)
1
4. Define a (7.9)
c 1 (¢(P) ) - - h (P) - f (¢(P)) - af(P) = h a-1 ad2 d1 - a _!_ h (P) - ad h (¢ - 1 (P)) - __9_ ad1 2 = + h (¢(P)) + h (¢ - 1 (P)) a 1
(: 2': ( 1
Hence if we take
(
1
-
L (P) d 1 ad2 ) h 1 --
-�
then and our assumption that d 1 d2 conclude that
) ( � �J h(P) + C
from (7.8).
a 1 -0 d 1 ad2 4 ensures that a > 1, so for this choice of a we
1 - - >
1)
-
'
7.1. Dynamics of Rational Maps on Projective Space
401
f (¢(P)) ;::: af(P) Applying this estimate to the points P, ¢( P), ¢2 ( P) , 0 0 0 , ¢n - l ( P), we obtain the fundamental inequality
for all P E A N (Q) and all n ;::: 00 Similarly, we define
g(P) and take
1f3d - 1d - -h(¢(P)) = -h(P) (3 - 1 l 2
(3
c
=
(70 10)
(7 o l l )
dl d2 + J(d l d2 ) 2 - 4dl d2 0 2dl
Then an analogous calculation, which we leave to the reader, shows that g satisfies
g(¢ - 1 (P)) ;::: (3g(P) from which we deduce that for all P E A N (Q) and all n ;::: 00
(70 12)
We compute
a- n f (¢n+ l (P)) + (3- n g (¢- n - l (P)) ;::: f ( ¢(P)) + g( ¢ - 1 (P)) from (70 1 0) and (7.12),
( d11 h(¢(P)) - a1d2 h(P) - a - ) c
=
( d2
+ 2_ h (¢- 1 (P)) -
>
1
h(P) - ___S!_ (3 - 1 __ ) f3dl 1
from the definition (7 09) and (7 0 1 1) of f and g,
1(3 --)1 1a --1 + ( 1 - -ad1-2 - -f3d1-1 ) h(P) - ( 1 +
C
from (7°8)0
Using the definition of f and g and rearranging the terms, we have proven the in equality
Now suppose that P E A N ( Q) is a periodic point for ¢0 Then h ( ¢k ( P)) is bounded independently of k, so letting n -+ oo in (7. 1 3) yields
7. Dynamics in Dimension Greater Than One
402
(a(3 - 1)C - (1 1 1 ) h(P) ' ad2 f3d1 (a - 1)((3 - 1) where we are using the fact that a 1 and (3 1. Our assumption that d 1 d2 4 also ensures that 1 - _ad1_2 - _(3d1_1 - v1 - d14d2 0' so the height of P is bounded by a constant depending only on ¢. This completes the proof of the first assertion of Theorem 7. 19 under the assumption that d 1 d2 4, and the second is immediate from Theorem 7.29(£), which says that for any given number field, JP>N (K) contains only finitely many points of bounded height. In order to deal with the case d1 d2 ::; 4, i.e., d1 d2 2, we use Theorem 7.1 0, which tells us that ¢2 is regular and has degree di . Similarly, deg( ¢ - 2 ) d§. Hence >
_ _ _ _
>
>
>
>
>
=
=
=
from what we have already proven, the periodic points of ¢2 form a set of bounded height, and since it is easy to see that Per(¢) Per( ¢2 ) , this completes the proof in all cases. D =
Remark 7 .20. We observe that Theorem 7.19 applies only to regular maps. It cannot
be true for all affine automorphisms, since there are affine automorphisms whose fixed (or periodic) points include components of positive dimension. For example, the affine automorphism ¢(x, y) = (x, y + f(x)) fixes all points of the form (a, b ) satisfying f(a) = 0. Of course, this map ¢ is not regular, since one easily checks that Z(¢) Z(¢- 1 ) { [0, } =
=
1,0] .
Definition. Let ¢ V V be a morphism of a (not necessarily projective) vari ety V. A point P E Per(¢) is isolated if P is not in the closure of Pern (¢) "'- {P} for all n 2: 0. In particular, if Pern (¢) is finite for all n, then every periodic point is isolated. :
__.
Conjecture 7.21. Let ¢ : AN __. AN be an affine automorphism ofdegree at least 2 defined over Q. Then the set ofisolatedperiodic points of¢ is a set ofbounded height in AN (Q).
A classification theorem of Friedland and Milnor [ 176] says that every automor phism ¢ : A A of the affine plane is conjugate to a composition of elementary maps and Henon maps. Using this classification, Denis [131] proved Conjecture 7.21 in dimension 2. (See also [287, 288].)
2 2 __.
7.2
Primer on Algebraic Geometry
In this section we summarize basic material from algebraic geometry, primarily hav ing to do with the theory of divisors, linear equivalence, and the divisor class group (Picard group). This theory is used to describe the geometry of algebraic varieties and the geometry of the maps between them. We assume that the reader is familiar
7.2. Primer on Algebraic Geometry
403
with basic material on algebraic varieties as may be found in any standard textbook, such as [ 1 86, 1 97, 198, 205]. This section deals with geometry, so we work over an algebraically closed field. Let K = an algebraically closed field, V = a nonsingular irreducible projective variety defined over K, K(V) = the field of rational functions on V. 72 1 .
.
Divisors, Linear Equivalence, and the Picard Group
In this section we recall the theory of divisors, linear equivalence, and the divisor class group (Picard group). Definition. A prime divisor on V is an irreducible subvariety W C V of codi mension 1 . The divisor group of V, denoted by Div(V), is the free abelian group generated by the prime divisors on V. Thus Div(V) consists of all formal sums
I:: nwW, w
where the sum is over prime divisors W C V, the coefficients nw are integers, and only finitely many nw are nonzero. The support of a divisor D = I: nw W is ID I =
u
W with
w.
n w#O
If W is a prime divisor of V, then the local ring at W is the ring Ov,w = { f E K(V) : f is defined at some point of W } . It is a discrete valuation ring whose fraction field is K (V). Normalizing the valuation so that ordw ( K(V)* ) = Z, we say that ordw(f) = order of vanishing of f along W. Then f vanishes on W if ordw (f) Definition.
the divisor
2::
1, and f has a pole on W if ordw (f)
:::;
-1.
Let f E K(V)* be a nonzero rational function on V. The divisor off is
(f) = 2::::: ordw (f)W E Div(V) . w A principal divisor is a divisor of the form (f) for some f E K (V). The principal divisors form a subgroup ofDiv(V). The divisor class group (or Picard group) of V is the quotient group p· Div(V) lc ( V ) - (principal divisors) _
7. Dynamics in Dimension Greater Than One
404
Two divisors D1 , D2 E Div(V) are linearly equivalent if they differ by a principal divisor, D1 D2 + ( f ), i.e., if their difference is in the kernel of the natural map =
Div(V)
-----+
Pic(V).
We write D1 "' D2 to denote linear equivalence. The next proposition follows directly from the definitions and the fact that every nonconstant function on a projective variety V has nontrivial zeros and poles. Proposition 7.22. There is an exact sequence 1
-----+
K*
-----+
K(V)* � Div(V)
-----+
Pic(V)
-----+
0.
Remark 7.23. The exact sequence in Proposition 7.22 is analogous to the fundamen
tal exact sequence in algebraic number theory, 1
-----+
. (umts )
-----+
(multiplicative ) ( fractional) ( ideal class) group Ideals group -----+
.
-----+
-----+
1
.
Let ¢ : V V' be a morphism of nonsingular projective varieties and let W' c V' be a prime divisor such that ¢(V) is not contained in W'. Then ¢ - l (W') breaks up into a disjoint union of prime divisors, say Definition.
----*
Let f E K(V') be a uniformizer at W', i.e., ordw' (f) by ¢ is defined to be the divisor ¢* W '
=
1 . Then the pullback of W'
T
=
2:::: ordw, (f o ¢)Wi E Div(V) . i=l
More generally, if D'
=
2::: nw' W' E Div(V'), the pullback of D' is the divisor
¢ * D'
=
L' nw, ¢* (W' ), W
provided that all of the terms with nw' -=/- 0 are well-defined. Thus ¢* D' is defined if and only if ¢(V) rj_ ID'I· There is also a way to push divisors forward. Let ¢ : V V' be a morphism of nonsingular projective varieties, let W C V be a prime divisor, and let W' ¢(W). If dim W' = dim W, then the function field K(W) is a finite extension of the function field K(W') via the inclusion ¢* : K(W') ¢* ( ! ) = f 0 ¢, K(W), and we define the pushforward of W by ¢ to be the divisor Definition.
----*
=
'------+
7.2. Primer on Algebraic Geometry
405
¢* W = [K (W) : K (W')] W' E Div(V') .
If dim W ' < dim W , we define ¢. W = 0 . And in general, for an arbitrary divisor D = 2::: n w W E Div(V), the pushforward of D is ¢* D =
L nw ¢* (W).
w Example 7.24. If ¢ : V V' is a finite map, then ¢ * ¢* D' = deg(¢)D' for all D' E Div(V'). Proposition 7.25. Let ¢ : V V' be a morphism ofnonsingular projective vari ---+
---+
eties. (a) Every D' E Div(V') is linearly equivalent to a divisor D" E Div(V') satisfY ing ¢(V) � I D" I· (b) If D' and D" are linearly equivalent divisors on V' such that ¢* D' and ¢* D" are both defined, then ¢* D' and ¢* D" are linearly equivalent. (c) Using (a) and (b), the map ¢ * : { D' E Div(V') : ¢(V) � I D' I }
__,
Div(V)
extends uniquely to a homomorphism ¢* : Pic(V' )
Pic(V) . Example 7 .26. A prime divisor W of IP'N is the zero set of an irreducible homoge neous polynomial F E K[X0, . . . , XN ] . We define the degree ofW to be the degree of the polynomial F and extend this to obtain a homomorphism __,
deg (l: n w w) l: n w deg(W). w w It is not hard to see that a divisor on IP'N is principal if and only if it has degree 0, so the degree map gives an isomorphism deg : Pic(IP'N ) Any hyperplane H c IP'N is a generator of Pic(IP'N). Example 7.27. A prime divisor of!P'N x IP'M is the zero set of an irreducible bihomo geneous polynomial F E K[X0, . . , XN , Yo , . . . , YM ] · We say that F and W have bidegree ( d, e if F satisfies F(o:Xo , . . . , o:XN , ,8Yo , . . . ,8YM ) o:d ,8e F(Xo, . . . , XN , Yo, . . . , YM ) . The bidegree map can be extended linearly to give an isomorphism bideg : Pic(IP'N x IP'M ) x Let p1 : IP'N x IP'M IP'N and p2 : IP'N x IP'M IP'M be the two projections and let H1 be a hyperplane in IP'N and H2 a hyperplane in IP'M . Then Pic(IP'N x IP'M ) is generated by the divisors and =
Z.
.
)
,
---+
=
---+
Z Z.
7. Dynamics in Dimension Greater Than One
406 722 .
.
Ample Divisors and Effective Divisors
Definition. A divisor D I: nw W is said to be effective (or positive) if nw 2: 0 for all W. We write D 2: 0 to indicate that D is effective. The base locus of a divisor D, denoted by Base( D), is the intersection of the support of all of the effective divisors in the divisor class of D, =
Base(D)
=
n
lEI.
E�D E?_O
Notice that any divisor is a difference of effective divisors, D
=
L
nw W -
W with n w >O
Definition.
space
L ( -nw ) W
W with nwN ( Q) is a measure of the arithmetic complexity of P . Similarly, the degree of a finite morphism ¢ measures the geometric complexity of¢. Thus an enlightening interpretation of(7. 14) is that it translates the geometric statement "¢ has degree d" into the arithmetic statement "h (¢(P) ) is approximately equal to dh(P)." A natural way to define a height function on an arbitrary projective variety is to fix an embedding ¢ : V JP>N and define h v (P) to equal h (¢(P)) . Unfortunately, different projective embeddings yield different height functions. But letting H denote a hyperplane in JP>N , one can show that if the divisors ¢* H and 'lj;* H are linearly equivalent, then the height functions attached to ¢ : V ]p>N and 'ljJ : V JP>M differ by a bounded amount. More intrinsically, the projective embedding ¢ determines the divisor class of the very ample divisor ¢* H. This suggests assigning a height function to every divisor on V. The Weil height machine provides such a construction. It is a powerful tool that translates geometric facts described by divisor class relations into arithmetic facts described by height relations. As such, the Weil height machine is of fundamental importance in the study of arithmetic geometry and arithmetic dynamics on algebraic varieties of dimension greater than 1. '----+
'----+
'----+
Theorem 7.29. (Weil Height Machine) For every nonsingular variety V/Q there exists a map
h v : Div(V) ----+ { functions V(Q )
___,
JR},
D
f--.>
h v,v,
with thefollowing properties: (a) (Normalization) Let H c JP>N be a hyperplane and let h : JP>N (Q) ___, lR be the absolute logarithmic heightfunction on projective space defined in Section 3 . 1. Then
h(P) + 0( 1 ) (b) (Functoriality) Let ¢ : V ___, V' be a morphism ofnonsingular varieties defined over Q and let D E Div(V'). Then hrN, H (P)
h v,q, •v (P)
=
=
h v', D (¢(P) ) + 0(1)
for all P E V(Q).
7. Dynamics in Dimension Greater Than One
408
(c) (Additivity) Let D, E E Div (V) . Then hv, D + E (P)
=
hv,D(P) + hv,E (P) + 0(1)
for all P E V(Q).
(d) (Linear Equivalence) Let D, E E Div(V) with D linearly equivalent to E. Then hv,D(P)
=
hv,E(P) + 0(1)
for all P E V(Q).
(e) (Positivity) Let D E Div(V) be an effective divisor. Then hv,D(P) � 0(1)
for all P E V(Q) "- Base( D).
That is, hv, D is bounded belowfor all points not in the base locus of D. (f) (Finiteness) Let D E Div(V) be ample. Then for all constants A and B, the set
{ P E V(Q) : [Q(P) : Q] ::; A and hv, D (P) ::; B} is finite. In particular, if V is defined over a number field K and if L / K is a finite extension, then
{ P E V(L) : hv,D (P) ::; B} is a finite set.
(g) (Uniqueness) The height functions hv, D are determined, up to 0(1), by the properties of (a) normalization, (b) functoriality, and (c) additivity. (It suffices to assumefunctorialityfor projective embeddings V '-+ lP'N.) Proof See [76, Chapter 2], [205, Theorem B.3.2], or [256, Chapter 4].
D
Remark 7.30. All of the 0(1) constants appearing in the Weil height machine (The
orem 7.29) depend on the various varieties, divisors, and morphisms. The key fact is that the 0 ( 1) constants are independent of the points on the varieties. More pre cisely, Theorem 7.29 says that it is possible to choose functions hv,D, one for each smooth projective variety V and each divisor D E Div(V), such that certain prop erties hold, where those properties involve constants that depend on the particular choice of functions hv, D. In principle, one can write down particular functions hv, D and determine specific values for the associated 0 ( 1) constants, so the Weil height machine is effective. In practice, the constants often depend on making the Nullstel lensatz effective, so they tend to be rather large. Remark 7 .3 1. Many of the properties of the Weil height machine may be succinctly summarized by the statement that there is a unique homomorphism . (V) h v : PIc
such that if cp : V
'-+
-----+
{functions V ( K) lR} {bounded functions V ( K) lR} ---+
-=------"-:-::--:-::---,---.:...'c:-:-: ..., =::-..:..._ -:::-:---+
lP'N is a projective embedding, then hv,q,• H
=
h + 0(1).
7.3. The Weil Height Machine
409
lP'N be a morphism of degree d and let H E Div(lP'N) be a hyperplane. Then ¢* H "' dH, so Theorem 7.29 allows us to compute hJJ>N,H (¢(P)) hJJ>N, ¢ • H (P) + 0 ( 1 ) = hJJ>N ,dH (P) + 0 (1) = dhJJ>N,H (P) + 0 (1). This formula is Theorem 3 . 1 1 . Example 7.33. Let V be a subvariety of lP'N x lP'M, say ¢ : V lP'N x lP'M. Continuing with the notation from Examples 7.27 and 7.28, the height of a point P = [x, y] E V with respect to the divisors ¢*pi H1 and ¢*p?,H2 is given by h v,¢·p�H1 (P) = hJJ>N,H1 (Pl ¢(P) ) = h(x) , h v,¢·p�H2 (P) = hJJ>M,H2 (P2¢(P)) = h(y). Example 7.34. This example uses properties of elliptic curves; see Sections 1 .6.3 and 6.3. Let E be an elliptic curve given by a Weierstrass equation. Then the x coordinate on E, considered as a map x E lP'1 , satisfies x* ( oo ) = 2(0), so we have hE, (O) (P) = 21 hJP 1 ,(oo ) (x(P) ) + 0 ( 1 ) . Note that the height hJP1 ,( oo ) i s just the usual height on lP'1 from Theorem 7.29(a). Now let d 2 2, let [d] : E E denote the multiplication-by-d map, and let E[d] = {P E E : [d]P 0}. The map [d] is unramified and
Example 7.32. Let ¢ : lP'N
--+
=
These projections are maps of degree 2. To see this, choose a generic point a E lP'2 . Then
7.4. Dynamics on Surfaces with Noncommuting Involutions
411
P1 1 ( a) = { (a, y) E lP'2 x lP'2 : L (a, y) = Q(a, y) 0} =
consists of two points (counted with multiplicity), since it is the intersection of the line L ( a, y) = 0 and the conic Q( a, y) = 0 in lP'2 . And similarly, p2 is a map of degree 2. In general, a degree-2 map between varieties induces an involution on the do main given by switching the two sheets of the cover. In our situation the maps p 1 and pz induce involutions L 1 and Lz on SA,B · Explicitly, if P = [a, b] E SA,B, then L 1 ( P) [a, b'] is the point satisfying =
and similarly, L2 ( P) = [a', b] is the point satisfying These involutions are uniquely determined as nonidentity maps SA,B ----+ SA,B sat isfying Pl 0 L I = Pl and P2 0 L2 = P2 · We note that L I and Lz are rational maps on S, i.e., they are given by rational functions. To see why this is true, observe that b and b' are the intersection points in lP'2 of the line and the conic L (a, y) = O
and
Q(a, y) = 0.
Thus each of b and b' can be expressed as a rational function in the coordinates of the other. The following example will help make this clear, or see Exercise 7.25 for explicit formulas to compute L 1 and L2 . Example 7.36. We illustrate the involutions on SA,B using the example L (x , y) = XoYo + X1Y1 + X2 Y2 , Q( x , y) x6y6 + 4x6 YoYl - x6 yi + 7x6YlY2 + 3xoXI Y6 + 3xoXI YO Yl + XoXI Y� + xi y6 + 2xi yi + 4x i YlY2 - XoX2 Yi + 5x oX 2 YOY2 - 4X1X 2 Yi - 4x l X 2 YO Y2 - 2X � YOYl + 3x � y� . =
The point P = ( [1, 0, 0] , [0, 7, 1] ) is in S(Q). In order to compute L 1 ( P) , we substi tute the value x [1, 0, OJ into L and Q and solve for y. Thus =
L ( [1, 0, 0] , y) = Yo = 0 and Q ([1, 0, 0] , y) = Y6 + 4YoYl - Yi + 7YIY2 = 0,
so the solutions are y = [0, YI , Y2 ] , where Y1 and Y2 are the roots of the polyno mial -yr + 7YIY2 = 0. One solution is Y1 = 7, which gives the original point P, and the other solution is Y1 = 0, which gives L 1 ( P) = ([1 , 0, 0] , [0, 0, 1] ) . Next we compute L2 ( P) . To do this, we substitute y = [0, 7, 1 ] into L and Q to obtain
7. Dynamics in Dimension Greater Than One
412
L (x, [0, 7, 1]) Q (x, [0, 7, 1]) Substituting x 2
=
= =
7xi + x 2 0, xoXI - 49xo x2 + 126xi - 196xi X2 + 3x� 0. =
=
- 7XI into the second equation gives
Q([xo , XI, -7xi], [0, 7, 1] ) = 344xo xi + 1645xi = 0. The solution X I 0 gives back the original point P. The other solution is [x0, xi] [1645, -344], and then setting x2 = - 7xi = 2408 gives =
{2 (P)
=
{2 ([1, 0,
OJ, [0, 7, 1])
=
=
([1645, -344, 2408], [0, 7, 1]) .
We could continue this process, but the size of the coordinates grows very rapidly. Indeed, the y-coordinates of ( I ( (2 (P) ) are already integers with 12 to 13 digits. Remark 7.37. The surface S described by (7. 1 5) is an example of a K3 surface. Formally, a K3 surface is a surface S of Kodaira dimension 0 with the property that H I ( S, 0s) = 0. However, all of the information that we will need is contained in the explicit equations (7. 1 5) defining S. The reader desiring more information about the geometric properties of K3 surfaces might consult [40, 44, 1 78, 298]. The dynamics of K3 surfaces with nontrivial automorphisms are studied by Cantat [93] and McMullen [296]. Remark 7.38. The collection ofK3 surfaces SA,B is a 43-parameter family, since the coefficients (A, B) vary over JID8 x JID35 . However, many of the surfaces are isomor phic. For example, we can use elements ofPGL3 to change variables in each of two factors of JID2 . This reduces the dimension of the parameter space by 16, since PG L3 has dimension 8. Further, the surface SA,B really depends only on the ideal gener ated by the bilinear form L(x, y) and the biquadratic form Q(x, y), so the surface does not change if we replace Q(x, y) by Q(x, y) + L(x, y) · M(x, y) for an arbitrary bilinear form M (x, y ). The space of such M is 9-dimensional, so we see that the isomorphism classes of K3 surfaces SA,B constitute a family of dimension at most 8
.._,_,
A EIP's
+
35 - .._,_, 8 - .._,_, 8 - .._,_, 9 = 18 .
.._,_,
B EIP'35
PGL3
PGL3
M
One can prove that these are the only isomorphisms between the various SA , B , so there is an 18-parameter family of isomorphism classes of nonsingular sur faces SA,B · There are severa1 linear, quadratic, and quartic forms that come up nat urally when one is working with the surface SA,B. We define linear and quadratic forms by setting Definition.
7.4. Dynamics on Surfaces with Noncommuting Involutions
Lj (x) L¥ (Y) Q'kc (x) Q¥j (y)
= the coefficient of yj in L (x, y), = the coefficient of xi in L (x, y) , = the coefficient ofyk YC in Q (x, y) , = the coefficient of Xi Xj in Q (x, y) .
413
(7. 16)
This notation allows us to write the bilinear form L and the biquadratic form Q as L (x, y)
2
=
L Lj (x) yj
j=O
=
L2 L¥ (y) xi , i=O
k
Then for each triple of distinct indices i , j, E {0, 1, 2} we define quartic forms G% = ( Lj )2 Qfi - Lf LjQfj + ( Lf )2 Qjj , G% = ( Lj )2 Q¥i - L¥ LjQ¥j + ( L n 2 Qjj ,
HfJ = 2Lf LjQ'kk - Lf Lk,Qjk - Lj Lk,Qfk + ( Lk )2 Q 'fj , H0 = 2 L¥ LjQ%k - L¥ L%Qjk - LjL%Q¥k + ( L% )2 Q¥j ·
(7. 17)
For some choices of A and B, there may be points on the surface SA,B at which � 1 or �2 is not well-defined. The next proposition, which provides a crite rion for checking whether � 1 and �2 are defined at a point, shows how the quartic forms (7 .17) naturally appear. Proposition 7.39. Let P = [a, b] E SA,B · (a) The involution �1 is defined at P unless
G� (a) = Gf(a) = G� (a) = Hg1 (a) = Hg2 (a) = Hf2(a) = 0.
(b) The involution � is defined at P unless 2
Proof By symmetry, it is enough to prove (a). The map � 1 is defined at P = [a, b] if and only if the fiber P1 1 (a) consists of exactly two points. That fiber is the set of points [a, y] satisfying L (a, y) = Q (a, y) = 0,
so as long as these two polynomials are not zero, the y values are given by the intersection of a line and a conic in IP2 . If a line and a conic intersect properly, then they intersect in exactly two points, counted with multiplicity. Further, given one solution y = b, the coordinates of the second solution b' are rational functions of b and the coefficients of L (a, y) and Q (a, y) . Hence � 1 is a morphism1 except in the following two situations: 1 We leave for the reader to check that everything works in a neighborhood of points where the line L (a, y) = 0 is tangent to the conic Q(a, y) = 0.
7. Dynamics in Dimension Greater Than One
414 • •
L(a, y) is identically 0. L( a, y) = 0 is a line that is contained in the set where Q( a, y) 0. =
With the notation defined by (7 .16) and (7 .17), we use the bihomogeneity of Q to write
and then we eliminate the variable y0 by substituting L0y0 After some algebra, we obtain an identity of the fonn
=
L - Lfy1 - L'!jy2 .
(L0) 2 Q G'fl y i + Hf2Y1 Y2 + Gfy� + L{ Q00L + (L'Q Q01 - 2LfQ'Oo ) YI + (L'O Qo2 - 2L'fl Q'Oo ) Y2 } · Since we will be interested in studying points [x, y] satisfying L(x, y) = 0, we write this identity, and the analogous ones obtained by eliminating y1 or y2 , as congruences =
in the polynomial ring Z [Aij , Bijk£, xi, Yjl · Thus
L0(x) 2 Q(x, y) = G'!j (x)yi + Hf2 (x) YI Y2 + Gf (x)y� (mod L(x, y)), (7. 1 8) Lf (x) 2 Q(x, y) G2 (x)y5 + Hg2 (x) Yo Y2 + G0(x)y� (mod L(x, y)), (7. 19) L2(x) 2 Q(x, y) = Gf (x)y5 + Hg1 (x) Yo Yl + G0(x)yi (mod L(x, y)). (7.20) =
Suppose first that L( a, y) is identically 0. Substituting x that the quadratic fonn
=
a into (7. 1 8), we find
is identically 0. Hence G2(a) = Hf2 (a) = Gf(a) = 0. Similarly, substitut ing x = a into (7. 19) and (7.20) shows that all of the other values Gf(a) and Hij (a) are equal to 0, which completes the proof in this case. We may now suppose that L( a, y) is not identically 0. Then the assumption that L 1 is not defined at [a, b] implies that the line L(a, y) = 0 is contained in the zero set of Q( a, y ). If Lf(a) L'!j (a) = 0, then the definition (7. 1 7) of G0 shows that G0 (a) = 0. And if Lf(a) and L'!j(a) are both nonzero, then we let =
b'
= [O, L2(a), -Lf (a)]
and note that [a, b'] E Sa,b · Hence A similar argument shows that also Gf(a) = G2(a) = 0. Next we evaluate (7. 18), (7. 19), and (7.20) at x = a and use the fact that we now know that G0(a) = Gf(a) G2(a) 0. This yields =
=
7.4. Dynamics on Surfaces with Noncommuting Involutions
415
Hf2 (a) Y1Y2 = H�2 (a) Yo Y2 = H�1 (a) Yo Yl = 0 for all y = [yo, Yl , Y2 ] satisfying L (a, y) = 0. (7.21) We will prove that Hf2 (a) = 0; the others are done similarly. If there is a point on the line L(a, y) = 0 with Y1Y2 i- 0, then (7.2 1) immediately implies that H'f2 (a) = 0. So we are reduced to the cases in which the line L( a, y) = 0 is either y 1 = 0 or Y2 = 0. If it is the line y1 = 0, then L( a, y) = cy 1 for some constant c i- 0, so L0 (a) = L'!j (a) = 0, and similarly if it is the line Y2 = 0, then L0 (a) = Lf (a) = 0. In either case, the definition (7. 17) of Hf2 yields
Hf2 (a) = 2 Lf (a)L� (a)Q00 (a) - L� (a)L0 (a) Q�0 (a) - L� (a)L0 (a) Q�0 (a) + L0 (a) 2 Q�2 (a) = 0.
D
Example 7.40. We illustrate Proposition 7.39 using the surface described in Exam
ple 7.36. The polynomials G'k and Htj for this example are given in Table 7 . 1 . Propo sition 7.39 says that L 1 is defined at P = [a, b] provided that at least one of the six polynomials G0 , Gf , G'!j , H01 , H02 , H'f2 does not vanish at a. For convenience we say that a point a E lP'2 is degenerate if
Our first observation is that and Hence there are no degenerate points with a0 = 0. (We assume that K does not have characteristic 2.) Thus if there exists a degenerate point a, we can dehomogenize a and write it as a = [1 , a 1 , a2 ]. We use a tilde to indicate the dehomogenization x0 = 1 of the G'k and Htj polynomials. So for example,
Now suppose that a = (a 1 , a2 ) is a degenerate point. Then x 1 = a 1 is a common root of the polynomials
Hence if we take resultants with respect to the x2 variable, then x 1 both of the polynomials
R 1 (x2 ) = Resx, (G� (x l , x2 ) , H�1 (x 1 , x2 )) , R2 (x2 ) = Resx, (G� (x 1 , x 2 ) , H�1 (x 1 , x2 )) . Explicitly, these polynomials are
=
a 1 is a root of
7. Dynamics in Dimension Greater Than One
416
eOx = XoX 31 - 7x20 x 1 x2 - 4X31 X2 - x20 x22 + 5X21 x22 - XoX32 - 4X 1 X32 ex1 = X3o X 1 - x02 x22 + 7XoX 1 X22 + X21 X22 e2x = -x04 - 4X3o X 1 + 3XoX31 + X41 - X3o X2 - 4X2o X 1 X 2 + 2XoX 1 X22 2 22 + 4XoX 1 X22 + 4X21 X22 - 2X42 H0x1 = 2x02 x 21 - 7Xo3 X2 - 4XoX21 X 2 + 4X0X H02 = -7xgx 1 - 4xo xi - 2xgx 2 - 4x6x 1 x 2 + 6xo xix2 - 4xix 2 - 2x6x� - 8xo x 1 x� + 2x 1 x� Hf2 = 7x6 + 4x6 xi - 4xgx2 - 6x6x 1 x 2 + 10xo xix2 + 2xix2 + 2xo x� eg = - 2YoYi + 4YiY2 + Y6Y� + 4YoY1Y� + 5yi y� + 4Y1 Y� ei = -2yJ y1 + YoYiY2 - Y6Y� + 4YoY1Y� - YiY� + 7Y1Y� e� = Y6 - 3yJ y1 + 4YoYi - y{ + 4Y6Y1Y2 + 7YiY2 - YoY1Y� H/!n = -4Y6Yi + 4YoYiY2 + YiY2 + 7y6 y� + 4YoY1Y� + Yi H1{2 = 4YoYi - y{ + 2yJ y2 + Y6Y1Y2 + 6YoYiY2 + 8yoY1 Y� - Y1 Y� H{2 = -4y6 yr + YoYi - 7yJ y2 - 6y6Y1Y2 + 8yoYiY2 - 2YiY2 + 14yr y� - YoY� Table 7.1 : The polynomials GZ and H;j for the surface in Example 7 .36.
R 1 (xi) = 4xF + 1 12x i 1 + 1160xi 0 + 5112xi + 7052x� - 227lxi + 18573x� + 2160xr + 16053x{ - 7304xi + 1045xi - 49x 1 , R2 (x l ) 4xi 6 + 80xi5 + 600xi 4 + 2064xi 3 + 2548xF - 3616xi 1 - 14216xi 0 - 10892xi + 9856x� + 21708xi + 15648x� + 1000xf - 13986x{ - 10462xi - 3124xi - 412x 1 - 18, and the assumption that R 1 and R2 have a common root implies that their resultant =
must vanish. However, when we compute it, we find that
Res(R1 , R2 ) = 198929 . . . 3830147072 1.99 1087. �
·
Hence � 1 is defined at every point of SA,B unless the characteristic of K divides this (large) nonzero integer Res ( R 1 , R2 ). We can use other resultants to reduce the list of possible bad characteristics. For example, let R0(x 2 ) = Resx 1 ( G0, H01 ). Then �1 is everywhere defined unless both Res(R0, R2 ) and Res(R1 , R2 ) vanish. We compute
gcd (Res( R0, R2 ), Res( R 1 , R2 )) = 439853213743020234882809856 1 = 2 7 . 3 6 . 317 . 14521485737273461, (7.22)
7.4. Dynamics on Surfaces with Noncommuting Involutions
417
which proves that L1 is everywhere defined unless p is one of the four primes appear ing in (7.22). We sketch a similar calculation for t2 • Let To ( Y2) T1 (Y2) T2 ( Y2)
Then Res(T0, T2)
�
= = =
Resy, ( G5 (Yl , Y2), Hlfl ( Yl , Y2)) , Resy, (Gi (Yl , Y2), Hlf1 (Yl , Y2)) , Resy, (G� ( Yl , Y2), Hlf1 (Y l , Y2)) .
2.57 1 097 and Res (T1 , T2) 2.75 1 0 1 1 4, and ·
�
gcd (Res(T0 , T2) , Res (T1 , T2))
=
·
2 1 6 507593 2895545793631. ·
·
Hence L2 is everywhere defined unless the characteristic p of K is one of the three primes appearing in this factorization. A map p V W of degree 2 between varieties always induces an involu tion L V V, but in general L is only a rational map, it need not be a morphism. This distinction is quite important. For example, height functions transform well for morphisms, but not for rational maps. We now show that for most choices of (A , B), the involutions on SA,B are morphisms. :
:
--->
--->
Proposition 7.41. There is a proper Zariski closed set z if(A , B) � Z, then the involutions
c
lP'8
X
lP'35 such that
are morphisms. Proof According to Proposition 7.39, the involution t1 is defined on all of SA , B
provided that the system of equations
(7.23) has no solutions in lP'2 . A general result from elimination theory (see [198, 1.5.7A]) says that there are polynomials h , . . , fr in the coefficients of .
such that the equations (7.23) have a solution if and only if h fr 0 . The coefficients of G0 , . , H'[2 are themselves polynomials in the coefficients of L(x, y) and Q(x, y), so we can write each fi as a polynomial in the variables A and B. There are then two possibilities: = · · · =
=
. .
•
•
The set of fi consists only of the zero polynomial, and hence every sur face SA,B has some point at which L 1 is not defined. The set of fi contains at least one nonzero polynomial fl , and then t 1 is well defined on SA,B provided that fl (A , B) # 0 .
7. Dynamics in Dimension Greater Than One
418
In order to eliminate the first case, it suffices to write down a single surface SA,B for which the system of equations has no solutions in JP'2 • We gave such a surface in Example 7.40, at least provided that the characteristic of K is not equal to 2, 3, 317, or 14521485737273461. Similarly, the above argument and Example 7.40 show that L2 is defined on a Zariski open set of (A, B) as long as the characteristic of K is not equal to 2, 507593, or 2895545793631. This completes the proof of Propo sition 7.41 except for fields having one of these six characteristics. We leave as an exercise for the reader to find other examples to cover the remaining cases. D Remark 7.42. See Exercise 7.28 for a less computational proof of Proposition 7.4 1 . 742 .
.
Divisors and Involutions o n SA B ,
In this section we study how the involutions L 1 and L2 act on divisors on SA,B. Later we use this information to study how iterates of the involutions act on points. This prompts the following definitions. Definition. Let L 1 and L2 be the involutions of the surface SA,B defined by (7 . 1 5). These involutions generate a subgroup (possibly all) of Aut(SA,B) · We denote this subgroup by A. Then for any point P E SA,B, the A-orbit of P is the set
A(P)
=
{7,b(P) : 7,b E A} .
Let H E Div(JP'2 ) be a line. As described in Example 7.28, pulling back using the two projections gives divisors on JP'2 x JP'2 , and and the Picard group oflP'2 x JP'2 is isomorphic to ';f} via (By abuse of notation, we write n 1 H1 + n2 H2 for its divisor class.) We note that n 1 H1 + n2 H2 is a very ample divisor on JP'2 x JP'2 if and only if both n 1 and n2 are positive. Next we define two divisors D1 , D2 E Div(SA,B) using the two projections of SA,B to lP'2 , and Wehler [448] has proven that the Picard group of a general surface SA,B satisfies Pic(SA,B) � ';!} and that D 1 and D2 are generators,2 but for our purposes it will not matter if Pic(SA,B) is larger than Z2 , we will simply use the part of Pic(SA,B) generated by D 1 and D2 . We now compute the action of L1 and L2 on D1 and D2 .
2 What this means is that the set of coefficients (A, B) E JP'8 x JP'35 for which Pic(SA,B) is strictly larger than :Z2 forms a countable union of proper Zariski closed subsets of JP'8 x JP'35. This implies that most (A, B) in JP'8(C) x JP'35(C) have Picard group :Z2 , but it does not directly imply that there are any such values in IP'8(1Qi) x JP'35(1Qi), since IQI is countable.
7.4. Dynamics on Surfaces with Noncommuting Involutions
419
Proposition 7.43. Let D 1 = p � H and D2 = p2H. The involutions � 1 and �2 act on the subspace of P i c(SA,B) generated by D 1 and D2 according to the following rules:
��D 1 = D 1 , � i D2 4D l - D2 , =
�; D 1 = -D 1 + 4D2 , �;n2 = D2 .
(7.24) (7.25)
Proof The involution � 1 switches the sheets of the projection p 1 , so it is clear that P1 � 1 = Pl · This allows us to compute o
This proves the first formula in (7.24). Next we observe that for any P E SA,B, the two points in the set p1 1 (p 1 ( P)) are P and � 1 (P). Thus if we start with a divisor on SA,B, use Pl to push it down to lP'2 , and then use p1 to pull it back to SA,B, we get back the original divisor plus its translation by �I · In other words, Using this formula with D = D2 allows us to compute (7.26)
The divisor P I *P2 H on lP'2 is linearly equivalent to some multiple of H. For simplic ity, let H be the line y2 = 0. Then p2H is the curve in lP'2 x lP'2 (lying on SA,B) given by the equations
Y2 = 0 . We solve the linear equation L = 0 to express y0 and YI as linear functions of [x0, X I , x2 ], and then substituting into the quadratic equation Q 0 yields a homogeneous equation of degree 4 in xo, X I , x2 . So when we use P h to push p2H down to lP'2 , we get a curve of degree 4 in lP'2 . Hence =
where this is an equality in Pic(lP'2 ). Substituting into (7.26) yields the second for mula in (7.24). By symmetry, or by repeating the above argument, the two formulas in (7.25) are also true. D Remark 7 .44. It is not hard to prove that the only relations satisfied by compositions of �1 and �2 are �r = 1 and �§ = 1. In other words, A is isomorphic to the free
product of the groups of order 2 generated by �I and �2 . An alternative description is that A is isomorphic to an infinite dihedral group. See Exercise 7 .3 1 .
7. Dynamics in Dimension Greater Than One
420 7.4.3
Height Functions on SA ,B
In this section we use the Weil height machine to translate the divisor relations and transformation formulas from Proposition 7.43 into relations among height func tions. We recall from Example 7.33 that the height functions associated to the divi sors D 1 and D2 are given at a point P = [x , y] E SA,B by h v , (P) = hp r H (P) h H (P1 P) = h(x) , h v2 (P) = hp�H (P) = hH (P2P) = h(y). =
(7.27)
Proposition 7.45. Assume that SA,B is defined over a numberfield K. Let
a = 2 + V3 and define functions h + , h- : SA,B (
h + ([x , yl) = -h(x) + ah(y)
K)
---+
JR. by theformulas
h - ([x, yl ) = ah(x) - h(y) .
and
Then h+ and h- transform according to thefollowing rules:
h - o [ 1 = a- 1 h + + h- [2 = ah+ +
h + o [ 1 = ah- + h + o [ a- 1 h - +
2
0(1), 0(1),
=
0(1), 0(1).
o
Remark 7.46. Before starting the proof of Proposition 7.45, we pause to explain
why the number a and the functions h+ and h- arise naturally. Consider a two dimensional real vector space V with basis elements D1 and D2, where we view V as a subspace of Pic(SA,B) 0 R Then the formulas (7.24) and (7.25) in Proposi tion 7.43 tell us how i and 2 act on V. In terms of the given basis, they are linear transformations that act via the matrices [
[
( 1 -41 )
[1 = 0 *
and
[2 *
=
( -14 01) .
We now look for a new basis { E1 , E2 } for V with the property that [i and [2 inter change the basis. More precisely, we ask that for some constants a, b, c, d. This problem can be solved directly, but it is easier to observe that [2[i E1 = adE1 and [2[i E2 bcE2 . Thus E1 and E2 must be eigenvectors for the linear transformation [2[i, whose matrix is =
( 1 -14 ) (-14 01) = ( 15 -14 ) .
[2 [ 1 = 0 * *
It is easy to check that
-4
and
7.4. Dynamics on Surfaces with Noncommuting Involutions
421
are a pair of independent eigenvectors with eigenvalues a 2 and a- 2 respectively. This explains the appearance of a, and then one checks that these eigenvectors satisfY L2*E1 = a -lE2 ,
(7.28)
It is then natural to define height functions h+ and h- corresponding to the divi sors E1 = -D1 + aD2 and E2 = aD 1 - D2 , since the divisor relations (7.28) and the Weil height machine should then yield corresponding relations for the height functions. Proof ofProposition 7 .45. Having given the motivation, we commence the proof of
Proposition 7.45, which is a formal calculation using the additivity and functorial ity of height functions (Theorem 7 .29(b,c)) and the transformation formulas (7 .24) and (7.25) in Proposition 7.43. Note that h(x) = hD, ([x, yl) and h(y) = hD2 ([x, yl)
from (7.27). We compute h + L1 = -hD, L 1 + ahD2 L1 = -h,�D1 + ah,� D2 + 0(1) = - hD, + ah4D1 - D2 + 0(1) ( -1 + 4a)h D1 - ahD2 + 0(1) = a2 hD, - ahD2 + 0(1) = ah- + 0(1). 0
0
0
=
by definition of h +, from Theorem 7.29(b), from Proposition 7.43, from Theorem 7.29(c), since a 2 = 4a - 1,
Similarly h + o L2 = -hD, L2 + ah D2 L2 = -h,� D 1 + ah ,� D2 = -h - D, +4D 2 + ahD2 = hD, + ( -4 + a)hD2 = hD, - a- 1 hD2 + 0(1) = a- l h- + 0(1). o
o
by definition of h-, from Theorem 7.29(b), from Proposition 7.43, from Theorem 7.29(c), since a 2 = 4a - 1,
This proves the transformation formulas for h+ . The proof for h- is similar and is D left for the reader. We can use Proposition 7.45 and the general theory of canonical heights (Theo rem 3.20) to construct two heights on SA,B that are canonical with respect to both L 1 and L2 . Proposition 7.47. Let SA,B be defined over a number field K. There exist unique functions
7. Dynamics in Dimension Greater Than One
422
and satisfYing both the normalization conditions and
h,- = ahD1 - hD2 + 0(1)
(7.29)
and the canonical transformation formulas
h,- o �1 = a- 1 h,+ ,
h,+ o �1 = ah- , h+ o �2 = a - 1 h- ,
h,- o �2 = ah+ .
(7.30)
Proof Let ¢ = � 1 o �2 be the composition of the two involutions on SA,B and let
and be the functions defined in Proposition 7.45. Then the transformation formulas in Proposition 7.45 allow us to compute 2 h+ o ¢ = h + o �1 o �2 + 0(1) = ah - o �2 + 0(1) = a h + + 0(1) . The constant a 2 satisfies a 2 13.93 > 1, so we may apply Theorem 3.20 to the functions ¢ and h+ to deduce the existence of a unique function h+ satisfying �
h,+ = h + + 0(1). and Repeating this construction with ¢- 1 = �2 o �1 , we find that h- o ¢- 1 = h - o �2 o �1 + 0(1) = ah+ o �1 + 0(1) = a2h- + 0(1). Applying Theorem 3.20 to the functions ¢- 1 and h-, we find that there is a func tion h- satisfying
and
h,- = h- + 0(1).
The functions h,+ and h,- that we have just constructed satisfy (7.29). In order to check the transformation formulas (7.30), we first note that h,+ o �1 = h+ o �1 + 0(1) = ah- + 0(1) = ah- + 0(1). In order to get rid of the 0(1 ), we compose both sides with ¢- n and use the formula
to compute anh,+ 0 �1 = h,+ o cpn o �1 = h,+ 0 �1 o cp-n = ah- o ¢-n + 0(1) = an+lh,- + 0(1).
Divide both sides by an and let n ____, oo to obtain the desired result h,+ o �1 = ah- . This proves the first of the transformation formulas (7.30). The others are proven similarly.
7.4. Dynamics on Surfaces with Noncommuting Involutions
423
Finally, in order to prove uniqueness, suppose that g+ and g- are functions sat isfying (7.29) and (7.30). Then and similarly g- o cjJ- 1 = a2 g-. Hence g + and g- have the same canonical proper ties as h,+ and h,- , so the uniqueness assertion in Theorem 3.20 tells us that g+ = h,+ D and §- = h,-. Remark 7 .48. In practice, it is infeasible to compute the canonical heights h + and h to more than a few decimal places using their definition as a limit. As with the other canonical heights studied in Sections 3.4, 3.5, and 5 .9, it is possible to decompose h,+ and h- as sums oflocal heights that may then be computed using rapidly convergent series (cf. Exercise 5 .29). See [89] for details. 744 .
Properties and Applications of Canonical Heights
.
The next proposition describes various useful properties of the canonical height functions h+ and h- and their sum. As an application, we prove that there are only finitely many K-rational points with finite A-orbit. This is the analogue for the K3 surfaces SA,B of Northcott's Theorem 3 . 1 2 on preperiodic points of mor phisms on lP'N and of Theorem 7.19 on periodic points of regular affine automor phisms. Proposition 7.49. Let SA,B be defined over a numberfield K, let h, + and h,- be the canonical heightfunctions constructed in Proposition 7.47, and let
(a) The set
{ P E SA,a(K) : h(P) :::; C}
isfinite. (N.B. This is not true ifwe replace h by either ofthe heights h + and h-, see Exercise 7.35.) (b) Let P E SA,a(K). Then
h,+ (P) = 0
-¢===>
h,- (P) = 0
-¢===>
h(P) = 0
-¢===>
P hasfinite A-orbit.
(c) There are onlyfinitely many points P E SA,B ( K) with finite A-orbit. Proof (a) Using the properties of h, + and h,-, we find that
h = h+ + h-
( -hv1 + ahv2 ) + (ahv1 - hv2) + 0(1) = (a - l) (hv1 + hv2) + 0(1). =
As noted earlier, the heights hv 1 and hv2 are given by
by definition of h, from Proposition 7.47,
7. Dynamics in Dimension Greater Than One
424
hD 1 ([x, yl)
=
h(x)
and
hD2 ([x, yl)
=
h( y) ,
where h(x) and h(y) are the standard heights ofx and y in J!D2 . Hence h ( [x, yl)
=
(a - 1) ( h(x) + h( y)) + 0 (1),
so if h ( [x, yl) is bounded, then both h(x) and h(y) are bounded. (Note that a sat isfies a > 1, which is crucial for the argument to work.) This completes the proof of (a), since Theorem 3.7 tells us that lP'2 (K) contains only finitely many points of bounded height. (b) Since h = h+ + h- and both h+ and h- are nonnegative, it is clear that
h(P) = 0 ====;. ft+ (P) ft- (P) = 0. Suppose next that ft+ (P) = 0. Let ¢ = �1 �2 as usual. Then h(¢n (P)) = ft+ (¢n (P)) + h - (¢n (P)) = a2n ft + (P) + a - 2n h - (P) = a - 2n h- (P). =
o
The righthand side is bounded (indeed, it goes to 0) as n ---t oo, so we see that {¢n (P) n 2 0} is a set of bounded h-height. It follows from (a) that it is a finite set. Since ¢ is an automorphism, we deduce that P is periodic for ¢. We now perform a similar calculation using ¢ - n , :
h (¢-n (P))
=
h+ (¢-n(P)) + h- (¢-n(P))
a - 2n ft+ (P) + a2n h - (P) a2n h- (P). The lefthand side is bounded, since P is periodic for ¢, so letting n ---t oo implies that ft- (P) 0. This proves that h + ( P) 0 implies h- ( P) = 0, and a similar argument gives =
=
=
=
the reverse implication, which completes the proof that
In order to study A-orbits of points, we make further use of the formula Suppose first that P has finite A-orbit. Then h ( ¢n (P)) is bounded, since it takes on only finitely many values. Letting n ---t oo in (7 .3 1) and using the fact that a > 1, we deduce that h+ (P) = 0. Finally, suppose that h(P) 0. Then ft+ (P) ft- (P) = 0, so (7.3 1 ) tells us that h(¢n (P)) = 0 for all n E Z. In particular, { ¢n (P) n E Z} is a set of bounded h-height, so (a) tells us that it is a finite set. But the A-orbit of P is equal to =
=
:
7.4. Dynamics on Surfaces with Noncommuting Involutions
425
0
so A(P) is also finite.
The canonical height functions on SA,B can also be used to count the number of points of bounded height in an A-orbit, as in our next result. See Exercise 7.2 1 for an analogous (conditional) estimate for regular affine automorphisms of JP'N . Proposition 7 .50. Let SA,B be defined over a numberfield K, andfor anypoint P [x , y] E SA,s (K), let h(P) be the heightfunction
h(P) Also let a let
=
=
hv, + D2 (P)
=
=
h (x) + h (y) .
2 + V3 as usual. Fix a point Q E SA,B ( K) with infinite A-orbit and p,(Q)
=
#{ 7jJ E A : 7/J(Q)
be the order of the stabilizer ofQ. Then
#{ P E A(Q) : h (P)
:S;
B}
=
1
loga JL (Q)
=
Q}
(,
B2 , h + ( Q )h- ( Q )
)
+ 0(1 ) as B --+ oo,
where the 0(1 ) constant is independent of both B and Q.
The key to proving Proposition 7.50 is the following elementary counting lemma. Lemma 7.51. Let a, b
>
0 and u > 1 be real numbers. Then as t --+ oo,
where the 0(1) constant depends only on u. Proof We start by writing the real number logu ( jFJ/a ) as the sum of an integer
and a fractional part,
logu
.
y{b� = m +
r
.
wtth m E Z and l r l
::;
1
2·
(The reason that we do this is because the function aux + bu - x has a minimum at x = logu ( jFJ/a ) ) Then replacing n by n + m in the expression aun + bu-n yields Hence It thus suffices to prove that if c, d E lR are both between u - l 1 2 and u 1 1 2 , then
7. Dynamics in Dimension Greater Than One
426
# { n E Z : cun + du - n :::; t} = log(t2 ) + 0(1)
as t ---> oo .
(7.32)
We note that if n � 0 , then and similarly if n :::; 0 , then
logu(cun + du - n ) = -n + logu(cu2n + d) = -n + 0(1). Here the 0(1) bounds depend only on u, since by assumption c and d are bounded in terms of u. Therefore #{ n E Z : cun + du- n :::; t} = # { n E Z : logu(cun + du - n ) :::; logu(t) } = #{n E Z : lnl + 0(1) :::; logu(t)} = 2 logu(t) + 0(1). This is the desired inequality (7.32), which completes the proof of Lemma 7.5 1 . D ProofofProposition 7.5 0. We do the case that J-l( Q ) = 1 and leave the similar case J-l( Q) = 2 to the reader. (It is easy to check that J-l( Q ) :::; 2; see Exercise 7.29.) Let ¢ = �1 o �2 · Every element of A is given uniquely as an alternating compo
sition of � 1 's and �2 's, so A splits up as a disjoint union
Our assumption that J-l( Q) = then implies that the A-orbit of Q is a disjoint union
1
(7.33)
Let h,+ and h,- be the canonical height functions constructed in Proposition 7.47 and let h = h,+ + h,-. We note that Proposition 7.47 tells us that h,+ o ¢ = ah + and h- o ¢ = a - 1 h-. This allows us to compute #{P E O¢(Q) : h(P) :::; B } since J-l(Q) = 1, = # { n E Z : h(¢n Q) :::; B } definition of h, = # { n E Z : h,+ (¢n Q) + h,- (¢n Q) :::; B } = #{ n E Z : a2n h,+ ( Q) + a- 2n h- ( Q) :::; B} from Proposition 7.47,
� � log"
(
h+ (
Q��- (Q) )
H J(i)
from Lemma 7.5 1 .
Further, i f we replace Q with � 1 ( Q), then we get exactly the same estimate, since Hence using the decomposition (7.33), we find that
Exercises
427
#{ P E A ( Q) : h(P) :::; B } = #{ P E 0 ( Q) : h(P) :::; B } + #{ P E O( t1Q ) : h(P) :::; B} � log.
(
/,+ (
��- (Q) )
+
0(1).
Finally, in order to replace the canonical height h with the naive height h , we note that
h = h+ + h - = (ahD1 - hD2 ) + ( -hD1 + ahD2 ) + 0(1) = (a - 1) (hD 1
+
hDJ + 0(1) = (a - l)h + 0(1).
Thus
#{ P E A ( Q) : h( P) :::; B} = #{ P E A(Q) : h(P) :::; (a - 1 )B + 0(1) } , and replacing B with (a - 1 ) B affects only the 0(1) , since a > 1 .
D
Exercises Section 7 . 1 . Dynamics of Rational Maps on Projective Space 7.1. Let a, b, c, d, e E IC and let ¢ : A 3
-->
A 3 be given by
(7.34) cj; (x, y, z) = (ax + by2 + (cx2 + dz)2, ey + (ax + by2)2, dz + cx2) . (a) Prove that ¢ is invertible if and only if ade =I= 0. (b) Prove that ¢ is a regular automorphism if and only if abcde =I= 0. (c) Clearly (0, 0, 0) is a fixed point of ¢. Let b - 1 d = 1 and e = 1 - t3 . Prove that (0, t, t) is also a fixed point of¢. Hence there are infinitely many maps ¢ E Q[x, y, z] of the form (7.34) such that Fix(¢) n A a (Q) contains at least two points. (d) Let b = - 1 and d = 1 . Find all of the (complex) fixed points of ¢. If a, b, c, d, e E K , =
,
,
describe the field K (Fix(¢)) . What are its possible Galois groups over K? (Hint. It is easier to do the computations if you set e = 1 - t3 .) (e) Suppose that ¢ E JR [x, y, z] and that d = 1 and b > 0. Prove that ¢ has only one real fixed point, i.e., show that Fix(¢) n A3 (JR) = { (0, 0, 0)}. In particular, ¢ has only one rational fixed point.
7.2. Let ¢ : A3 --> A3 be the map
¢(x, y, z) = (x2z, xy, yz). (a) Calculate the indeterminacy locus of ¢. (b) What are the values of .
.
1Ill I llf (x,y,z ) E Z3 h(x,y,z)�oo
h(¢(x, y, z)) h(x, y, ) Z
and
. sup hm 3
(x,y,z) EZ h(x,y,z) � oo
h(¢(x, y, z)) h(x, y, z) ?
Exercises
428 (c) Same question as (b), but with the points (x, y, z ) E
Z3 restricted to satisfy xyz -!= 0.
7.3. Let ¢ : AN --+ A N be a regular affine automorphism and let n 2: 1 . Prove that Pern ( ¢) is a discrete subset of AN (C), and that counted with appropriate multiplicities,
(This is Theorem 7.10(c). Hint. Rewrite ,pNn (P) = P as ¢f2n (P) ,p -( N-f 2 ) n (P) , show that the homogenizations of ¢f2n and ,p < N - f2 ) n have the same degree, and use Bezout's the orem to count the number of solutions.) =
7.4. Let ¢ : A3 --+ A3 be the map ¢ ( x , y , z ) = ( y, z , x2 ) . (a) Find an explicit expression for ,pn (x , y , z ) . (There may be more than one case.) (b) Calculate the dynamical degree of ¢,
dyndeg(¢) = lim deg(¢n )l / n . n �oo (See Remark 7.14 for a discussion of the dynamical degree.) (c) Let dn = deg(¢n ). Compute the generating function l: > o dn Tn and prove that it is nin Q(T). (d) Prove that PrePer(¢) c { P E A3(QJ) : h(P) 0}. (e) Let li dyndeg(¢) and P E A3(QJ). Find real numbers b > a > 0 such that =
=
for all (sufficiently large) integers n. (f) With notation as in (d), if P � PrePer( ¢), prove that
7 .5. Let ¢
:
IP'3 --+ IP'3 be the rational map
(a) Prove that ¢ is a birational map, i.e., find a rational map 'ljJ so that ¢ o 'ljJ and 'ljJ o ¢ are the identity map at all points where they are defined. (b) Compute Z ( ¢) and Z ( ¢ - l ) . Where do they intersect? (c) * Let dn deg( ¢n ). Prove that the generating function l: > o dn Tn is not in Q(T). n The map in this exercise and the map in the previous exercise are examples of monomial maps, see [ 1 99]. =
7.6. Let u, a 1 , . . . , aN , h , . . . , bN E
QJ with u -!= 0. We proved in Lemma 7.17 that
Prove that this inequality need not be true if u
=
0.
429
Exercises
7.7. This exercise generalizes Theorem 7.15. Let (/;! , . . . , rPt : A N ---> A N be affine automor phisms with the property that
Z (¢ ! ) n Z (¢2 ) n
· ·
·
n Z(¢t ) = 0.
Let di = deg(rPi) for 1 ::::; i ::::; t. Prove that there is a constant C = C ( ¢ 1 , . . . , rPt ) so that for all P E A N (Q), 1 1 h (¢ 1 ( P ) ) + h ( ¢2 ( P) ) + · · · + d1 h ( ¢t ( P) ) 2: d2 d1 t
-
-
h ( P ) - C.
7.8. Let ¢1 , ¢2 : A2 ---> A 2 be the maps
and (a) Prove that ¢ 1 and ¢2 are jointly regular. (b) Let a a positive integer and P = (0, a ) E A2 (1Ql). Prove that
This proves that the lower bound in Theorem 7. 1 5 cannot be improved in general for jointly regular affine morphisms. (c) ** Can the lower bound in Theorem 7. 1 5 be improved for jointly regular affine auto morphisms, i.e., if we add the requirement that ¢ 1 and ¢2 be invertible, although not necessarily inverses of one another? 7.9. Let ¢ : A N ---> AN be an affine automorphism (not necessarily regular) and let d 1 deg ¢ and d2 = deg ¢ - 1 . Prove that
.
mm
{
h (¢ ( P) ) h ( ¢- 1 (P) ) d1
'
d2
}
> h ( P)
- d1d2
+
0( 1 )
"'or a!I P E ' N (rn,). 1'
""'
""'
Exercises on Integrability and Reversibility
The notions of integrability and reversibility play an important role in classical real and com plex dynamics. Their algebraic analogues lead to dynamical systems with interesting arith metic properties, which we explore in Exercises 7. 1 0-7 . 14. An affine automorphism ¢ : A N ---> A N is said to be algebraically reversible if there is linear transformation g E GL N satisfying Definition.
l = 1,
det(g) = - 1 ,
and
The terminology is meant to reflect the idea that conjugation by the involution g has the effect of reversing the flow of the map ¢. 7.10. Assume that ¢ is reversible, say ¢9 = ¢ - 1 . Let '/ = ¢ o g. Prove that '/2 is the identity map. Thus g and '/ are both involutions, so a reversible map can always be written as a composition ¢ = '/ o g of two, generally noncommuting, involutions.
Exercises
430 7.11. Let a "f. 0, let f (y ) be a polynomial of degree d � 2, and let
¢(x , y) = (y , ax + f(y) )
be the associated Henon map. Suppose that ¢ is reversible. Prove that ¢ and its reversing involution g E GL have one of the following forms: 2 (a) a = 1, g(x , y) = (y , x) . (b) a = 1, f satisfies f( - y) = f(y). g(x , y) = ( -y, -x), f satisfies f( - y) = -f(y). (c) a = - 1 , g(x , y) = ( - y , -x), 7.12. The real and complex dynamics of reversible maps are in some ways less chaotic than nonreversible maps. Similarly, reversibility (and integrability) appear to have a significant effect on arithmetic dynamics. For an affine automorphism ¢ : A N -> AN , we let
Cp ( ¢) = number of distinct orbits of ¢ in A2 (IFP ) . For each of the following Henon maps, compute Cp(¢) for all primes 2 < p < 100 (or further) and make a graph of p versus Cp ( ¢): (a) ¢ (x , y) = (y , x + y2 ). (b) ¢ (x , y) = (y , 2x + y2 ). (c) ¢ (x , y ) = (y , -x + y3). Do you see a difference in behavior? Try plotting the ratio Cp ( ¢) /p . (Notice that Exercise 7. 1 1 says that the maps in (a) and (c) are reversible, while the map in (b) is not reversible.) Definition. Let ¢ : A N -> A N be a rational automorphism, by which we mean that ¢ is a rational map (but not necessarily a morphism) and that there is an inverse rational map ¢ - 1 : AN -> A N such that ¢ o ¢ - 1 is the identity map wherever it is defined. The map ¢ is said to be algebraically integrable if there is a nonconstant rational function I : A N -> A1 satisfYing I o ¢ = I. 7.13. Let ¢ : A2 -> A2 be the rational map ¢ (x , y) =
(
y , -x -
:: )
y
1 1
.
(a) Prove that ¢ is a rational automorphism. (b) Let ¢ and ¢- 1 be the extensions of ¢ and ¢- 1 to maps lP'2 -> lP'2 . Compute Z ( ¢) and Z ( ¢- 1 ), the sets of point(s) where ¢ and ¢- 1 are not defined, and verifY that Z ( ¢) n Z (¢ - 1 ) = 0. (c) Prove that ¢ is integrable by the function I ( x , y) = x2 y + xy2 + x 2 + y2 + x + y.
In other words, verifY that ! o ¢(x , y) = I ( x , y). (d) Prove that for all but finitely many values of c E C, the level curve I (x , y) = c is an elliptic curve. Find the exceptional values of c for which the level curve is singular. 7.14. This exercise generalizes Exercise 7.13. Let a, b, c, d, e E K and define a rational map ¢ : P2 -> lP'2 (using dehomogenized coordinates on A 2) by ¢ (x , y) =
(
y , -x -
Prove that ¢ is integrable by the function I (x , y) = ax2 y2 + b ( x2 y + xy2 )
by2 + dy + e ay2 + by + c
)
.
+ c (x2 + y2 ) + dxy + e (x + y) .
Exercises
431
7.15. Assume that ¢ is integrable by the function I. For each c E K, the set J(x) = c is called a level set of ¢. (a) Prove that ¢ maps each level set to itself. Thus the dynamics of ¢ may be studied by investigating the behavior of the iterates of ¢ on the lower-dimensional invariant level sets that give a foliation oflP'N . (b) * Let N = 2 and assume that ¢ has infinite order, i.e., no iterate of¢ is the identity map. Prove that the level sets of ¢ are curves of genus 0 or 1 .
The following result will be helpful in doing Exercise 7 . 1 6.
7.52. ([172, Proposition 4.2]) Let ¢ : lP'N --+ lP'N be a morphism ofdegree d ;::: 2 and let V C lP'N be a completely invariant hypersurface, i.e., ¢ - 1 ( V ) = V = ¢ ( V) . Then V Theorem
has at most N
+ 1 irreducible components.
7.16. Let ¢ = [¢0, . . . , ¢ N ] be a morphism ¢ : lP'N --+ lP'N of degree d given by homogeneous polynomials ¢i E C[Xo , . . . , XN ] . We say that such a map is a polynomial map if its last coordinate function is equal to X'/v. Equivalently, ¢ is a polynomial map if the inverse image of the hyperplane H = {XN = 0} is simply the hyperplane H with multiplicity d. (a) Assume that ¢ is a morphism and suppose that there is an n ;::: 1 such that the iterate ¢n is a polynomial map. Prove that ¢n is already a polynomial map for some n ::=:; N + 1. This generalizes Theorem 1 .7. (Hint. Use Theorem 7.52.) (b) Show that (a) need not be true if we assume only that the map ¢ : lP'N --+ lP' N is a rational map of degree d. More precisely, prove that for all d ;::: 2 and all n ;::: 2, there exists a finite rational map ¢ : lP'N --+ lP' N of degree d such that ¢n is a polynomial map, but ¢i is not a polynomial map for all 1 ::=:; i < n. Exercises on Canonical Heights for Regular Affine Automorphisms
Exercises 7. 1 7-7.22 describe Kawaguchi's construction [230] of canonical heights for regular affine automorphisms assuming the validity of Conjecture 7 . 1 8, which is presently known only in dimension 2 [230, 413]. Let ¢ : A N --+ AN be a regular affine automorphism of degree at least 2 defined over Q and let d 1 = deg(¢)
and
We assume that Conjecture 7. 1 8 is true, i.e., we assume that there is a constant C = C ( ¢) ;::: 0 such that for all P E AN (Q),
11 h(¢(P) ) + 12 h(¢- 1 (P) ) ;::: ( 1 + d11dJ h(P) - C. (7.35) For any point P E AN (Q), Kawaguchi defines canonical heightfunctions by the formulas (7.36) sup �h(¢- n (P) ) , h+ (P) lim--+sooup d�h(¢n (P) ) , h- (P) = lim--+CXJ n d2 n (7.37) h(P) = h+(P) + h- (P). 7.17. Assuming that (7.35) is true, prove that the canonical height functions h + , h- , and h Assumption :
=
1
defined by (7.36) and (7.37) have the following properties: (a) h,+(P) ::::; h(P) + 0(1) and h,- (P) ::::; h(P) + 0(1). (b)
h(P) + 0(1) ::::; h(P) ::::; 2h(P) + 0(1) .
Exercises
432 (c) (d)
h,+(P) :2': 0 h,+(P) 0
and
=
h, - (P) 2 0 and h(P) :2': 0. h, - (P) 0 ¢==;> h ( P) 0 =
P E Per( ¢).
=
(Hint. Before proving (d), you may find it advantageous to do the next exercise.)
7.18. Assuming that (7.35) is true, prove that the canonical height functions satisfy the fol lowing transformation formulas:
(7 .38) (7.39) 7.19. Suppose that
and are two functions satisfying (7.39), and suppose further that
h' h" + 0 ( 1 ). =
Prove that h' =
h".
7.20. Let ¢ : AN --> A N be a regular affine automorphism satisfying (7.35) and let P E AN ( Q) be a wandering point for ¢, i.e., P is not a periodic point. Prove that
Hence the constant 1 + larger constant.
1 d 1 d2
appearing in the inequality (7.35) cannot be replaced by any
7.21. Let ¢ : A N --> AN be a regular affine automorphism satisfying (7.35) and let P E AN {Q) be a wandering point for ¢. We define the (two-sided) orbit-counting function of P to be
Prove that N¢ ,
p (T)
=
( log
1 1 -- + __ d1 log d2
) log T - ( loglogh+ (P) d1
+
log h - (P) log d2
)
+ 0( 1) ,
where the 0 ( 1 ) constant depends only on the map ¢ and is independent of both the point P and the number T. 7.22. Let ¢ : AN --> A N be a regular affine automorphism satisfying (7.35). Define sequences (An) and (En) by the formulas
and Prove that
Exercises
433
(Hint. Verify that An and En satisfy the linear recurrences Ao = 0, d1 A; - (1 + d1d2) A;- 1 + d2 A;- 2 = 0, Eo = 0, d2 E; - (1 + d1 d2) E; - 1 + d1 E;- 2 0, =
and use a telescoping sum argument.) 7.23. ** Let ¢ : AN ___. AN be an automorphism defined over Q and denote the dynamical degree of ¢ by 8(¢ ) = lim deg(¢n) 1 1 n.
n �oo
We associate to ¢ the number
Remark 7. 1 6 tells us that S ( ¢) satisfies
S(¢) :::; 2. If ¢ is regular, then 8(¢) = deg(¢) and 8(¢ - 1 ) = deg(¢- 1 ) . If in addition ¢ satisfies assumption (7.35), then Exercises 7. 1 8 and 7.20 imply that
S(¢)
=
1+
1
8(¢) 8(¢- 1 )
(a) Do there exist automorphisms ¢ : AN ___. A N of degree at least 2 satisfying S ( ¢) = 1? What ifwe require that ¢ be algebraically stable? (See Remark 7.13.) (b) Do there exist automorphisms ¢ : AN ___. AN of degree at least 2 satisfying S ( ¢) = 2 ? (c) What are the possible values of S ( ¢) for automorphisms of A N ? (d) What are the possible values of S ( ¢) for algebraically stable automorphisms of AN ? 7.24. Let K be a field that is complete with respect to a nonarchimedean absolute value and let ¢ : IP'N ( K) ___. IP'N ( K) be a morphism. Prove that ¢ is an open map, i.e., the image of an open set is an open set.
7.4. Dynamics on Surfaces with I nvolutions 7.25. Let G';, and H;*j be the quartic forms defined by (7 . 1 7). Prove that the following algo rithm computes L1 and L2 . (a) Let [x , y] E SA,B and write L 1 ([x , yl) = [x , y']. Then Section
[yo G(! (x) , -yoHa1 (x) - y1 G(! (x), -yo Ho2 ( x) - y2G(! ( x) ] y' = [ -y1 Ho1 (x) - yo Gf (x), y1 Gf (x), -y 1 H1 2 (x) - y2Gf ( x) ] [ -y2Ho2 (x) - Yo G� (x), -y2Hf2 (x) - y1 G� ( x), y2G� ( x) ] Let [x , y] E SA,B and write L 1 ([x , yl) = [x', y]. Then
{
(b)
{ x' =
if yo if y1
::/= 0, ::/= 0,
if y2 ::/= 0.
[xo G!j (y ) , - xo Hg1 (y ) - x1 G!j (y ) , - xo Ho2 (y ) - x 2Gij (y) ] ifxo ::j= 0, [ - x1 Hg1 (y) - xo Gi (y) , x1 G]' (y ) , - x 1 H12 (y ) - x 2Gi (y) ] ifx 1 ::/= 0, [ - x 2HJ2 (y ) - xo G§ (y ) , - x2Hf2 (y ) - x1 G� (y ) , x2G� (y)] if x2 ::/= 0.
434
Exercises
7.26. The K3 surface given in Example 7.36 contains the following 1 2 points of small height:
([0, 1, 1], [1, 1, -1]), ( [1 , o, OJ , [O , o, 1]) , = ( [0 , 1 , 0], [0, 0 , 1]) , P4 ( [1, 0 , -1 ], [0 , 1 , OJ) , P1 P2 Pa
=
=
=
P5 = P6 = P7 = Ps =
( [o, 0 , 1], [0, 1, 0l), ( [o, o, 1 ], [1, o, ol) , ( [3, 1, 3], [-3 , 3, 2l), ([1, 0 , 0], [0 , 7 , 1l),
Pg = Pw =
( [8, 6, 9], [-6, 5 , 2l), ( [1 , o, -1], [9, 1, 9l), Pn = ( [3, 8 , 11], [1, 1, - 1l), P1 2 ( [12, 1, -20], [2, -4 , 1l) . =
(a) Which of these 12 points lie in the same A orbit? How many distinct A orbits do they generate? (b) Which of the points in the list are fixed by a nontrivial element of A? (c) The list includes all points in S (Q ) having integer coordinates at most 40. Extend the computation to find all points in S(Q) having integer coordinates at most 100. (Hint. Loop over x with Jx; J :::; 1 00, substitute into L and Q, eliminate a variable, and check whether the resulting quadratic equation has a rational solution.) 7.27. For each of the primes in the set
{2 , 3 , 317 , 507593 , 2895545793631 , 1452148573727346 1} find an example of a surface SA , B defined over lFP such that L 1 and L2 are defined at every point of SA , B (JFp). (These examples can be used to complete the proof of Proposition 7.4 1 .) 7.28. This exercise sketches a noncomputational proof of Proposition 7.41 using more ad vanced methods from algebraic geometry. (a) Let S and S' be nonsingular projective K3 surfaces and let ¢ : S --> S' be a birational map, i.e., a rational map with a rational inverse. Prove that ¢ is a morphism. (Hint. Find a surface T and birational morphisms 'ljJ : T -. S and '1/J ' : T --> S' so that ¢ o 'ljJ = '1/J' [ 1 98, V.5.5]. Do this so that 'ljJ is a minimal number of blowups and let E be an excep tional curve of the last blowup. Deduce that q( E) is a curve C on S' . Then show that the intersection of C with the canonical divisor on S' satisfies C Ks' :::; E · Ks = -1, which contradicts the fact that Ks' = 0, since S' is a K3 surface.) (b) Prove that there is a proper Zariski closed set Z c lP'8 x lP'3 5 such that for all (A, bf B) ¢: Z, the surface SA , B is nonsingular. (Hint. Elimination theory says that the set of (A, bf B) E lP'8 x lP'3 5 such that SA , B is singular is a Zariski closed set. Thus it suffices to find a single (A, bf B) for which SA , bf s is nonsingular.) (c) Combine (a) and (b) to prove Proposition 7.4 1 . ·
7.29. Let P E SA , B with infinite A-orbit. Prove that the A-stabilizer of P,
{'1/J E A : '1/J( P)
=
P},
has order either 1 or 2. 7.30. This exercise describes intersections on the surface SA , B . For the basics of intersection theory on surfaces, see, for example, [ 1 98, V § 1 ] . (a) Let D1 = p'{ H and D2 = p2 H b e the usual divisors i n Pic(SA,s). Prove that
and
435
Exercises (b) Let a: = 2 + v'3 and define divisors E + and E - in Pic(SA,a) E + = - D 1 + a: D2 and E- = a: D1
0 1R
by the formulas
- D2 .
Prove that �'i.E± = a: ± 1 E 'f and �2E± = a: 'f 1 E 'f . (c) Prove that E + E + = E - · E - = 0 and E+ · E - = 12a:. ·
7.31. (a) Prove that under composition, the involutions � 1 , �2 E Aut(SA,a) satisfy no re lations other than �i = �� = 1 . Thus A is the free product of the subgroups generated by �1 and �2 . (Hint. Use Exercise 7.30. Apply a composition of � 1 's and �2 's to E+ + B and intersect with E+ .) (b) Show that A is isomorphic to the infinite (discrete) dihedral group 'Doo
=
V tj i , j E Z} 2 {t = 1 and ts = s- 1 t} :
via the map
7.32. Let P E SA,B be a point whose A-orbit A(P) is an infinite set. Prove that A(P) is Zariski dense in SA,B · (Hint. If A(P) is not dense, find a curve C c SA,a fixed by some nontrivial element 'lj; E A and consider the intersection of C with the divisors E + and E defined in Exercise 7.30.) 7.33. Let 1> = �1
o
� 2 and fix a nonzero integer n. Prove that the set
is a finite set. (Hint. If the set is infinite, find a curve C C SA,B fixed by 1>n and consider the intersection of C with the divisors E+ and E - defined in Exercise 7.30.) 7.34. Let D E Pic(SA,a) be the divisor D = D 1 + D2 = p'J.H + p2H. Let ¢> = �1 o �2 and a: = 2 + V3 as usual. Prove that hD (c/>n P) + hD (c/> - n P) = hD (P) + 0(1) for all P E SA,a (K) and all n ?: 0. a:2n + a:- 2n
(The 0(1) constant depends on the surface SA,B, but is independent of both P and n. )
7.35. Let SA,B be defined over a number field K and let h,+ and h, - be the canonical height functions constructed in Proposition 7.47. Assuming that SA,a (K) is an infinite set, prove that there is a constant C such that both of the sets
and are infinite. This shows that Proposition 7.49 is not true if h is replaced by either h + or h- . 7.36. Let SA,B be defined over a number field K, let h,+ and h, - be the canonical height func tions constructed in Proposition 7.47, and let h = h, + + h, - . Fix a point Q E SA,a(K). (a) Prove that the product h,+ (P) h - (P) is the same for every point P E A(Q). This prod uct measures, in a certain sense, the arithmetic complexity of the A-orbit of Q. Notice how h + ( Q) h - ( Q) naturally appears in Proposition 7.50 counting points of bounded height in the A-orbit of Q.
Exercises
436 (b) Prove that
(Here a: = 2 + J3 as usual.)
7.37. Let SA,B be defined over a number field K and let Q E SA,B (K) be a point whose A orbit A(Q) is infinite. Further, let h = f-t+ + h-, and define a height zetafunction for the A orbit of Q by the series
Z(A(Q), s)
(a) (b) (c) (d)
=
, 1- .
L
h(Q)s Prove that the series defining Z(A(Q) , s ) converges on the half-plane Real(s) > 0. Prove that Z (A( Q), s) has a meromorphic continuation to the entire complex plane. Find the poles of Z(A(Q) , s). Find the residues of Z (A(Q), s) at its poles. PEA(Q)
7.38. Let SA,B be defined over a number field K and let P E SA,B(k) be a point whose A orbit A(P) is Galois-invariant, i.e., if Q E A(P) and cr E Gal(k/ K), then cr(Q) E A(P). Prove that P satisfies one of the following conditions: (a) P E SA,B (K). (b) A( P) is finite. (c) [K(P) : K] = 2. If P satisfies condition (c), prove that there exist a 1jJ E A and an index j E {1, 2} such that P] ('I/J(P) ) E 1P'2 (K).
Let V c lP'N biquadratic form, 7.39.
x
lP'N be a variety given by the vanishing of N - 1 bilinear forms and one L 1 (x , y )
LN - 1 (x, y ) = Q(x , y ) = 0 , N and let P1 : V ---+ lP'N and P2 : V ---+ lP' be the usual projection maps p1 ( x, y ) = x and P2 (x, y ) = y. (a) Prove that P1 and P2 are generically 2-to-1, so they induce involutions t 1 : V ---+ V and L2 : V ---+ V. In other words, there are rational maps L1 and t2 such that LI and t� are the identity map wherever they are defined. (b) If N 2: 3, prove that L1 and L2 are not morphisms. 7.40. Let a E
= · · · =
K * . The Markoffequation
Ma : x2 + l + z2
=
axyz
defines an affine surface in A..3 . (a) Prove that there are involutions L1 2 , L1 3 , and L23 of Ma defined by the formulas
L 12 (x, y, z) = (x, y, axy - z) , L1 3 (x, y , z ) = (x, axz - y , z) , L23 (x, y, z) = (ayz - x, y, z). Explain how these involutions correspond to natural double covers Ma by projection maps A..3 ---+ A..2 .
---+
A..2 induced
Exercises
437
(b) Prove that the involutions �ij do not extend to morphisms on the projective variety Ma = { x2 w + y 2 w + z 2 w = axyz} C lP'3 , and determine the points at which they fail to be defined. (c) Find a birational map lP'2 --+ Ma defined over Q. (A birational map between projective varieties V and W is a rational map from V to W that is an isomorphism from a Zariski open subset of V to a Zariski open subset of W.) In particular, this implies that Ma (Q) contains many points. (d) Prove that every point in M3 ('£) with positive coordinates can be obtained by start ing with the point ( 1 , 1, 1) and applying the involutions �ij . (Hint. Define the size of a positive integral point P = ( x, y , z ) to be the largest of its coordinates and prove that if P f. ( 1 , 1 , 1), then at least one of �ij (P) has size strictly smaller than the size of P.) (e) Let a be a positive integer. Prove that if a f. 1 and a f. 3, then Ma (Z) = 0. (Hint. Use the same type of descent argument as suggested in (d).) (f) A normalized Markoff triple is a point (x, y, z) E M3 ('£) with x :::; y :::; z. Let N(T) = #{normalized Markoff triples (x, y, z) with z :::; T } . Prove that there are positive constants c 1 and c2 such that (7.41 )
More precisely, prove that there is a constant c such that N(T) = c(log T) 2 + O ( (log T) (log log T) 2 ) .
(7.42)
(g) ** Let ( X 1 , Y 1 , 21 ) and ( x2 , Y2 , 22 ) be normalized Markoff triples. Prove that if z1 = 22 , then also X 1 = x2 and Y1 = Y2 · (This is known as the unicity conjecture for Markoff numbers.) Exercises on K3 Surfaces with Three Involutions
Exercises 7.41-7.44 ask you to explore a family ofK3 surfaces that admit three noncommuting involutions. These hypersurfaces Sc c lP'1 x lP'1 x lP' 1 are described by the vanishing of a trihomogeneous polynomial of degree 2, Q(x, y, z) =
L
O X0 (2), 1 62 map X1 (3) ----> Xo (3), 230 points solve moduli problem, 159 quadratic polynomial, 1 56 quotient, 1 6 1 Xo (4) , Xo (5) , X0 (6), 230 xl (1 ) xl (2) , xl (3) are rational, 1 57 X 1 (4) , X1 (5) , X1 (6), 230 dynatomic divisor, 1 50 dynatomic field, 1 23 action of Galois, 1 24 ,
Galois group is subgroup of wreath product, 1 25 quadratic polynomial, 123 ramification, 1 29, 1 3 1 unit, 1 29 dynatomic polynomial, 39, 1 48, 1 49, 1 8 1 associated multiplier polynomial, 225, 227 dehomogenized, 1 49 discriminant, 226, 227 for z2 + c, 1 56 generalized, 227, 229 homogeneous of degree vd (n), 1 8 1 is a polynomial, 1 5 1 , 226 of z + 1/ z, 224 of z2 + bz, 1 82 order at P (aj,(n)), 1 5 1 , 225, 226 quadratic map, 39 reducible, 235 resultant of two, 1 65, 226, 227 dynatomic 0-cycle, 1 50, 226 is positive, 226 Effective divisor, 406 positivity of height, 408 eigenvalue, 42 1 involution, 438 elementary symmetric polynomial, 87, 1 80 elimination theory, 4 1 7 elliptic curve, 30, 336, 409 addition law, 3 1 automorphism group, 340, 34 1 explicit description, 342 canonical height, 409 complex multiplication, 32, 341 complex uniformization, 33, 1 27 critical values of quotient, 353 defined over K, 337 degree 2 map is even, 382 degree map on divisors, 339 degree of an endomorphism, 349 degree of an isogeny, 340 determined by critical values of double cover, 347 discriminant, 30, 337, 383 division polynomial, 362, 383 divisor group of, 339
483
Index elliptic curve (continued) divisor is principal iff degree and sum are zero, 339, 409 divisor of a function, 339 divisor on, 338 dual isogeny, 340, 352 duplication formula, 338 duplication map, 32, 35 1 , 370 endomorphism ring, 3 1 , 340, 341, 378 field generated by torsion, 344, 363 flexible Lattes map, 355 fundamental domain, 33, 1 27 geometric points, 337 global minimal Weierstrass equation, 343, 37 1 good reduction, 58, 62, 342, 343, 362, 383 good reduction implies reduction is homomorphism, 342 group law, 337 explicit formulas, 338 is algebraic, 338 Haar measure, 1 27 Heeke correspondence, 369 height on, 409 ideal class group maps to CM, 350 identity element, 337 integer point, 372 integral j-invariant, 370 invariant differential, 344, 359, 367 inverse of a point, 33 7 is a pair (E, 0), 337 is abelian group, 338 isogeny, 339 is homomorphism, 340 is unramified, 340, 353 isomorphic to Picard group, 339 isomorphism class, 337 j-invariant, 337 kernel of endomorphism, 343 Lattes map, 32, 97, 35 1 preperiodic points, 4 1 , 352 level curve of integrable map, 430 £-series, 409 minimal Weierstrass equation, 222, 342 modified group law, 382
morphism is isogeny and translation, 339 m-torsion subgroup, 343 multiplication map, 3 1 , 1 1 2, 340, 383 effect on height, 409 kernel, 343 noncommutative endomorphism ring, 378 nonconjugate Lattes maps, 354 in characteristic 2, 383 nonsingular, 337 normalized embedding of endomorphism ring, 345 point at infinity, 336 potential good reduction, 3 7 1 principal divisor, 339 has degree zero, 339 quasiminimal Weierstrass equation, 37 1 , 385 quotient by automorphism group, 346, 365 rational point, 337 reduction modulo p, 342 reduction of torsion points, 62, 343, 364 subgroup of K -rational points, 338 summation map on divisors, 339 supersingular, 378 Szpiro conjecture, 221 Tate s algorithm , 342 torsion equidistributed, 1 27 torsion point, 4 1 torsion subgroup, 32, 343, 352 twist, 1 98, 341 , 374 uniform bound for integer points, 372 uniform boundedness of torsion, 97, 369 universal cover, 346 Weierstrass r function, 34, 345 Weierstrass class, 237 Weierstrass equation, 30, 336, 409 elliptic modular curve, 1 63 endomorphism ring elliptic curve, 340, 341, 378 kernel, 343 multiplicative group, 325, 378 noncommutative, 378 '
484 endomorphism ring (continued) normalized embedding in IC, 345 order in quadratic imaginary field, 341 order in quatemion algebra, 341 entropy, 6 algebraic, 397 equicontinuity, 3, 22, 254, 264, 265, 27 1 , 313 Fatou set i s maximal open, 22 is not open condition, 40, 254 on a metric space, 22 topological, 306 equidistribution, 6 dynamical Galois, 128 Galois of a sequence, 128 ofpreperiodic points, 128 of small height points, 128 torsion on elliptic curve, 1 27 equi-Lipschitz, 3 1 3 equivalence relation from group action, 234 Eremenko 's theorem, 378 ergodic theory, 6 essential singularity, 23 etale map, 353 Euclidean distance versus chordal metric,
1 19 Euler totient function, 1 37, 1 64, 1 82 iteration of, 6 exact period, I , 1 8, 38, 1 50 condition for larger formal period, 1 5 1 , 1 65, 226 rational map has points of different, 1 54, 1 55, 1 96 exceptional set, 1 6, 1 7, 266 exercise hard ( textbf*), 7 unsolved (**), 7 extension formula, 83, 84 Faltings' theorem, 1 07, 442 family of maps, 1 7 1 , 1 94 Fatou domain, 240 Fatou set, 22, 255 all oflP' 1 , 26, 59, 239 Berkovich, 306, 3 1 1 canonical height of points in, 1 3 8
Index classical contained in Berkovich, 306 classification of connected components, 28 complement of Julia set, 22 component at infinity is completely invariant, 40 contains attracting periodic points, 40 contains nonrepelling periodic points, 256 empty, 26, 35, 361 Herman ring, 28 is completely invariant, 23, 40, 266 is nonempty, 256, 285, 3 1 4 is open, 22 Julia set is boundary, 25 local canonical height of points in, 141 n o wandering disk domains, 284, 285 no wandering domains, 28 nonarchimedean, 254 number of connected components, 27 of an iterate, 24, 255 of commuting maps, 378
of (z 2 - z)/p, 283 of zd , 22 parabolic component, 28 polynomial cannot have Herman ring, 28 recurrent critical point is periodic, 284, 3 1 7 set of disk components, 283 Siegel disk, 28 Fermat's little theorem, 2, 252, 275 Fermat's sum of two squares theorem, 200, 236 field complete, 240 completion, 241 extension of complete, 24 1 finite, 2 of characteristic p, 37 separable extension, 37 valued, 44, 240 field of definition, 206 abelian variety, 2 1 7
Index field of definition (continued) cocycle is coboundary, 209 contains field of moduli, 208 covering map, 2 1 7 intersection of all i s field of moduli, 236 not equal to field of moduli, 208, 2 1 3, 236 of algebraic family, ! 59 of morphism, 89 of point, 86, 1 22, 145 of quotient variety, 1 6 1 field of moduli, 1 77, 207 abelian variety, 2 1 7 cocycle, 209 contained in field of definition, 208 covering map, 2 1 7 equals field of definition for polynomials, 2 1 5 equals field o f definition i f degree is even, 2 1 5 from image i n M d , 207 is intersection of all fields of definition, 236 not equal to field of definition, 208, 2 1 3, 236 of a point, 1 45 filled Julia set, 74, 140 Green function, 1 40 fine moduli space, 1 60, 1 72, 230 finite field dynamical system on, 2, 5 function field over, 5 finite invariant set, 1 6, 266 finite subgroup of PGL 2 , 1 97 fixed point, 1 8 attracting in Berkovich space, 324 flexible Lattes map, 382 Galois conjugates, 1 80, 1 84 Galois group, 427 index summation formula, 20, 38, 255, 3 1 4, 380, 382 Lattes map, 34 multiplier, 1 8, 47, 1 80 at infinity, 3 8 neutral, 322 no Type-IV, 321 nonrepelling, 255, 3 1 4 in Fatou set, 256
485 of affine automorphism, 427 of Chebyshev polynomial, 332, 380 in characteristic p, 3 8 1 ofType I, II, III, IV, 322 rational, 427 rational map, 149 real, 427 repelling in Julia set, 256 residue index, 38 rigid Lattes map, 366, 3 84 summation formula, 20, 38, 255, 3 1 4, 380, 382 symmetric polynomial of multipliers, 1 80 totally ramified, 16, 1 7, 37 flexible Lattes map, 1 87, 355 composition of, 382 degree, 357 fixed points, 382, 384 good reduction, 362, 383 multiplier, 358, 366, 382, 384 spectrum, 356 multiplier at oo, 383 periodic points, 358, 382, 384 FOD, see field of definition foliated space, 6 FOM, see field of moduli formal derivative, 12, 37, 3 13 formal group, 67 formal multiplier spectrum, 1 82, 1 83 of z2 + bz, 1 82 of z d , 1 82, 1 83 formal period, 1 8, 39, 150 algebraic family induces map to dynatomic curve, ! 59 algebraic family of points of given, 159 condition for smaller primitive period, 1 5 1 , 1 65, 226 moduli problem, 158 vd (n ) points of formal period n, 1 50, 224 formal power series, 5 periodic points over ring of, 78 forward orbit, 1 , I 08, 280, 353 fractional ideal, 350 exact sequence, 404 height formula, 136
486 fractional linear transformation respects reduction, 50 free product of cyclic groups, 4 1 9, 435 Frobenius map, 3 5 1 cyclotomic field, 1 30 periodic points are attracting, 78 function analytic, 244 equicontinuous family, 254, 264, 265, 27 1 , 306, 3 1 3 equi-Lipschitz family, 3 1 3 holomorphic, 244 meromorphic, 244 uniformly continuous family, 39, 254, 3 1 3 uniformly Lipschitz family, 39, 254, 256, 264, 265, 27 1 , 3 1 3, 3 1 5 function field, 5 absolute values on, 44 dynamics, 5 functoriality ofheight, 407, 42 1 fundamental domain, 33, 1 27 Galois cohomology, 205 Galois conjugate, measure supported on, 128 Galois equidistributed sequence, 128 Galois equidistribution, 128 Galois group action on dynatomic field, 1 24 action on PGL2 , 203 action on projective space, 122 action on rational map, 207, 2 1 1 cyclotomic field, 1 29 of dynatomic field, 1 23, 1 25 of field generated by periodic points, 122 of fixed point, 427 orbit of conjugate point, 1 37 profinite topology, 2 1 1 subgroup associated to rational map, 207 subgroup of wreath product, 1 25 Galois invariance of height, 85 Galois-invariant orbit on K3 surface, 436 Galois theory, 1 22 Gauss norm, 296 associated Gauss point, 297 is sup norm, 296
Index on ICp[z], 296 on Tate algebra, 298 Gauss point, 297 fixed by polynomial, 321 is Julia set of good reduction map, 307, 322 line segment to Type-IV point, 322 of Berkovich disk of radius R, 30 1 Gel'fond metric, 306 Gel'fond topology, 300, 322 base of open sets, 300 not induced by path metric, 306, 323 restricts to ICp topology, 3 0 1 , 304, 323 general linear group, 10 generalized dynatomic polynomial, 227 for quadratic map, 229 generalized Weierstrass equation, 336 generating function for degree of rational map, 397, 428 generic fiber, 77 generic rational map dynamical unit, 1 46 not very highly ramified, 23 1 genus, 1 5, 37 of Xo (n), 1 64 of X1 ( 4), X1 (5) , X1 (6 ), 230 of X1 (n), 1 64 geometric invariant theory, 1 76 geometric point, 337 geometric quotient, 179 GL2 , 10 global minimal model, 222 implies trivial Weierstrass class, 223 over Q, 236 global minimal Weierstrass equation, 343, 371 difference from quasiminimal, 385 global S-minimal model, 237 good reduction, 43, 58, 2 1 8 abelian variety, 442 attracting periodic points, 64 Berkovich Julia set of map with, 307, 3 1 1 , 322 composition of maps, 59, 77 degree of reduction is same, 58 elliptic curve, 342, 362, 383
Index good reduction (continued) finiteness of periodic points, 66 from bad reduction, 2 1 8 Green function of map with, 289, 294, 3 1 8 iff resultant is unit, 58, 2 1 8 implies reduction is homomorphism, 342 is everywhere nonexpanding, 59 Julia set is empty, 59, 239 local canonical height for, 1 03, 1 4 1 map on Berkovich disk, 32 1 , 322 nonrepelling periodic points, 64 of ¢(P) is J (P) , 59 of a rational map, 58 over power series ring of characteristic p, 78 period of reduced point, 62, 66, 78 periodic points reduce to periodic points, 6 1 potential, 3 7 1 scheme-theoretic characterization, 59, 6 1 , 77 separable, 64 twists with outside S, 237 v-adic distance between periodic points, 69 graph, 1 50, 2 1 6, 226 Green function, 287, 288 algorithm to compute, 3 1 8 complex, 140 defining properties, 289 homogeneity properties, 289 is continuous, 289 is Holder continuous, 3 1 8 modified, 3 1 8 of bad reduction map, 294, 3 1 8 of good reduction map, 289, 294, 318 relation to local canonical height, 291 sum is canonical height, 290, 31 S group, 2 action, 234 additive, 30 algebraic, 325 commutative algebraic, 375 dynamical system on, 2 law on elliptic curve, 338
487 multiplicative, 29 multiplicative scheme, 29 periodic points form subgroup, 8 variety PGL2 , 1 70 wreath product, 125 H 1 , 202, 236 Haar measure, 1 27 haif a loaf, 133 hard problem ( textbf*), 7 harmonic function, local canonical height is, 140 Hausdorff space, Berkovich disk is, 300 Heeke correspondence, 369 height, 4, 8 1 absolute, 85 absolute logarithmic, 93 additivity, 408, 42 1 bounded for ample divisor is finite set, 408 canonical, 97, 99, 1 00, 287, 409 on K3 surface, 42 1 , 426, 435, 436 effect of morphism of degree d, 90, 1 36, 398, 407, 409 effect of rational map, 388 effectivity of constants, 408 elliptic curve, 409 effect of multiplication map, 409 equals one iff root of unity, 88, 1 00 field extension formula, 84 finitely many points of bounded, 86, 407, 423 for linearly equivalent divisors, 408 fractional ideal formula, 1 36 functoriality, 407, 42 1 Galois invariance of, 85 h([u, a , b] ) ::=; h([u, a] ) + h([u, b] ), 398, 428 in affine space, 397 inequality for jointly regular affine morphisms, 397, 429 inequality for nonregular affine automorphism, 429 inequality for regular affine automorphisms, 399, 43 1 infinitely many points of bounded, 435
488 height (continued) is at least one, 84 is well-defined, 84 Lehmer's conjecture, 1 00, 1 0 1 , 1 3 8 logarithmic, 93 lower bound for, 1 00, 1 0 1 , 1 37, 1 38, 22 1 machine, 407 map on Picard group, 408 Misiurewicz points have bounded, 1 67 number of points ofbounded, 1 35, 1 36, 441 on K3 surface, 425 of a rational map, 143 of a rational point, 82 of an algebraic point, 84 of isolated periodic points is bounded, 402 of polynomial and its roots, 1 36, 1 38 of wandering point for regular affine automorphism, 432 on K3 surface, 420 on M d , 237 on JP'N X JP'M, 409, 420 on projective space, 407 positivity, 408 product h + h- of canonical, 435 transformation rules on K3 surface, 420 uniqueness, 408 Wei!, 89, 407 zeta function, 436 height machine, 407 Henon map, 390, 402 number of orbits over a finite field, 430 reversible, 430 Hensel's lemma, 48 Herman ring, 28 Hilbert basis theorem, 90 Hilbert Nullstellensatz, 90, 92 Hilbert theorem 90, 1 99, 205, 207 Holder continuity of Green function, 3 1 8 holomorphic function, 244 coefficients uniquely determined, 312 distance between image disks, 268
Index dynamics of, 23 equicontinuous family, 254, 264, 313 equi-Lipschitz family, 3 1 3 family omitting one point, 264, 3 1 5 Fatou set, 24, 255 is continuous, 253 is open, 253 Julia set, 24, 255 Lipschitz bound, 248, 254, 272 maximum modulus principle, 250 Monte! theorem, 264, 271 , 3 1 5 Newton polygon, 249 norm, 247 norm of derivative, 252, 3 1 3 order of, 244 product is holomorphic, 3 1 2 reciprocal i s holomorphic, 3 1 3 roots determined by Newton polygon, 249 sends disks to disks, 252, 305 uniformly continuous family, 254, 313 uniformly Lipschitz, 254, 264, 265, 271 , 3 1 3, 3 1 5 zeros are isolated, 244 homogeneous coordinates, 10 homogeneous ideal, 89 homogeneous polynomial, 10 representation of integer by, 105 homogeneous seminorm, 303, 305 homogeneous space, 6 homogenization, 389 Hsia kernel, 323 Hsia's theorem, 264, 265, 271 , 273 Hurwitz formula, 13, 2 1 7 hyperbolic component ofMandelbrot set, 1 66 hyperbolic map, 3 1 7 nonarchimedean, 279, 285, 3 1 7 Icosohedral group, 1 97 ideal algebraic set attached to, 90 attached to an algebraic set, 90 Hilbert basis theorem, 90 homogeneous, 89 radical, 89 ideal class group, 350
489
Index ideal class group (continued) exact sequence, 404 map to CM elliptic curves, 350 identity element of an elliptic curve, 337 imaginary field, quadratic, 341 immediate basin of attraction, 3 1 0, 3 I I inclusion-exclusion, 1 48, 224 indeterminacy locus, 389, 427 dimension, 394 disjoint, 394 of involution on K3 surface, 4 1 3 of iterate, 394 indifferent periodic point, 1 9, 47, 326 one implies infinitely many, 3 14 induced rational map, 389 inertia group, 344 infinite dihedral group, 4 1 9, 435 integer point counting on Markoff variety, 437 cutoff for in orbits, 375 in orbit, 3, 108, 1 09, 1 1 2, 142, 143, 1 45 in orbit of z + 1 / z, 8 on lP' 1 minus 3 points, 1 06, 1 08 on elliptic curve, 372 on Markoff variety, 437 orbits with many, I 1 0, 1 I I , 143, 375 uniform bound in orbits, I 12, 372, 385 value of rational function, 1 43 integrable map, 429, 430 level curve is elliptic curve, 430 level set, 43 1 McMillan family, 443 QRT family, 443 intersection theory, 434 invariant differential, 344, 359, 367 is holomorphic, 345 is translation-invariant, 345 transformation for [m] , 345 invariant measure, 307 invariant set, 1 6 completely, 1 7, 266 contains Berkovich Julia set, 3 1 1 Fatou and Julia sets are completely, 23 finite, 1 6, 266 inverse function theorem, 1 1 5, 1 44
over rCp, 3 1 2 inverse of point on elliptic curve, 337 inversion maps disk toj, 321 involution, 410 action o n divisor, 4 1 8, 4 1 9, 438 as linear transform on Picard group, 420 composition is reversible affine automorphism, 429 degenerate point, 4 1 5 eigenvalue of, 42 1 , 438 formulas to compute, 433 indeterminacy locus, 4 1 3 induced by degree 2 map, 4 1 1 , 436, 437 is morphism on most K3 surfaces, 417, 434, 438 noncommuting, 4 1 0 not a morphism, 436, 437 on Markoff variety, 436 quotient by, 410 relation between noncommuting, 4 1 9, 435 surface in lP'N x lP'N with two, 436 three noncommuting, 437 irrational number approximated by rational number, 104 irrationally neutral periodic point, 19, 47 isogenous elliptic curves, 339 isogeny, 339 degree, 340 dual, 340, 352 is homomorphism, 340 is unramified, 340, 353 isolated point, 402 Julia set contains no, 25 of bounded height, 402 periodic, 402 preperiodic, 8 isospectral family, 1 86, 1 88, 356, 362 iterate, coordinate functions of, 149, 226 iteration commutes with conjugation, 1 1 itinerary, 258 as map on sequence space, 260 left shift, 260 Jacobian variety, 369 ]-invariant, 22 1 , 337 in terms of cross-ratio, 348
Index
490 jointly regular affine morphisms, 397, 429 height inequality, 397, 429 Julia set, 22, 255, 284 algebraic points, 40 all of!P' 1 , 26, 35, 3 6 1 backward orbit i s dense, 25 Berkovich, 306 backward orbit dense in, 3 1 1 for bad reduction map, 307, 3 1 1 for good reduction map, 307, 3 1 1 , 322 has empty interior, 3 1 1 is connected or infinitely many components, 3 1 1 boundary is completely invariant, 23 boundary ofFatou set, 25 canonical height of points in, 1 3 8 chaotic behavior, 22 Chebyshev polynomial, 30, 4 1 classical contained i n Berkovich, 306 closure of repelling periodic points, 41 compact, 275 complement of Fatou set, 2 2 connected, 26, 1 65 contained in closure of periodic points, 273 contains repelling periodic points, 40, 256 disjoint from postcritical set closure, 280 effect of strictly preperiodic critical points, 26 equal to :Z.p, 275 equals closure of repelling periodic points, 27, 274, 3 1 1 filled, 74, 1 40 Green function for filled, 1 40 has empty interior, 267 in two disks, 257 is closed, 22 is closure of backward orbit, 267 is completely invariant, 23, 40, 74, 266 is empty for good reduction map, 26, 59, 239
is nonempty, 25 is perfect set, 23, 25, 267, 307, 3 1 1 is uncountable, 267, 307, 3 1 1 local canonical height of points in, 141 n o critical points, 285 nonarchimedean, 254 nonempty interior, 26 of an iterate, 24, 255 of Chebyshev polynomial, 336 of commuting maps, 378 of Lattes map, 35, 3 61 2 of (z - z)/p, 262, 283, 3 1 5 2 of z - 2, 40 of z d , 22 of z · · · ( z d + 1) /p, 3 1 5 open set orbit omits at most one point, 266 orbit of open subset, 27 periodic points dense, 23, 263, 3 1 5 smallest closed completely invariant set, 267 strictly expanding map on, 279, 3 1 7 support of canonical measure is, 307 topologically transitive map, 263, 315 totally disconnected, 23, 26 with no critical points, 279 -
K3 surface, 4 1 0, 4 1 2 action o f involution o n divisors, 4 1 8, 4 1 9, 438 A orbit, 4 1 8 arithmetic complexity of orbit, 435 canonical height, 42 1 , 423, 426, 435, 436 infinitely many points of bounded, 435 normalization conditions, 422 product h + h-, 435 transformation formulas, 422 zero iff finite orbit, 423 degenerate point for involution, 4 1 5 dimension of family of, 4 1 2, 438 divisors on, 4 1 8 eigenvalue of involution on divisors, 421 , 438 finitely many periodic points, 435
Index K3 surface (continued) finitely many points of bounded canonical height, 423 Galois-invariant orbit, 436 H1 (S, Os) = 0, 412 height functions on, 420 height transformation rules, 420 height zeta function, 436 homogeneous forms associated to, 4 1 2, 433 in !P' 1 X !P' 1 X IF'\ 437 indeterminacy locus of involution, 413 intersection theory, 434 involution, 4 1 1 as linear transform on Picard group, 420 example, 4 1 1 formulas, 433 is morphism on most, 4 1 7, 434, 438 Kodaira dimension, 4 1 2 nonsingular, 438 number of points ofbounded height, 425 orbit is Zariski dense, 435 Picard group of general, 4 1 8 projections to IP'2 , 4 1 0 relation between noncommuting involutions, 4 1 9, 435 stabilizer of point on, 425, 434 with three involutions, 437, 438 Kawaguchi's theorem, 399, 43 1 Kodaira dimension, 412 Kronecker's theorem, 88, 1 00 Krylov-Bogolubov theorem, 127 Kummer sequence, 205 £-adic representation, 344 Lang's height lower bound conjecture, 1 0 1 , 221 Lang's integer points conjecture, 1 1 2 Lattes diagram, 365 Lattes map, 32, 97, 1 86, 1 87, 3 5 1 affine minimal, 372, 385 commuting, 378 composition of, 382 critical point of 1r , 384 critical values, 353
491 cutoff for integral points in orbit, 375 defining equation, 32, 3 5 1 degree, 34 fixed points, 34, 366, 3 82, 384 flexible, 355 for multiplication-by-( ! + i), 32 for multiplication-by-2, 32, 3 5 1 , 370 on j = 0 curve, 3 5 1 , 3 8 1 o n j 1728 curve, 3 5 1 good reduction, 362, 383 higher-dimensional, 377 in characteristic p, 366 integral j-invariant, 370 isospectral family, 362 iterate, 3 82 multiplier, 34, 1 86, 358, 366, 382, 384 at oo, 383 of fixed point, 382 spectrum, 362 summation formula, 382 nonconjugate, 354 in characteristic 2, 383 to polynomial, 3 8 1 periodic points, 1 86, 358, 366, 382, 384 all attracting, 3 62 all nonrepelling, 362 expanding, 35 in Julia set, 35, 361 padically p-adically, 3 6 1 postcritical set, 353 postcritically finite, 353 preperiodic points, 32, 41, 352 projection to E;r, 365 reduced diagram, 365 rigid, 364 set of is monoid, 382 torsion points map to preperiodic points, 4 1 translation fixed by r 365 ' uniform bound for integral points in orbit, 372, 385 uniform boundedness of periodic points, 13 7 lattice, 33 =
492 Laurent series, 245 coefficients uniquely determined by function, 3 1 2 of Schwarzian derivative, 232 product of, 3 1 2 least period, see exact period Lefschetz principle, 20, 365 left shift map, 259, 3 1 4 as itinerary map, 260 backward orbit dense, 3 1 4 backward orbit equidistributed, 3 1 5 continuous, 259 is topologically transitive, 259 Lipschitz, 259 periodic points, 259 are dense, 3 1 4 properties of, 259 topologically transitive, 3 1 4 uniformly expanding, 259 Lehmer's conjecture, 1 00, 1 3 8 dynamical, 1 0 1 , 1 3 8 level curve of integrable map, 430 level set of an integrable map, 43 1 Lie group, 6 lift, 287 from lFp to Zp, 48 of affine morphism, 3 89 of rational map, 287, 389 line bundle, metrized, 41 0 line segment from Gauss point to Type-IV point, 322 in Berkovich disk, 298 of Newton polygon, 249 linear conjugation, 1 1 , 173 commutes with iteration, I I linear equivalence, 403, 404 divisors give same height, 408 pullback preserves, 405 linear fractional transformation, 10 action of Galois, 203 chordal sup norm, 269 distance between images, 269, 3 1 6 effect on chordal metric, 76 image of unit disk, 3 1 7 Lipschitz constant, 36 move three points, 36 on lP'N, 226 resultant of, 76
Index linear group general, 1 0 projective special, 1 75 special, 17 5 Lipschitz, 1 1 , 24, 36, 56 equi-, 3 1 3 holomorphic function, 248, 254, 272 shift map, 259 uniform family, 39, 254, 256, 264, 265, 271 , 3 1 3, 3 1 5 local canonical height, 1 02, 291 associated to a divisor, 320 associated to an eigendivisor class, 104 computation of, I 03 existence proof, 291 for a polynomial map, 103, 140, 141 good reduction case, 103, 141 Green function and, 291 is harmonic, 1 40 Laplacian is invariant measure, 104 normalized, 141 of point in Fatou or Julia set, 141 properties of, 102, 29 1 , 320 sum to global canonical height, 1 03 , 293 transformation law, 320 local degree, 83 local ring, 67, 79 at a divisor, 403 normalized valuation, 403 locally compact, 74, 239, 243, 304 «=v is not, 239, 268, 294, 295 locally constant function, 3 1 8 locus of indeterminacy, 389 affine automorphism, 392 affine morphism, 392 algebraically stable morphism, 396 jointly regular affine morphisms, 397, 429 logarithmic distance, 102 logarithmic height, 93 absolute, 93 logarithmic singularity, 140 long exact sequence, 205 Liiroth's theorem, 1 58 Lyubich measure, 307
Index Mahler measure, 1 3 8 Mandelbrot set, 26, 1 65 analytic description ofMisiurewicz points, 229 bulb, 1 66 complement conformal to unit disk exterior, 1 67, 168 component of interior, 1 66 in disk of radius 2, 1 66 is connected, 1 67 Misiurewicz point, 1 66 analytic description, 1 68 dense in, 1 66 in boundary, 1 66 root of hyperbolic component, 1 66 spider algorithm, 1 68 uniformization, 1 67, 1 68 Manes's theorem, 235 Manin-Mumford conjecture, 1 27 dynamical, 127 Markoff equation, 436 integer points, 437 Markoff triple, 437 unicity conjecture, 437 variety is rational, 437 Masser-Oesterle conjecture, 372, 373 maximum modulus principle, 248, 250, 252, 253, 268, 296, 298, 305, 321 false if K not algebraically closed, 252 for rational functions, 250 Mazur's theorem, 97 Mazur-Kamienny-Merel theorem, 369 McMillan family of integrable maps, 443 McMullen's theorem, 1 87, 367 M2, 1 88 2 completion equals lP' , 1 94 explicit formula for a- 1 , a-2 , 1 89 is a rational variety, 232 is a scheme over Z, 1 89 isomorphic to A ? , 1 88 locus of polynomial maps, 232 point with Aut(¢) = S3, 234 subvariety with Aut = {l 2 , 235 M d , 1 74 affine coordinate ring, 1 76
493 coordinate ring contains a-i n ) and * (n ) 1 83, 232 0" ; ' field of moduli, 207 generic map not very highly ramified, 23 1 has dimension 2d - 2, 1 76 height, 22 1 , 23 7 is a rational variety?, 232 is a scheme over Z, 1 79, 1 86 is a unirational variety, 232 is a variety, 1 75 is affine, 1 76 is complex orbifold, 1 76 is connected, 1 76 is integral, 1 76 is nonsingular?, 193 M2 � A 2 , 1 89 M3 embedded by a-i n )?, 232 map from Ratd , 1 74, 1 75 multiplier 1 is Zariski closed subset, 230 not embedded by a-i n) , 1 86, 1 87, 356, 367 rational points, 1 77 semistable completion, 178 stable completion, 1 78 subvariety with nontrivial automorphisms, 1 99, 234 twists have same image, 1 98, 22 1 M�, 1 78 has natural quotient property, 1 79 is a scheme over Z, 179 is geometric quotient, 1 79 is quasiprojective, 178 isomorphic to M�s iff d is even, 1 78 M �8, 1 78 is a scheme over Z, 1 79 is categorical quotient, 1 79 is projective, 1 78 isomorphic to M� iff d is even, 178 measure, 6, 36 Brolin, 307 canonical on Berkovich space, 1 29, 306 Dirac, 128 Haar, 1 27 invariant, 104, 1 27, 307 Lyubich, 307
494 measure (continued) spherical, 36 support on Galois conjugates of point, 128 weak convergence, 1 28 Merel's theorem, 97, 369 meromorphic function, 244 coefficients uniquely determined, 3 12 dynamics of, 23 equicontinuous family, 254, 265, 313 equi-Lipschitz family, 3 1 3 family omitting two points, 265, 315 Fatou set, 24, 255 is continuous, 253 is open, 253 Julia set, 24, 255 Laurent series, 245 Monte! theorem, 265, 3 1 5 order of, 244 pole, 244 product is meromorphic, 3 1 2 uniformly continuous family, 254, 3 13 uniformly Lipschitz, 254, 3 1 3 zero, 244 metric space, 3 chaotic map, 3 continuous function, 22 equicontinuous collection of functions, 22 equicontinuous map, 3 Fatou set, 22 Julia set, 22 normal family of maps, 25 ofsequences, 258, 3 14 path in, 306, 323 SN is a, 258, 3 14 uniform continuity, 39 uniform Lipschitz, 39 metrized line bundle, 4 1 0 minimal discriminant, 22 1 minimal model, 220, 222 global S-, 237 minimal resultant, 220, 22 1 , 237 exponent at p, 220 is PGL2 (K)-invariant, 22 1
Index is resultant times power, 222 of conjugate, 2 1 9 valuation, 2 1 9 minimal Weierstrass equation, 222, 342 global, 343, 37 1 , 385 quasi, 3 7 1 , 385 Tate's algorithm, 342 Misiurewicz point, 1 66 analytic description, 1 68, 229 dense in Mandelbrot set, 1 66 has bounded height, 1 67 in boundary of Mandelbrot set, 1 66 minimal polynomial, 230 no neutral cycles, 1 66 of type (m, n ) , 1 66 repelling critical orbit, 1 66 spider algorithm, 1 68 subhyperbolic, 1 66 Mobius inversion formula, 1 48, 224 Mobius JL function, 39, 1 48, 1 49, 1 64, 224 modified Green function, 3 1 8 modular curve dynatomic, 1 57, 1 58, 1 6 1 elliptic, 1 63, 369 Heeke correspondence, 369 modular Jacobian, 369 moduli space, 4, 26, 1 47 coarse, 1 60 degree of symmetric polynomial map, 1 87, 232, 367, 384 fine, 1 60, 1 72, 230 isospectral family, 1 88, 356 M 2 S;< it? , 1 88 M2 S;< IP'2 , 1 94 map defined by symmetric polynomials of multipliers, 1 87, 232, 367, 384 of rational maps of degree d, 17 4 scheme over Z, 1 79, 1 86, 1 89 moduli, field of, 207 Mobius transformation, see linear fractional transformation monoid of Lattes maps, 3 82 monomial map, 29, 95, 325, 428 automorphism group, 234, 327 in characteristic p, 380 commuting maps, 326, 380 dynamical unit, 328
Index monomial map (continued) in characteristic p, 380 multiplier of periodic point, 326 periodic point, 326 preperiodic point, 326 twist, 206, 328, 380 monomial, number of given degree, 1 36 Monte! theorem, 25 nonarchimedean, 264, 265, 27 1 , 28 1 , 3 1 1 , 3 1 5 ofHsia, 265, 271 ofRivera-Letelier, 3 1 1 on Berkovich space, 3 1 0, 3 1 1 with moving targets, 271 morphism, 89, 388 affine, 375, 388 algebraic family of, 1 7 1 bijective =fo- isomorphism, 1 93 Bogomolov conjecture, 1 29 canonical height associated to, 99, 287 canonical measure, 1 27 defined over K, 89 degree of, 389 degree of composition, 392 dynamical degree, 397, 428 dynamically affine, 376 effect on absolute value, 288 effect on height, 90, 1 36, 398, 407, 409 functoriality of height for, 407, 421 Green function, 288 induced on family, 1 7 1 invariant measure, 1 27 involution on most K3 surfaces is, 4 1 7, 434, 438 is open map, 433 jointly regular, 397, 429 local canonical height, 1 02 over a variety, 1 7 1 pullback of divisor, 404 pullback of linearly equivalent divisors, 405 pushforward of divisor, 404 Morton-Silverman Conjecture, 96, 135, 368 for zd , 1 37 for Lattes map, 370 moving targets, 27 1
495 m-torsion subgroup of elliptic curve, 343 multiplication map by 2 on QjZ, 1 68, 229 on elliptic curve, 3 1 , 340, 383 effect on height, 409 kernel of, 343 multiplicative group, 29, 30 endomorphism, 29, 325 endomorphism ring, 378 preperiodic points, 326 quotient by { ±1 }, 29, 328 torsion, 326 multiplicative group scheme, 29 multiplier, 1 8, 1 9, 47, 1 80 at infinity, 3 8 Lattes map, 383 chain rule used to calculate, 1 9, 47 Chebyshev polynomial, 4 1 condition for unequal formal and primitive periods, ! 5 1 , 226 conjugacy invariant, 1 9 differential one-form definition, 1 9 equal to one, 225, 230 flexible Lattes map, 1 86, 358, 382 Galois conjugates, 1 80, 1 84 integral over Q[Rat d ], 1 80, 1 85 is PGL invariant, 1 8, 1 80, 1 85 2 minimal polynomial, 225, 227 not equal to one, 4 1 of a Lattes map, 34 of critical periodic point, 38 of degree two rational map, 1 90, 233 of fixed point of Chebyshev polynomial, 332, 380 in characteristic p, 3 8 1 of fixed point of Lattes map, 382 of z2 + bz, 1 82 rigid Lattes map, 366, 384 root of unity, 1 5 1 , 226 symmetric polynomial of, 1 80, 1 83, 1 87, 232, 367 multiplier polynomial, 225, 227 resultant with cyclotomic polynomial, 226 multiplier spectrum, 1 82, 1 87 flexible Lattes map, 356 formal, 1 82 isospectral family, 1 88, 356, 362
496 multiplier spectrum (continued) of Lattes map, 1 86 of z2 + bz, 1 82 of zd , 1 82 Natural numbers, 258 n1h -dynatomic polynomial, 1 49, 1 8 1 homogeneous of degree vd (n), 1 8 1 Neron local height, 1 04 nested sequence of closed disks, 295, 322 associated seminorm, 297, 322 equivalent, 322 neutral fixed point, 322 neutral periodic point, 1 9, 47 Newton polygon, 248, 249, 252, 275, 276 determines roots, 249 line segment, 249 n-multiplier spectrum, 1 82 formal, 1 82 of zd , 1 82 no wandering domains disk conjecture, 284 disk theorem, 284, 285 theorem, 28, 276 nonarchimedean absolute value, 44, 83, 242 morphism is open map, 433 nonarchimedean analysis, 242 nonarchimedean Cauchy residue formula, 314 nonarchimedean disk component has simple form, 278 nonarchimedean hyperbolic map, 279, 285, 3 1 7 nonarchimedean inverse function theorem, 3 1 2 nonarchimedean chordal metric, see v-adic chordal metric noncommutative monoid of Lattes maps, 382 noncommuting involutions, 4 1 0 three, 437 nondegenerate automorphism, 226 nonrepelling cycles, number of, 27 nonrepelling fixed point, 255, 3 1 4 nonrepelling periodic point, 362 in Fatou set, 256 nonsingular elliptic curve, 337 nonsingular K3 surface, 438
Index norm, 247, 288 Gauss on Cv[z], 296 of a holomorphic function, 247 of derivative, 252, 3 1 3 semi-, 296 sup, 288 normal family, 25 normal forms lemma, 1 90, 233 normalized coordinates, 49 normalized Markoff triple, 437 normalized rational map, 5 1 , 75 normalized valuation, 44, 249 Northcott's theorem, 94, 407, 423 false for rational maps, 1 36 notation, 7 n-period polynomial, 1 49, 1 8 1 homogeneous of degree dn + 1 , 1 8 1 Nullstellensatz, 90, 92, 408 number field algebraic closure, 85 completion of, 83 extension formula for absolute values, 83, 84 product formula, 83 ring of integers, 83 ring of S-integers, 83 numerical criterion for (semi)stability, 1 78 0(1), 93 octahedral group, 1 97 one coboundary, 20 1 , 202, 209 trivial when extended, 203 one cocycle, 201 , 202 corresponds to twist?, 203 determines field of moduli, 209 extension from Aut(¢) to PGL2, 203 of twist, 20 I open branch of Berkovich disk, 300 open disk, 242 closed under addition, 243 image of D(O, 1) by PGL2, 3 1 7 in JP' 1 , 243 is closed, 243 is maximal ideal, 243 rational radius, 243, 278 standard collection, 278 open map, 10, 253, 3 1 2
Index open map (continued) morphism is, 433 rational map is, 1 0, 24 open set base of for Gel'fond topology, 300 disjoint union of disk components, 277, 3 1 7 disk connected, 3 1 7 iterates omit at most one point, 266 orbit, 1 backward is dense in Julia set, 267 finite iff canonical height zero, 423, 43 1 Galois-invariant on K3 surface, 436 height zeta function, 436 integral points, 3, 1 08, 1 09, 142, 1 43 construction of many, 1 1 0, I l l , 143, 375 cutoff for, 375 uniform bound for, 1 12, 372, 385 integrality estimate for points in, 1 12, 145 number over a finite field, 430 of critical points, 280 is bounded, 26, 1 65 is unbounded, 26 of Galois conjugate, 137 of open set omits at most one point, 266 product of multipliers, 19, 47 S-integral points in, 1 43 two sided, 432 under collection of rational maps, 1 43 Zariski dense on K3 surface, 435 orbit-counting function, 432 orda, 1 2 order in quadratic imaginary field, 341 in quatemion algebra, 341 of a holomorphic function, 244 of a meromorphic function, 244 ordp , 82 of a polynomial, 2 1 8 Ostrowski's theorem, 44, 3 1 2 p, Weierstrass function, 34, 345 p-adic absolute value, 44
497 p-adic Cauchy residue formula, 3 1 4 p-adic hyperbolic map, 279, 285, 3 1 7 p-adic inverse function theorem, 3 1 2 parabolic component, 28 path-connected, 304 Berkovich disk is, 300 same as disk-connected over C, 277 path metric, 306, 323 does not induce Gel'fond topology, 323 Pell equation, 106 perfect set Berkovich Julia set is, 307, 3 1 1 Julia set is, 267 period exact, 150 formal, 1 50 of a point, 1 , 1 8 primitive, 1 50 period polynomial, 149, 1 8 1 dehomogenized, 149 for z 2 + c, 156 generalized, 227 homogeneous of degree dn + 1, 1 8 1 multiple root, 230 order at P (ap (n ) ), 1 5 1 , 226 periodic disk component, 283 periodic domain, 28 periodic point, 1 algebraic family induces map to dynatomic curve, 1 59 algebraic family of with given formal period, 1 59 algebraic properties, 3 attracting, 1 9, 47, 326, 362 for good separable reduction, 64 iff reduction is critical, 78 in Berkovich space, 3 1 0 in Fatou set, 22, 40 basin of attraction, 3 1 0 canonical height zero, 99, 43 1 Chebyshev polynomial, 4 1 closure contains Julia set, 273 closure of repelling equals Julia set, 274, 3 1 1 condition for unequal formal and primitive periods, 1 5 1 , 1 65, 226 critical, 38
498 periodic point (continued) critical point with reduction a, 78 cycle of, 1 50, 226 cyclotomic, 95 dense for shift map, 3 14 differential one-form definition of multiplier, 1 9 discrete, 428 dynamical unit, 69, 7 1 , 72, 79, 1 32, 1 33 dynamics on finite set, 7 dynatomic field, 123 equation of, 1 22 equidistribution, 6 every is attracting, 78 exact period, 1 8, 38, 1 50 field generated by, 1 22 finitely many on K3 surface, 435 finitely many rational, 8 finiteness for regular affine automorphism, 400 finiteness over number field, 66, 94, 407 finiteness over Q, 3 flexible Lattes map, 358, 382 form subgroup, 8 formal period, 1 8, 39, 1 50 moduli problem, 1 5 8 Galois theory o f field generated by, 1 22 generalized dynatomic polynomial, 227, 229 generalized period polynomial, 227 immediate basin of attraction, 3 1 0 index summation formula, 20, 38, 255, 3 14, 380, 382 indifferent, 1 9, 47, 326 irrationally neutral, 1 9, 47 isolated, 402 Julia set is closure of nonrepelling, 27 Lattes map, 1 86 least period, 1 8 multiplier, 1 8, 19, 47 at infinity, 3 8 Lattes map, 35, 1 86 of critical, 38 power map, 326 nonrepelling, 362
Index for good reduction, 64 in Fatou set, 256 number of, 39 given period, 1 22 nonrepelling cycles, 27 of commuting maps, 8, 3 78 of power map, 326 of quadratic polynomial map, 96 of regular affine automorphism, 394, 428 of (z2 - z)/p, 263 of z2 + 1, 8 of z2 - 1, 8 of z d + a over finite field, 8 of z · · · (z - d + 1)/p, 3 1 5 on subvariety, 127 one indifferent implies infinitely many, 3 1 4 p-adic, 361 prime period, 18 primitive, 1 22 field generated by, 1 23 primitive period, 1 8, 1 50 product of multipliers in orbit, 1 9, 47 rational map can have infinitely many rational, 136 rational map has points of different periods, 1 54, 1 55, 1 96 rationally neutral, 1 9, 47 in hyperbolic component, 28 reduction of, 61, 62, 66, 78 regular affine automorphism, 400 relative abelian extension, 1 26 repelling, 1 9, 47, 326, 361 in Berkovich space, 3 1 0 i n Julia set, 22, 40 limit of attracting, 274 repelling in Julia set, 256 residual degree, 3 1 0 rigid Lattes map, 366, 384 set of is Galois-invariant, 1 23 shift map, 259 summation formula, 20, 38, 255, 3 14, 380, 382 superattracting, 1 9, 47 Type-1, 3 1 0 Type-11, 3 1 0 Type-IV, 321
Index periodic point (continued) uniform boundedness conjecture, 96, 1 35, 368 for zd , 1 3 7 for Lattt�s map, 370 v-adic distance, 69 periodic subvariety, 1 27 permutation group, 1 24 permutation polynomial, 5 Pern (¢), 1 8 PGL2 , 1 0 action of Galois, 203 automorphism group is subgroup, 234 cocycle gives twist of!Il>1 , 2 1 1 , 2 1 5 conjugation is algebraic group action, 1 73 equivalence, 1 95, 233 extension of cocycle to Aut ( 4>) , 203 finite subgroups, 1 97 invariant functions on Ratd , 1 75 is group variety, 1 70 is Rat 1 , 1 70 map from PSL2 , 1 75, 23 1 minimal resultant is invariant, 22 1 PGL2 (K) equivalence, 1 95, 233 quotient of Ratd by, 17 4 PGLn , 226 map from PSLn , 23 1 Picard group, 339, 403 action of involution, 4 1 8, 4 1 9, 438 degree map, 405 eigenvalue of involution, 42 1 , 438 exact sequence, 404 generated by hyperplane, 405 height map on, 408 intersection theory for K3 surface, 434 isomorphic to elliptic curve, 339 linear transform induced by involution, 420 of general K3 surface, 4 1 8 of lP'2 x lP'2 , 4 1 8 ofJP'N X JP'M , 405, 406, 409 pullback homomorphism, 405 point absolute value, 90, 288 at infinity, 10, 1 1
499 on elliptic curve, 336 field of definition, 86, 1 22 polynomial action on Berkovich, 305 radius of in Berkovich space, 296 reduction modulo a prime, 48 sup norm, 90, 288 pole, 244, 403 polynomial absolute value of, 9 1 , 288 bidegree, 405, 4 1 0 bihomogeneous, 405, 4 1 0 bounded seminorm o n ring of, 296 cyclotomic, 1 48, 224 degree of, 389 dynatomic, 1 48, 1 49, 1 8 1 elementary symmetric, 87 Gauss norm, 296 height of roots, 1 36, 1 38 homogeneous, 1 0 homogeneous ideal, 89 local canonical height, 1 40 normalized, 5 1 number of monomials, 1 36 ordp, 2 1 8 permutation, 5 resultant of two, 53 sup norm, 91, 288 trihomogeneous, 437 polynomial map, 1 7 cannot have Herman ring, 28 canonical measure on Berkovich space, 306 Chebyshe� 29, 95, 329 in characteristic p, 3 8 1 commuting, 378 complete invariance of component at infinity, 40 condition for unequal formal and primitive periods, 1 65 cyclotomic preperiodic points, 95 dynamical unit, 7 1 , 132 of generic monic, 1 46 effect on absolute value, 288 filled Julia set, 74, 140 finitely many rational periodic points, 8 fixes Gauss point, 321 FOM = FOD, 2 1 5
500 polynomial map (continued) good reduction, 3 2 1 , 322 has primitive periodic points of different periods, 1 55 iff ill ( ¢) = 2, 232 iterate is a, 1 7, 37, 43 1 iterates of open set omit at most one point, 266 Latt(�s map not conjugate to, 3 8 1 local canonical height, 1 03, 1 40, 141 monomial, 29, 95, 325 multipliers not equal to one, 4 1 nontrivial automorphism group, 234 number of nonrepelling periodic cycles, 27 on Berkovich space, 304 power, 95, 325 power map, 29, 95, 234, 32 1 , 325 in characteristic p, 380 on Berkovich space, 32 1 radius of image disk, 321 sends disks to disks, 305, 3 1 2 wandering disk domains exist over Cp, 286 polynomial type rational map, 1 43 positive divisor, see effective divisor positivity of height, 408 postcritica1 set, 280, 353 potential good reduction, 371 potential theory on Berkovich space, 323 Hsia kernel, 323 power map, 29, 95, 325 automorphism group, 234, 327 in characteristic p, 380 commuting maps, 326, 378, 380 dynamical unit, 328 in characteristic p, 380 multiplier of periodic point, 326 on Berkovich space, 321 periodic point, 326 preperiodic point, 326 twist, 206, 328, 380 power series, 244 coefficients uniquely determined by function, 3 1 2 converges i ff an ---> 0 , 244 distance between image disks, 268 family omitting one point, 264, 3 1 5
Index formal, 5 maximum modulus principle, 250 Monte! theorem, 264, 27 1 , 3 1 5 Newton polygon, 249 norm of derivative, 252, 3 1 3 periodic points in ring of, 78 product of, 3 12 sends disks to disks, 252, 305 Tate algebra, 297 preperiodic disk component, 283 preperiodic disk domain, 284, 285 preperiodic domain, 28, 276 preperiodic point, 1 canonical height zero, 99 critical, 26, 1 66 cyclotomic, 95 dynamics on finite set, 8 finitely many, 38 finiteness over number field, 94, 407 Galois equidistribution, 128 generalized dynatomic polynomial, 227, 229 generalized period polynomial, 227 isolated, 8 large ramification index implies is, 37 Lattes map, 32, 352 of abelian group is torsion, 2, 4 1 , 326 of commuting maps, 8, 38, 378 of multiplicative group, 326 of power map, 326 on subvariety, 1 27 reduction of, 6 1 Type-IV, 321 uniform boundedness conjecture, 96, 1 35, 368 for z d , 137 for Lattes map, 370 preperiodic subvariety, 1 27, 1 28 prerequisites, 7 prime divisor, 403 degree, 405 local ring, 403 prime is sum of two squares, 200, 236 prime period, see exact period primitive period, see exact period primitive periodic point, 1 22
Index primitive periodic point (continued) dynatomic field, 123 field generated by, 123 set of is Galois-invariant, 1 23 principal divisor, 320, 339, 403 has degree zero, 339 probability measure, 1 27 Borel, 127 canonical, 127 canonical on Berkovich space, 306 invariant, 127 product formula, 83, 84, 290 profinite topology, 2 1 1 projection map, 287 of K3 surface to lP'2 , 4 1 0 projective general linear group, 1 0 projective line automorphism group, 10 Berkovich, 302 change of variables, 1 1 chordal metric, 1 1 , 35, 45, 144, 243 construction of, 1 0 integer points on, 1 06, 108 inversion is isometry, 35 linear fractional transformation, 10 morphism of over a variety, 1 7 1 moving three points, 1 0, 36, 5 1 not locally compact over Cp, 294 over a variety, 1 7 1 point at infinity, 1 0, 1 1 rational map, 1 0 Riemann sphere, 1 0 Riemann-Hurwitz formula, 1 3 , 2 1 7 spherical measure, 36 twist, 200, 2 1 1 , 236 twist trivial iffC(K) # 0, 2 1 5 universal map of degree d, 1 7 1 projective space absolute value of a point, 90 action of Galois group, 1 22 compact iff K locally compact, 243 cross-ratio, 7 1 , 79, 1 32, 348 degree map on divisors, 405 field of definition of point, 86, 1 22 finitely many points of bounded height, 86, 407 height, 82, 84, 407 hyperplane generates Picard group, 405
501 monomial map, 428 morphism, 89, 388 is open map, 433 moving N + 2 points, 75 nondegenerate automorphism, 226 normalized coordinates, 49 number of points of bounded height, 1 35, 1 36, 44 1 probability measure, 127 product of two, 405, 406, 409, 4 1 8 K3 surface in, 4 1 0 surface with involutions in, 436 projection map to, 287 rational map, 89, 388 Segre embedding, 406, 409 projective special linear group, see PSL PSL, 175 map to PGL, 175, 23 1 stable locus, 176 Puiseux series, 5 pullback divisor, 404 linearly equivalent, 405 pushforward divisor, 404 QRT family of integrable maps, 443 quadratic formula, 261 quadratic imaginary field, 341 class number, 350, 368 fractional ideals in, 350 ideal class group, 350 quadratic map algebraic family induces map to dynatomic curve, 1 59 algebraic family of, 1 59, 1 94, 230 associated multiplier polynomial, 227 automorphism group contains J.L2 , 235 automorphism of order two, 96, 235 bifurcation point, 165 bound for periodic points, 96 canonical height, 137, 138 conjugation to z 2 + c , 156 degenerate, 1 94 dynamical unit, 1 34, 145, 146 dynatomic curve, 1 56, 1 57, 1 6 1 dynatomic field, 123 ramification, 1 3 1
Index
502 quadratic map (continued) dynatomic polynomial, 39 explicit formula for a 1 ( ¢), 1 8 1 generalized dynatomic polynomial, 229 local canonical height, 1 4 1 Mandelbrot set, 26, 1 65 Misiurewicz point, 1 66 minimal polynomial, 230 orbit of critical point, 26, 1 65 parameter space has dimension three, 1 68 polynomial iff a1 ( ¢) = 2, 232 rational periodic points, 1 3 7 reducible dynatomic polynomial, 235 space of conjugacy classes has dimension one, 1 68 with preperiodic critical point, 1 66 quadratic twist, 198 rational map, 200 quasiminimal Weierstrass equation, 371 difference from global minimal, 385 quatemion algebra, 341 , 378 quotient curve, 1 6 1 critical values, 353 dynatomic, 1 6 1 elliptic, 346 quotient variety, 41 0 by finite group, 1 6 1 by infinite group, 1 6 1 , 1 74 categorical, 1 79 field of definition, 1 6 1 geometric, 1 79 Rat d modulo PSL2, 1 75 rational points on, 1 77 semistable, 1 79 stable, 1 79 Radical of an ideal, 89 radius ofBerkovich disk, 297, 301 ofBerkovich point, 296 of Type-IV point is positive, 296 ramification field generated by torsion on elliptic curve, 363 in dynatomic field, 1 29, 1 3 1
in field generated by torsion on elliptic curve, 344 ramification index, 12, 36, 340 automorphism invariant, 234 divisibility by p, 37 effect on chordal metric, 1 1 5, 144 large implies preperiodic, 37 of iterate, 1 1 7 of Lattt:s projection, 384 sum of over inverse image of a point, 15 ramification point, see critical point Rat 1 is PGL2, 1 70 Rat 2 , 1 70, 1 74 normal forms lemma, 1 90, 233 subvariety with Aut = JL 2 , 235 Rat d , 1 69 affine coordinate ring, 1 69, 1 74 coordinate ring contains ai n) and ;(n) 1 83 232 generic map not very highly ramified, 23 1 geometry of boundary, 1 70 in PSL2-stable locus, 1 76 is a rational variety, 232 is subset of JID 2 d + 1 , 1 69 map induced by Schwarzian derivative, 232 map to M d , 1 74, 1 75 multiplier 1 is Zariski closed subset, 230 PGL2-invariant functions on, 175 quotient by PGL2 , 1 74 quotient by PSL2 is a variety, 1 75 semistable rational maps, 1 78 stable rational maps, 1 78 subvariety with nontrivial automorphisms, 1 99, 234 universal map over, 1 7 1 Ratd, 1 78, 1 79 Ratds, 1 78, 1 79 rational automorphism, 430 rational function, 9 divisor, 320, 403 finitely many nonzero residues, 3 14 formal derivative, 1 2, 3 1 3 induces rational map, 1 0 integer values of, 1 06, 1 08 '
'
'
Index rational function (continued) maximum modulus principle, 250 pole, 403 principal divisor, 320, 403 residue, 3 1 3 Schwarzian derivative, 23 1 space of associated to a divisor, 406 Taylor series, 1 2 zero, 403 rational map, 1 0, 89, 388 action of automorphism on critical point, 234 action of Galois, 207, 2 1 1 acts on X1 (n ) , 1 6 1 affine minimal, 1 12, 372, 385 algebraic entropy, 397 algebraically integrable automorphism, 430 Arzela-Ascoli Theorem, 25 associated field extension, 207 associated Galois subgroup, 207 automorphism group, 1 96 contains J.L2, 235 is finite, 1 96 of conjugate, 1 96, 234 order two, 235 S3 , 1 97 {z ± 1 } , 205, 234 automorphism of order two, 96, 1 97, 1 98, 200 az + b/z, 96, 2 3 5 backward orbit, 1 09, 1 42 bad reduction, 77, 239 bicritical, 233 Bogomolov conjecture, 1 29 can have infinitely many rational periodic points, 1 3 6 canonical height, 99, 287 lower bound, 22 1 canonical measure, 127 on Berkovich space, 306 cocycle gives twist?, 203 coefficients well-defined up to homogeneity, 1 69 commuting, 378 have same canonical height, 13 7 have same preperiodic points, 38 with polynomial, 378 completely invariant set, 1 7, 266
503 contains Berkovich Julia set, 3 1 1 condition for unequal formal and primitive periods, 1 5 1 , 1 65, 226 conjugation is algebraic group action, 173 coordinate functions of iterate, 149, 226 critical point, 1 2, 284, 353 distinct modulo p, 237 periodic, 38 critical value, 353, 382 cutoff for integral points in orbit, 375 cyclic automorphism group, 1 97, 204 defined at P, 89 degenerate quadratic, 1 94 degree, 10, 89, 388, 389 of composition, 392 of regular, 394 degree generating function, 397, 428 determined by 2d + 2 coefficients, 1 69 domain of quasiperiodicity, 3 10, 311 dynamical degree, 397, 428 dynamical unit, 7 1 , 72, 79, 1 32, 133 of generic, 1 46 dynamically affine, 376 dynatomic field, 1 23 dynatomic polynomial, 39, 1 48, 1 49, 1 8 1 is a polynomial, 1 5 1 , 226 effect on absolute value, 288 effect on height, 90, 388, 398, 407, 409 equicontinuous, 22 equicontinuous family, 254, 265, 313 equi-Lipschitz family, 3 1 3 etale, 353 even degree implies FOM = FOD, 215 expanding on average, 3 6 family omitting two points, 265, 315 Fatou set, 22, 24, 255
504 rational map (continued) field of definition, 206, 236 field of moduli, 1 77, 207, 236 not field of definition, 208, 2 1 3, 236 finitely many preperiodic points, 38 finitely many with bounded height and resultant, 23 7 fixed point, 1 49 formal n-multiplier spectrum, 1 82 Galois equidistribution, 1 28 generalized dynatomic polynomial, 227, 229 generalized period polynomial, 227 generic not very highly ramified, 23 1 global minimal model, 222 over Q, 236 global minimal resultant, 220 global 5-minimal model, 237 good reduction, 58, 59, 2 1 8, 239, 362, 383 Berkovich Julia set, 307, 3 1 1 , 322 good reduction =? same degree, 58 graph, 2 1 6 Green function, 2 8 8 has d + 1 fixed points, 1 49 has dn + 1 points of period n, 1 50 has vd (n) points of formal period n, 1 50, 224 has nonrepelling fixed point, 255, 3 14 has primitive periodic points of different periods, 1 54, 1 55, 1 96 height of, 143 in M d , 221 , 237 Herman ring, 28 homogenization, 389 hyperbolic, 279, 285, 3 1 7 image of disk component may not be disk component, 283 indeterminacy locus, 389, 427 induced, 389 induced map on disk components, 283 integer points in orbit, 3, 1 08, I 09, 1 42, 143
Index integrality estimate for points in orbit, 1 1 2, 145 invariant measure, 1 27 inverse function theorem, 1 1 5, 1 44, 3 12 involution, 4 1 0, 436 is a morphism if. . . , 89, 388 is continuous, I 0, 24, 253 is open, 10, 24, 253, 3 12 isospectral family, 1 86, 1 88, 362 iterate is a polynomial map, 1 7, 37, 43 1 iterates of open set omit at most one point, 266 iteration of regular is regular, 394 itinerary of a point, 258, 260 Julia set, 22, 24, 255, 284 lift of, 389 to affine space, 287 linear conjugation, I I , 1 73 minimal model, 220 minimal resultant, 23 7 is PGLz (K)-invariant, 221 of conjugate, 2 1 9 monomial, 428 Monte! theorem, 25, 265, 3 1 5 most have no automorphism s, 1 99, 234 most have no wildly critical point, 284 multiplier, 1 8 at infinity, 3 8 of fixed point, 1 8, 47, 1 80 of periodic point, 1 9, 47 spectrum, 1 82, 1 87 summation formula, 20, 38, 255, 3 14, 380, 382 new coordinate functions of conjugate, 2 1 8 no critical Julia points, 279 no twists if trivial automorphism group, 1 98 no wandering disk domains, 284, 285 no wandering domains, 28, 276 nonrepelling periodic point in Fatou set, 256 nonrepelling periodic points, 64 normal family, 25
Index rational map (continued) normal form for degree two, 1 90, 233 normalized form, 5 1 , 75 n-period polynomial, 1 8 1 number of critical points, 14, 37 number of nonrepelling periodic cycles, 27 number of periodic points, 39 of polynomial type, 1 43 omitting three points, 25 on Berkovich space, 305 orbit of collection of, 1 43 orbit with many integral points, 1 10, 1 1 1 , 143, 375 parabolic component, 28 period polynomial, 1 49 periodic point, 394 field generated by, 1 22 index summation formula, 20, 38, 255, 3 1 4, 380, 382 periodic subvariety, 1 27 PGL2-equivalence, 1 95, 233 PGL2 ( K ) -equivalence, 1 95, 233 polynomial if has totally ramified fixed point, 1 7 postcritical set, 353 power map, 29, 95, 234, 32 1 , 325 in characteristic p, 380 preperiodic subvariety, 1 27, 128 primitive periodic point, 1 22 product of primes of bad reduction, 22 1 pullback of divisor, 404 linearly equivalent, 405 pushforward of divisor, 404 quadratic is polynomial iff (Tl ( ¢) = 2, 232 quadratic twist, 200 ramification index, 12, 36 of iterate, 1 17 ramification point, see critical point recurrent point, 284, 3 1 0 reducible dynatomic polynomial, 235 reduction modulo a prime, 52 repelling periodic point in Julia set, 256 resultant, 56, 58, 75, 1 12, 2 1 8, 372
505 of composition, 77 of conjugate, 76 of iterate, 1 84 Riemann-Hurwitz formula, 1 3, 2 1 7 self similarity, 1 96 semistable, 179 separable, 37 separable reduction, 64 set of (Rat d ), 1 69 Siegel disk, 28 S-integer points in orbit, 1 43 space of conjugacy classes of, 1 74 stable, 179 strictly expanding on Julia set, 279, 317 symmetric polynomial of multipliers, 1 83, 1 87, 232, 367 Taylor series, 1 90 totally ramified fixed point, 37 totally ramified point, 12 twist, 1 97 same image in M d , 1 98, 22 1 with good reduction outside S, 237 uniform bound for integral points in orbit, 1 12, 372, 385 uniformly continuous family, 254, 3 13 uniformly Lipschitz, 254, 256, 3 1 3 universal of degree d, 1 7 1 valuation of minimal resultant, 2 1 9 very highly ramified, 23 1 , 284 wandering disk domains exist over ICp, 285, 286 Weierstrass class, 223, 237 wildly critical point, 284 with Aut(¢) = S3, 234 with (nz in Aut(¢), 205, 234 with same minimal discriminant, 22 1 with two critical points, 233 z + b/z, 1 97, 1 98 rational numbers, 7 absolute values on, 44, 3 1 2 rational radius, 243, 248, 252, 278, 295, 317 rational variety, 232 rationally neutral periodic point, 1 9, 47 in hyperbolic component, 28
Index
506 Raynaud's theorem, 127 real fixed point, 427 realizable sequence, 6 recurrent point, 3, 284, 3 1 0 critical i n Fatou set i s periodic, 284, 317 critical i n Julia set, 284 reduction good, see good reduction modulo p on elliptic curve, 342 of a point modulo a prime, 48 of a rational map, 52 of periodic point, 6 1 , 62, 66, 78 of torsion on elliptic curve, 62, 343, 364 relation to v-adic distance, 49 respected by fraction linear transformation, 50 separable, 64 regular affine automorphism, 39 1 , 394, 427 algebraically stable, 396 canonical height, 43 1 zero iff finite orbit, 43 1 degree, 394 of inverse, 394, 395 height inequality, 3 99, 43 1
height of wandering point, 432 indeterminacy locus, 394 iterate is regular, 394 number of periodic points, 428 orbit-counting function, 432 periodic points, 394, 400, 428 regular function on Rat d , 1 69, 1 74 repelling periodic point, 1 9, 47, 326, 3 6 1 closure equals Julia set, 274, 3 1 1 dense in Julia set, 23, 263, 3 1 5 in Berkovich space, 3 1 0 in Julia set, 22, 40, 256 limit of attracting, 274 of Type II, 3 1 0 one implies infinitely many, 274 representation of Galois on elliptic curve torsion, 344 residual degree, 3 1 0 residue abstract, 3 1 3 at simple pole, 3 1 3 of a rational function, 3 1 3
rational function has finitely many nonzero, 3 14 residue fixed-point index, 38 residue formula, 3 1 4 resultant, 53, 1 07 chordal metric formula, 76 determinant formula, 54 homogeneity of, 54, 2 1 9 measures expansion, 56, 76 minimal, 2 1 9 nonvanishing defines Ratd , 1 69 normalized, 77 of a composition of maps, 77 of a linear fractional transformation, 76 of a rational map, 56, 58, 75, 1 12, 2 1 8, 372 of conjugate, 76 of coordinate functions of iterate, 227 of division polynomial, 383 of dynatomic polynomial, 1 65 of iterate, 1 84 of quadratic polynomial, 75 product of differences of roots, 53, 75 properties of, 53
transformation formulas, 75, 1 73, 219 unit iff good reduction, 58, 2 1 8 with a linear polynomial, 75 reversibility, 429 reversible affine automorphism, 429 composition of involutions, 429 Henan map, 430 number of orbits over a finite field, 430 Riemann sphere, see projective line Riemann surface, 3 7 Riemann-Hurwitz formula, 37 V - E + F formula, 37 Riemann-Hurwitz formula, 13, 65, 1 09, 1 17, 2 1 7, 340, 347, 379 application to finite invariant set, 1 6, 266 for curves, 1 5 , 37 is local-global formula, 1 5 weak version, 1 5 Riemann-Roch theorem, 2 14, 337
Index rigid analytic connectivity, 276 rigid analytic space, 298 rigid LaW:s map fixed points, 366 good reduction, 383 periodic points, 366 ring of integers, 83 ring of S-integers, 83 Ritt's theorem, 378 Rivera-Letelier classification theorem, 3 1 0 Fatou domain classification, 240 strong Monte! theorem, 3 1 1 root ofholomorphic function determined by Newton polygon, 249 root of hyperbolic component of Mandelbrot set, 1 66 root of unity iff height one, 88, 1 00 multiplier, 1 5 1 , 226 primitive, 148 Roth's theorem, 1 04, 1 06, 1 20 is ineffective, 1 07 over number fields, 1 08 Schanual's theorem, 1 35, 44 1 scheme, 59, 303 good reduction, 59, 6 1 , 77 moduli, 1 79, 1 86, 1 89 multiplicative group, 29 Schwarzian derivative, 23 1 O' d , N , 1 87, 232 degree unknown for d = 3, 1 8 8 embeds M 2, 1 88 Segre embedding, 406, 409 self similarity, 1 96 seminorm action of polynomial map, 304 action of rational map, 305 associated to each Berkovich point type, 297 associated to nested sequence of disks, 297, 322 is norm for Type-II,-III,-IV points, 297, 321 on Cp [z], 296 properties, 321 set of bounded is Berkovich disk, 297
507 two variable homogeneous, 303, 305 semistable, 178 locus ofPSL2 action on IP'2d+l , 1 78 numerical criterion, 1 78 quotient, 1 79 rational map, 1 79 separable field extension, 37 separable rational map, 37 number of critical points, 37 sequence space, 258, 3 1 4 backward orbit dense, 3 1 4 backward orbit equidistributed, 3 1 5 left shift, 259, 3 1 4 metric on, 258, 3 14 shift map periodic points dense, 3 14 shift map topologically transitive, 314 Shafarevich conjecture, 442 Sheshadri's theorem, 1 79 shift map, 259, 3 1 4 as itinerary map, 260 backward orbit dense, 3 1 4 backward orbit equidistributed, 3 1 5 continuous, 259 is topologically transitive, 259 Lipschitz, 259 periodic points, 259 periodic points dense, 3 14 properties of, 259 topologically transitive, 3 1 4 uniformly expanding, 259 Siegel disk, 28 Siegel's theorem 1 integer points on IP' minus 3 points, 1 06, 1 1 0 integrality of points on elliptic curves, 107 is ineffective, 1 07 over number fields, 108 ai (¢), 1 80 explicit formula for a 1 ( ¢), 1 8 1 is in Q[Md], 1 80 (J( n ) 1 83 explicit formula for a1 , a2 , 1 89 for Lattt:s map, 1 86 in Q[u1 , a1 for Rat2 , 1 89 integral over Q[Ratd], 1 85 is in Q[Md], 1 83 '
'
508
a-;n ) (continued)
isomorphism M2 � A2 , 1 88 of z2 + bz, 1 83 of z d , 1 83 n ) (;.(' , 1 83 for Lattt:s map, 1 86 in Q[o-1 , 17] for Rat2, 1 89 is in Q[M d ], 1 83, 232 of z2 + bz, 1 83 of z d , 1 83 simple algebraic group, 377 SL2, 1 75 small model ofBerkovich disk, 323 solenoid, 6 special fiber, 77 special linear group, 1 75 sphere V - E + F formula, 1 3 spherical measure, 36 spider algorithm, 1 68 stabilizer of point on K3 surface, 425, 434 stable locus, 1 76 2 of PSL2 action on lP' d+ 1 , 1 78 numerical criterion, 178 quotient, 1 79 rational map, 1 79 standard collection of disks, 277 strange attractor, 391 Sullivan no wandering domains theorem, 28, 276 sum of two cubes, 1 4 1 sum of two squares, 200, 236 S-unit in finite extension, 1 32 sup norm chordal, 269 Gauss norm is, 296 of a holomorphic function, 247 of a point, 90, 288 of a polynomial, 9 1 , 288 superattracting periodic point, 19, 47 supersingular elliptic curve, 378 support of a divisor, 403 surface involution induced by degree 2 map, 436, 437 K3, 41 0, 4 1 2 Kodaira dimension, 4 1 2 symbolic dynamics, 258, 3 14
Index backward orbit dense, 3 14 backward orbit equidistributed, 3 1 5 shift map periodic points dense, 3 1 4 shift map topologically transitive, 314 symmetric group, 1 24 rational map with Aut() = S3, 234 wreath product with cyclic group, 125 symmetric polynomial elementary, 1 80 of multipliers, 1 80 symmetric polynomial of multipliers, 1 83, 1 87, 232, 367 Szpiro conjecture, 221 Szpiro-Tucker theorem, 442 Tamely ramified critical point, 37 Tate algebra, 297 Gauss norm, 298 Tate construction of canonical height, 97 Tate module, 344 £-adic representation, 344 Tate's algorithm, 342 Taylor series, 12, 63, 67, 1 90 telescoping sum, 9 8 tetrahedral group, 197 Thue's theorem, 105 is ineffective, 1 07 over number fields, 1 08 Thue-Mahler theorem, 108 topological equicontinuity, 306 topological group, 8 topological space disk component, 2 77, 3 1 7 disk connected set, 3 1 7 dynamical system on, 3 equicontinuity, 306 generalized disk, 277, 3 1 7 path connected, 304 uniquely path connected, 304 topological wandering point, 108 topologically transitive map, 259, 263, 315 shift is, 3 14 topology, 6 on Berkovich disk, 299 profinite, 2 1 1
Index torsion point elliptic curve, 4 1 field generated by on elliptic curve, 344, 363 of abelian group is preperiodic, 2, 4 1 , 326 of multiplicative group, 326 torsion subgroup, 2 elliptic curve is equidistributed, 1 27 of elliptic curve, 32, 343, 352 torus, 33 totally disconnected, 43, 239, 277 Cp is, 239, 276, 294 Julia set, 23, 26 totally ramified, 1 2 fixed point, 1 6, 1 7, 3 7 totient function, 1 37, 1 64, 1 82 iteration of, 6 tree, Berkovich disk is, 298 triangle inequality, 44, 87 nonarchimedean, 44, 70, 242 ultrametric, 44, 242 uniform version, 9 1 trihomogeneous polynomial, 437 twist associated 1-cocycle, 201 cocycle corresponds to, 203 elliptic curve, 1 98 general theory, 1 99 none if Aut(¢) is trivial, 198 nth root, 204 of an object, 1 99 of Chebyshev polynomial, 336 of elliptic curve, 34 1 , 374 oflP'1 , 200, 2 1 1 , 236 trivial iff C(K) =f. 0, 2 1 5 o f power map, 206, 328, 380 of rational map, 1 97, 200 quadratic, 1 98, 200 same image in M d , 1 98, 22 1 set of injects into H 1 , 202, 236 with good reduction outside S, 23 7 twisted action, 2 1 1 Type-I point, 295 as nested sequence of disks, 322 attracting fixed point, 324 attracting periodic, 3 1 0 fixed point, 322
509 intersection of orbit with open set, 324 seminorm is not norm, 297 Type-II point, 295 as nested sequence of disks, 322 fixed point, 322 repelling periodic, 3 1 0 seminorm i s norm, 297, 321 Type-III point, 295 as nested sequence of disks, 322 fixed point, 322 seminorm is norm, 297, 321 Type-IV point, 295 fixed point, 322 has positive radius, 296, 3 21 is nested sequence o f disks, 295 line segment to Gauss point, 322 no fixed in A13 , 321 seminorm is norm, 297, 3 21 Ullmo's theorem, 1 29 ultrametric inequality, 44, 242 is an equality, 45, 242 uncountable Julia set, 267 Berkovich, 307, 3 1 1 unicity conjecture for Markoff numbers, 437 uniform bound for integer points on elliptic curve, 372 uniform boundedness for abelian varieties, 97 for elliptic curves, 97, 369 for preperiodic points, 96, 1 35, 368 of Lattes map, 370 of z d , 1 37 uniform continuity, 39, 254, 3 1 3 i s open condition, 254 uniformization of Mandelbrot set, 1 67, 168 uniformly Lipschitz, 39, 254, 256, 264, 265, 27 1 , 3 1 3, 3 1 5 is open condition, 254 uniquely path connected, 304 uniqueness of height, 408 unirational variety, 232 unit cyclotomic, 69, 72 dynamical, 69, 7 1 , 72, 79, 1 32, 133 in dynatomic field, 1 29
Index
510 unit group exact sequence, 404 universal cover, 346 unramified dynatomic field, 1 3 1 field generated by torsion on elliptic curve, 344, 363 isogeny map is, 340, 353 unsolved problem (**), 7 v-adic chordal metric, 45, 1 44, 243 effect of linear fractional transformation, 76 invariant maps for, 46 is a metric, 45 periodic points, 69 relation to reduction, 49 resultant measures expansion, 56 valuation, 44 discrete, 44 normalized, 44, 249, 403 valued field, 44, 240 complete, 240 completion, 241 disk, 242 extension of complete, 241 ultrametric inequality is an equality, 45 variety, 2, 89, 147, 1 59, 376 divisor, 403 double cover, 4 1 0 involution, 4 1 0 K 3 surface, 41 0 periodic, 127 preperiodic, 127, 128 quotient, 1 6 1 , 174 quotient by involution, 4 1 0 rational, 232 twist, 200, 202, 2 1 1 , 2 1 5, 236 unirational, 232 vector space associated to a divisor, 406 V - E + F formula, 1 3, 37 very ample divisor, 406 very highly ramified, 23 1 , 284 Veselov's theorem, 43 1 Wandering disk component, 283 wandering disk domain, 284, 285 exist over
