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(a.) = a. d • (c) G = JR NI ZN and if>(a. ) = do. mod ZN. 0.6. (a) Describe Per(if>, lQ) for the function if>(z) = Z 2 + 1. (b) Describe Per] if>, lQ) for the function if> (z) = Z2 - 1. (c) Let if>(z) E Z[z] be a monic polynomial of degree at least two. Prove that Per( if>, lQ) is finite. (Hint. First prove that Perf if>, lQ) C Z.) (d) Same question as (c), but now if>(z) E lQ [z] has rational coefficients and is not assumed to be monic. 0.7. Let if>(z) = z + liz and let a. E lQ'. Prove that 0 (0.) number of points that it can contain?
n Z is finite. What is the largest
Chapter 1
An Introduction to Classical Dynamics Classically, the subject of discrete complex dynamical systems involves the study of iteration ofpolynomial and rational maps on C or pI (C). The local theory of analytic iteration dates back to the 19th century, but the modem theory of global complex dynamics starts with the foundational works of Fatou [166, 167] and Julia [223] in 1918-1920. From that beginning has arisen a vast literature. Our goal in this book is to study number-theoretic questions associated to iteration of polynomial and rational maps. A standard technique in number theory is to attempt to answer questions related to a number field K by first studying analogous questions over each completion K; of K . In particular, the archimedean completions of K lead back to class ical dynamics over lR or Co In this chapter we describe some of the fundamental concepts and theorems in complex dynamics. These theorems prov ide both tools and templates for our subsequent study of dynamics over complete nonarchimedean fields in Chapters 2 and 5 and over number fields in Chapters 3 and 4. This chapter provides sufficient background and motivation for reading the remainder of this book , but the reader wishing to learn more about the beautiful subject of complex dynamics might look at one or more of the following texts: • •
A . Beardon [43], Iteration ofRational Functions. L. Carleson and T. Gamelin [95], Complex Dynam ics.
•
R. Devaney [132] , An Introduction to Chaotic Dynamical Systems.
•
1. Milnor [302] , Dynamics in One Complex Variable.
1.1 Rational Maps and the Projective Line A rational fun ction ¢(z) E C( z) is a quot ient of polynom ials
¢( z) = F(z) = ao + aIZ +
G(z)
bo + bIZ +
9
+ adz d + bdzd
10
1. An Introduction to Classical Dynamics
with no common factors. 1 The degree of¢ is deg¢ = max{degF,degG}. So if we write ¢ as above with at least one of ad and bd nonzero, then ¢ has degree d. We generally consider maps with deg ¢ :::: 2. A rational function of degree d induces a rational map of the complex projective line ]P1(C), (1.1) The map ¢ is a ramified d-to-l covering. The complex projective line is also known as the Riemann sphere, since topologically, and indeed real analytically, ]pI(C) is isomorphic to 8 z. Two very important properties of rational maps (1.1) are that they are continuous and open. 2 There are many equivalent (and useful) ways to construct the projective line, including and
]p1(C) ~
1, then it will expand. This observation prompts the following definitions.
Definition. Let a be a periodic point for a rational function ¢ E
q z), and let A", (¢)
be the corresponding multiplier. Then a is called
superattracting
if A", (¢) = 0,
attracting
if IA", (¢) I < 1,
neutral
ifIA",(¢)1 = 1,
repelling
if IA", (¢) I > 1.
(Neutral periodic points are also sometimes called indifferent.)
Remark 1.12. A periodic point is superattracting if its orbit contains a critical point, so Corollary 1.2 implies that a map of degree d can have at most 2d - 2 superattracting cycles. We will state a much stronger result below (Theorem 1.35(a)). Remark 1.13. A neutral periodic point is called rationally neutral if its multiplier is a root of unity, that is, if A", (¢ ) k = 1 for some k ?: 1. Otherwise, it is called irrationally neutral. 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.
20
1. An Introduction to ClassicalDynamics
Theorem 1.14. Let K be an algebraically closedfield and let ¢(z) E K(z) be a rational function ofdegree d ~ 2. Assume that for all P E Fix(¢). Then
1
L
PEFix(¢)
A()=1. 1- P ¢
Remark 1.15. The condition that Ap (¢) # 1 is equivalent to the condition that the fixed point P has multiplicity 1, so Theorem 1.14 holds provided ¢ has d + 1 distinct fixed points. If some fixed point has Ap (¢) = 1, then a similar, but more complicated, formula involving a higher-order index is true; see Exercise 1.17 or [302, §12]. Remark 1.16. More generally, we can apply Theorem 1.14 to an iterate ¢n to obtain a relation among the multipliers of the n-periodic points of ¢ provided that those multipliers are not equal to 1. Example 1.17. Let ¢(z) = z2
a=
+ c. The fixed points of ¢(z) are
1+vT=4C 2
(3
=
1- V1- 4c 2 '
and
00,
with corresponding multipliers Ao;{¢) = 2a, A{3(¢) = 2{3, and Aoo(¢) = O. (Note that the map ¢ is totally ramified at 00, so in particular 00 is a critical point.) The sum in Theorem 1.14 is
1 1 1 1 1 --+--+--= + +1=1. 1-2a 1-2{3 1-0 -VI-4c VI-4c ProofofTheorem 1.14. We prove the theorem under the assumption that K has characteristic 0, in which case the Lefschetz principle (see, e.g., [410, VI §6]) says that we can embed a sufficiently large subfield of K into C. It thus suffices to prove the theorem for K = C. (The case of characteristic p can be proven by reduction modulo p from the characteristic 0 case or by developing the theory of residues in characteristic p as described in Exercise 5.10.) To simplify the proof, we make a change of variables (i.e., conjugate ¢ by an element ofPGL 2(C» so that ¢(oo) # 00. For any rational function 'ljJ(z) E C(z), or more generally for any meromorphic function, the Cauchy residue theorem says that
L
~s('ljJ(z) dz) = O.
PEIl>! (iC)
For P = a E C in the affine plane, the residue is given by the usual formula
~~('ljJ(z) dz) = ~ 27rZ For P = 00, we substitute z with respect to w at a = O.
1
Iz-al=€
'ljJ(z) dz
for sufficiently small
E
> O.
= 1/wand use the same formula to compute the residue
21
1.3. PeriodicPoints and Multipliers We apply the Cauchy residue formula to the rational function
1
7/;(z) = ¢(z) - z ' whose poles are the fixed points of ¢. For any 0: E Fix( ¢) (note that 0: i=- (0) we observe that
¢(z) - z = {¢(o:) + A",(¢)(z - 0:) + O(z - 0:)2} - z =
{o: + A,:,(¢)(z - 0:) + O(z - 0:)2} - z
Taylor expansion, since ¢(o:) = 0:,
= (z - 0:) {Aa(¢) - 1 + O(z - o:)}.
Hence under the assumption that Aa (¢) i=- 1, the function q,(zl-z has a simple pole at z
= 0: and its residue Res ( z=a
is given by"
1
dz ) = _1 dz ¢(z)-z 21l"i Iz-al=E(z-0:){A a(¢)-1+0(z-0:)} 1
The differential q,(;)_ z has no other poles in Co In order to compute its residue at 00, we substitute z
= w- 1 and use d(w- 1 ) = -w- 2dw to compute 1
Res(
dz )=Res( d(w- ) )=Res( -dw ). z=oo ¢(z) - z w=o ¢(w- 1 ) - w- 1 w=o w(w¢(w- 1 ) -1) The assumption that ¢( (0) i=- 00 implies that the function w¢( w- 1 ) vanishes at w = 0, so this last differential has a simple pole at w = 0 and its residue is equal to
Res
w=o
-dW) -1 = lim w ( (w(w¢(w1) w---.o w(w¢(w1) -
1) -
1)
)
=
1.
Substituting these residue values into the Cauchy residue formula yields
o=
"L..J Res ( d Z ) = P ¢(z)-z
PEP! (iC)
L aEFix( q,)
L
aEFix(q,)
Res ( d Z ) + Res ( d Z ) z=a ¢(z)-z z=oo ¢(z)-z 1
A() +1. a ¢ - 1
Finally, we move the sum to the other side, changing A-I to 1 - A, to complete the proof of Theorem 1.14. 0 4In general, if'l,b(z) has at most a simple pole at o, then Res", ('l,b(z) dz) equals limz~", (z - n)'l,b(z).
22
1. An Introduction to Classical Dynamics
1.4 The Julia Set and the Fatou Set The notion of equicontinuity is central to dynamics. We begin by recalling the classical definition of continuity. A function ¢ : (5 1, P1) ---. (5 2, P2) between metric spaces is continuous at a E 51 iffor every f > 0 there exists a 0 > 0 such that
We extend this notion to a collection of functions from 51 to 52 by requiring that a single 0 work for every function in . This means that if {3 is chosen sufficiently close to a, then ¢( a) and ¢({3) will be close to one another for every map ¢ in .
Definition. Let (51' pd and (52, P2) be metric spaces, and let be a collection of maps from 51 to 52. The collection is said to be equicontinuous at a point a E 51 if for every e. > 0 there exists a 0 > 0 such that
P1(a,{3) < 0
===}
P2(¢a,¢{3) < f
for every
e E .
The collection is equicontinuous on a subset U C 51 if it is equicontinuous at every point of U. For an individual map ¢ ; 5 ---. 5 from a set 5 to itself, we say that ¢ is equicontinuous at a if the collection of iterates {¢n ; n 2 I} is equicontinuous at a.
Definition. Let ¢ be a map from a metric space to itself. The Fatou set of¢, denoted by F( ¢), is the maximal open set on which ¢ is equicontinuous. The Julia set of ¢, denoted by .:J( ¢), is the complement of the Fatou set. Note that equicontinuity is not, in general, an open condition (see Exercise 1.22), but that F(¢) is an open set by definition. Informally, one might say that points in the Julia set .:J(¢) tend to wander away from one another as ¢ is iterated, so ¢ behaves chaotically on its Julia set. Nearby points in the Fatou set F (¢ ), on the other hand, tend to stay together, so the behavior of ¢ on F( ¢) is considerably easier to analyze. Much of complex dynamics is devoted to studying the Fatou and Julia sets of rational maps. For further reading, see the textbooks cited at the beginning of this chapter and the two survey articles [74, 235].
Example 1.18. Let ¢(z) E C(z) be a rational map. If a is an attracting periodic point of ¢, then points close to a are all attracted to the points in the orbit of a, from which it is easy to conclude that they are in tile Fatou set. Similarly, it is easy to check that repelling periodic points are in the Julia set. See Exercise 1.27. The behavior near the neutral points is more complicated.
Example 1.19. Consider the map ¢ ; lP'1(C) ---. lP'1(C) given by ¢(z) = z": Then n ¢n(z) = zd , so lim ¢n(a) n-+oo
=
{o
00
if lal if lal
< 1, > 1.
1.4. Tbe Julia Set and tbe Fatou Set
23
It is easy to see that if lal 1: 1, then a E F (~,d(P(())P'(() = dp'(d().
(1.8)
But we know that p(z) is a meromorphic function on CI L, or equivalently, it is a meromorphic function on C with the periodicity property
p(z + w) = p(z)
for all wE L.
Differentiation gives the same relation p' (z + w) = p' (z) for the derivative. (Notice the analogy with the function e21ri z , which is holomorphic on C and invariant under the translations z f-t z + n for n E Z.) We evaluate at z = d(, use the fact that d( == ±( (mod L) to get p'(d() = p'(±() = ±p'((), and substitute into (1.8) to obtain the relation
35
Exercises ¢~ ,d(P(())p'( ( ) =
dp' (d( ) = ±dp' (().
Finally, canceling p' (O yields ¢~ d(P(( )) = ±d. In particular, (almost) every fixed point of ¢ E,d is expanding and th~s is in the Julia set. But much more is true. The relation (1.7) shows immediately that
so ¢'E,d = ¢E,dn. Thus the periodic points of ¢E,d of period dividing n are precisely the fixed points of ¢E,dn , and their multipliers are (almost) all equal to ±dn . This proves that (almost) every periodic point of ¢E,d is expanding , hence in the Julia set. Finally, we observe that the set of points in C j L satisfying
z ==
±~ z
(mod L )
for some n
= 1,2 ,3 , .. .
is dense in Cj L, so their image using the double cover p : Cj L -+ jp'l (C) is dense in jp'1(C). We have thus shown that .J (¢E,d) = jp'1(C). This proves the following theorem, which was published by Lattes in 1918 and provided the first known examples of rational maps with empty Fatou set. (See also [300, §6) for a discussion of earlier work by Schroder and Bottcher on elliptic functions and their associated rational maps, as well as an 1815 paper of Babbage in which he uses semiconjugacy to study the periodic points of certain maps.)
Theorem 1.43. (Lattes [260]) Let E be an elliptic curve, let d ~ 2 be an integer, and let ¢ E,d : jp'1(C) -+ jp'1 (C) be the rational map of degree d2 characterized by (1.6). Then and
Exercises Section 1.1. Rational Maps and the Projective Line
1.1. Define a mapping from the x y-plane (which we identify with C) to the unit sphere S2 in ]R3 by mapping z E C to the point z" E S2 such that the line through z and z" goes through the point (0, 0, 1). Notice that z" ---+ (0,0 ,1) as z ---+ 00, so if we set 00* = (0,0,1), we obtain a bijection jp'1(C) = CU {00} --->S2, Z f--> Z*, as illustrated in Figure I. I. Prove that with this identification, the chordal metric on jp'l (C) is given by p(z , w) = ~ I z* - w*l.
1.2. (a) Prove that the inversion map z ,... z -l is an isometry Ofjp'l (C) for the chordal metric (i) by a direct calculation using the formula for p; and (ii) by using the identification jp'1(C) ~ S2 described in Exercise I. I. (b) Let a, b E C satisfy lal 2 + Ibl2 = 1, and let j( z ) = (az - b)j (bz + a). Prove that P(J (z ),j(w)) = p(z ,w) for all z, w E jp'1(C) . (Hint. Although this can be proven directly, an alternative method is to show that j corresponds to a rigid rotation of S2, hence preserves chordal distances. For a nonarchimedean analogue, see Lemma 2.5.)
36
Exercises
1.3. Let ¢ : jp'l (C) -. jp'l (C) be a rational map. (a) Prove that there is a constant C (¢) such that ¢ satisfies a Lipschitz inequality for all a, (3 E jp'l (C).
p( ¢( a), ¢((3)) :::; C( ¢ )p(a, (3)
(b) Find an explicit formula for the Lipschitz constant C(f) in the case of a linear map
f(z)
= (az + b)/(cz + d) E PGL2(C).
1.4. Let (a, a', a") and ((3, (3', (3") be triples of distinct points in jp'l. Prove that there exists a unique automorphism f E PGL 2(C) satisfying
f(a)
= (3,
f(a')
= (3',
f(a")
= (3".
(Hint. First do the case that one ofthe triples is (0,1,00).)
Section 1.2. Critical Points and the Riemann-Hurwitz Formula
1.5. Let ¢( z) E q z) be a rational function. In the text we defined the ramification properties of ¢ at a point a E jp'l (C) provided that a -I 00 and ¢( a) -I 00. In general, let f E PGL2(C) be any linear fractional transformation, let ¢f = ¢ 0 I, and let (3 = I ( a ). If (3 -I 00 and ¢f ((3) -I 00, we say that ¢ is ramified at a if (¢f)' ((3) = 0 and we define the ramification index of ¢ at a to be
rIo
ea(¢)
r
= e{3(¢f).
(a) Assuming that none of a, (3, ¢(a), ¢f ((3) are equal to 00, prove that
Conclude that the ramification points of a rational map ¢ : jp'l -. jp'l are independent of the choice of coordinates, i.e., the vanishing of ¢' (a) is independent of the choice of f. (b) Show that the ramification index e a (¢) is well-defined, independent of the choice of f. (c) Under what conditions is it true that the derivative ¢' (a) is independent of the choice of f?
1.6. Let ¢( z) E
q z) be a rational function of degree d, say given by
¢(z) = F(z) = ao + alZ + + adzd. G(z) bo + bIZ + + bdzd (a) Prove that ¢ is ramified at z = 0 if and only if aobl = albo. (b) Prove that ¢ is ramified at z = 00 if and only if adbd-l = ad-Ibd. (c) Find a similar criterion for eo(¢) ::::: 3. Same question for Coo (¢) ::::: 3. 1.7. Let ¢ : jp'l -. jp'l be a rational function of degree d ::::: 2. Using the standard spherical measure on jp'l (C), normalized so total area is 1, prove that
1
1(¢n)'(z)1 dM(Z)
rv
n
d
as n -. 00.
1I'1(C)
Conclude that "on average," the map ¢ is expanding. 1.8. (a) Let C be a compact Riemann surface with 9 holes. Prove that if C is triangulated using V vertices, E edges, and F faces, then V - E + F = 2 - 2g. The number 9 is called the genus of C.
Exercises
37
(b) Let ¢> : C 1 ---> C2 be a finite map of degree d of compact Riemann surfaces, and let gi be the genus of C«. Give a topological proof of the Riemann-Hurwitz formula,
2g1 - 2 = d(2g2
-
2) +
L
ep(¢» - 1.
P EC ,
1.9. Let ¢> E iC( z ) be a rational map of degree d :::': 2. (a) Prove that 00 is a totally ramified fixed point of ¢> if and only if ¢> E iC[z]. (b) Let a E pI (iC) with a :f 00 . Prove that a is a totally ramified fixed point of ¢> if and only if (z - a)d / (¢>(z) - a) is in iC[z]. (c) Let a E pI (iC) be arbitrary. Prove that a is a totally ramified fixed point of ¢> if and only if there exists a linear factional transformation f E PGL2(iC) such that f-l (a) = 00 and ¢>f (z) E iC[z]. 1.10. Let K be a field of characteristic p > 0 and let ¢>(z) E K (z) be a rational function. (a) Show that the following are equivalent : (i) ¢>(z) ~ K(zP). (ii) The formal derivative ¢>' (z) is not identically zero. (iii) The field extension K (z)/ K (¢>(z)) is separable. If these conditions hold, we say that ¢> is separable. (b) Assume that ¢> is separable . Recall that the ramification index is defined by the formula e o (¢» = ordo (¢>(z ) - ¢>(a )) . Let T o (¢» = ordo (¢>' (z )) . If a is a critical point of ¢>, then we say that ¢> is tamely ramified at a if p f eo (¢» , and ¢> is wildly ramified at a ifp I eo (¢» . Prove that T o (¢» :::': eo (¢» - 1, with equality if and only if ¢> is either unramified or tamely ramified at a . (c) Prove that
2d - 2 :::':
L
To
(¢» :::':
oE II" (C)
L
(eo (¢» - 1).
o EI" (C)
(Hint. Mimic the algebra ic proof of Ihe Riemann-Hurwitz formula , Theorem 1.1.) (d) Deduce that a separable map has at most 2d - 2 critical points . (e) More generally, if the rational function ¢> is wildly ramified at t points , prove that ¢> has at most 2d - 2 - (p - l )t critical points.
1.11. Let K be a field of characteristic p > 0 and let ¢>(z) E K (z) be a rational function. (a) Suppose that ¢> is separable and that ¢>n E K[z] for some n :::': 1. Prove that ¢>2 E K[z]. Thus Theorem 1.7 is true in characteristic p for separable maps. (b) Let p = 2 and ¢>(z) = 1 + Z - 2. Prove that ¢>2 ~ K[z] and that ¢>3 E K[z]. Thus Theorem 1.7 need not be true in characteristic p for inseparable maps. (c) Suppose that ¢>2 ~ K[z] and that ¢>n E K[z] for some n :::': 3. Prove that ¢> has the form ¢> = (az q + b)/ (cz q + d), where q is a power ofp.
1.12. Let ¢> : pI ---> p I be a rational map of degree d :::': 2, and suppose that there is a point a E pI and an e > 0 such that
eo (¢>n ) :::': (1 +
fr
for infinitely many n :::': 1. Prove that a is a preperiodic point for ¢>, and that the eventual period of a is at most (2d - 2)/ f.
Exercises
38 Section 1.3. Periodic Points and Multipliers
1.13. We defined the multiplier of ¢ at a fixed point 0: to be the derivative ACt (¢) = ¢' (0:). This definition obviously must be modified if 0: = 00. In order to be compatible with Proposition 1.9, we must set Aoo(¢) = (¢f)'(f-l(oo)) for all f E PGL2(C). Show that taking f (z) = 1/z leads to the definition
Prove that Aoo (¢) is finite and may be computed without taking a limit by showing that the fraction on the righthand size is a rational function in iC(z) with no pole at z = 0, so it may be evaluated at z = o. 1.14. Let ¢ : jp'l ~ jp'l be a rational map. (a) Let P E jp'l bea point of exact period n for ¢ and suppose that ¢k(p) that n I k. (b) Prove that Per n (¢) is the disjoint union of Per;; (¢) over all min.
=
P. Prove
1.15. Let ¢ : jp'l ~ jp'l be a rational map of degree at least 2. (a) Let n > m ~ O. Prove that {p E jp'l : ¢n(p) = ¢m(p)} is a finite set. (b) Suppose that ¢l and ¢2 are rational maps of degree at least 2 and suppose that ¢l and ¢2 commute with one another, i.e., ¢l 0 ¢2 = ¢2 0 ¢l. Prove that PrePer(¢l)
= PrePer(¢2).
(Hint. Use (a) and apply Exercise 0.2.)
1.16. Suppose that 0: is a periodic point of ¢. Prove that the orbit of 0: contains a critical point of ¢ if and only if the multiplier ACt (¢) vanishes. 1.17. This exercise generalizes the multiplier sum formula described in Theorem 1.14. Let ¢(z) E iC(z) be a nonconstant rational map and let 0: E C be a fixed point of C. The residue fixed-point index of ¢ at 0: is the quantity
1 t( ¢, 0:) = -2.
1
1l"t
Iz-Ctl=-
z
_dz¢( )' z
where the integral is any sufficiently small loop around 0:. (a) IfACt (¢) =11, prove that 1 t(¢, 0:) = 1 _ ACt(¢)" (b) Prove that in all cases, the index t(¢, 0:) is independent ofthe choice oflocal coordinate at 0:. (c) Prove the index summation formula
L CtEFix(,p)
t(¢, 0:)
= 1.
39
Exercises (d) Suppose that An (¢J)
= ¢Jt (0:) = 1 and that ¢J" (0:) =I O. Provethat ~(¢J, o:)
=
¢J"' (o:) / 6 . (¢J"(0:) /2) Z
In other words, if the series expansionof ¢J( z) around z
¢J(z)
= 0: + (z -
= 0: has the form
0:) + A(z - o:) z + B( z - 0:)3
+ ...
with A
=I 0,
then ~ ( ¢J,o:) = BIA z .
(e) The polynomial¢J( z) = z + Az z + z3 has fixedpoints {O, -A, 00 }. The point at infinity is superattracting, so Aoo (¢J) = O. Use the formula in (d) and the summation formula to compute L A(¢J) , and then check your answer by computing LA (¢J) directly.
q z) be a rational function of degree d ~ 2. (a) Prove that #Pern(¢J) ~ d" + 1. (b) Prove that # Pern(¢J) --> 00 as n --> 00 . (c) Prove that Per~* (¢J) is nonempty for infinitelymany values of n. (d) * More precisely, prove that ifPer~* ( ¢J) is empty, then (n , d) is one of the pairs
1.18. Let ¢J( z) E
(n, d) E {(2, 2), (2, 3), (3, 2), (4, 2)}. 1.19. Let ¢J(z) E iC[z] be a polynomial of degree d ~ 2.
(a) Ifm I n, prove that the polynomial ¢Jm(z) - z dividesthe polynomial ¢In(z) - z. (b) Let f.l be the Mobius f.l function. Prove that for every n ~ 1, the rational function
~ (z)
=
II (¢Jm(Z) - zt
(n/ m)
min
is a polynomial. The polynomial ~( z ) is a dynamical analogue of the classical cyclotomic polynomial TImln (zm - l )JL(n/m). (c) Compute the first few polynomials ~ (z) for the quadratic polynomial ¢Jc(z) = ZZ+ c. (d) Let 0: be a root of ~ ( z ) . Prove that rj>"(o:) = 0:. Thus roots of ~ ( z ) are in P ern (rj» . (e) Let ¢Jc(z) = ZZ + c. Find a value of c such that ; (z) has a root whose exact period is strictly smaller than 2? (t) For n = 3 and n = 4, find all valuesof 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 function ¢J( z ) = az I (ZZ + b). Verify directly the formula in Theorem 1.14 (cf. Example 1.17). Section 1.4. The Julia Set and the Fatou Set
1.21. Let (S l, Pl) and (S z, pz) be metric spaces. A collection of maps from S , to Sz is said to be uniformly continuous iffor every € > 0 there exists a 8 > 0 such that
The family is said to be uniformly Lipschitz if there is a constant C PZ (¢J(O:), ¢J({3 )) ~ C· PI(0:, {3)
= C ( E .
Exercises
40
(a) If q, is uniformly continuous, prove that q, is equicontinuous at every point of 81. (b) If q, is uniformly Lipschitz, prove that q, is uniformly continuous. (c) Let q, = {4>n} n::o: 1 and suppose that for every point a E 8 1 , the limit
¢(a)
= n---;oo lim 4>n(a)
exists. If q, 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> : 8 --+ 8 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 proofthat the Fatou and Julia sets of 4> are completely invariant, that is, 4>-1 (F) = F = 4>(F) and 4>-I(J) = J = 4>(J). (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 4>( z) E qz] be a polynomial map, and let F 00 be the connected component of F( 4» containing the point 00. Prove that 4>-1 (Foo ) = F oo = 4>(Foo ) . In other words, for a polynomial map, the connected component of 00 is completely invariant for 4>. 1.25. Let 4>( z) = 1 - 2/ Z2 be the rational map from Example 1.31. (a) Let fez) = II z. Prove that (4)1)' (0) = 0 and conclude that 00 is a critical point of 4>. (b) Let fez) = zl(z - 1). Prove that (1/)'(0) = 0 and conclude that 0 is a critical point of 4>. 1.26. Let 4>(z) E Q(z) be a rational map of degree d 2: 2 with coefficients in Q, and let J (4); Q) = J (4)) n jp'1(Q) be the set of points in the Julia set whose coordinates are algebraic numbers. (a) Is J( 4>; Q) Galois-invariant? (b) Is J(4); Q) infinite? More generally, is J(4); Q) dense in J(4))? (c) ** Does J (4); Q) contain points that are not preperiodic? More generally, does J (4); Q) contain infinitely many nonpreperiodic points Pi such that the orbits of the Pi are disjoint?
Section 1.5. Properties of Periodic Points 1.27. Let a be a periodic point ofa 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 J (4>). 1.28. Let 4>(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 4>, compute their multipliers, and show directly that J(4)) is equal to the closure of Perre). (Hint. Write z = ei t + e- i t = 2cos(t), so 4>(z) = 2 cos(2t). Prove that 4>n(z) = 2 cos(2nt) and use this formula to study the dynamics of 4>.)
Exercises
41
q
1.29. Let 4>( z) E z] be a polynomial of degree d ;::: 1, and suppose that every a E Per n (4)) has multiplier Aq,(a) f. 1. Prove that the equation 4>n(z) = z has simple roots, and deduce that Per n (4)) contains exactly d" + 1 distinct points (including the point 00).
Section 1.6. Dynamical Systems Associated to AlgebraicGroups 1.30. This exercise describes algebraic properties of the Chebyshev polynomials Td (w). (a) Prove that T 2(w) = w 2 - 2, T3(w) = w 3 - 3w, and T4(W) = w 4 - 4w 2 + 2. (b) Prove that Td(Te(w)) = Tde(w) for all d, e;::: O. (c) Prove that Td(-w) = (-l)dTd(w). Thus T d 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 Tl (w) = w. (e) Prove that the generating function of the Chebyshev polynomials is given by 00
l: Td(W)X
d
=
d=O
2 -wX X X2' l-w +
(f) Prove that the dth Chebyshev polynomial is given by the explicit formula
1.31. This exercise describes dynamical properties of the Chebyshev polynomial Ts (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:l(U) = [-2,2]. (b) Prove that limn~oo T:t(w) = 00 for all points wE 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 ofTd and compute their multipliers. Observe that .:J(Td) is the closure ofthe (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 A tors = {P E A : nP = 0 for some n ~ I} be the torsion subgroup of A, let d ;::: 2 be an integer, and let 4> : A --+ A be the map 4>(P) = dP. Prove that PrePer( 4» = A tors. (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 : ]p'l --+ ]p'l be the rational map associated to multiplication-by-d on E, as described in Section 1.6.3. Prove that PrePer(4)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 C p of an algebraic closure of Qp. The nonarchimedean nature of the absolute value on Qp and C p 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 of nonarchimedean dynamics must deal with the fact that Qp is totally disconnected and far from being algebraically closed, while C p 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 dynamics 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:
• lal
20, and
lal =
0 if and only if a =
43
o.
44
2. Dynamics overLocalFields: Good Reduction
• la/31 = lal . 1/31 for all a, /3 E K . • la + /31 :::; lal + 1/31 for all a, /3 E K
(triangle inequality).
A valuedfield is a pair (K, 1 . 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, estimate
I . I) be a valuedfield. If the absolutevalue satisfies the stronger la + /31 :::; maxjjc], 1/31}
for all a, /3 E K,
(2.1)
then the absolute value is nonarchimedean or ultrametric. The associated valuation is the homomorphism v : K*
----t
It satisfies
v(a +
v(a) = -log lal.
JR,
m~ mini v(a), v(/3)}.
The valuation v is discrete if v (K*) is a discrete subgroup of JR, in which case the associated normalized valuation (sometimes denoted ord.i) is the constant multiple of v chosen to satisfy ord, (K*) = Z. Example 2.1. The field Q has the usual real absolutevalue
lal oo = max{a, -a}. For each prime p it also has a p-adic absolutevalue definedas follows. Every nonzero rational number a has a unique factorization of the form a
=±
IT
pep(a)
with ep(a) E Z.
P prime
Then
lal p
= p-ep(a).
The p-adic absolute values are nonarchimedean. Two absolute values are said to be equivalent if there is a constant r > a such that lail = lal2 for all a E K. 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 k be a field and let K = k(T) be the field of rational functions. Then each a E k determinesan absolute value on K associatedto the valuation ord., that gives the order of vanishing of f(T) E k(T) at T = a. The degree map deg : k(T)* ---- Z is also a valuation. If k is algebraicallyclosed, this yields the complete set of absolutevalues on k(T) that are trivialon k, up to equivalence. More generally, if k is not algebraically closed, there is a valuation corresponding to each monic
45
2.1. The Nonarchimedean Chordal Metric
irreducible polynomial in k[T]. In this book we are primarily interested in the number field scenario, i.e., the field Q 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 k[T] and its extensions and completions. We now prove the elementary, but extremely useful, fact that if 10:1 the ultrametric inequality (2.1) is actually an equality.
I:-
Lemma 2.3. Let K be a field with a nonarchimedean absolute value let a, /1 E K. Then
Proof We suppose that 1/1lv
1/1lv. The strict inequality
lal v = I(a
+ /1) - /1 lv ::; maxi la + /1l v, l/JIv}
implies that the maximum on the right is [o + /1lv, so we find that [o ], ::; la + /1lv' The opposite inequality is also true , since [o + /1l v ::; m ax{ la lv , 1/1l v} = lal v . 0 Recall that the chordal metric on by the formula
pI
(C), which we now denote by Poo, is defined
for points PI = [X I , YI ] and P2 = [X 2 , Y2] in p I (iC). In the case of a field K having a nonarchimedean absolute value I . Iv, it is convenient to use a metric given by a slightly different formula. Definition. Let K be a field with a nonarchimedean absolute value I . Iv, and let PI = [Xl, YI ] and P2 = [X2, Y2] be points in PI (K ). The u-adic chordal metric on pI (K ) is
It is clear from the definition that Pv(PI , P2) is independent of the choice of homogeneous coordinates for PI and P2. The first thing to check is that Pv is indeed a metric. In fact, it is an ultrametric; that is, it satisfies the nonarchimedean triangle inequality. Proposition 2.4. The u-adic chordal metric has the following properties. (a) 0 ::; Pv (PI , P2 ) ::; 1. (b) Pv(PI , P2) = 0 if and only if PI = P2. (c) Pv(PI ,P2) = Pv(P2,PI ). (d) Pv(PI ,P3 ) ::; max{PV(P 1 ,P2),Pv(P2,P3 )} .
46
2. Dynamics over Local Fields: Good Reduction
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 IXIYZ
X zYIl v :::; max{I XIYzl v, IXzYIl v}
-
< max{IXIl v, IYIlv} m ax{IXzl v, IYzl 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 coordinates. It is the analogue of the fact that the classical chordal metric is invariant under linear fractional tran sformations that define rigid rotations of the Riemann sphere (cf. Exercise 1.2).
Lemma 2.5. Let
R = {a E K : lal v be the ring ofintegers ofK , and let tion ofthe f orm
f ([X YJ) ,
i.e., f
E
aX + bY cX +dY
=
f : pI
-t
:::;
I}
pI be a linear fractional transforma-
with a, b, e, d E R and ad - be E R* ,
PGLz(R ). Then Pv(J(PI ), f (Pz)) = Pv(PI , Pz)
(N.B. 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 X i or Y, inR*. Then max{IXil v , IYilv} = 1,so
Pv(PI, PZ ) = IX1 YZ -XzYIl v. Further, the identities
d(aXi + bYi) - b(eXi e(aXi + bYi) - a(eXi
+ dYi) = (ad - be)Xi , + dYi) = - (ad - be)Yi,
and the fact that max{ IXi lv, IYi Iv} = 1 and lad - belv = 1 immediately imply that max{l aXi
+ bYilv, leXi + dYilv } =
1.
Thus
Pv(J(Pd , f (Pz ))
=
I(aX I + bY1) (eX
Z
+ dYz ) - (aXz + bYz )(eX 1 + dYdl v
= I(ad - be)(XIYz - XzYI )l v
= Pv(PI, Pz), where we have again made use of the fact that lad - 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 Pi = [Xi, Yi] with Xi, Yi E R and at least one of Xi or Yi in R*. Then
max{IXil v, IYilv} = 1 and Pv(Pi, Pj) = IXiYj - XjYilv, as usual. If IX2 1v > 1Y21v, we apply the map f = Y/X to the three points. This preserves the chordal distance (Lemma 2.5) and allows us to assume that
IX2 1v ::; 1Y21v = 1. Next we apply the map f = (Y2 . X - X 2 . Y)/Y to the three points. Lemma 2.5 again tells us that the chordal distance is preserved (note that 1Y21v = 1), and we are reduced to the case that P2 = [0,1]. Finally, we compute
IXI Y3 - X3 YI lv ::; max{ IXI Y3Iv, IX3 YIlv} ::; max{IXIl v, IX3Iv}, which is exactly the desired inequalitywhen P2 is the point [0, 1].
o
2.2 Periodic Points and Their Properties In Section 1.3 we described various properties of periodic points for rational maps definedover C. Virtuallyall of these definitions make sense, mutatis mutandis,when we work over any field with an absolute value. In this section we briefly recall the relevantmaterial. Let K be a field with an absolute value I . I and let ¢( z) E K (z) be a nonconstant rational map. The multiplier of ¢ at a fixed point 0: E K is the derivative
An (¢) = ¢' (0: ) . (See Exercise 1.13 for the case 0: = 00.) The multiplier is well-defined, independent of the choice of coordinateson pI; see Proposition 1.9. More generally, if 0: E pI (K) is a point of exact period n for ¢, then 0: is a fixed point of ¢n and we definethe multiplier of ¢ at 0: to be
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 neighborhoodof a periodic point 0:. The periodic point 0: is called superattracting
if An (¢) = 0,
attracting
if IAn (¢) I < 1,
neutral (or indifferent)
if IAn (¢) I = 1,
repelling
if IAn (¢) I > 1.
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.
48
2. Dynamics over Local Fields: Good Reduction
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 f(x) == 0 (mod p) to solutions in Zp. Then, using information gathered from many primes, one is sometimes able to deduce results for a global field such as Ql. Our principal objects of study are maps ¢ : r 1 --t r 1 . In this section we study the behavior of such maps under reduction modulo a prime. We work over a discrete valuation ring, so we set the following notation:
K
I . Iv R p
R*
k
a field with normalized discrete valuation v : K* = cV(x) for some e
--*
Z.
> 1, an absolute value associated to v.
{a E K : v(a) :::: O}, the ring of integers of K. = {a E K: v(a) :::: I}, the maximal ideal of R. = {a E K : v(a) = O}, the group of units of R. = Rf», the residue field of R. =
reduction modulo p, i.e., R
--t
k, a
f---t
a.
Before studying reduction of maps, we consider the problem of reducing points modulo p. This is as easy in r N as it is in r 1 , so we look at the general case. Let
P = [XO,X1,'" ,XN] E rN(K) be a point defined over K. We cannot immediately reduce the coordinates of P modulo p, since some of the coordinates might not be in R. However, since the coordinates of P are homogeneous, we can replace them with
P
= [exo, eX1,"" eXN]
for any e E K*. Choosing e to be highly divisible by p, we can ensure that every ex, is in R. However, if we overdo this process and end up with every ex, in the prime ideal p, then when we reduce modulo p, we end up with [0,0, ... ,0], which does not represent a point of projective space. The trick is to "clear the denominators" of the Xi'S as efficiently as possible. To do this, we choose an element a E K* satisfying
v(a) = min{ v(xo), v(xt}, ... ,V(XN
n.
(2.2)
For example, a could be the Xj having minimal valuation. Then a- 1xi 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~] Er 1 (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 numbers 0: - 1Xi is a unit. Hence P is a well-defined point in p I (k). We say that P = [xo , . . . , XN] has been written using normalized coordinates if
min{v(xo),v(xd, . .. , V (XN)} = 0, in which case
P is simply
[xo, ... , XN]'
iJ5]
E p I (Ql) . We can reduce P Example 2.6. Consider the point P = [I~ ' ; 5' ~~ , modulo 11 without any modification , since every coordinate is an l l-adic integer and not all coordinates vanish modulo 11. Thus P = [1,10,7,9] (mod 11). However, if we want to reduce P modulo 3, then we first need to divide all of the coordinates by 3,
P _ [3 -
9 24 27] _ [1 3 8 9] 14 ' 35 ' 49' 245 - 14' 35' 49' 245 '
and then P = [2,0,2,0] (mod 3). Similarly, in order to compute P modulo 7, we first multiply the coordinates by 49,
P _ [~ ~ 24 ~] _ [21 63 24 27] -
and then
P=
14 ' 35 ' 49 ' 245
-
2 ' 5 ' 1 ' 5
'
[0,0,3,4] (mod 7).
It may appear that the reduction P depends on the choice of 0: . We now check that this is not the case.
Proposition 2.7. Let P = [xo , . .. , XN ] E pendent of the choice of 0: satisfy ing (2.2).
pN (K).
Proof Suppose that 0: and {3 both satisfy (2.2). Then tion, so 0:{3- 1 E R*. This allows us to compute
[------ -----0:
- I
Then the reduction P is inde-
0:
and {3 have the same valua-
------ ] [------ ------ ------------ ------ ------ ] [------xo,{3------ ,{3------ ]
-I - 1 Xo , O: XI, ... , O: X N
_ -
_
-
- 1 - I -1 - 1 - 1 -I 0:{3 0: xo,o:{3 0: X I, .. . , o:{3 0: XN -1
{3
-I
XI, ...
- 1
XN
·
Hence the reduction of P modulo p is independent of the choice of 0: satisfying (2.2).
o We next prove an easy and useful lemma that relates reduction modulo p to v-adic distance .
Lemma 2.8. Let PI and P 2 be po ints
in ]P'I (K) .
Then
2. Dynamics over Local Fields: GoodReduction
50
Proof Write PI = [Xl, YI ] and P2 = [X2 , Y2 ] using normalized coordinates, so in particular Pv(pl, P2 ) = IXI Y2 - X 2 Yl lv ' Suppose first that PI = P2 • This means that there is a U E k* such that Xl = uX2 and Y"l = UY"2. In other words, there is a U E R* such that that Xl =' uX 2 (mod p) and YI =' uY2 (mod p). Hence
which completes the proof that Pv (PI, P2 ) < 1. Next suppose that Pv (PI , P2 ) < 1, which implies that X IY2 =' X 2YI (mod p). If X IX2 ¢. 0 (mod p), then Xl, X2 E k*, so we have
On the other hand, if X IX2 =' 0 (mod p), then the fact that the coordinates are normalized and the equality X I Y2 =' X 2 YI (mod p) imply that Xl=' X 2 =' 0 (mod p), 0 so PI = P2 = [0, 1]. This completes the proof of the lemma. As a first application, we show that fractional linear transformations in PGL 2(R) respect reduction modulo p.
Proposition 2.9. Let P, Q E pl(K) and J E PGL2(R). Then
P = Q if and only if J(P) = J(Q). Proof We combine Lemmas 2.5 and 2.8. Thus
P= Q
~
Pv(P, Q) < 1
from Lemma 2.8,
~
Pv(J(P),J(Q)) < 1
from Lemma 2.5,
~
~
~
J(P) = J(Q)
from Lemma 2.8 again.
o
Example 2.10. Let J = (~~), so J .,. PGL 2 (Z3). Consider the points P = [7,5] and Q = [4,2] in pI (Q3)' They satisfy P = Q = [1,2] in pI (IF3), but
J(P) = [45,75] = [3,5] =' [0,1] J(Q) = [24,36] = [2,3] =' [1,0]
(mod 3), (mod 3),
~
so J(P) f- J(Q). This shows the necessity of the condition J E PGL 2(R) in Proposition 2.9.
It is easy to see that if K is a field and PI, P2 , P3 are distinct points in pI (K), then there is an element ofPGL 2 (K) that moves them to the points 0, 1,00. (See Exercise 1.4.) The next proposition gives a stronger result for points whose reductions are distinct. It is especially useful because Lemma 2.5 says that the nonarchimedean chordal metric is invariant for maps in PGL 2(R), so changing coordinates via an element ofPGL 2(R) does not change the underlying dynamics.
2.3. Reduction of Pointsand Maps Modulo p
51
Proposition 2.11. Let PI , P2 , P3 E pI (K ) be points whose reductions PI , P2 , P3 are distinct. Then there is a linear fractional transformation f E PGL2 (R) such that
f(Pd = 0, Proof Write Pi = [Xi, Yi] with normalized coordinates. If v(X I ) > v(YI ) , we begin by applying the map f = Y/X E PGL2 (R) to each of the three points, so we may assume that v(X d ::; v(Yd . Since the coordinatesare normalized, this implies that v(YI ) = 0, so YI is a unit. We next apply the map
to the three points. Having done this, we see that PI = [0,1]. Next consider the point P3 . Since P3 i- PI = [0,1], we see that v(X3 ) = 0, so we can apply the map X /(Y3X - X 3 y) E PGL2(R) to the three points. This fixes PI and sends P3 to [1 ,0], i.e., P3 gets sent to 00. Finally, since P2 is equal to neither PI = [0,1] nor P3 = [1 ,0]' we see that v(X2 ) = v(Y2 ) = 0. Applying the map Y2X/ X 2Y E PGL2(R) to the three points then fixes PI and P3 and sends P2 to [1 ,1]. D Havinglookedat the reductionof a point, we next tum to the problem of reducing a rational map modulo p. Let ¢ : pI -; pI be a rational map of degree d defined over K, so ¢ is given by a pair of homogeneous polynomials of degree d,
F (X ,Y ),G(X,Y ) E K [X ,Y ]. Note that the map ¢(X, Y ) = [F (X ,Y ),G(X,Y )] does not change if F and G are each multiplied by a nonzero constant c E K *, since the coordinates are homogeneous.
Definition. Let ¢ : pI -; pI 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, if F , G E R[X, Y] and at least one coefficient of For G is in R*. Equivalently, ¢ = [F, G] is normalized if
F(X, Y) = aoXd + aIXd-1y + ... + ad_lXyd- I
+ adyd
and satisfy min { v(ao) ,v(ad , . . . ,v(ad), v(bo), v(b1), .. . ,V(bd) } = 0.
(2.3)
Givenany representation ¢ = [F,G], it is clear that one can alwaysfind some e E K * such that reF, eGl is a normalizedrepresentation. Further, the element e is unique up to multiplicationby 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)] - y d,b-0 X d + b- 1 X d- 1y + ... + b-dy d] . - Xd + al - X d- 1y + ... + ad = [ao 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 ¢ : pI (k) ----. pI (k). Further, the reduced map ¢ is independent of the choice of F 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 a E K* and consider the rational map
°
If a E R*, then ii -=I- and the reduced map ¢a(X, Y) = liiX d, yd] is again a rational map of degree d. However, if v(a) > 0, then ii = 0, so ¢a(X, Y) = [0, yd] = [0, 1] is a constant map! Similarly, if v(a)
< 0, then a-I = 0, so
is again a constant map, but not to the same point!! To summarize, the reduction of the map ¢a(X, Y) = [aX d, yd] separates into three cases,
[iiX d,Y dj ¢a =
{
[0,1] [1,0]
ifv(a) =0, ifv(a) > 0, 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, ifv(a) = 0, then it is easy to see that for any P E pl(K),
¢(p) = ¢(F),
and hence by induction,
¢n(p) = ¢n(F).
Thus the ¢ orbit of F yields valuable information about the ¢ orbit of P. However, ifv(a) -=I- 0, then ¢(F) is constant, independent of P, so the ¢ orbit of F contains no information. We formalize this notion in Section 2.5 after a preliminary discussion of the theory of resultants.
53
2.4. The Resultant of a Rational Map
2.4 The Resultant of a Rational Map A rational map c/J : !pI -; !pI is given by a pair of homogeneous polynomials c/J =
[F(X, Y) , G(X , Y)]
having no nontrivial common roots. However, if we reduce the coefficients of F and G modulo some prime , they may acquire common root s 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 ofthe coefficients of F and G . 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 section only a brief introduction to the theory of resultants . The reader desiring further information might consult [105, Section 3.3], [112, Chapter 3], [259, Section V 10], 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 A (X , Y ) = aoX n + a l X n- 1y + B (X , Y ) = boX m + b X m- 1y +
+ an_ lXy n - 1 + any n , + bm_ 1Xym- l + bmy m
1
be homogeneous polynomials ofdegrees n and m with coefficients in afield K . There exists a polynomial
in the coefficients of A and B , called the resultant of A and B , with the following properties: (a) Res(A , B ) = 0 if and only if A and B have a common zero in!p 1 (K). (b) If aobo =I- 0 and if we f actor A and B as n
A = ao
II (X -
m
Qy)
i= l
then
B = bo
and n
Res(A , B ) = aob~
m
II II (Qi -
II (X - ,6jY) , j=l ,6j).
i=l j=l
(c) There exist polynomials
F l , G l , F2 , 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
Fl (X , Y )A(X , Y ) + G 1 (X , Y )B(X , Y ) = Res(A , B )x m+n- l , F (X ,Y )A(X , Y ) + G (X ,Y )B (X ,Y ) = Res(A , B )y m+n- 1 . 2
2
Notice that in the first equation. the variable Y has been eliminated, and similarly X has been eliminated in the second equation.
2. Dynamics overLocal Fields: Good Reduction
54
(d) The resultant is equal to the (m + n) x (m + n) determinant ao al a2 . . . an ao al a2 ... an ao al a2 ... an
Res(A, B) = det
m
ao al a2 . .. an
bo bl bz bm bo bi b2 . •. •..•. • bm bo bl b2 •. • •. • • .. bm
bo bl b2
••••••. ••
n
bm
In particular, Res(A, B) is homogeneous of degree m in the variables ao, ... , an and simultaneously homogeneous ofdegree n in the variables bo, ... , bm . Proof We begin by showing that the following three conditions are equivalent. (i) A(X, Y) and B(X, Y) have a common zero in jpl(k). (ii) A(X, Y) and B(X, Y) have a common (nonconstant) factor in the polynomial ring K[X, Y J. (iii) There are nonzero homogeneous polynomials C , DE 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.
(2.4)
The equivalence of (i) and (ii) follows immediately from the fact that the greatest common divisor of A and Bin K[X, Y] vanishes at exactly the common zeros of A and B in jpl (k). (What we are really using here, of course, is the fact that the ring of homogeneous polynomials K[X, Y] is a principal ideal domain.) It is also clear that (ii) implies (iii), since if A and B have a common factor F, then we simply choose C and D using the formulas A = F D and B = FC. Finally, to prove that (iii) implies (i), we suppose that (2.4) is true. Ifwe factor both sides of (2.4) into linear factors in k[X, Yj, then A(X, Y) has n factors, while D(X, Y) has at most n - 1 factors. Therefore A(X, Y) shares at least one linear factor with B(X, Y) in k[X, YJ, and hence they have a common zero in jpl (k). We next multiply out equation (2.4), treating the coefficients of C and D as unknowns. This gives a system of m + n homogeneous linear equations in the m + n variables Co , ... , Cm - I , do, . .. , d n - I , 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 deg(A) = 3 and deg(B) = 2, then equating coefficients in (2.4) gives the system oflinear equations
2.4. The Resultant of a Rational Map coao COal COa2 COa3
55
= dob o,
+ ClaO = + Clal = + Cla2
dob l dob2
+ dlb o, + dlb l + d2bo, dlb2
=
Cla3 =
+ d2bl,
d2b2,
whose associated matrix
ao -bo al ao -b l -bo a2 al -b2 -b l -bo a3 a2 -b2 -b, 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 JP'1 (k), which proves (a). In order to prove (c), we write and
xjyn-l-j
B
for
0
~
j -period of P, which, in our current notation, says that m divides n.
63
2.6. Periodic Points and Good Reduction Replacing ¢ by ¢m and m by 1, we are reduced to th~ that
F is a fixed point
of ¢. Having done this, we note that AJ,(F) is equal to ¢'(P). If ¢(P) = P, then ti = m and we are done. We thus assume that ¢(P) -I- P. To simplify notation, we use Proposition 2.11 to find a transformation f E PGL2(R) with f([O, 1]) = P. Replacing P and ¢ with f-1(p) and ¢I = 0 ¢ 0 f, respectively, we may assume that P = [0,1]. Dehomogenizing z = X/Y, we write ¢ in the form
r:'
1 d ¢(z) = aoz + a1 zd- + bozd + b1zd- 1 +
+ ad-1 z + ad + bd-1Z + bd
with coefficients aQ' ... , b« E R and at least one coefficient in R*. The fact that [0, 1] is a fixed point of ¢ says that
¢(O) = ad/bd == 0 (mod p), so ad E p and bd E R*. (We are also using the fact that ¢ has good reduction, of course.) Multiplying numerator and denominator by bd1, we may thus write ¢( z) in the form
¢(z)
=
ad + ad-1 Z + 1 + bd- 1Z +
+ a1 zd- 1 + aozd + b1zd- 1 + bozd
The first couple of terms of the Taylor expansion of ¢ around z case may be obtained by simple long division, look like
¢(Z)=p,+AZ+
=
0, which in this
A(z) 2 B( )z l+z z
(2.6)
with
A(z), B(z)
E
R[z],
A = ¢'(O),
and
p, =
ad E p.
A simple induction argument using (2.6) shows that
(2.7) In particular, since ¢n(o) = 0 and p, E p, we find that 1 + A + A2
+ ... + An-1 == 0
(mod p),
(2.8)
The analysis now splits into two cases. First, suppose that r ::::: 2, or equivalently,
A¢. 1 (mod p). Then formula (2.8) implies that An == 1 (mod p), so we find that r divides n. If n = r, the proof is complete. Otherwise, we replace ¢ by ¢T and n by ii]«. By an abuse of notation, we continue to write
¢(Z)=p,+AZ+
A(z) 2 B( )z l+z z
with the understanding that the values of p" A, A(z), and B(z) may have changed. The principal effect of replacing ¢ by ¢T is that we are now in the situation that
64
2. Dynamics over Local Fields: Good Reduction
>. == 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,
p, = ¢(O)
== 0 (mod p),
>. = ¢'( O) == 1 (mod p).
and
We are further assuming that ¢(O) =1= 0 (otherwise, we are done), so (2.8) becomes
n
== 1 + >. + >. 2 + ... + >. n - I == 0
(mo d p).
Thus n is divisible by p. We replace ¢ by ¢P and n by n ip. Ifnow ¢(O) = 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
for some e 2 1.
=m
or
n
= mr
or
n
= mrp" 0
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 k(z) is separable ifit is not in k(zP). See Exercise 1.10 for details.) Corollary 2.23. Let ¢ : lP'I -> lP'I be a rational map that has good reduction. (a) Every p eriodi c point of ¢ is nonrepelling. (b) If the reduction ¢ is separable, then ¢ has only finitely many attracting periodic points. Proof (a) Theorem 2.18 tells us that ¢n has good reduction . Let P be a periodic point of ¢ of exact period n. 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
The fact that ¢ n has good reduction implies that bo E R*, since otherwise z = 0 would be a common root of P and G. Hence
so I>'p (¢) Iv :S 1, which shows that P is a nonrepelling point for ¢. (b) Again let P be a periodic point for ¢ of exact period n, and let m be the period of the reduced point P. Then we have equivalences
2.6. Periodic Points and Good Reduction
P is attracting
65
¢:=:}
IAp(¢)lv = 1(¢n)'(p)lv < 1
¢:=:}
(¢n)'(p)
= 0
¢:=:}
(¢n),(p)
= 0
¢:=:}
¢'(P) . ¢'(¢P) . ¢'(¢2 P)
¢:=:}
(¢'(P) . ¢'(¢P) . ¢'(¢2 P)
¢:=:}
(¢m)'(p)n/m = 0
¢:=:}
OJ, (P) contains a critical point.
¢,(¢n-I P)
= 0
¢,(¢m-I p))n/m = 0
The fact that ¢ is separable implies that a version of the Ri~mann-Hurwitz formula (Theorem 1.1) is valid; see Exercise 1.1O. Hence the map ¢ _has finitely many (precisely, 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 P. Further, we know that the multiplier AJ, (P) = (¢m)'( 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 0 conclude that ¢ has only finitely many attracting periodic points.
Remark 2.24. The separability assumption in Proposition 2.23 is necessary, as is shown by the example ¢( z) = z", all of whose periodic points are attracting. See also Exercise 2.16. E Z2[Z] 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 (Q2) be a periodic point of exact period n 2:: 2. In the notation of Theorem 2.21, n = mr2 e , where m is the period of P in lP'1 (IF 2) and r is the order of AJ, (P) in lF~. But lP'1 (IF 2) has only three points, and the fact that ¢ is a polynomial means that the point at infinity is not in the orbit of i; so either m = lor m = 2. Similarly, we note that F has only one element, so r = 1. It follows that n = 2 8 for some s 2: O. Similarly, if ¢(z) E Z3[Z] is a polynomial of degree d 2: 2 with leading coefficient a 3-adic unit, and if P E lP'1 (Q3) is a periodic point of exact period n 2: 2, then we find that n = mr3 e with 1 ~ m ~ 3 and 1 ~ r ~ 2. Thus n = 2 t . 3U for some o ~ t ~ 2 and some u 2:: O. Finally, let ¢(z) E Z[z] be a polynomial of degree d 2: 2 whose leading coefficient 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 (Q) satisfies both n = 2 8 and n = 2t . 3U with t ~ 2. This proves that n is either 1, 2, or 4. The examples ¢(z) = Z2 with z = 0 and ¢( z) = Z2 - 1 with z = 0 show that n = 1 and n = 2 are possible. Can you find an example with n = 4? See Exercise 2.20 for a stronger version of this example.
Example 2.25. Let ¢(z)
z
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 argument to give a general result that, although not the strongest possible bound using these methods, is sufficient for many applications.
2. Dynamics over Local Fields: Good Reduction
66
Corollary 2.26. Let K be a numberfield, let ¢ : pI -+ pI be a rational map defined over K, and let p and q be primes of K such that ¢ has good reduction at both p and q and such that the residue characteristics of p and q are distinct. Then the period n ofany periodic point of ¢ in pI (K) satisfies
n::; (Np2 - 1)(Nq2 - 1), where NP and Nq denote the norms of p and q respectively. In particular, the set Per( ¢, K) of K -rational periodic points is finite. (For an alternative proofofthe finiteness ofPer( ¢, K) using the theory ofheight functions, see Theorem 3.12.) Proof Using the obvious notation, we have m p = (periodof¢(F) mod p)::; #pl(JF p ) = Np
+ 1,
r p = (period of AJ(F) mod p) ::; #JF; = Np - 1, and similarly for m q and r q- Let p and q denote the residue characteristics of p and q, respectively. Then Theorem 2.21 says that
Since p and q are distinct primes, it follows that n ::; m p . r p . m q . r q
::;
(Np + l)(Np - l)(Nq + l)(Nq - 1),
which is the first part of the corollary. The finiteness ofPer( ¢, 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 various ways on the rational map. See for example [52, 87, 90, 91, 162, 171,312, 325, 326, 328, 332, 353, 355, 358, 359, 361, 454]. However, none of these articles achieves the uniformity predicted by a conjecture that we discuss in Chapter 3 (Conjecture 3.15). This conjecture asserts that for a number field K of degree D and a rational map ¢ E K(z) of degree d 2 2, the number of K-rational preperiodic points of ¢ should be bounded solely in terms of D and d. If K is a discrete valuation ring of characteristic 0, then it is possible to bound the exponent e appearing in the formula n = mrp" in Theorem 2.21.
Theorem 2.28. (Zieve [454], see also Li [266] and Pezda [355]) We continue with the notation and assumptions from Theorem 2.21. Wefurther assume that K has characteristic and we let v : K* --* Z be the normalized valuation on K. If the period n of P E pI (K) has the form n = mrp", then the exponent e satisfies
°
2.6. Periodic Pointsand Good Reduction p
e- l
67
2v(p)
< - -. - p -1
(2.9)
Further, ifp = 2, then the upper bound may be replaced with v(p)/(p - 1). (Note that v(p) is the ramification index ofp in K.) Remark 2.29. Let R be the local ring of K and let F be a formal group defined over R. (See, e.g., [410, chapter IV] for basic material on formal groups.) Theorem 2.28 is a close analogue of the fact [410, IY.6.1] that the torsion in the formal group F (R) consists entirely of p-power torsion, and that if a E F (R) has exact order p" , then pe- l :::; v(p) /(p - 1).
o,
Example 2.30. Let q be a power of an odd prime p, let ( E be a primitive qth root of unity, let K = Qp((), and let ¢( z) = 1 + (z - z", The maximal ideal of K is P = (1 - (), so ¢ = 1 + z - zq has good reduction. The point a = 1 is a point of exact period q for ¢, since cP (1) = ( j, while a is clearly a fixed point of ¢. Further, ¢'(a) = 1. Hence in the notation of Theorem 2.21, we have n = q, m = 1, r = 1, and e is determined by q = p" , On the other hand, the extension K / Qp is totally ram ified, so the normalized valuation v : K * ->t Z satisfies v(p)
= [K
: Qp]
= q(l -
l /p )
= pe-l(p -
1).
Thus the inequality (2.9) in Theorem 2.28 becomes pe-l :::; 2pe- l, which shows that the power of p cannot be improved. We do not give the proof of Theorem 2.28, but are content to prove the following special case, which serves to indicate some of the combinatorial issues that arise.
Theorem 2.31. Let p ;::: 5 be a prime, let K = Qp, or more generally, an unramified extension of Qp, let ¢ : 1P'1 ---+ IP'I be a rational map defined over K with good reduction, and let P E 1P'1 (K) be a periodic point with the property that and
¢'(F) =
1.
Then ¢(P) = P . (In the notation of Theorem 2.21, this theorem asserts that ifm = = 1, then e = 0 and n = 1.)
r
Proof As in the proof of Theorem 2.21, we begin by moving the periodic point to the origin and dehomogenizing cjJ. Recall that during the proof of Theorem 2.21, we used a first-order Taylor expansion (2.6) for ¢(z) around z = O. In order to bound the exponent e, we need to use the second-order expans ion
cjJ(z) = J.l + AZ + uz
2
A(z)
3
+ l+ z B (z ) z ,
(2.10)
where
A(z), B(z) E R[z],
J.l E p,
A = cjJ' (O) == 1 (mod p) ,
and
v E R.
2. Dynamics overLocal Fields: Good Reduction
68
(In general, to prove a sharp estimate for the exponent as in Theorem 2.28, one needs to consider a longer Taylor expansion. But for the unramified case that we are considering, the second-order expansion (2.10) suffices.) Using the results already proven, it suffices to show that if f-l -f:. 0, i.e., if ¢(O) -f:. 0, then ¢P(O) -f:. O. This will then imply by induction that ¢Pe (0) -f:. 0 for all e 2: 1, contradicting the assumption that 0 is a periodic point of ¢. During the proof of Theorem 2.21 we gave a simple formula (2.7) for ¢i(O) modulo f-l2. In a similar manner we use (2.10) to find a formula modulo f-l3. To derive this formula, we write ¢k(O) = f-lak + f-l2 vbk, substitute into (2.10), and do some algebra to obtain f-lak+1
+ f-l2vbk+l = ¢(f-lak + f-l2 vbk) == f-l + A(f-lak + f-l2 vbk) + vf-l2a~ (mod f-l3) == f-l(1 + Aak) + f-l2v(a~ + Abk) (mod f-l3).
This yields the recurrences and Starting from al = 1 and b1 = 0, it is now a simple matter to find formulas for ak and bk and check them by induction. The end result is
1'(0)
=~(~ >,;) + 1'2v(~ >,H-'
(to
>,;
r)
(2.11)
».
We are going to apply (2.11) with k = p. Consider the sum L We know that A == 1 (mod p), and we are assuming that p is unramified in K, so A = 1 + cp for some c E R. Assuming that c -f:. 0, i.e., that A -f:. 1, we compute
Note that the final congruence is true because every binomial coefficient (~) with 1 ::; i < p is divisible by p (and we are assuming that p -f:. 2). We also observe that the congruence is true even for A = 1, although the intermediate calculation is incorrect. We perform a similar calculation for the more complicated sum in (2.11), but this time we are interested only in the value modulo p, so we can replace A by 1:
(2.13) Note that for the last step we are using the assumption that p 2: 5. Substituting (2.12) and (2.13) into the iteration formula (2.11) (with k yields
= p)
69
2.7. PeriodicPoints and Dynamical Units
¢P(O) == j.t(p + ap2) + j.t2vbp
(mod j.t3),
where a and b are in R . The fact that p is unramified in K means in particular that p divides u, so ¢P(O) == j.tp (mod j.tp2). Now using our assumption that j.t the proof of the theorem.
i- 0, we deduce that ¢P(O) i- 0, which completes 0
2.7 Periodic Points and Dynamical Units Let ( be a primitive pth root of unity. Then the expression
is a unit in the cyclotomic field iQ((), a so-called "cyclotomic unit." Similarly, if (1 and ( 2 are roots of unity of relatively prime orders, then the difference (1 - (2 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
the proposition actually says something stronger than the simple assertion that certain pairs of points have distinct reduct ions.
Proposition 2.32. Let ¢ (z ) E K (z) be a rational fun ction of degree d ?': 2 with good reduction. (a) Let P E Pl(K) be a point ofperiod n for o. Then
Pv(¢ip,¢jp) = Pv(¢i+kp, ¢i +kp ) forall i , j ,k
E Z,
wherefor i < 0 we use the periodicity ¢nP = P to define ¢i P. (b) Let P E pI (K) be a po int of exact period n for ¢. Then
Pv(¢iP, ¢i P ) = Pv(¢P, P ) for all i ,j
E Z satisfying gcd (i -
i . n) = 1.
(c) Let Pl , P2 E p l (K) be periodic points for ¢ ofexact periods nl and n2. respectively. Assume that nl n2 and n 2 nl. Then
t
t
70
2. Dynamics over Local Fields: Good Reduction
Proof (a) Proposition 2.14 and the good-reduction assumption imply that for all Q, R E pI (K).
Pv(Q, R) 2: Pv(¢Q, ¢R) Applying this repeatedly yields
Pv(Q, R) 2: Pv(¢Q, ¢R) 2: Pv(¢2Q, ¢2R)
2: Pv(¢3Q,¢3R) 2: ... 2: pv(¢nQ,¢nR). If we now make the further assumption that Q and R are points of period n, then pv(¢nQ, ¢nR) = Pv(Q, R), so all of the inequalities must be equalities. This proves that if ¢n( Q) = Q and ¢n(R) = R, then
Pv(Q, R) = Pv(¢kQ, ¢k R)
for all k E Z.
Substituting Q = ¢i P and R = q) P for the given point P of period n completes the proof of (a). (b) We know from (a) that the distance Pv(¢iP,q)P) depends only on the difference i - j. We use this fact and the nonarchimedean triangle inequality (Theorem 2.4(d» to estimate
Pv(P, ¢k P) :S max{pv(p, ¢P), Pv(¢P, ¢2P), ... , p(¢k-I P, ¢k pn =Pv(P,¢P).
(2.14)
In a similar fashion we obtain the estimate
Pv(P, ¢P) :S max{pv(P, ¢k P), Pv(¢kP, ¢2kP), Pv(¢2k P, ¢3kP), ... , pv(¢(m-I)k P, ¢mkP), pv(¢mkP, ¢P)} = max{pv(P,¢kp),pv(¢mkp,¢Pn.
(2.15)
If we now make the further assumption that gcd(k, n) = 1, then we can find an integer m satisfying mk == 1 (mod n). Then ¢mkP = ¢P, so (2.15) becomes
Pv(P, ¢P) :S max{pv(P, ¢k P), Pv(¢P, ¢pn = Pv(P, ¢k P).
(2.16)
Combining (2.14) and (2.16) yields
Pv(P, ¢P)
= Pv(P, ¢k P)
In particular, if gcd( i - j, n)
for all k satisfying gcd(k, n)
= 1.
= 1, we can use ttis formula and (a) to compute
Pv(¢ip,cpJp)
= Pv(p,¢i-jp) = Pv(P,¢P).
(c) Suppose that Pv(PI, P2) < 1, or equivalently from Lemma 2.8, suppose that = P2 . Let m be the exact period of and let r be the order of its multiplier AJ,(PI ) , Then Theorem 2.21 tells us that both nl and n2 are in the set
PI
A
{m, mr, mrp, mrp2, mrp3, .. .}.
It follows that either nl divides n2, or that n2 divides nl, contradicting the assump0 tion on nl and n2. Therefore Pv(PI, P2) = 1.
71
2.7. Periodic Points and Dynamical Units
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.
Theorem 2.33. (Narkiewicz [325]) Let ¢(z) E R[z] be a polynomial of degree d 2: 2 whose leading coefficient is a unit in R. Let a E K be a periodic point of ¢ of exact period n (with n 2: 2), and let i, j E 7l, be integers satisfying gcd(i - j, n) = 1. Then ¢i a - ¢Ja R* ----E . ¢a-a Proof The assumption that ¢(z) is a polynomial with unit leading coefficient implies that ¢ has good reduction, since
Res(aoX d + aIXd-Iy + ... + ady d, yd)
=
ago
We next observe that every periodic point of ¢ is integral over R, since it is a root of an equation ¢n (z) - z = O. Finally, we note that the distance between integral points a, (3 E R is given by
pv(a,(3)
= pv([a, 1], [(3, 1]) = la - (3lv'
Using this formula and Proposition 2.32(b), we compute
¢ia - ¢Ja I I ¢a-a v
= I¢ia - ¢Jal v = Pv(¢i a, ¢Ja) = 1. I¢a-al v
pv(¢a,a)
o
We create units from periodic points of arbitrary rational maps using a cross-ratio construction.
Definition. Let PI, P2 , P3 , P4 E pI (K), and choose homogeneous coordinates Pi = [Xi, Yi] for each point. The cross-ratio of PI, P2 , P3 , P4 is the quantity
Notice that K(PI, P2 , P3 , P4 ) is independent of the choice of homogeneous coordinates for the points.
Theorem 2.34. (Morton-Silverman [313, Theorem 6.4(a)]) Let ¢ E K(z) be a rational map of degree d 2: 2 with good reduction. Let P E pI (K) be a periodic point for ¢ ofexact period ti, and let i and j be integers satisfying
gcd(j, n) = gcd(i - 1, n) = gcd(i - j, n) = 1. Then
72
2. Dynamics over Local Fields: Good Reduction
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
o
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 ( n indicates a primitive nth root of unity and the condition gcd( m , n ) ensures that (m - ( n is a unit.
=
1
Theorem 2.35. Let
jp'l. Prove that there is a constant u E k* such that
uF=F' Deducethat the reducedmap ¢ ; jp'l (k) representation.
uG=G'.
and -> jp'l (k)
does not dependon the choice of normalized
Section 2.4. The Resultant of a Rational Map 2.5. (a) Let A(X, Y) = aoX + al Y be a linear polynomialand let B(X, Y) be an arbitrary homogeneouspolynomial. Prove that Res(A, B) = B( -ai, ao). (b) Let A(X, Y) = aoX 2 + alXY + a2y2 and B(X, Y) = bOX 2 + blXY + b2y 2 be quadratic polynomials. Prove that
= (2aob2 -
4 Res(A, B)
alb l
+ 2a2bo)2 - (4aoa2 - ai)(4bob2 - bi).
(Of course, in characteristic 2 one needs to cancel 4 from both sides before using this formula!) 2.6. Withnotation as in the statementof Proposition2.13, prove that the resultant of A and B is related to the roots of A and B by n
Res(A, B)
= a~b~
'rn
i=] j=l
2.7. Let
n
II II (a; - (3j) = a~ II et«; 1) = (_I)""nb~ II A((3j, 1).
A(X, Y) = aoX n + alXn-ly + B(X, Y) = boX"" + blX""-ly +
i=l
j=l
+ an_lXy n- l + any n, + b""_IXy""-1 + b""Y"" ,
be homogeneous polynomials of degrees nand m and let a, (3", fI E K be arbitrary. (a) Prove that
Res(A(aX
+ (3Y, ,X + flY), B(aX + (3Y, ,X + flY)) = (afl - (3,)""n Res (A(X, Y), B(X, Y)).
(Hint. Use Proposition2.13(b).) (b) Suppose that m = n = d. Prove that
Res ( aA(X, Y)
+ (3B(X, Y), ,A(X, Y) + ss;x, Y)) = (afl - (31')d Res (A(X, Y), B(X, Y)).
(Hint. Use Proposition2.13(d).)
Exercises
76 (c) Continuing with the assumption that m las
= n = d, define new polynomials by the formu-
= M(aX + /3Y,1X + 8Y) - /3B(aX + /3Y,1X + 8Y), = -1A(aX + /3Y,1X + 8Y) +aB(aX + /3Y,1X + bY).
A*(X, Y) B*(X, Y) Prove that
Res(A*, B*)
= (a8 -
/31)d
2+d
Res(A, B).
(d) Suppose that m = n = d and that Res(A, B) =1= 0, so ¢> = [A,B] is a rational map ¢> : 1P'1 ----> 1P'1 of degree d, and suppose further that a8 - /31 =1= 0, so the map
f
= [aX + /3Y,1X + 8Y]
is a linear fractional transformation, Let ¢>* = [A*, B*]. Prove that ¢>* Use (c) to prove that ¢>* : 1P'1 ----> 1P'1 is a rational map of degree d.
=
r:'
0
¢>
0
f.
2.8. Let f E PGL 2(K) be a linear fractional transformation, and write
f(z) in normalizedform, (a) Prove that Res(f) (b) Prove that
= ad -
=
az + b ez+d
be.
p(J(P),f(Q))::; IRes(f)1-
1p(p,Q)
for all P, Q E 1P'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 ¢>(z) E K(z) be a rational map, and let ¢>f = ¢> 0 f. (a) If Res (f) E R*, prove that Res(¢>f) = Res(¢». (b) If Res(f) is not a unit, find a formula or an inequality relating the valuations of the three quantities Res(f), Res (¢», and Res (¢>f).
rio
2.10. Let ¢> : 1P'1 ----> 1P'1 be a rational map defined over a field K with a nonarchimedean absolute value I . Iv, and consider the statement
sup Pl,P2EI"1(K) PFP P2
pv(¢>(P1 ) , ¢>(P2)) Pv(P1 , P2)
1
(2.18)
(a) Prove that (2.18) is true for the map ¢>(z) = az", where a E R. This example shows that Theorem 2.14 cannot be improved in general. (b) Prove that (2.18) is not true for the map
over the field K 2.11. Let ¢> : 1P'1 (C)
= Qp by computing both sides. (Youmay assume that p =1= 2.) ---->
1P'1 (C) be a rational map of degree d.
77
Exercises (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 !(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(j, g)D
2
.
(b) Use (a) to give an alternative proof ofTheorem 2.18(b). (c) For any rational map ¢ : lP'1 ~ lP'\ define ov(¢) = v(Res(¢))j deg(¢), so ov(¢) is a kind of normalized resultant of ¢. Prove that Ov satisfies the composition formula
2.13. Show that the good-reduction assumptions in Theorem 2.18 are necessary by constructing 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) -I- ¢(?). (b) ~ratio~al ~aps ¢ : lP'1 ~ lP'1 and'l/J : lP'1 ~ lP'1 such that ¢ has good reduction and
¢ 0 'l/J -I- ¢ 0 'l/J. (c) Same as (b), except now 7/J is required to have good reduction and bad reduction.
rP is allowed to have
2.14. Let p 2: 5 be a prime and define rational maps ¢(z) =
2
2 2
3
+p Z p3z2 + P Z
and
4
+P p3 Z + 1
'l/J(z) = p z
(a) Prove that ¢f has bad reduction modulo p for all! E PCL2(Qp). (N.B. We are allowing! to have coefficients in Qp and/or the determinant of! to be divisible by p.) (b) Prove that 'l/Jf has bad reduction modulo p for all ! E PCL 2 (Qp). (c) Prove that the composition 'l/J 0 ¢ has good reduction at p.
2.15. Let ¢ : pI ~ 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 ¢R: lP'k ~ lP'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 restriction of rPR to the special fiber oflP'k.
78
Exercises
2.16. Let K IQp be a p-adic field. (a) Prove that every periodic point of the map rjJ(z) = zP is attracting. (b) More generally, suppose that 'ljJ(z) E K(z) has good reduction, and let rjJ(z) Prove that every periodic point of rjJ is attracting. (c) Is (b) true without the assumption that 'ljJ has good reduction?
= 'ljJ(zP).
2.17. Let KIQp 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., rjJ' (a) = 0, and suppose that a E Per (¢). More precisely, suppose that the reduced point a satisfies ¢m (a) = a. Prove that there is an attracting periodic point (3 of rjJ of period m such that lim rjJnm(a) = (3. n~oo
Generalize to the case of a rational map rjJ( 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 JIll! (K) and an integer m ~ 1 such that the following limit exists:
T
=
lim rjJmn(p).
(2.19)
n~oo
Prove that rjJm(T) = T, i.e., T E Perm (rjJ). (This is true for any complete field K, so for example, it holds for Qp,Cp, JR, 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 ofa finite collection of polynomials. If V c JIl>N (K) is an algebraic set, the ideal attached to V is the ideal
I (V ) =
ideal generated by all hOmOgeneous) polynomials f E K[Xo, . .. , X N] . ( such that f (P ) = 0 for all P 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 .Ji is the same as the algebraic set attached to I . Theorem 3.10. (Hilbert's Nullstellensatz) Let I and J be homogeneous ideals properly contained in K [Xo , . . . , X N]. Then
V (I) = V( J)
if and only if
Vi = /J.
Proof Suppose that .Ji = /J. Let P E V (I) and f E J. Then rEI for some n ~ 1, so r(p) = 0, so f(P) = O. This is true for every f E J, so P E V(J). This proves that V(I) c V(J), and the opposite inclusion follows by interchanging the roles and I and J . Hence V(I ) = V(J ). This proves the trivial direction of the theorem. For a proof of the nontrivial converse , see [198, I.1.3A] or [259, Section X.2]. 0 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 rf't power. Theorem 3.11. Let ¢ : JIl>N (K) -7 JIl> M(K) be a morphism ofdegree d. Then there are constants C1 , C2 > 0, depending on ¢, such that
for all P E JIl>N (K ). (Infact, the upper bound H(¢(P )) ~ C 2 H (P )d is validfor rational maps provided we restrict attention to po ints P at which ¢ is defined.) Proof We begin with some notation. For any point P = [xo , . . . ,x N] with coordinates in K and any absolute value v E M K , we write
(This assumes that we have fixed particular homogeneous coordinates for P.) Similarly, we define the absolute value of a polynomial
3.2. Height Functions and Geometry
f(X 0,
... ,
91 iN
~ aio ...iN x: 0'" X N '"
X) N =
io,o.. ,iN
to be
If Iv = . max laio ...iN lv, 'l,o,···,'l,N
and if ¢ = [fo, ... , f M] is a collection of polynomials, we let
1¢lv = max IfJlv. J Notice that the height of a point P E lP'N (K) may now be written in the compact form
H(P) =
II
(
l/[K:lyl2+E
forall - E Q.
(3.21)
Y
-
Combining (3.20) and (3.21), we find that
lyld-2-E
< CI n,
By assumption, d 2:: 3, so this shows that there are only finitely many possible values for y. Finally, we observe that for each y, the equation G(x, y) = b implies that there are at most d possible values for x. Therefore the equation G(x, y) = b has only 0 finitely many integer solutions 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 rational/unction with at least three distinct poles in lP'1 (C). Then
{o: E Q : ¢(o:) E'1.} 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
F(z)
¢(z) = (Z2 _
ov:
where D > 1 is a squarefree integer and F (z) E '1. [z] is a polynomial of degree 2d. If we now take any solution (u, v) E '1. 2 to the Pell equation u 2 - Dv 2 = 1, then ¢( ulv) = v 2d F( ulv) E '1.. The Pell equation has infinitely many solutions, so there are infinitely many rational numbers u I v E Q with ¢( u I v) E '1.. Proof Write
¢ = [F(X, Y), G(X, Y)] using homogeneous polynomials F(X, Y), G(X, Y) E '1. [X, Y] of degree d having no common factors. For any fraction a = alb E Q written in lowest terms, we have
3.6. Diophantine Approximation
107
F (a, b) ¢(a ) = G(a, b)' so ¢(a) E Z if and only ifG (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 homogeneous polynomials Ii,gl , 12 ,g2 E Z[X, Y] satisfying
h (X, Y )F(X, Y) + gl (X ,Y )G(X , Y) = 12 (X ,Y )F (X , Y) + g2( X ,Y )G(X , Y) =
RX 2d- l , Ry 2d- l .
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 Ra2d- 1 and Rb2d- 1 . However, a and bare 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}c U {~ E Q : G(a, b) = D} .
(3.22)
DI R
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 provide a method for computing an allowable constant K(to , a) in terms of to and a. Thus our proofs of Thue's and Siegel 's theorems (Theorems 3.35 and 3.36) are also ineffective. 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 finitenes s result on integral points in orbits (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 C pN be a smooth projective curve defined over Q and let ¢ : C -+ pI be a nonconstant rational function on C. If C has genus 9 ~ 1, Siegel proved that C (Z) = {P E C(Q) : ¢(P) E Z} is a finite set. For 9 ~ 2, this is superseded by Faltings' theorem that C(Q) is finite, but for elliptic curves (g = 1), the set C(Q) is often infinite. In this case, Siegel greatly strengthened the qualitative statement that C(Z) 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.
Theorem 3.39. (Siegel [403,404]) Let E / Q 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 jp'l(Q)
with a(P) ,b(P) E Z and gcd(a(P),b(P)) = 1.
Then
log la(P) I = l.
lim PEE (Q) h(¢ (P ) )-oo
loglb(P) I
Proof See [410, IX.3.3]. For a general version on curves of arbitrary genus 9 2:: 1,
see for example [205, D.9.4], although as noted above, if 9 2:: 2, then Faltings [164, 0 165] proves that C(Q) is finite. For the convenience of the reader, we conclude this section by stating general versions of Theorems 3.34, 3.35, and 3.36 for number fields. Proofs of these, and even more general, results may be found in [205, Part D] or [256, Chapter 7]. We set the following notation: K a number field; S a finite set of absolute values on K; Rs the ring of S-integers of K. Theorem 3.40. (Roth) Let E > O. For each v E S, extend v to tc in some fashion, and choose an algebraic number a v E tc. Then there is a constant /'l, > 0, depending on K. S. E, and {a v } v ES, such that
v
IImin{lz-a vlv,1}n 2:: H (: )2+f vES
forall zEK.
K
Theorem 3.41. (Thue-Mahler) Let G(X, Y ) E K [X , Y ] be homogeneous of degree d with at least three distinct roots in pi (iC), and let B E K . Then there are only finitely many (X, Y) E R~ satisfying G(X, Y ) = B. Theorem 3.42. (Siegel) Let ¢( z) E K (z ) be a rational function with at least three distinct poles in f( a ) E R s.
3.7
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 Q and Z, but the result holds quite generally for rings of S-integers in number fields; see [411] or Exercise 3.38. We begin with a definition. Definition. A wandering point for a rational map ¢ : jp'i -+ pi is a point P E jp'i whose forward orbit G¢(P) is an infiniteset. Thus every point is either a wandering point or a preperiodic point? 2The 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 V of P and an integer n o such that rj>i(V ) n cf>J (V) = 0 for all i > j ~ no. Our definition coincide s with the topological definition if we use the discrete topology.
3.7. Integral Points in Orbits
109
Theorem 3.43. Let ¢( z ) E Q(z ) be a rational map ofdegree d ~ 2 with the property that ¢2(z) tf: Q[z]. Let Q E Q be a wandering pointfor ¢. Then the orbit
contains only finitely many integer points.
Theorem 3.43 is an immediate consequence of the following elementary geometric result combined with Siegel's Theorem 3.36 concerning integral values ofrational functions . Proposition 3.44. Let ¢ E q z ) be a rational fun ction of degree d ~ 2 satisfying ¢2(z) tf: C[z]. so Theorem 1.7 impli es that no iterate of ¢ is a polynomial map. Then
Ifd ~ 3. then #¢-3(00) ~ 3. Proof. We give a pictorial proof using the Riemann-Hurwitz formula . Suppose that ¢ is a rational map with # ¢ - 3(00) ::; 2. There are four possible pictures for the backward orbit of 00, as illustrated in Figure 3.1.
• - - Q - - p - - 00
• --........ •
.>
·-··---
Q - - p - - oo
(A)
(B)
Q--........
Q/~
p - - 00
(C)
Q - - P--........
• - - Q/--
P/~
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 - # ¢- l (p )).
(3.23)
PE p !
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 00 must be distinct, since P = 00 would mean that 0, and suppose that
lanl ;::: Ibnl H
€
for infinitely many n ;::: O.
(3.26)
This means that ¢n(a) = an/b n is very large, so ¢n(a) is close to 00. It follows that a is quite close to one of the points in the inverse image ¢ -n ( 00), say a is close to (3 E ¢ -n (00). But a is in Q, so one can hope that if n 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 n, so the constant in Roth's theorem changes with each new value of n. A more sophisticated idea is to use a fixed (large) integer m and apply Roth's theorem to the rational point ¢n-m (a) and a nearby point in ¢ -m (00). Note that
so the height of ¢n-m (a) is much smaller than the height of ¢n(a). We fix an integer m satisfying dm > 6/f (we will see later why this is a good choice for m) and we let (3 be the point in ¢ -m ( (0) that is closest to ¢n-m (a). How close is (3 to a? It turns out that this depends on the ramification index of ¢ at various points. Ifwe 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 C l , C 2 , ... may depend on ¢, a, and m, but they do not depend on n: n l ~ ; : : I b 1= l¢n(a)llanl an
ClP(¢n(a), (0), definition of P, since I¢n(a) I > 1, = ClP(¢n(a), ¢m((3)), since ¢m ((3) = 00, ~ C2P(¢n-m(a),(3), assuming ¢ is unramified, ~ C31¢n-m(a) - (31, definition of p, ~
>
C4
- H(¢n-m(a»3'
Cs
~ H(¢n(a))3/d""
Cs
Roth's Theorem 3.34 (with exponent 3), property of heights (Theorem 3.11),
3.8. Integrality Estimates for Points in Orbits
115
since m satisfies d"
> 6/E.
Takinglogarithms,we have proventhat log lanl
::;
2 -log(C5 )
for all n satisfying(3.26), i.e., lanl 2:
E
Ibnll+€.
We reiterate that the constant C 5 depends only on ¢, a, and m; it does not depend on n. Hence if n satisfies (3.26), then an, and also b«, are bounded. It follows that the ¢-orbit of a contains only finitely many points satisfying(3.26), and therefore . log lanl hmsuP I Ib I ::;l+E. n--->oo og n
(3.27)
Repeating the argument with the rational map 1j¢( 1/ z) and the initial point 1/00 yields . log Ibnl lim sup I I I::; 1 + E. (3.28) n--->oo og an Since (3.27) and (3.28) hold for all E > 0, we conclude that the limit in (3.25) is 1. This would complete the proof of Theorem 3.48 except for the unfortunate fact that rational maps 1P'1 ---. 1P'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.
Lemma 3.51. Let ¢ : 1P'1(C) ---. 1P'1(C) be a rational map oj degree d 2: 2, let p : 1P'1 (C) X 1P'1 (C) ---. IR be the chordal metric as defined in Section 1.1, and for Q E 1P'1 (C), let (3.29) eQ (¢) = max eQ' ( ¢) Q'E-l(Q)
be the maximum ofthe ramification indices ojthe points in the inverse image ojQ. Then there is a constant C = C( ¢, Q), depending on ¢ and Q, such that
min
Q'E-l(Q)
p(P, Q')e Q (
::;
Cp(¢(P), Q)
Jorall P E 1P'1(C).
(3.30)
In other words, if ¢(P) is close to Q, then there is a point in the inverse image oJQ that is close to P, but ramification affects how close.
Proof We dehomogenize using a parameter z such that Q -I 00 and 00 ~ ¢-l(Q). This means that we can write Q = (3 and P = a and that we are looking for the (3' E ¢-1((3) that is closest to oo. Writing¢(z) = F(z)/G(z) as a ratio of polynomials, the set ¢-1((3) is precisely the set of roots of the polynomial F (z) - (3G (z). If we factorthis polynomialover C as
3. Dynamics over Global Fields
116
with distinct 131, ... ,13r, then 13i E ¢ -1 (13) has ramification index ei. For notational convenience, we write e = 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 00, and writing a = z(P), we may assume that a is quite close to at least one of the points 13k in ¢-1(13). For example, we may require that a satisfy and This implies in particular that for all i
=I k.
Hence
IF(a) - 13G(a) I = Iblla - 1311 e 11a - 132I e2 .. ·Ia - 13rl er =
Iblla - 13kl e k II la- 13il e i i#
: : : Iblla - 13kl e k II ~113k - 13il e i if'k
=
C 1 1a - 13kl
ek
,
where the constant C 1 is positive and depends only on ¢ and 13. Further, the exponent satisfies ek ~ e, so we obtain the estimate
The fact that ¢(a) = F (a) / G(a) is close to 13 and the assumption that G(13) implies that G(a) is bounded away from 0, so dividing by G(a) yields
=I 0
This in turn implies the same estimate for the chordal metric, since we can estimate 1¢(a)1 using 1131 and we can estimate lal using l13kl. Therefore
o 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
Lemma 3.52. Let 1J : p I - pI be a rationalmap ofdegree at least 2 and let Q E pI be a point such that Q is not a totally ramifiedfixed point of 1J2. Then
lim eQ(1Jm) = O. (deg 1J) m
m -+ oo
(See Lemma 3.51for the definition of eQ(1J).) Proof The proof of this geometric result uses the Riemann-Hurwitz Theorem 1.1 in a manner similar to the way that we have used it in the past. Let d = deg(1J), fix an integer m, and let P E 1J - m ( Q). We use the fact that the ramification index is multiplicative,
To ease notation, we write We consider several cases. Case I. The points P, cP(P), cP2(P) , ... , cP=-l(p) are distinct. Note that this covers the case that Q is a wandering point. The Riemann-Hurwitz formula allows us to estimate
ep(1Jm) = eOeIe2 ' " em- I
'" < _ ( eo + el +m
+ em_ I ) m
= C eo - 1) + (el :'S
(2d~ 2 + 1) m
-
arithmetic-geometric inequality
1 ~+' " + (em- I - 1) + 1) m Riemann-Hurwitz formula.
(3.31)
This estimate is clearly much stronger than the stated result, since the righthand side of (3.31) has a finite limit as m - 00. Indeed, it is an elementary exercise to verify that (1 + t/m)m :'S e t is valid for all t 2: 0, so
ep(1Jm) :'S e2d -
2
for all m 2: 1 provided P,1J(P), ... ,1Jm - I (P) are distinct.
(N.B. ee is a ramification index, while the e in e2d -
2
is the number 2.71828 . .. !)
Case II. P is purely periodic of period k ~ 3, and m ~ k. The assumption that 1Jk(p ) = P implies that e, = e",i(p)(1J) depends only on i modulo k. Writing m = qk + r with 0 :'S T < k, we use the multiplicativity of ramification, applied to 1Jk, to estimate
118
3. Dynamics over Global Fields
Note that P, ¢(P), ... , ¢k-l(P) are distinct, so we can apply Case I to each of the ramification indices on the righthand side of (3.32). This yields
q ep(¢m)::; (2d;2 +l)k (2d;2
+lr
2d - 2 )m-r 2d- 2 ::; ( - k - + 1 e
applying (3.31) to ¢k and ¢r, using (1 + t/ry
::; e',
)m e2d- 2.
2d - 2 ::; ( - k - + 1 Finally, we observe that
~ (2d; 2+ 1) = 1- (1 - ~)
(1 _~) : ; ~
for all d
~
2 and k
~
3.
This proves that e p (¢m) ::; e2d- 2 (~d) m.
Case III. P is purely periodic of period k ~ 2, and m ~ k: We use the assumption that Q is not a totally ramified fixed point of ¢2 to deduce that at least one of e p (¢) and e¢(P) ( ¢) is strictly smaller than d - 1, so
ep(¢2) = ep(¢)e¢(p)(J) = eOel ::; d2 - d. Now using the fact that ¢2(p) = P, we see that em is even and equal to el if m is odd, so
ep(¢m) = eOele2··· em-l
(3.33)
= e¢",(p)(¢) is equal to eo ifm
multiplicativity of ramification index,
ifm is even, (e oe 1 )m/ 2 - { (eoel)(m-l)/2 eo ifm is odd,
(d2 - d)m/2 ifm is even, 2 ::; { (d - d)(m-l)/2d ifm is odd, =
(1-~) Lm/2J d":
Case IV. P is preperiodic. If P is periodic, we are already done from earlier cases. Suppose that P is not periodic and let j ~ 1 be the smallest integer such that ¢1 (P) is periodic. Let R = ¢1 (P) and let k be the exact period of R. Then P, ¢(P), ... , ¢1-1(p) are distinct and R = ¢1 (P) is periodic, so from Case I we have
ep(¢m) = ep(¢J) . e¢j(p) (¢m- j )
::;
e2d- 2eR(¢m- j )
::;
e2d- 2eR(¢m).
We also note that R is not a totally ramified fixed point of ¢2, since ¢-l (R) contains at least the two distinct points ¢k-l(R) and ¢1-1(p). (This is where we use the assumption that P is not itself periodic.) Hence we can apply Case II or III to eR(¢m), which gives the desired estimate for ep(¢m).
3.8. Integrality Estimates for Pointsin Orbits
119
In all four cases we have proven estimates for e p (qr) that imply that
This completes the proof of Lemma 3.52. More precisely, we proved that there are constants CI and C2, depending only on d and with C2 < 1, so that for all m 21. And if P is a wandering point, we have shown that e p ( qr) is bounded by a constant 0 depending only on the degree of ¢. We next prove an elementary lemma that relates the chordal metric p(x, y) to the Euclidean distance [z - yl.
Lemma3.53. Let p be the chordal metric on pI (C). Thenfor all x, y E C C pI (C), 1
p(x,y):S "2 p(y, oo)
p(x,y)
Proof Note that p(z, 00) = 1/ Jlzl2 have
p(x, y) =
2Ix-yl·
p(y 00)2
+ 1, so directly from the
Ix -yl Jlxl2+ IJlyl2 + 1 = Ix -
'2
definition of p we
Ylp(x, oo)p(y, 00).
Then the triangle inequality for p in the form p(x, 00) + p(x, y) 2 p(y, 00) (see (1.2) in Section 1.1) yields
p(x, y) 2
Ix -
yl{p(y, 00) - p(x, y) }p(y, 00).
Making the further assumption that p(x, y) equality.
:S
~p(y, 00), we obtain the desired in0
ProofofTheorem 3.48. We now have the tools needed to complete the proof of Theorem 3.48. Writing ¢n(a) = an/b n as usual, our initial goal is to prove that the set
is finite. We observe that for n E N(¢, a, f), the point ¢n(a) is close to 00, since
By assumption, ¢2(z) ~ qz], so 00 is not a totally ramified fixed point of ¢2. This allows us to apply Lemma 3.52, which gives us an integer mo = mo(d, f) such that eoo(¢m) :S dm for all m 2 mo. (3.35)
i
P
Having fixed an m 2 mo, we apply Lemma 3.51 to the map ¢m and the points ¢n-m(a) and Q = 00 to obtain a point f3n E ¢-m(oo) satisfying
=
120
3. Dynamics over Global Fields (3.36)
where C 1 = C1 (qr
, 00) depends
only on m and ¢. Hence if we define
then N( ¢, a, E) is the union of N( ¢, a, E, (3) for (3 E ¢-m( 00), so it suffices to prove that the set N( ¢, a, E, (3) is finite for each (3 E ¢-m (00). Note that if N( ¢, a, E, (3) is infinite, then (3.34) and (3.36) imply that lim
nEN(,p,a,E,(3)
p(¢n-m(a), (3) = O.
(3.37)
n-oo
In other words, ¢n-m(a) approaches (3 in jp'1(C) as n --t 00. We consider first the case that (3 i- 00. The limit (3.37) tells us that
p(¢n-m(a), (3) :S
~p((3, 00)
for all but finitely many n E N(¢, a, E, (3). (3.38)
Note that the inequality (3.38) allows us to apply Lemma 3.53 with x = ¢n-m(a) and y = (3, which yields
where as usual C 2 > 0 depends only on ¢ and m. We are now ready to repeat the calculation from page 114, 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
lanl
E
from (3.34),
2: p(¢n(a),oo) n
2: C3P(¢n-m(a),(3t",W 2: C4 1¢n- m(a ) _ (3l
e oow n
> C5 - H(¢n-m(a))3e oo(,pm)
>
C6 - H( ¢n(a)) 3e oo(,pm)ld m
C6
>
- H(¢n(a))'/2
)
from (3.36),
)
from (3.39), Roth's Theorem 3.34 with exponent 3, from Theorem 3.11, which says that
h(P)
=
d-mh(¢m(p))
+ 0(1),
from (3.35),
C6 lanlE/2
ci
/ E is bounded for n E N(¢, a, E, (3), and the same is true Hence lanl :S of Ibnl since Ibnl H E :S lanl, so ¢n(a) takes on only finitely many values for
121
3.8. Integrality Estimates for Pointsin Orbits
n E N(¢, a, E, (3). But by assumption, a is a wandering point for ¢, so we conclude that N( ¢, a, E, (3) is finite in the case that (3 ¥- 00. Suppose now that (3 = 00. Then a similar, but elementary, argument not requiring Roth's theorem yields
1
from (3.34),
lanl' 2: p(¢n(a), 00)
2: C7P(¢n-m(a), (3t'x((z) = z2 + 1. The sets of primitive 2nd , 3rd , 4th , and 6th periodic points are given, respectively, by the roots of the polynomials * ¢2(Z) - Z 2 ¢>2(Z) = ¢(Z) _ Z = Z
~(~! ~
¢>;(z)
=
¢*(z) 4
= ¢4(Z) -
¢*(z)
=
6
: Z
¢2(Z) - Z
+ Z + 2,
= z6 + z5 + 4z4 + 3z3 + 7z2 + 4z + 5,
= Z12 + 6zlO + z9 + 18z8 + 4z7 + 33z6 + 8z5 + 40z4 + 9z3 + 30z2 + 6z + 13,
(¢6(z) - z)(¢(z) - Z)
(¢3(Z) - Z)(¢2(Z) - Z)
= Z54
_ Z53
+ 27z52 -
25z51 + ...
-
13750z
+ 45833.
As this simple example clearly shows, it is infeasible to study dynatomic fields via explicit equations except for very small values of n.
124
3. Dynamics over Global Fields
The Galois group Gn,, so a that W is injective.
= 1, which proves D
The above discussion shows that the Galois group Gn,r/> of the dynatomic extension Kn,r/>/K roughly splits up into two pieces, a permutation piece determined by a permutation action on the orbits, and a cyclic piece determined by the action within each orbit. The permutation piece can be quite complicated, but one might hope that the dynamics of ¢ can be used to study the cyclic piece. We now try to make these vague remarks more precise. Let
G~,r/>
= Ker(7r: Gn,r/> ---. Sr) = {a
E Gn,r/>: a(Oj)
= OJ for all 1 :::; j:::; r},
and let
K~,r/> = fixed field of G~,r/>' Then
More precisely, if we fix a particular primitive n-periodic point P E Per~*(¢), then the extension K~,q'> (P) / K~,q'> is a cyclic extension of order dividing n whose Galois group is determined by the homomorphism
i : Gal(K~,r/>(P)/ K~,q'»
'-t
Z/nZ,
satisfying
a(P) = ¢i a (P).
(3.43)
This formula gives us some control over the Galois group of the relative abelian extension K~,r/> (P) / K~,r/>'
Remark 3.57. If L = K(a) is any Galois extension and if a E Gal(L/K) is any automorphism, then a(a) can always be expressed as a polynomial in a. Thus there is nothing special, a priori, in the fact that the action of Galois in (3.43) is given by a polynomial. However, ifwe fix ¢(z) of degree d and take n very large compared to d, then the extension K~,r/> (P) / K~,r/> is interesting because there is an automorphism a described by a polynomial (or rational function) ¢ of small degree compared to the degree of the extension.
3.10
Equidistribution and Preperiodic Points
There are many theorems and conjectures concerning the distribution of torsion points and points of small height on elliptic curves and abelian varieties. In this section we describe, without proof, some dynamical analogues. For further details, see Zhang's survey article [453] and the papers listed in its references.
3.10. Equidistribution and Preperiodic Points
127
We begin with the Manin-Mumford conjecture, which asserts that if X is an irreducible subvariety of an abelian variety A such that X n A tors is Zariski dense in X, then X is a translate by a torsion point of an abelian subvariety of A. The original Manin-Mumford conjecture was proven by Raynaud [367, 368]. Replacing torsion points by preperiodic points leads to a dynamical conjecture, where the dynamical analogue of a translate of an abelian subvariety of A is a preperiodic subvariety oflP'N as in the following definition.
Definition. Let ¢ : IP'N
----* IP'N be a morphism. A subvariety X C IP'N is a periodic variety for ¢ if there is an integer n ::::: 1 such that ¢n(x) = X. The subvariety is a preperiodic variety for ¢ if ¢m(x) is periodic for ¢ for some m ::::: O.
Conjecture 3.58. (Dynamical Manin-Mumford Conjecture) Let ¢ : IP'N a morphism defined over C and let X C IP'N be a subvariety. Then X
is Zariski dense in X
----*
IP'N be
n PrePer( ¢)
if and only if the subvariety X
is preperiodic for ¢.
The set of torsion points on an elliptic curve E, or more generally an abelian variety, is equidistributed with respect to the natural (Haar) measure on E(C). More precisely, we identify E(C) = C/ L as described in Section 1.6.3, and then for any open set U C C lying in a fundamental domain for E(C) we have
. #(E[n] n U) }~ #E[n]
=
Area(U).
A deeper equidistribution result of an arithmetic nature says that the Galois conjugates of torsion points are equidistributed. In order to state a dynamical analogue for morphisms ¢ : IP'N ----* IP'N, we need a 4>- invariant measure on IP'N (C) as described in the following proposition.
Proposition 3.59. Let ¢ : IP'N
----* IP'N be a morphism of degree d defined over C. There is a unique probability measure f-L¢ on IP'N (C) satisfying
and We call f-L¢ the canonical ¢-invariant probability measure on IP'N (C). Proof For a general construction that covers both archimedean and nonarchimedean base fields, see [453]. We also mention a standard result in dynamics (the KrylovBogolubov theorem [226, Theorem 4.1.1]), which says that any continuous map ¢ : X ----* X on a metrizable compact topological space X admits a Borel probability measure f-L¢ satisfying ¢* (f-L¢) = f-L¢, i.e., f-L¢ ( ¢ -1 (A)) = f-L¢ (A)
for every Borel-measurable subset A of X.
D
128
3. Dynamics over Global Fields
For the remainder of this section we fix an algebraic closure Q ofQ and an embedding Q '---+ Co So when we speak of a number field K and its algebraic closure K, we assume that they come with compatible embeddings into Co Definition. Let KjQ be a number field. For any algebraic point P E lP'N(K), let C(PjK) denote the set of Galois conjugates of P, i.e.,
C(PjK) = {a(P) E lP'N(K): a E Gal(KjK)}, and let
op denote the Dirac measure on lP'N (C) supported at P, s
(U)={l
°
p
ifPEU, if P ~ U.
We associate to P E lP'N (K) the discrete probability measure
1 Itp
= #C(PjK)
I:
oQ
QEC(PjK)
supported on the Galois conjugates of P. Definition. Let KjQ be a number field and let PI, P2 , P3 , ... E lP'N (K) be a sequence of points with algebraic coordinates. Fix a probability measure It on lP'N (C). We say that the sequence {Pd c- I is Galois equidistributed with respect to It if the sequence of measures J-tPi converges weakly' to J-t. We are now ready to state a dynamical equidistribution conjecture for Galois orbits of preperiodic points on lP'N, and more generally for points of small height. Conjecture 3.60. (Dynamical Galois Equidistribution Conjecture) Let KjQ be a numberfield, let ¢ : lP'N ---. lP'N be a morphism ofdegree d ~ 2 defined over K, and let PI, P2 , P3 , ... E lP'N (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 {Pdi:2:1 is Galois equidistributed in lP'N (C) with respect to the canonical d-invariant probability measure J-t¢.
(b)
If lim.,., h¢(Pi) =
00 0, then the sequence {Pdi:2:1 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 O. A version of the conjecture is known provided that the sequence of points satisfies a somewhat stronger Zariski density condition as in the following theorem. 3Recall that a sequence of measures J.Li on a compact space X converges weakly to J.L if for every Borel-measurable set U, the sequence of values J.Li(U) converges to J.L(U) as i -+ 00.
3.11. Ramification and Units in Dynatomic Fields
129
Theorem 3.61. (Yuan [450]) Let ¢ : JIDN ----+ JIDN be a morphism of degree d 2: 2 defined over K and let P1 , P2 , P3 , . .. E JIDN (K) be a sequence ofpoints satisfying the following two conditions: (a) Every infinite subsequence of {Pi }i>l is Zariski dense in JIDN. (b) hq,(Pi ) ----+ 0 as i ----+ 00. (In the terminology of [453], sequences with property (a) are called generic and sequences with property (b) are called small.) Then the sequence {Pdi>l is Galois equidistributed with respect to the canonical o-invariant probability measure !Lq, onJIDN(C).
Proof The proof is beyond the scope of this book. See Yuan [450] for a general version over archimedean and nonarchimedean base fields and algebraic dynamical systems on arbitrary projective varieties. Earlier results and generalizations are given by Autissier [15, 16], Baker-Ih [24], Baker-Rumely [28], ChambertLoir [98], Chambert-Loir-Thuillier [99], Favre-Rivera-Letelier [169] and SzpiroUllmo-Zhang [432]. 0 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. Conjecture 3.62. (Dynamical Bogomolov Conjecture) Let ¢ : JIDN ----+ JIDN be a morphism of degree d 2: 2 defined over a number field K and let X C JIDN be an irreducible subvariety that is not preperiodic. Then there is an f > 0 such that the set
{p
E
X(K): hq,(P) < f}
is not Zariski dense in X. Notice that Conjecture 3.62 implies Conjecture 3.58, since the set of points with h¢(P) < E 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.
3.11 Ramification and Units in Dynatomic Fields In Section 3.9 we used periodic points to construct field extensions Kq"n 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 nth root of unity and a E Gal(Q/Ql). Then
130
3. Dynamics over Global Fields
IT(( ) =
(i(a)
fora unique jfo] E (7L./ n7L.)* .
This defines an isomorphism
j : Gal(Q (( )/ Q )
-----t
(Z/ nZ)*,
IT t------t
j (IT ),
expres sing the action of IT on Q(() as a polynomial action I/>(z) = zi . Now let p be a prime not dividing n , let p be a prime of Q (() lying above p, and let ITp E Gal(Q / Q ) be the corresponding Frobenius element. The definition of Frobenius says that
IT p ( () == C/
(mod p).
(3.44 )
However, the n th roots of unity remain distinct when reduced modulo p, so the congruence (3.44) implies an equality in Q((),
IT 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 , although the final results are not as complete as in the classical case . Let p be a prime ideal of the ring of integers of K g,n' let p be the residue characteristic of p, and let q be the norm of p. We assume throughout that p f n. Choose a prime ideal liJ in K g,n(P) lying above p. Assuming that p does not ramify in Kg,n(P), the associated Frobenius element ITp is determined by the condition ITp(a)
== a q
(m od
liJ)
for all a in the ring of integers of Kg,n(P), On the other hand , by construction the action of ITp on P E Per~* (I/» is given by by formula
The next proposition allows us to characterize the primes at which the extension
K g,n(P)/ K g,n may be ramified. Proposition 3.63. Let K be a number field, let I/> E K (z) be a rational map of degree d 2: 2, let P E Per~* (I/> , K) be a point in pI (K) ofexact p eriod n, and let p be a prime ofK satisfying the following three conditions:
I/> has good reduction at p. p does not divide n . If A (P ) i- 1, then p does not divide A (P ) - 1. Then
P mod p has exact p eriod n . In particular, the set {p : P mo d p has p eriod strictly less than n }
is a finite set ofprimes of K.
(3.45) (3.46 ) (3.47)
131
3.11. Ramification and Units in Dynatomic Fields
Proof Let p be a prime of K satisfying (3.45), (3.46), and (3.47). Let m be the exact period of F and let r be the order of AJ,(F) in lF~. Theorem 2.21 tells us that either n = m or n = mr, since (3.46) rules out powers of p appearing in n. If Aq,(P) = 1, then also AJ,(F) = 1, so r = 1 and n = m. On the other hand, if Aq,(P) i= 1, then (3.47) tells us that
But AJ,(Ft
= i
by definition of r, so nlm cannot equal r. Hence n
=
m
III
all cases, which shows that F mod p has exact period n for all primes p satisfying (3.45), (3.46), and (3.47). 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
D
~~~R
Corollary 3.64. Let K be a number field, let c/>( z) E K (z) be a rational map of degree d 2: 2, and let Kn,q, be the nth dynatomic field for c/>. Let S be the set of primes p of K such that either c/> has bad reduction at p or p divides n or p divides the quantity
II
(Aq,(P) -1).
(3.48)
PEPer~*(q,)
Aq,(P)#l
Then Kn,q,1 K is unramified outside ofS. Proof The field extension Kn,q,1 K is generated by the points of Per~* (c/». Proposition 3.63 tells us that if p ~ S, then those points remain distinct when reduced modulo primes lying over p. Hence the extension Kn,q,1 K is unramified at p. D Example 3.65. We continue studying the quadratic polynomial c/>(z) = z2 + 1 from Example 3.55 (see also Example 2.37). The polynomial c/>(z) has everywhere good reduction, so if we assume for the moment that none of its periodic points have multiplier equal to 1, then the quantity (3.48) in Corollary 3.64 is equal to
II
(Aq,(OO) -1)
q,~(a)=O
=
II
((c/>n),(oo) - 1)
= Res(c/>~(z),
(c/>n),(z) - 1).
q,~(a)=O
Denoting this resultant by D. n (c/> ), it is not hard to compute the values of D. n (c/» for small values of n. We obtain
D.2(c/» = 72 , D.3(c/» = (33 . 11)3, D.4(c/» = (32'11 . 13·41)4, D.5(c/»
= (33.7.83.331.140869)5,
D.6(c/» = (34.5.7.23.73.223.2251. 347495839)6.
132
3. Dynamics over Global Fields
The field K rjJ,2 is generated by the roots of ¢2 (z) = z2 + z + 2, so we have explicitly KrjJ,2 = K( A). For higher values of n, we can check the primes dividing I::i. n (¢ ) by computing directly the discriminant of the polynomial ¢~(z), whose roots generate KrjJ,n/K. Thus for example, Disc(¢j) = _3 6 . 113 and Disc(¢4') = 34 .11 4 .13 3.41 4 . The local theory of units described in Section 2.7 allows us to construct units (or at least 8-units) in dynatomic fields Kn,rjJ and their composita. For convenience, we make two definitions.
Definition. Let 8 be a finite set of places of K and let L / K be a finite extension. We say that u E L is an 8-unit ifit is an 8 L -unit, where 8 L is the set of places of L lying over 8.
Definition. Let ¢(z) E K(z) be a rational map. We write 8rjJ for the set 8rjJ =
M'K U {primes at which ¢ has bad reduction}.
Thus ¢ has good reduction at all primes in the localized ring Rs. In particular, if
¢(z) =aozd+,.·+ad is a polynomial, then the finite primes in 8rjJ are the primes where some a; is nonintegral together with any primes for which ao is not a unit. Having set this notation, we now state globalized versions of the dynamical unit theorems from Section 2.7.
Theorem 3.66 (Global version of Theorem 2.33). Let ¢(z) E K[z] be a polynomial of degree d 2': 2, let 0: E Per~* (¢) with n 2': 2, and fix integers i and j satisfying gcd(i - j, n) = 1. Then ¢i(o:) - ¢J(0:) ¢(o:) - 0:
is an 8rjJ-unit in KrjJ,n' For the next result, we recall that the cross-ratio of four points Pi, P2 , P3 , P4 in lP'1 is the quantity
Theorem 3.67 (Global version of Theorem 2.34). Let ¢( z) E K (z) be a rational function ofdegree d 2': 2, let P E Per~* (¢), andfix integers i and j satisfying gcd(j, n)
Then is an SrjJ-unit in KrjJ,n'
= gcd(i - 1, n) = gcd(i - i, n) =
1.
133
3.11. Ramification and Units in Dynatomic Fields
Theorem 3.68 (Global version of Theorem 2.35). Let ¢(z ) E K (z) be a rational function ofdegree d ;::: 2, let m and n be integers with m 1n and n 1m, and let
P
= [x, y] E Per;; (¢)
Q = [x', y'] E Per~* ( ¢ ) .
and
Denote by S p the set ofplaces Sp
=
{v E M~ : v(x)
> 0 and v (y ) > o}
U {v E M~ : v (x )
< 0 or v (y ) < o} ,
and similarly for S Q. Then x y' - x' y is an (8 U Sp U SQ) -unit in the compositum K ,mK ,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 ,mK ,n. Further, since Gal(K,n / K ) frequently contains an element a whose action on points P E Per~* (¢) is characterized by the equation a(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
We compare this to the construction of cyclotom ic units in cyclotomic fields. Let q be a prime power and let ( be a primitive qlh root of unity. Then the cyclotomic field Q( () contains the units for 2
< i ::; q; 1 with gcd(i , q).
The Galois group Gal (Q(() /Q) is the set of elements a t characterized by
at (()=(t
withO Sand h : S ---> ~ arc cont inuous maps. Prove that h : S ---> ~ is also a continuous map . (Hint. Show that the functio~s h", (P) = d: " h (cP m(P)) for m = 1,2, 3, .. . are continuous and converge uniformly to h.) 3.15. Let h be the canonical height for cP(z) = Z2 + c. For any given e E Q , there is a minimum nonzero value for h<jJ (z) as z ranges over nonpreperiodic points in Q . Find that minimum value for: (a) e = O.
Exercises
138
(b) c = -2. (c)c=-1. (d) ** arbitrary c E Z. (e) ** arbitrary cEQ. (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 h¢ in tenus of c.) 3.16. Let ¢(z) = Z2 - 1 . Analyze the canonical height for points in the following sets. (a) 8¢( 00), the attracting basin of 00, where we recall (Exercise 2.1) that the attracting basin of 00 for a polynomial rjJ is the set
8¢(00) =
{a E C:
lim rjJn(a) = n~oo
oo} U [oo}.
(b) F¢" 8¢ (00). (Note that all ofthe points in this set are eventually attracted to the attracting 2-cycle 0
..!.-..
-1.)
(c) .:T¢. (See also Exercise 3.27.) 3.17. Let rjJ : IP N ---+ IP N be a morphism defined over a number field K, and for any point P E IP N (K), let D(P) = [K(P) : Q]. Prove that there is a constant C = C(¢) > 0 such that 11,
¢
(P)
> dC D1( P ) 2
for all P E IPN(K) with P
-
d 'F
PrePer(A..). 'f/
(Hint. Use the estimate for the number of points of bounded degree and height given in Exercise 3.3.) This is a very weak version of the dynamical Lehmer Conjecture 3.25. 3.18. ** Let cEQ with c # 0, -2 and let rjJ(z) = z2+ c. Prove that there exist K = K(C) and N = N(c) > 0 such that K
A
h¢(a) 2: [Q(a) : Q1N
>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)
= aox d + a1x d- 1 + ... + ad E Z[x]
be the minimal polynomial of a, normalized so that ao Factor f(x) over Cas f(x) = ao TI(x - ai). Prove that
H(a)d = ao
>
0 and gcd(ao, ... , ad)
1.
II max{ 1, lail} = 11 10glf(e27rio)IdB. d
i=l
0
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 tenus 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.
Exercises
139
3.20. Let ¢(z) E Q(z). Write a program to estimate h¢(a) directly from the definition. Use your program to compute the following heights to a few decimal places. (See also Exercises 3.30 and 5.31.) (a) ¢(z)=z2-1anda=~. (b) ¢(z) = Z2 + 1 and a = ~. (c) ¢(z) = 3z 2 - 4anda = 1. 1 (d) ¢(z)=z+-anda=1.
(e) ¢(z) =
3z 2
z
Z2 _
3.21. Let ¢(z) =
1 1 and 00= 1.
Z2 -
Z
+ 1. The ¢-orbit of the point 2 is called Sylvester's sequence [428],
CJ¢(2) = {2,3, 7,43,1807,3263443,10650056950807, 113423713055421844361000443, ... }. (a) Prove that Sylvester's sequence satisfies ¢n+l(2) = 1 + ¢O(2)¢1(2)¢2(2)¢3(2) ... ¢n(2). (b) A rough approximation gives h¢(2) ~ 0.468696, so ¢n(2) is approximately equal to 2. e°.468696n 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 Z2 + c.) 2
(c) Deduce that there is a real number H such that ¢n(2) is the closest integer to H" for all n 2 O. 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 (xnk:>o be a sequence of positive real numbers with the property that Xo
> 1+8
(a) Prove that the sequence (b) Prove that the limit
and Xn
is strictly increasing and that H
=
lim
x;jd
X n ---. 00
as n ---.
00.
n
n~oo
exists. (Hint. Take logs and use a telescoping sum to show that the sequence is Cauchy.) (c) Prove that n2 I 8 IH - X n ::; d _ 1 for all n 2 O. 3.23. Let d
2 2 and let ¢(z) E Z[z] be a monic polynomial of degree d, say ¢(z)
=
zd + az d- 1
+ ...
E
Z[z].
Prove that for every E > 0 there is a constant C = C(¢, E) such that for all a E Z satisfying 1001 2 C we have for all n 2 O.
Exercises
140 Section 3.5. Local Canonical Heights
The general theory oflocal canonical heights is developed in Section 5.9. However, the theory becomes much simpler if 1> 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. Exercises 3.24-3.30 ask you to develop some of the theory oflocal canonical heights for polynomials. 3.24. Let 1>(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
tc.,
(b) The function
a
1---+
>-¢,v(a) -logmax{lalv, I}
is continuous on tc, and has a finite limit as lal v -+ 00. 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>-¢,v(a).
vEM K
3.25. Let 1>(z) E K[z] be a polynomial and let v be an absolute value on K. Prove that the local height >-¢,v as defined by (3.49) has the following properties. (a) >-¢,v(a)~OforallaEKv. (b) >-¢,v (a) = a if and only if I¢n (a) Iv is bounded, equivalently, if and only if a is in the v-adic filled Julia set K; (1)) of 1> (see Exercise 2.1). 3.26. Let 1>(z) E qz] be a polynomial with complex coefficients and let B¢(oo) be the attracting basin of 00 for 1>. (See Exercises 2.1 and 3.16 for the definition of B¢ (00).) Prove that the local height >-¢ : ([ -+ IR as defined by (3.49) has the following properties: (a) >-¢ is a real analytic function on B¢( 00) -, {oo}. (b) The function >-¢ is harmonic on B¢ (00) -, {00}. In other words, writing z function >-¢ is a solution to the differential equation
= x + iy, the
on the open set B¢ (00) -, {oo}. It satisfies the boundary conditions that >-¢ vanishes on the boundary of B¢ (00) and has a logarithmic singularity at z = 00, i.e.,
>-¢(z) -log Izi
is bounded as z
-+
00.
(c) >-¢ is the unique function that has the properties described in (b). In classical terminology, the function >-¢ is the Green function for the filled Julia set lC(¢)
jp'1(iC) -, B¢(oo).
=
Exercises
141
3.27. Let 4>(z) = Z 2 - 1. Analyze the local canonical height '\ ",.v for points in: (a) the attracting basin B1> (00) of 00. (b) F1> " B1' (oo). (c) :1",. (See also Exercise 3.16 .)
3.28. Let 4>(z ) E Z[z] be a monic polynomial of degree d ~ 2 and let a rational number written in lowest terms. Prove that
= alb
E Q be a
with equality if and only if a is in the filled Julia set K:(4» of 4>. 3.29. Let K IQ be a number field and let 4>(z) E K (z ) be a ration al map of degree d ~ 2. For each finite place v E M~ , write 4>(z) = Fv(z)/Gv(z) as a ratio of polynomials that are normalized for v, i.e., the coefficients of F; and G» are v-adic integers and at least one coefficient is a v-adic unit. Let ,\~~': be the local canonical height, as described in Theorem 3.27, normalized using G» in (3.16). (a) Prove that ,\~~': is well-defined, independent of the decomposition 4>(z ) as a ratio of vnormalized polynomials. For nonarchimedean places v this serves to pin down a specific function ,\¢,': that depend s only on 4> and v (cf. Remark 3.28). (b) Give an example to show that it may not be possible to write 4>(z ) F (z )/G(z ) such that F and G are simultaneously normalized for all finite places v E M~ . (c) More generally, prove that every 4> E K (z ) can be written as F(z )/G (z ) with F and G simultaneously normalized for all finite places v E M~ if and only if K has class number 1.
=
3.30. Suppose that function
4>( z) E K (z) has good reduction at a finite place v E M K. Prove that the '\¢,v(a ) = log max{ lal v , 1}
for a E K;
is a local canonical height by showing that it has the required propertie s, Section 3.6, Diophantine Approximation 3.31. Prove Dirichlet's Theorem 3.33, which says that for every a E lR" Q there are infinitely many x] y E Q satisfying
I~ -al ~
y\'
(Hint. Look at the numbers ya - LyaJ for 0 ~ y ~ A. They all lie in the interval [0,1]. Divide the interval into A equal pieces and use the pigeonhole principle.) 3.32. Let B be a nonzero integer. (a) Prove that every solution (x , y) in integers to the equation (3.50)
(0:) n Rs is a finite set. (This exercise generalizes Theorem 3.43.)
3.39. Let K be a number field, let S C M K be a finite set of absolute values on K that includes all the archimedean absolute values, and let Rs be the ring of S-integers of K. Let ¢(z) E K(z) be a rational function of degree d ~ 2 satisfying ¢2(Z) .;. K[z]. (a) Let n = 4 if d = 2 and let n = 3 if d ~ 3. Prove that the set
{o
E
K: ¢n(o:) E Rs}
is finite. (b) Give an examplewith (n, d) = (3,2) and oo nonperiodicfor ¢ such that the set in (a) is infinite. (c) Same question as in (b) with n = 2 and d arbitrary. (d) Repeat (b) and (c) with oo a fixedpoint of ¢.
3.40. Let ¢l, ... ,¢r E Q( z) be rational functions of degree at least 2, and let be the collection of rational functions obtained by composing an arbitrary finite number of ¢l, ... , ¢r. Note that each ¢i may be used many times. For example,if r = 1, then is simply the collection of iterates {¢l}. Also note that in general, compositionof functions is not commutative, so if r ~ 2, then is likely to be a very large set. (a) A rational map ¢ E K(z) is said to be of polynomial type if it has a totally ramified fixed point. Prove that for such a map, there is a linear fractional transformation f E PGL2(K) such that ¢f E K[z]. Further, if ¢ is not conjugate to Zd, prove that one can take f to be in PGL2(K). (b) Assume that contains no maps of polynomial type. Prove that there are finite subsets 1 and 2 of satisfying and (Here ¢ denotes the compositionof ¢ with every map in .) (c) Let 0: E Q. The -orbit of 0: is the set 0(0:) = {¢(o:) : ¢ E }. Continuing with the assumptionthat contains no maps of polynomialtype, prove that the -orbit of 0: contains only finitely many integers, i.e., prove that 0 (0:) n Z is a finite set.
3.41. Definethe (logarithmic) height ofa rational map ¢(z) E Q(z) by writing
¢(z)
= F(z) = ao + alZ + G(z) bo + blz +
+ adzd + bdzd
with F(z), G(z) E Q[z] and setting
h(¢)
= h([ao, al, ... , ad, bo, bl, ... , bdl).
Prove the followingquantitative version of Proposition3.46. For all d ~ 2 there is a constant 0 = Oed) such that for all integers N rational map ¢( z) E Q( z) of degree d with the followingproperties:
~
0 there exists a
Exercises
144 • rji (z ) i C[z]. • 0 is a wandering point for . • 0, (0), 2(0), 3(0) , . . . , ¢ N (0) E Z. N
• h( ¢) S; C . d . Try to do quantitatively better than this, for example, replace the height estimate with one of the form h( ¢) S; C · fJ N for some s < d.
3.42. Recall that a rational map ¢ (z ) E Q(z) is affine minimal ifits resultant Res(2 is not a polynomial, we have conjectured (Conjecture 3.47) that the size of O(a) n Z is bounded solely in terms of the degree of ¢. For each integer d ~ 2, let
C (d)
= sup { # (O (a) n Z) :
(z) E Q(z), ¢2(Z) i Q[z], } ¢ is affine minimal, and . a E ]P'N (Q) " P rePe r(¢)
Thus Conjecture 3.47 says that C(d) is finite. (a) Prove that for every integer N ~ 0 there exists an affine minimal rational map ¢(z) E Q(z) such that ¢2 (z) i Q[z], such that 0 is not a preperiodic point for ¢, and such that
are all integers. In particular, C(d) - t 00 as d - t 00 . (b) Prove that the function ¢(z) in (a) can be chosen to have degree L(N - 1)j2J. Hence C(d) ~2d+2 .
(c) Let ¢>(z) be the function ¢ (z ) = 481z
5
-
Z5 -
4
2565z + 9385z 465z 4 + 2185z 3
3 -
-
2
14955z + 12094z - 3720 . 6975z 2 + 8254z - 3720
Trace the orbit of O. How many integers do you find? (d) Find a rational function as in (a) of degree 2 and with the property that ¢i (0) is an integer for all 0 S; i S; 6. Hence C (2) ~ 7, which improves the bound from (b). Can you find infinitely many such functions? Can you prove that C(2) ~ 8? (e) Find a rational function as in (a) of degree 3 and with the property that ¢i(O) is an integer for all 0 S; i S; 8. Hence C(3) ~ 9, which improves the bound from (b). Can you find infinitely many such functions? Can you prove that C(3) ~ 1O? Section 3.8. Integrality Estimates for Points in Orbits
3.43. Let K; be a field complete with respect to the absolute value v, let ¢ : ]P'i - t ]P' i be a rational map of degree d ~ 2 defined over K v , and let p v : ]P'l (K v ) x ]P'l(K v ) - t 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 ]P'i (K v ) there is a constant C« = C v (¢ , Q) such that
Here e Q(¢) is as defined in Lemma 3.51.
145
Exercises
3.44. Let K be a number field and let 1> : pI --4 IP'I be a rational map of degree d 2:: 2 defined over K . Prove that there is a finite set of absolute values S C MK such that for all v 1- Sand all Q E 1P'1 (K), the constant Cv ( 1>, Q) in Exercise 3.43 may be taken equal to l. 3.45. Let 1>(z) E Q(z) be a rational map of degree d 2:: 2 with 1>2(z) 1- Q[z], let a E Q, and write 1>n(a ) = anlb n E Q as a fraction in lowest terms as usual. Prove that
3.46. Let KIQ be a number field, let v E M K be an absolute value on K , and let 1>( z) E K (z) be a rational function of degree d 2:: 2. Suppose that a E 1P'1 (K) is a wandering point for 1> and that 'Y E 1P'1 (K ) is any point that is not a totally ramified fixed point of 1>2. Prove that . -log pv (1)n(a), 'Y) lim =0. n -oo h( 1>n(a)) Takingfirst 'Y = 00 and then 'Y to number fields.
= 0, explain how to use this result to generalize Theorem 3.48
Section 3.9. Periodic Points and Galois Groups 3.47. Let P E IP'N(K) be a point in projective space and let K (P ) be its field of definition. Prove that K (P ) = fixed field of {u E Gal(KIK ) : u{P ):=: Pl. In mathematical terminology, this says that the field of moduli of P is a field of definition for P . Wediscuss fields of moduli and fields of definition for (equivalence classes of) rational maps in Chapter 4.
Section 3.11. Ramification and Units in Dynatomic Fields 3.48. Let 1>{z) = Z2 + c and let
; (z)
= 1>3(Z) -
z 1>(z) - z
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 ;; (z) and assume that the field Q(c, a) is an extension of Q( c) of degree 6 (i.e., ;;(z) is irreducible in Q(c) [z]). Theorem 2.33 implies that and are units. Compute these units explicitly as elements of Q[c, o]. (c) Prove that there is a field automorphism a : Q (c, a) --4 Q(c, a) characterized by the fact that it fixesQ (c) and satisfies u(a ) = 1>(a) = a 2 +c. (Note that in general, Q( c, a) will not be a Galois extension of Q (c).) Prove that u 3 is the identity map. (d) Compute the units a (u 1) and a (U2). (e) Compute the units u 2(uI) and U2(U2). (f) Express u(uI), U(U2), u 2(uI), and U2(U2) in the form ±ui u~ .
Exercises
146
3.49. Let ¢>( z ) = Z2 + c and let q>; (z ) be as in the previous exercise. (a) Prove that q>;(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 ge(z) E Q[e , z] such that the factorization q>3(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 K; ofQ(e). (c) Let Q be a root of ge(z). Use the results of the previous exercise to construct units in Q (e) (Q). 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 Z d + al z d- l + ...+ ad , where al , .. . , ad are indeterminates. Let p be a prime and let Q E Per;*(¢» . Theorem 3.66 says that the elements
U i ,j
=
¢>i(Q) _ qJ(Q) ¢>(Q) _ Q '
0:::; j < i < p,
are units. What is the rank of the group that they generate? Is the answer the same for the polynomial ¢>(z) = Z2 + c? 3.51. ** Let ¢>(z ) be a generic rational function of degree d ~ 2, let p be a prime, and let p E Per; *(¢». Theorem 3.67 describes how to construct units K,(p, ¢>p,¢>ip, qJp). What is the rank of the group that they generate? Is the answer the same for the rational function ¢>(z) = (Z2 + b)/(bz 2 + I) ?
Chapter 4
Families of Dynamical Systems Most of our work in previous chapters has focused on a single rational map ¢(z) and the effect its iterates have on different initial values. We now shift focus and consider the effect of varying the rational map ¢( z). In order to do this, we study the set of all rational maps. This set turns out to have a natural structure as an algebraic variety, as does the set of rational maps modulo the equivalence relation defined by PGL 2 conjugation. There are many threads to this story. The specific topics that we touch upon in this chapter include: 1. 2. 3. 4. 5. 6.
Dynatomic polynomials and fields generated by periodic points. The space of quadratic polynomials (the simplest nontrivial case). Rational maps that are PGL 2-equivalent over k, but not over K (twists). Field of defintion versus field of moduli for rational maps. Minimal models for rational maps. Moduli spaces of rational maps (with marked periodic points).
All of these topics have close analogues in the geometric and arithmetic theory of elliptic curves and abelian varieties. For purposes of comparison, we list the correspondences: 1. 2. 3. 4. 5. 6.
Division polynomials and fields generated by torsion points. The space of elliptic curves (the simplest abelian varieties). Abelian varieties isomorphic over K, but not over K (twists). Field of definition versus field of moduli for an abelian variety. Minimal Weierstrass equations and Neron models. Elliptic modular curves and moduli spaces of abelian varieties.
In this chapter, we let K be a perfect field and fix an algebraic closure k of K, although much of what we do is also valid for nonperfect fields if we use instead a separable closure of K. We write Gal(k/ K) for the absolute Galois group of K. It is convenient to use the profinite group Gal(k/ K), but we observe that it generally suffices to work with finite extensions and finite Galois groups, since the coefficients of any particular rational map ¢(z) E k (z) lie in a finite extension of K. 147
148
4. Families of Dynamical Systems
4.1 Dynatomic Polynomials Let ¢( z) E K [z] be a polynomial. Then the fixed points of ¢ are the roots of ¢( z) - z (plus the point at (0), and more generally, the points of period n for ¢ are the roots of ¢n(z) - z. However, the polynomial ¢n(z) - z may have roots of period smaller than n, since if ¢k(a) = a and kin, then also ¢n(a) = a. It is natural to try to eliminate these points of strictly smaller period and focus on the points of exact period n. Example 4.1. We recall an analogous situation. The roots of the polynomial z" - 1 include all nth roots of unity, not only the primitive ones. The nth cyclotomic polynomial is defined using an inclusion-exclusion product, nth
cyclotomic polynomial
=
II(zk -
l)lL(n/k).
(4.1)
kin It is the polynomial whose roots are the primitive Mobius function J-l defined by J-l(l) = 1 and
ei
nth
roots of unity. Here J-l is the
ec)_{(-lY ifeI=···=er=l, . a If any e; 2: 2.
J-l (PI .. ·P r
(4.2)
See [216, Section 2.2] or [11, 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 polynomial, using the fact that the (complex) roots of z" - 1 are distinct and the following basic property of the J-l function (Exercise 4.2):
L J-l (~) = {I ~f n = 1, a
kin
(4.3)
Ifn> 1.
Taking our cue from the example provided by the cyclotomic polynomials, we might define the nth dynatomic polynomial by the formula
n(z)
= II(¢k(z) - zt(n/k). kin
However, it is not clear that n(z) is a polynomial, since ¢n(z) may have multiple roots, as shown by the following example. Example 4.2. Let ¢(z) be the polynomial
¢(z) Then
=z
2
-
3 4.
4.1. Dynatomic Polynomials
149
¢(z) - z = z2 - Z -
~ = ( z - ~)
3 2 4 3 ¢ (z) - z = z - - z2 - z - - = 2
¢2(Z) - Z = ¢(z) - z
16
(
(z+
~) ,
3 z- - ) 2
(
1 z+ - ) 2
3
'
(Z +~) 2 2
Thus ¢2(z) - z vanishes with multiplicity 3 at the point z = - ~ , and although it is true that the ratio ~g/~: 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 ¢(z). We now tum to general rational maps ¢(z) E K(z) and develop tools that are useful for studying their periodic points. Let ¢( z) be a rational function of degree d and write ¢ = [F(X, Y), G(X, Y)] using homogeneous polynomials F and G. Then the roots of the polynomial
Y F (X ,Y ) - XG (X, Y ) in ]F'l 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 ¢n of ¢ and assign multiplicities to the n- periodic points.
Definition. Let ¢( z) E K (z) be a rational function of degree d, and for any n 2: 0, write
¢n = [Fn( X ,Y ),Gn(X,Y )] with homogeneous polynomials Fn , Gn E K [X,Y ] of degree d" , (See Exercise 4.9 for a formal inductive definition of Fn and G n . ) The ti-period polynomial of ¢ is the polynomial q" n(X, Y ) = YFn(X,Y ) - XGn (X,Y ). Notice that q"n (P ) = 0 if and only if ¢n(p) = P, which justifies the name assigned to the polynomial q"n. The n1hdynatomic polynomial of ¢ is the polynomial'
q"* n(X , Y) = II(YFk (X, Y ) - XG k (X, y))!"(n/k) =
kin
II q" k" (X y)!"(n/k) kin
where JL is the Mobius function . If ¢ is fixed, we write n and ~. If ¢(z) E K [z] is a polynomial, then we generally dehomogenize [X ,Y] = [z,l] and write n (Z) and ~ ( z ) . All of the roots P of; ,n(X , Y ) satisfy ¢n(p ) = P , but we saw in Example 4.2 that ; ,n (X, Y ) may have roots whose periods are strictly smaller than n . Following Milnor [302], we make the following definitions. I See
Theorem 4.5 for the proof that ep; .n is indeed a polynomi al.
150
4. Families of Dynamical Systems
Definition. Let cjJ{ z) E K (z) be a rational map and let P E pI be a periodic point for cjJ .
• P has period n if n(P) =
a.
• P has primitive (or exact) period n if n(P) = m< n.
• P has formal period n if ~ (P) =
a and m{P ) =I a for all
a.
We set the notation Pern{cjJ)
= {p E pI: n {P ) = a},
Per~ { cjJ) = {P E pI : ~ {P ) = Per~*{cjJ)
= {p
E
a},
pI: n{P ) =
a and m{P) =I a for all 1 ~ m < n}.
Thus P er., (cjJ) is the set of points of period n, Per~ (cjJ) is the set of points of formal period n, and Per~*( cjJ) is the set of points of primitive (or exact) period n. Sometimes we treat these as sets of points with assigned multiplicities, e.g., if P E Per~ (cjJ) , the multiplicity of P is the order of vanishing of ~ at P. It is clear that primitive period n
==>
formal period n
==>
period n ,
but neither of the reverse implications is true in general.
Remark 4.3. The polynomial ,n is homogeneous of degree d" + 1, so counted with multiplicity, the map cjJ has exactly d" + 1 points of period n. And if we let IId(n ) = deg(; ,n{X, Y)) =
LJL ( ~) (dk + 1),
(4.4)
ki n
then counted with multiplicity, the map cjJ has exactly lid ( n ) points of formal period n. The number IId(n) grows very rapidly as d or n increases . See Exercise 4.3. The first few period and dynatomic polynomials for cjJ( z ) = z2 + c are listed in Table 4.1 on page 156. Notice how complicated n and ~ are, even for small values of n. For example, 6 has degree 54 as a polynomial in z and degree 27 as a polynomial in c.
Remark 4.4. Rather than using the homogeneous polynomials n and ~, it is sometimes more natural and convenient to consider instead the associated divisors in pI, especially in generalizing the theory to higher-dimensional situations. This is the approach taken in [313], where for any (nondegenerate) morphism cjJ : V --+ V of smooth algebraic varieties, the O-cycle ,n is defined as the pullback of the graph of cjJn by the diagonal map A : V --+ V x V. See Exercise 4.8. We now prove the important fact that the dynatomic polynomial ; ,n(X, Y) is indeed a polynomial.
4.1. Dynatomic Polynomials
151
Theorem 4.5. Let ¢( z) E K (z) be a rationalfunction ofdegree d jpI
~
2. For each P E
(K), let and
or, in terms ofdivisors in Div(lPi ), div(,n) =
L ap(n)(P)
and
div( ; ,n) =
p
L aj,( n)(P). p
(a) ;,n E K[X, Y], or equivalently,
aj,(n) ~ 0 for all n ~ 1 and all P E jpI. (b) Let P be a point ofprimitive period m and let >.( P) = (¢m)' (P) be the multiplier of P. Then P has formal period n. i.e., aj,(n) > 0, if and only if one of the following three conditions is true: (i) n = m. (ii) n = mr and >.( P) is a primitive r 'h root ofunity. (iii) n = mrp", >.(P) is a primitive r 'h root of unity, K has characteristic p, and e ~ 1.
In particular, ofn.
if K
has characteristic 0, then a j,(n) is nonzero for at most two values
Proof By the definition of~, we have the relation
aj,(n) =
L J.l(n/m)ap(m). min
We begin with a lemma that describes the value of a p (n) for fixed points.
Lemma 4.6. Let 'ljJ (z) E K (z) be a rational function of degree d ~ 2, let P E jpI (K) be a fixed point of'ljJ, let>' = >.p( 'ljJ ) = 'ljJ' (P) be the multiplier of'ljJ at P , and let
ap(N) = ordp(1jJ,N(X, Y)) be as in Theorem 4.5. Thenfor any N
>.N # >. # 1 and >.N =
~
2,
1
=}
1
=}
>. = 1 and N # 0 in K >. = 1 and N = 0 in K
=} =}
= ap(l) = 1,
(4.5)
ap(N) > ap(l) = 1, ap(N) = ap(l) ~ 2, ap(N) > ap(l) ~ 2.
(4.6)
ap(N)
(4.7) (4.8)
Proof Making a change of variables, we may assume that P = 0, and then the assumption that P is a fixed point of 'ljJ means that 'ljJ( z) has the form (4.9)
4. Families of Dynamical Systems
152
where 0: -I 0, e 2: 2, and O(ze+l) is shorthand for a function that vanishes to order at least e + 1 at O. Note that
z) =
ap(l) = ord(1P(z) z=o
I { e 2: 2
if),
.
-11,
If A = 1.
Weconsiderfirstthe case that A -11. Using the weaker form 1P(z) = AZ + O(z2) of (4.9) and iterating gives
so
ap(N) = ord(1P N (z) - z) z=o N = ord((A
z=o
-l)z + O(z2))
-{ -1
2: 2
if),N
-11,
if AN = 1.
This completes the proof of the lemma in this case. Next we consider the case A = 1. Then ap(l) = ord z=o(1P(z) iteration of (4.9) yields
z) = e,
and
(4.10) Hence
ap(N) = ord('¢N (z) - z) z=o e = ord(No:z
z=O
+ O(ze+I))
-{ =e=ap(l)
> e = ap(l)
if N -lOin K, if N
= 0 in K. D
This completes the proof of Lemma 4.6
Resuming the proof of Theorem 4.5, we observe that if the primitive period m does not divide n, then ap(n) = 0, and hence also ap(n) = O. We may thus assume that min. Since we will deal with several different maps, we write ap( 1.
Using the equality ap('ljJ, 1) = ap(qr, 1) = ap(c/J,m), this shows in Case I(a,b) that * ('" ) _ { ap(c/J,n ) 2 1 if n = m ,
ap 'f', n -
=
o
if n > m.
=
Case II. >..(P) 1 and N 0 in K. Let p be the characteristic of K and write N = peM with p f M . We observe that if kiM and 0 :::; i:::; e, then k 1= 0 in K and )..p ('ljJpi ) = ).. p('ljJ )pi = 1, so (4.7) of Lemma 4.6 applied to 'ljJpi tells us that
ap('ljJ ,pik) = ap('ljJpi , k) = ap ('ljJpi , 1) = ap('ljJ,pi). This allows us to compute
ap('ljJ,N) = LtL k iN
( ~) ap('ljJ ,k) = L
ttl
kiM i =O
= (LtL kiM
(pei~) ap (1/J,pi k ) P
(~)) (ttL (pe- i)ap(1/J,pi)) 2=0
= {a p(1/J,pe) - ap ('ljJ ,pe- l )
o
2 1 if M
= 1,
if M
> 1.
The fact that the value is positive when M = 1 follows from applying condition (4.8) of Lemma 4.6 to the difference ap (1/J pC- l ,p) - ap('ljJpe- l , 1). Hence in Case II, we have shown that ap(c/J,n) 2 0, and further, ap(c/J, n) 2 1 if and only if n = mp", Since Case II includes the assumption that )..(P ) = 1, this proves Theorem 4.5 in this case.
Case III. >..(P) =11 and >..(p)N = I. Let r be the exact order of )..(P ) in K*, so r lN and r > 1. Then (4.5) of Lemma 4.6
154
4. Families of DynamicalSystems
tells us that if r f k, then ap('ljJ , k) defining ap('ljJ ,N) as
ap('ljJ ,N ) = (L + L)1l kiN kiN rtk rlk =
= ap('ljJ , 1). This
allows us to write the sum
(~) ap('ljJ,k)
(L Il( ~) ap('ljJ ,l)) + (L Il( ~) ap('ljJ ,k)) kiN rtk
= (L Il kiN = L
kiN rlk
( ~) ap('ljJ , 1)) + (LIl ( ~) (ap('ljJ,k) kiN rlk
ap('ljJ ,
1)))
11 ( Nt ) (ap('ljJ , kr) - ap('ljJ, 1))
kl ~ r
=a p( 'ljJr ,N - {a p( 'ljJ, l) lr)
o
If N
ifN=r, if N i= r.
(4.11)
= r, so n = mr, then we have
ap(¢, n) = ap('ljJ, N ) = ap('ljJr , 1) - ap('ljJ, 1) = ap('ljJ,r) - ap('ljJ, 1) 2: 1 from (4.6) of Lemma 4.6. We may thus assume that N
ap(¢,n)
> r, so (4. 11) says that
= ap('ljJ,N) = ap('ljJr ,Nlr) .
We now observe that >"p('ljJr ) = >..p('ljJY = 1. Hence if N i= 0 in K, then we can apply Case I(b) to ap('ljJr ,Nlr), and if N = 0 in K, then we can apply Case II to ap('ljJr ,Nlr) . This compl etes the proof of Theorem 4.5. 0 As an easy application of Theorem 4.5, we now prove that a rational map
¢(z) E K (z) possesses periodic points of infinitely many distinct periods, i.e., the set of primitive-n periodic points Per~* (¢) is nonempty for infinitely many n . Corollary 4.7. Let ¢ (z ) E K( z) be a rational map of degree d 2: 2. Then fo r all prim e numbers f except possibly for d + 2 exceptions, the map ¢ has a point of primi tive period f. Proof We begin by discarding the finitely many primes f satisfying either of the followin g conditions:
• K has characteristic £. • There is some Q E Fix(¢) with >"(Q)
i= 1 and >..(Q)i! = 1.
155
4.2. Quadratic Polynomials and Dynatomic Modular Curves
The set Fix (¢) contains at most d + 1 points, so these condit ions eliminate at most d + 2 primes . take a point P E Pere(¢ ). If P does not For any of the remaining primes have primitive period then it must be a fixed point of ¢, since its period certainly divides e. Then our assumption s on eimply that either )..(P ) = lor )..(p )e i=- 1, and further that ei=- 0 in K. It follows from (4.5) and (4.7) of Lemma 4.6 that in both cases we have ap (e) = ap(I ). Hence
e,
e,
L PEPere(.p)n F ix(.p)
ap( e) =
L
ap( I) ::;
P EPer e(4))nFix(4>)
L
ap (l ) = d + 1.
P EF ix(4))
Thus the total multiplicity of all of the fixed points of ¢ in the e-period polynomial
is at most d + 1. However, the degree of ,e is de + 1, so we conclude that ,e has at least one root that is not fixed by ¢. Hence there exists at least one primitive periodic point. 0
e-
For many applications , Corollary 4.7 is sufficient, but a more detailed analysis yields a much stronger result. We state the full theorem in characteristic 0 and refer the reader to Pezda 's two papers [354, 357] for the more complicated description required in characteristic p. See also [161, 172, 173, 174] for higher-dimensional results of a similar nature.
Theorem 4.8. (LN. Baker [19]) Let ¢( z ) E K (z ) be a rational map ofdegree d :::: 2 defined over afield K ofcharacteristic O. Suppose that ¢ has no primitive n -periodic points. Then (n, d) is one ofthe pairs (2, 2), (2, 3) , (3, 2), (4, 2).
If ¢ is a polynomial, then only (2,2)
is possible.
Proof For the proof, which is function-theoretic in nature, see [19] or [43, § 6.8].
o 4.2
Quadratic Polynomials and Dynatomic Modular Curves
In this section we expand on the material from the previous section in the special case of the quadratic polynomial
¢(z ) = ¢c(z ) = z2 + C. For further material on the iteration of quadratic polynomials, see for example [17, 18,113,115,171,220,221 ,222,305,309,350,361 ,388].
4. Families of Dynamical Systems
156 1(C, Z) = Z2_ Z+ C 2 4 2 2(C, Z) = Z + 2cz - Z + (C + C) 2 3(C,Z) = Z8 + 4cz 6 + (6c + 2C)Z4 4(C, Z) =
+ (4c3 + 4c2)Z2 - Z + (C4 + 2c3 + C2 + c) Z16 + 8CZ 14 + (28c2 + 4C)Z12 + (56c3 + 24c2)ZlO 4 4 + (70c + 60c3 + 6c2 + 2C) Z8 + (56c5 + 80c + 24c3 + 8c2)Z6 2 6 5 4 3 + (28c + 60c + 36c + 16c + 4c )Z4 7 6 4 + (8c + 24c + 24c5 + 16c + 8c3) Z2 4 2 7 6 5 3 8 - Z + (C + 4c + 6c + 6c + 5c + 2c + C + c)
~ (C, Z )= Z2_ Z + C
2 (C , Z) = Z2 + Z + (C + 1)
+ Z5 + (3c + 1)Z4 + (2c + 1)Z3 + (3c2 + 3c + 1)z2 2 2 3 + (c + 2c + 1)z + (c + 2c + c + 1) 10 ~(c, z) = Z12 + 6cz + z 9 + (15c2 + 3C)Z8 + 4cz 7 + (20c3 + 12c2 + 1)Z6 + (6c2 + 2c)z 5 + (15c4 + 18c3 + 3c2 + 4C) z4 + (4c3 + 4c2 + 1)z3 4 4 2 3 2 3 5 + (6c + 12c + 6c + 5c + c)z2 + (c + 2c + c + 2c)z 5 4 3 2 6 + (c + 3c + 3c + 3c + 2c + 1) ~ (c, z ) = Z54 - Z53 + 27cZ52 + (- 26c + 1)z51 + (351c2 + 13c _ 1)Z50 + (- 325c2 + 12c)Z49 + (2925c3 + 325c2 - 24c + 1)Z48 + (- 2600c3 - 12c2 + 12c)z47 + ... + (_c 26 _ 12c25 _ 66c24 _ 226c23 - ... + 2c3 + c2 + 2c - 1)z + (C27 + 13c26 + 78c25 + 293c24 + 792c23 + ... + 3c3 + c2 - C + 1) ;;(c, z ) = Z6
Table 4.1: Period and dynatomic polynomials for ¢(z) =
4.2.1
Z2
+ c.
Dynatomic Curves for Quadratic Polynomials
Note that as long as the field K does not have characteristic 2, then any quadratic polynomial can be put into the form z2 + c by a simple change of variables working entirely within the field K. More precisely, if we let f( z) = (2z - B )/ (2A), then
¢ f( z) =
U- 1 0 ¢o J) (z ) = z2 +
(AC - ~B2 + ~B) .
(4.12)
The roots of the period polynomials n(z) = ¢~ (z ) - z and associated dynatomic polynomials ~ ( z ) are periodic points of the map ¢c(z ) = Z 2 + c. In order to investigate how the periodic points of ¢c(z) vary as a function of c, we observe that ~ (z) is a polynomial in the two variables z and c. Thus in studying
157
4.2. Quadratic Polynomials and Dynatomic Modular Curves
quadratic polynomial maps, it is natural to write ll>~ (c, z) and treat ll> ~ as a polynomial in Z[c, z]. (Table 4.1 lists the first few period and dynatomic polynomials for ¢c(z).) Treating ll>~ ( c, z) as a polynomial in two variables leads to a natural association polynomial ¢( z) E K [z] of } degree 2 with a point { P E K of formal period n
~
{ (c,Q) E K x K : ll> ~ (c,Q) =
O}.
Thus given a polynomial ¢(z) E K[z] of degree 2 and a point P of formal period n, we make a change of variables as in (4.12) so that ¢I (z) = z2 + c and set Q = 1-1(P). Note that this entire procedure takes place within the field K (which we assume has characteristic different from 2). Thus the solutions to the equation ll>~(y, z) = 0 parameterize pairs (¢, P), where ¢ is a conjugacy class of quadraticpolynomials and P is a point offormal period n for ¢. Further, the solution is K -rational if and only if ¢ and Pare K -rational, Definition. The dynatomic modular curve Y l (n) C A2 is the affine curve defined by the equation ll> ~ (y , z )
=0.
The normalization of the projective closure of Yl (n) is denotedby X I (n). Example 4.9. It is easy to see that X l (1) and X l (2) are rationalcurves. Indeed, the projective closures ofYl (l ) and YI (2) are smooth conics,
and
Xl (2) : z2
+ ZW + yw + w 2 = O.
It turns out that Xl (3) is also rational, but this is less clear from the equation in Table 4.1. In order to parameterize X l (3), suppose that ¢(z) = Az 2 + B z + C is any quadratic polynomial with a periodic point of primitive period 3. Conjugating by a linear map z ~ Q Z + /3, we may assume that the given 3-cycle has the form o-> 1 -> t -> 0 for some t. This gives the equations ¢ (1) =A+B+C=t,
¢(O) = C = 1,
Solving for A, B, C in terms oft yields ¢(z) =
t2
t+1 2 2 Z t -t
t3
-
-
+1
t2
-
t-t
2
Z
+ 1.
Now we apply the linear change of variables (4.12) to put ¢ into the form z2 + c. Thus letting I (z) = (2z - B )/(2A ), we find that
I ¢ (z) = z2
+
t6
-
4t 5
+ 9t 4 - 8t3 + 4t 2 -4t 4 + 8t 3 _ 4t 2
2t + 1
Our computation showsthat for every value of t
' ~
{O, I} , the point
4. Families of Dynamical Systems
158 t 6 - 4t 5 (
+ 9t 4 - 8t 3 + 4t 2 -4t 4 + 8t 3 - 4t 2
2t
+
1 '
_ t3 + t2 -2t 2 + 2t
1)
is a solution to the equation ; (y , z) = z6 + Z5
+ (3y + l )z4 + (2y + l )z3 + (3y 2 + 3y + l )z2 + (y2 + 2y + l) z + (y3 + 2y2 + y + 1) = O.
(You may check this directly using a computer algebra system .) We have thus constructed a nonconstant rational map s--«
~g
o ------+
t6 - 4t 5
+ 9t 4
8t 3 + 4t 2 --- 1)2
t 3 --- t 2 + 1 ) . , 2t (t --- 1) (4.13) General principles (Liiroth's theorem [198, IV.2.5.5]) tell us that Xl (3) is birational to pl. More concretely, we can prove that the map (4.13) has degree 1 by constructing its inverse . Thus let (c,b) be a root of ;. We set g(z) = (b 2 + c - b)z + b, so g sends 0 to band 1 to ¢(b). Thus the 3-cycle b ----t ¢(b) ----t ¢2(b) ----t 0 becomes the following 3-cycle for ¢ g:
t
(
1
-
2t
-
+1
-4t 2(t
~g
------+
b2
+ b+ 1 + c
~g
------+
O.
This gives the map (c, b) f----t b2 + b + 1 + c, which is inverse to (4.13) , a fact that can also be checked directly with a computer algebra system.
4.2.2 Dynatomic Curves as Modular Curves The curve YI (n ) and its completion X I (n) are modular curves in the sense that their points are solutions to the moduli problem of describing the isomorphism classes of pairs (¢, 0:), where ¢ is a polynomial of degree 2 and 0: E A I is a point of formal period n for ¢. Here two pairs (¢l , o:d and (¢2,0:2) are PGL 2-isomorphic if there is a linear fractional transformation f E PGL 2 satisfying and In order to state this more carefully, we define
Formal(n)
=
. ¢ E k[z], deg(¢ ) = 2, 0: E k} { (¢,o:) . 0: has formal period n for ¢
.
PGL 2-lsomorphism
.
We have demonstrated that the elements of Formal(n) are in one-to -one correspondence with the points of YI (n). But much more is true: the correspondence is algebraic 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 polynomials over V with a markedpoint offormal period n consists of a quadratic polynomial
'ijJ(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 ----t p} such that for all P E V(K), the point A(P) is a point offormal period n for the quadratic polynonomial
'ijJp(z) = A(p)z2
+ B(P)z + C(P)
K[z].
E
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]. Example 4.10. The pair
A(t)=t-l
and
is an algebraic family of quadratic polynomials over lP'1 -, {O, oo} with a marked point of formal period 2.
Theorem 4.11. Let K be a field ofcharacteristic different from 2. (a) The map
Yl(n)
---+
Formal(n),
(c, CY)
r---+
(z2
+ c, CY),
(4.14)
is a bijection ofsets. (b) Let V be a variety and suppose that the points ofV algebraically parameterize a family of quadratic polynomials 'ijJ together with a marked point A offormal period n. Then there is a unique morphism ofvarieties
with the property that
'f/(P) = ('ijJp(Z),A(P)) E Formal(n)
(c)
for all P E V(K),
(4.15)
where we use (4.14) to identify Formal(n) with Y 1 (n). is defined over the field K, then the morphism 'f/ is also defined over K.
If the family
Proof (a) We have shown this earlier in this section. (b) By definition 'ijJ has the form
'ijJ(z)=Az 2+Bz+C,
A,B,C E K[V],
with A not vanishing on V and A a morphism A : V ----t A l such that for all P E V(K), the point A(P) is a point offormal period n for the quadratic polynonomial
'ijJp(z)
=
A(p)z2
+ B(P)z + C(P)
E
K[z].
4. Families of Dynamical Systems
160
For any point P E V(K), let Ip(z) be the linear fractional transformation
2z - B(P)
fp(z) =
2A(P)
.
Note that Ip is well-defined for every P E V(K), since we have assumed that A is nonvanishing on V. We define a map T/ from V to A,.2 by the formula
T/(P) = (cp,ap)
with
{
cp
=
A(P)C(P) - ~B(p)2
p
=
A(P)A(P) + ~B(P).
Cx
+ ~B(P),
(4.16)
Note that T/ : V --+ A,.2 is a morphism, Le., it is given by everywhere-defined algebraic functions on V. We are now going to verify that the image of T/ is the curve Y1 (n). The computation that we performed in deriving formula (4.12) shows that
'l/;{:(z)
=
z2 + cpo
To ease notation, we let ¢p(z) = z2 + Cp . We also let n,P, ~,p, \lIn,P, and \lI~,p be the period and dynatomic polynomials for ¢p and 'l/;p, respectively. Note that the period polynomials are related by
\lI~~p(z) = Ip1o('l/;p(z)-z)olp = (fp1o'l/;plpt(z)-z = ¢p(z)-z = n,p(z). Hence the dynatomic polynomials also satisfy
(\lI~,p)fp = n,P.
(4.17)
We are given that A(P) E Per~('l/;p), which is equivalent to saying that A(P) is a root of \lI~,p. It follows from (4.17) that
I p 1(A(P)) = A(P)A(P) + ~B(P) is a root of ~ p, and thus is in Per~ (¢ p ). This proves that the image of the map T/ defined by (4.1'6) is contained in Y1 (n), so T/ is a morphism from V to Y1 (n). Further, this map T/ respects the identification of Ydn) with Formal(n) from (a), since it takes P to a pair (cp,ap) that is isomorphic to the pair ('l/;P(Z),A(P)) via the conjugation Ip E PGL2 (K ). Finally, it is clear from the construction that the map T/ is uniquely determined as a map (of sets) from V(K) to Formal(n), so it is the unique morphism V --+ Y1 (n) satisfying (4.15). (c) The definition (4.16) of T/ shows immediately that T/ 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 4.11 says that Y1 (n) is a coarse moduli space. In fact, the curve Y1 (n) is actually a fine moduli space for all n 2: 1; see Exercise 4.18. The underlying reason is that there are no nontrivial elements ofPGL 2 that fix a quadratic polynomial and its points of formal period n, i.e., the moduli problem has no nontrivial automorphisms.
4.2. Quadratic Polynomials and Dynatomic Modular Curves
4.2.3
161
The Dynatomic Modular Curves X 1(n) and Xo(n)
The curve YI (n), and by extension X I (n), has the interesting property that the rational map ¢ acts on the points of YI (n) via the map
(y, z)
f------7
(y, Z2
+ y) =
(y, ¢y(z)).
(4.18)
By abuse of notation, we will also use ¢ to denote the map (4.18). This map is welldefined, since if ex is a point of formal period n for the polynomial ¢y(z), then ¢y (ex) is also a point offormal period n for ¢Y' Further, the nth iterate ¢n is the identity map on YI (n) and Xl (n), so Aut(YI (n)) and Aut(X I (n)) contain subgroups of order n generated by ¢. In general if V is any algebraic variety and if G c Aut(V) is any finite group of automorphisms of V, then there exist a quotient variety W = VI G and a projection map n : V ----t W with the property that n(y) = n(x) if and only if there is agE G such that y = g(x). (See, for example, [321, II §7].) Further, if V is defined over a field K and ifG is K-invariant,2 then VIG is defined over K. The construction of VI G when V is a nonsingular curve is particularly simple. Each 9 E G induces an automorphism of the function field g* : K(V)
----t
K(V)
fixing K, so we may view G as a subgroup of Aut(K(v)1K). Then one shows that the fixed field K (V) G of G has transcendence degree lover K, so there exists a unique nonsingular projective curve WI K with function field K(W) = K(V)G. The inclusion K(W) c K(V) defines the projection n : V ----t W making W into the quotient of V by G. Finally, if G is K -invariant, one shows that W has a model defined over K. Remark 4.13. We note that taking the quotient of a variety by an infinite group of automorphisms, as we will need to do in Section 4.4, is considerably more difficult than taking the quotient by a finite group of automorphisms. Indeed, in the infinite case it often happens that the quotient does not exist at all in the category of varieties.
Definition. With notation as above, we let Yo (n) be the quotient of YI (n) by the finite subgroup of Aut(YI(n)) generated by ¢. Similarly, Xo(n) is the quotient of XI(n) by the finite subgroup of Aut(XI(n)) generated by ¢. By construction, the points of Yo (n) classify isomorphism classes ofpairs (¢, 0), where ¢ is a quadratic polynomial and 0 is the orbit of a point of formal period n. The moduli-theoretic interpretation of the projection map from YI (n) to Yo (n) is
2The elements in G are morphisms V - t V that are defined over some extension of K. The Galois group Gal(K/ K) acts on G by acting on the coefficients of the polynomials defining the maps in G. We say that G is K -invariant if each element of Gal( K / K) maps G to itself.
162
4. Families of Dynamical Systems
Example 4.14. We have seen in Example 4.9 that X I (2) and X I (3) are rational curves, i.e., they are isomorphic to pI , so their quotient curves X o(2) and X o(3) must also be rational curves. However, it is still of interest to make the quotient maps explicit. We do this for X I (2) and leave X I (3) as Exercise 4.19. The affine curve YI (2) has equation
YI (2) : z2 + Z + Y + 1 = 0, where a point (c, 0: ) E YI (2) corresponds to the quadratic map cPc (z ) = z2 + c and point 0: E P er; (cPc) . The automorphism cP ofYI (2) is given by (y, z) t---; (y, Z2 + y). From general principles we know that cP maps Y1 (2) to itself, or we can see this explicitly by the formula
(z2 + y)2 + (Z2
+ y) + y + 1 = (z2 + Z + Y + 1)(z2 - Z + Y + 1).
The affine coordinate ring of YI (2) is
q z,y] rv C[ ] (z2 + Z + Y + 1) = z, since we can express y in terms of z as y = - z2 - Z - 1. In terms of z alone, the automorphism cP ofYI (2) corresponds to the map on q z] given by
z
t------>
Z2 + Y = Z2 + (- Z2 - Z - 1) = - z - l.
So the affine coordinate ring of the quotient curve Yo(2) is the subring of C ]»] that is fixed by the automorphism z t---; - z - 1. Since the map YI (2) ~ Yo (2) has degree 2, we look for an invariant quadratic polynomial in z. Setting
Az 2 + B z + C = A( - z - 1)2 + B ( -z - 1) + C and equating coefficients, we find that A = B . Hence the invariant subring is
q z2 + z], which is the affine coordinate ring ofYo(2).
Suppose now that (cP, {0:,,8}) is a quadratic polynomial and an orbit consisting of two points . How does this correspond to a point on Yo(2) whose coordinate ring is z2 + z]? First, we use a PGL 2 conjugation as in (4.12) to put cP into the standard form cPc(z) = z2 + c. Then 0: and ,8 are related by
q
,8 = cPc(O:) =
0:
2 +c
Notice that this implies that 2 0: + 0: = (,8 - c)
and
0:
= cPc ((J) = ,82 + c.
+ (,82 + c) =
,82
+ ,8.
Letting c = -0: 2 - 0: - 1 and using the isomorphisms YI (2) ~ Al and Yo(2 ) ~ Al given earlier, we have the following commutative diagram: (c,a).-.a
4.2. Quadratic Polynomials and Dynatomic Modular Curves
163
Remark 4.15. Morton [307] describes an alternative method for finding an equation for the curve Yo(n ) and the map Yi (n) --+ Yo(n ). The roots of ~ ( e, z) = can be grouped into orbits, say 0'1 , .. . , a r are representatives for the different orbits, so
°
The points in each orbit all have the same multiplier, and we define a polynomial r
On(e, x) =
IT (x - A,p(ai ))' i =l
Then one can show that On (c, X) E Z [e, X] and that the equation
gives a (possibly singular) model for Yo(n). Using this model, the natural map from Yt{n ) to Yo(n ) is
(y, z)
t-------;
(y, ( ¢~) /(z) ).
See Morton 's paper [307] and Exercise 4.13.
Remark 4. 16. The dynatomic modular curves Y1 (n) and Yo(n ) defined in this section are analogous to the modular curves that appear in the classical theory of elliptic curves. Briefly, the elliptic modular curve y 1ell(n) classifies isomorphism classes of pairs (E, P), where E is an elliptic curve and PEE is a point of exact order n . Similarly, the elliptic modular curve y~ll ( n ) classifies pairs (E, C ), where E is an elliptic curve and C c E is a cyclic subgroup of order n. The group (Z/nZ)* acts on y 1ell (n ) via m
* (E , P) =
(E , [m]P)
for
m E (Z /nZ)*,
and the quotient ofy{lI (n ) by this action is y~lI (n ) . The reader should be aware that standard terminology is to write
for elliptic modular curves. But since in this book we deal exclusively with dynatomic 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 n superscripts such as y t (n) and y 1ell(n) to distinguish them. We showed earlier that X 1 (1), X 1 (2), and Xl (3) are all (irreducible) curves of genus 0, and similar explicit computations show that Xl (4) has genus 2 and Xl (5) has genus 14. See [171, 305, 309] and Exercise 4.20. The geometry of X 1 (n) and Xo(n ) for general n is described in the following theorem, which is an amalgamation of results due to Bousch and Morton.
4. Families of Dynamical Systems
164
Theorem 4.17. (a) The affine curve YI (n) defined by the equation
is nonsingular. (b) The dynatomic modular curves Xl (n) and Xo(n) are irreducible. (c) The projection map
(z, c) - > c, exhibits X I (n) as a Galois cover of p l. The Galois group is maximal in the sense that it is the appropriate wreath product, cf Section 3.9. (d) Let sp denote the Euler totient function (not to be conf used with the rational map 1» , let f..L be the Mobius function, and let K, be the function
(This is essentially half the degree of 4" n" cf Remark 4.3.) Then the genera ofX I (n) and X a(n) are given by the formulas
genus X j ln ) = 1 + n ; 3 K,(n) -
L
~
mK,(m)
(z) = z2 + c, whose moduli space is the cplane. the bifurcation points are also known as the roots ofthe hyperbolic compo nents ofthe Mandelbrot set M .
4.2. Quadratic Polynomials and Dynatomic Modular Curves
167
It is clear that every Misiurewicz point is contained in the Mandelbrot set M , since preperiodicpoints certainlyhave bounded orbit. This sufficesto prove an arithmetic result.
Proposition 4.22. The set ofMisiurewicz points is a set ofbounded (absolute) height in Q. More precisely, the height ofa Misiurewicz point 1 satisfies H (,) :S 2. Hence there are only finitely many Misiurewicz points defined over any given numberfield. Proof A Misiurewicz point C = 1 is the root of a polynomial of the form
for some m ~ 1 and n ~ 1. These are monic polynomials with coefficients in Z, so not only is 1 an algebraic number, it is an algebraic integer. Further, the minimal polynomialof 1 is a factor of Mm,n(c). On the other hand, every root of Mm,n(c) is in the Mandelbrotset M, since if c is a root of Mm,n(c), then 0 is preperiodicfor ¢e, so it certainly has bounded orbit. It follows from Proposition4.19 that every Galois conjugateof 1 has absolute value at most 2. Let K = Q( I) and let 11, ... Lr: be the full set of Galois conjugates of I ' Then i
II
HK(r ) =
r
max{1,lll vr" =
vEM K
II max{l, lril} :S 2
r
.
i= l
o
Takingthe r 1h root yields H (r ) :S 2.
We now describe an analytic characterization of Misiurewicz points. It depends on the following deep result giving an analytic uniformization of the complementof the Mandelbrot set. Theorem4.23. (Douady-Hubbard) There is a conformal isomorphismfrom the exterior of the unit disk to the complement ofthe Mandelbrot set,
e: {w E C : Iwl >
I} ~ C " M .
Proof It is not hard to show that for all sufficiently large z (dependingon c) there is
a consistent way to choose square roots so that the limit
We(z) = lim 2\/¢~(Z) n -> oo
converges and defines a holomorphic function We on some region Izi > R e . (See ehas an analytic continuation Exercise 4.15.) If c ~ M, then one can prove that W to C jp'~ 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 'l/J can be written in the form
'l/J = [F(X, Y), G(X, Y)], where F, G E K(V)[X, Y] are homogeneous polynomials with coefficients that are rational functions on V. The degree of'l/J is the degree of the homogeneous polynomials F and G.4 Any morphism .\ : V ---> W of varieties induces a natural morphism jp'~ ---> jp'tv, which, by abuse of notation, we also denote by .\. Thus
.\(P, t)
for (P, t) E jp'~ .
= (p, .\(t))
With this notation, the map (4.20) says that there is a natural morphism
over Ratd. The next proposition says that this ¢ is a universal family ofrational maps of degree d.
Proposition 4.31. Let V be an algebraic variety and let 'l/J : jp'~
------t
jp'~
be a morphism over V ofdegree d. Then there is a unique morphism 4More generally, if'lj! : 1I'~ is the degree of 'Ij!.
-+
1I'~ is a morphism of S-schemes, then 'Ij!*OpN (1) ~ OpN (d) and d
s
s
172
4. Families of Dynamical Systems
such that the inducedmap ,\ : p~
-->
Phatdfits into the commutative diagram ,p
pI
-------t
V
pI
V
lA
lA Phatd
<J>
-------t
(4.21 )
pI
Ra t .,
Remark 4.32. The commutative diagram (4.21) in Proposition 4.31 says that any algebraic family of degree d rational maps W : JIl'~ --> p~ factors through the Ratd family. Thus ¢ : Phatd --> Phatd is a universal family of rational maps of degree d. It is an example of afine moduli space. See [322, 323, 335] for further information about moduli spaces. Proof Let U c V be an affine open subset and write K[U] for its affine coordinate ring. The fact that 'IjJ is a morphism over V implies that it restricts to give a morphism e : JIl'h --> ph over U. This restriction of 'IjJ to ph has the form 'IjJ = [o:oX d + O:IX d- Iy
+ ... + O:d y d, f3oX d + f3 IX d- I y + with 0:0 , .. . , O:d, f30,
+ f3dy d], ,f3d E K[U].
In other words, the coefficients of the polynomials defining 'IjJ are regular functions on the affineopen set U . In particular, if there is a map ,\ : V --> Ratd making(4.21) commute, then it must be given by ,\ = [0:0"" , O:d, {3o, . . . , (3d] at any points of V at which the O:i and f3i are defined and the homogeneous polynomials in (4.21) have no common roots. Now given any point in V , we can find a neighborhood U of that point and O:i , f3i E K[U] as above such that 'l/J : ph --> ph is given by
'l/J ([x , y], t)
=
[o:o(t )x d + ...+ O:d(t)yd, f3o( t)x d + ...+ f3d(t )yd] for
([x, y], t) E ph. (4.22)
Using a natural notation, we abbreviate this by writing
W(P, t ) =
[Fo: (t ) (P), F,6(t ) (P)]
.
In order to make the diagram(4.21) commuteon p h c P~ , we are forced to define,\ on Uby
'\(t ) = [0:0(t),0:1(t), .. " O:d(t ),f30(t ), f3I(t )' ... ,f3d(t )] E p 2d + l .
(4.23)
Further, this ,\ will have the desired properties provided that its image lies in Rat d. However, the fact that 'IjJ is given by (4.22) for every point in Ph implies that for every t E U, the homogeneous polynomials Fo: (t ) and F,6(t ) have no common root in pl. It follows from Proposition 2.13(a) that their resultant
4.3. The Space Ratd of Rational Functions
173
Res(Fn , F(3 ) E K[U] is nonzero at every point in U. Hence the image of the map>. = [a, b] defined by (4.23) is in Ratd . We have now proven that every point in V has a neighborhood U for which there is a map>. : U ----t Ratd making (4.21) commute. Further, the maps on different U must agree on the intersection by the uniqueness discussed earlier. Fitting them together, we find that there is a unique map>. : V ----t Ratd making (4.21) commute . 0 An ongoing theme of this text has been the observation that a pair of rational maps ¢ l , ¢ 2 E Rat., determine (arithmetically) equivalent dynamical systems if they are PGL 2-conjugate, i.e., if there is a linear fractional transformation f E PGL 2 such that ¢2 = ¢{ = f -l ¢f. The conjugation action ofPGL 2 on Rat., is algebraic in the following sense.
Proposition 4.33. The map (4.24) is an algebraic group action ofPGL 2 on Rat., and is defined over Q. This means that the map (4.24) is both a morphism defined over Q and a group action.
Proof The proof is mostly a matter of unsorting the definitions. Let
f = [aX + {3Y" X + bY]
E PGL 2 C
jp'3
and
¢
= [Fa, Fb] E Ratd C
jp'2d+l .
Then
+ (3Y, I'X + oY) - {3Fb(a X + (3Y, I'X + OY), ,Fa(aX + (3Y ,I'X + oY ) + a Fb(aX + (3Y, I'X + OY)] .
¢ I ([X , YJ) = [oFa (aX -
(4.25) The homogeneity of Fa and Fb shows that (4.24) at least gives a well-defined rational map jp'3 X
jp'2d+ 1
---t
jp'2 d+l ,
(J,¢) ~ ¢/ .
Further, an elementary resultant calculation (Exercise 2.7) shows that the resultant of the two polynomials appearing in the righthand side of (4.25) is equal to
It follows that if f E PGL 2 and ¢ E R atd, then ¢I is a well-defined point in Ratd. This proves that the map (4.24) is a morphism. The fact that it is a group action is then a straightforward, albeit tedious, calculation. 0
174
4. Families of DynamicalSystems
Example 4.34. Inprinciple it is possible to explicitly write down the action ofPGL 2 on Ratd, but in practice the expressions become hopelessly unwieldy for even moderate values of d. As illustration, we describe the action for d = 2. Let j
= [aX + (3Y, 'YX + oY]
and
and set
Multiplying out both sides of (4.25) and equating the coefficients of X 2, XY, and y 2 yields the following formulas for the coefficients of Fal and Fbi:
a~ = a 20ao
+ a'Y0al + loa2 - a 2{3bo - a{3'Ybl - {3'Y2b2, a~ = 2a{3oao + (ao + {3'Y)oal + 2'Y02a2 - 2a{32bo - (ao + {3'Y){3b l - 2{3'Yob2, a~ = {32 0ao + {302 al + 03a2 - {33b o - {32 0bl - {302b 2, b~ = -a2'Yao - a'Y2al - 'Y3a2 + a 3bo + a2'Ybl + a'Y2b2, b~ = - 2a{3'Yao - ({3'Y + aohal - 2'Y20a2 + 2a 2{3b o + (ao + {3'Y )ab l + 2a'Yob2, b~ = _{32'YaO - {3'Y0al - 'Y02a2 + a{32b o + a{3ob l + a0 2b2.
4.4
The Moduli Space M d of Dynamical Systems
The intrinsic properties of the dynamical system associated to a rational map ¢ depend only on the PGL 2-conjugacy class of ¢, so it is natural to take the quotient of the space Rat., by the conjugation action ofPGL 2 .
Definition. The moduli space of rational maps of degree d on pI is the quotient space
M d = Ratd / PGL 2 , where PGL 2 acts on Rat., via conjugation ¢f = j-I¢j as described in Proposition 4.33. For the moment, the quotient space Md is merely a set, in the sense that for any algebraically closed field R,
We denote the natural map from Rat., to Md by
In order to endow Md with the structure of a variety, we observe that the action ofPGL 2 on Rat., induces in the usual wayan action ofPGL 2 on the ring of regular functions Q[RatdJ on Ratd. Thus an element R E Q[Ratd] is a function whose domain is the set {¢ : pI ----+ pI} of rational functions on pI, and for j E PGL 2 we define Rf E Q[RatdJ by the formula Rf (¢)
= R(¢f).
4.4. The Moduli Space
Md of Dynamical Systems
175
In this way we obtain a map
so it makes sense to talk about functions in Q[RatdJ that are invariant under the action ofPGL 2 . We write
for the ring ofPGL 2-invariant functions on Ratd. Remark 4.35. For technical reasons, it is sometimes advantageous to replace PGL 2 by a slightly different group. The special linear group is the subgroup of GL 2 defined by
8L 2 = {
(~ ~)
:
aJ -
~1 =
1} ,
and the projective special linear group P8L 2 is the image of 8L 2 in PGL 2 . The groups 8L 2 and P8L 2 act on Rat., via conjugation in the usual way. Geometrically, the actions ofPSL 2 and PGL 2 on Rat., are identical, since it is easy to see that the natural inclusion 8L 2 C GL 2 induces an isomorphism
However, if K is not an algebraically closed field, then the map
need not be an isomorphism; see Exercise 4.22. The following theorem provides the abstract quotient Ratd / PSL 2 with the structure of an algebraic variety. Theorem 4.36. There is an algebraic variety Md defined over Q and a morphism
(4.26) defined over Q with the following properties: (a) The map (4.26) is P8L 2-invariant, i.e., the following diagram is commutative:
P8L 2 x Ratd
fit
Ratd
(f,e/> )--->1/
) Ratd
l(i
C)
Md
In terms ofelements, (.)
=
(T
+ 2n)22n-l_2n (T - 2n)22n-l+2n (T + 22n).
).,EAn(t)
In particular, we see that every map ¢t has the same set ofmultipliers, so 0"; n) (¢t) does not depend on t. (A similar statement holds for ~~n) (¢t).) Hence no matter how
4.5. Periodic Points, Multipliers, and Multiplier Spectra
ai
187
n
k ) we use, the resulting map M 4 -+ A always compresses all of many functions the maps ¢t down to a single point. On the other hand, we will later prove that the ¢t are not PGL 2-conjugate to one another (see Theorem 6.46). Hence the image ofthe map C*" {_2J} ~ M 4, t r-----t (¢t),
is a curve in M 4 that is compressed to a single point in A k. An important theorem n says that aside from Lattes examples of this kind, the map on Md defined by the ) is finite-to-one.
ai
Theorem 4.53. (McMullen [294, §2]) Fix d 2: 2, andfor each N 2: 1 let (4.34)
ai
n
be the map defined using all ofthe functions n) with 1 ~ ~ N. If N is sufficiently large, then the map U d,N is finite-to-one on Md(C) except for certain families of Lanes maps that it compresses down to a single point. In particular, it isfinite-to-one if d is not a perfect square. Further, the same statement is true for the map U
*d,N .. M d
Ai
~ fl
defined using all ofthe functions ;;-~n) with 1 ~ n ~ N. (The flexible Lattes mapsfor which U d,N and U d,N' are not finite-to-one are discussed in detail in Section 6.5.)
McMullen's theorem says that aside from the flexible Lattes maps, the maps U d,N and U d,N' are finite-to-one onto their image. One might hope that they are actually injective ifwe avoid the flexible Lattes maps, but it turns out that this is far from true. Theorem 4.54. Define the degree of U d,N to be the number ofpoints in U d,~ (P) for a generic point P in the image U d,N(Md)' One can show that the degree of U d,N stabilizes as N -+ 00. We write deg(u d)for this value. Thenfor every E > 0 there is a constant CE such that for all d. In particular, the multipliers of a rational function ¢ E Rat., determine the conjugacy class of ¢ only up to DE (d~ -E) possibilities. Proof We will prove this in Chapter 6 using Lanes maps associated to elliptic curves with complex multiplication; see Theorem 6.62. 0
Definition. Following Milnor [300], we define the multiplier spectrum of a rational map ¢ to be the function A (¢) that to each positive integer ti assigns the set An(¢). In other words, A( ¢) is the set-valued function
A( ¢) : N
~
Sets with Multiplicities,
188
4. Families of Dynamical Systems
Two rational maps are said to be isospectral if they have the same multiplier spectrum. 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 A2 in (c) induces a bijection M 2(K) +--+ A2(K). (Note that Md(K) is not the same as Ratd(K)/ PSL 2(K ); see Remark 4.39 and Section 4.10.)
Remark 4.57. For some applications it is useful to have an explicit description of the map 2 (T = (iTl' iT2) : Rat2 ---7 A described in Theorem 4.56(b). Let
p(a, b)
= a§b6 - ala2bObl + aoa2br + arbOb2 - 2aOa2bob2 - aoa1b1b2 + a6b§
denote the resultant of Fa and Fb . Then, after some algebraic manipulation, one finds that iTl and iT2 are given by the expressions
= a{bo - 4aOa1a2bo - 6a§b6 - aOarbl + 4a6a2bl + 4a 1a2bob1
p(a, b)iTl("2 -I- 1, then there is an f E PGL 2 (C) such that
¢f (z) = z2 + A1 Z . A2Z + 1 Further, Res(z2 + A1Z, >"2Z + 1) = 1 - A1A2. (b) If A1A2 = 1, then Al = A2 = 1 and there is an
(4.35)
f E PGL 2(iC) such that
¢f (z) = z + ~ + ~.
(4.36)
z
Proof We recall that if a E Fix( ¢) has multiplier An, then the Taylor expansion of ¢ around a looks like
¢(z)
= ¢(a) + ¢'(a)(z - a) + O(z - a)2 = a + An(Z - a) + O(z - a)2.
Hence
= ¢(z) - z = (An - 1)(z - a) + O(z - a)2, so ¢ has multiplicity 1 at the fixed point a if and only if An i- 1. ,l(Z)
We also note the formal identity
(X _1)2 - (XY -1)(XZ -1)
= X(X + Y + Z
- 2 - XYZ)
4.6. The Moduli Space M 2 of Dynamical Systems of Degree 2
191
and apply it using the relation
from (a). This yields the useful formulas
(AI - 1)2 = (AIA2 - 1)(AIA3 - 1),
(A2 - 1)2 = (A2AI - 1)(A2A3 - 1),
(4.37)
(A3 - 1)2 = (A3AI - 1)(A3A2 - 1). We now start with a rational map
¢(Z)
=
2
aoz + alz + a2 boz2 + bIZ + bz
and change coordinates in order to put ¢ into the desired form. (a) For this part we assume that AIA2 f- 1, so (4.37) tells us that Al f- 1 and A2 f- 1. Thus the fixed points associated to Al and A2 have multiplicity I, so in particular they are distinct and we can find an element of PGL 2 (q that moves them to 0 and 00, respectively. After this change of variables has been made, the rational map ¢ satisfies ¢(O) = 0 and ¢( 00) = 00, so it has the form with Since ao f- 0, we can dehomogenize by setting ao = 1, and then a simple calculation yields
¢'(O) = al = Al b2 Thus
f
and
has the form
¢(z) = z2 + b2AIZ A2Z + b2
with
Finally, replacing ¢(z) by b:;I¢(b2z) yields the desired form (4.35), and we calculate
1 Al 0 Res(z2 + AIZ, A2Z + 1) = det A2 1 0 o A2 1
= 1-
AIA2,
which completes the proof of (a). (b) The proof of this part is similar. We begin by moving the fixed point associated to Al to 00, so the map ¢ has the form with The assumption that AIA2 = 1 combined with (4.37) tells us that Al we have bl = ao. Dehomogenizing ao = 1 puts ¢ into the form
= A2 = 1, so
192
4. Families of Dynamical Systems
Next we replace ¢(z) by ¢(z - b2 )
+ b2 , so now ¢(z)
¢(z) = Z2 +a1 z+ a2 z
looks like
with
(Of course, the values of a1 and a2 have changed.) Finally, replacing ¢(z) by ¢ (;a2 z)/ ;a2 gives ¢(z) the desired form
,1,( ) _ Z2 + a1Z + 1 _ z - z + a1 z
'f/
1 + -. z
(4.38)
We can compute the value of a1 by observing that ¢ has a double fixed point at 00 and that its other fixed point is at -all. (If a1 = 0, then there is a triple fixed point at 00.) Thus the third multiplier is
so we find that a1 = ~. (This is also correct if a1 of a1 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 a (¢1) = a (¢2). Thus and and then the relation (73 = (71 - 2 from (a) implies that also (73(¢d = (73(¢2). It follows that the set of multipliers of the fixed points of ¢1 and ¢2 are the same, say {>'1, >'2, >'3}, since they are the three roots (counted with multiplicity) of the polynomial
T 3 - (71 T 2 + (72 T - (73.
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 + >, 1z)/ (>'2 Z + 1), so they are PGL 2-equivalent to each other, and similarly if >'1>'3 -# lor if >'2>'3 -# 1. Weare left to consider the case >'1>'2 = >'1>'3 = >'2>'3 = 1. Then Lemma4.59(b) tells us that >'1 = >'2 = >'3 = 1 and that ¢1 and ¢2 are both PGL 2-equivalent to the rational map z + z-l. This completes the proof of (b,ii), so we turn to (b,iii), the surjectivity of a . Given (81,82) E A 2(C), we set 83 = 81 - 2 and factor the polynomial
Notice that the condition 83
= 81 - 2 gives the familiar relation (4.39)
4.6. The Moduli Space M 2 of Dynamical Systems of Degree 2 which in tum implies the usual formal identities (4.37) for A1'A2'A3. Suppose first that some Ai is not equal to 1, say A1 =I=- 1. Then A1A2 from (4.37), so we can set
193
=I=-
1
¢(Z)=Z2+ A1Z, A2Z + 1 since Res(z2 + A1Z, A2Z + 1) = 1 - A1A2 from Lemma 4.59(a). The fixed points of ¢ are 0:1 = 0, 0:2 = 00, and 0:3 = i=~~. Its multipliers at 0:1 and 0:2 are easily computed, and and the multiplier at
0:3
is
where we use (4.39) for the middle equality. Hence a1 (¢) = 81 and a2 (¢) = 82. We are left with the case A1 = A2 = A3 = 1, which corresponds to the values 81 = 82 = 3. It is easy to check that the rational map ¢(z) = Z + z~l has a triple-order fixed point at 00 and that all three multipliers are equal to 1, so O"(¢) = (3,3). (c) We briefly 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 0" : M 2 ----t A,2 is proper. This follows easily from the valuative criterion [198, 11.4.7] using the fact proven in (b) that every fiber O"~ 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 0" is finite, which can be proven using the fact that both M 2 and A,2 are affine varieties (cf. [416, Lemma 5.6]). Alternatively, one can show in general that a proper quasifinite map is finite ([299, I, Proposition 1.10]). Then one uses the fact that ft,2 is nonsingular and the bijectivity of 0" on complex points to prove that 0" is an isomorphism (cf. [416, Lemma 5.7]). Finally, it is clear from the explicit formulas for a1 and a2 in Remark 4.57 that 0" is defined over Q. (d) This is immediate from (c), which says that Q[M 2 J = Q[a1,a2], since we already know from Theorem 4.50(b) that a;n) and ;~n) are in Q[M 2 ]. (e) This is a consequence of the fact that the isomorphism 0" : M 2 ----t A 2 in (c) is defined over Q. 0 Remark 4.60. Regarding the proof sketch of Theorem 4.56(c), we observe that a morphism of irreducible varieties V ----t W may be bijective on complex points, yet not be an isomorphism. A simple example of this phenomenon is the cuspidal cubic map Thus in the proof that 0" : M 2 ----t A 2 is an isomorphism, it is crucial that 0" maps M 2 onto the nonsingular variety A 2 . It does not appear to be known in general whether Md is nonsingular.
194
4. Families of Dynamical Systems
The content of Theorem 4.56 is that the moduli space M 2 of dynamical systems of degree 2 is isomorphic to A 2 and an explicit isomorphism is provided by the pair of functions (0"1, 0"2) created from the multipliers of the fixed points of a rational map. From our general theory (Theorem 4.40), the space M 2 sits naturally inside two larger spaces M~ and M~8, but since d = 2 is even, these two spaces coincide and will be denoted by M 2 . We conclude this section with a description of M 2 •
Theorem 4.61. Let M2 = M~ = M~8 be the completion ofM 2 constructed using geometric invariant theory in Theorem 4.40. Then the isomorphism
extends to an isomorphism ir : M mutes:
2 -+ jp'2
such that the following diagram com-
M2~A2
1 M2 ~
1
(x,y)
[x,t,l]
jp'2
The points in M 2 (C) that are not in M 2 (C) correspond to degenerate maps of degree 2 that may be informally described as maps ofthe form (4.40) The point [A, BJ is uniquely determined up to reversing A and B. Proof See [302] and [416, Theorem 6.1 and Lemmas 6.2 and 6.3].
D
Remark 4.62. We expand briefly on what it means to say that the points in the set (M 2 "M 2)(C) correspond to the maps cPA,B given by (4.40). Let U C Al be a neighborhood of and let
°
be a rational map that is a morphism away from 0. We denote the image of t E U by cPt to help remind the reader that cPt is itself a map, i.e., cPt : jp'l -+ jp'l. Consider the composition
which by abuse of notation we again denote by cP. It is a rational map, and since U CA l and M 2 is complete, we see that it is a morphism from U to M 2 . In particular, the point cPo is a well-defined point in M 2(C) . If cPo E M 2(C) , there is nothing to say, so we suppose that cPo ~ M 2(C) . Then the second part of Theorem 4.61 means that possibly after choosing a smaller neighborhood of 0, there exists a morphism
195
4.7. Automorphisms and Twists
1 : U -----+ PGL 2 such that the conjugate map
4/
has the form
with ao, .. . , b2 regular functions on U and satisfying and
a1(0) , b2(0) not both O.
In other words, the degeneration of ¢/ at t = 0 has the form [AXY, XY + By 2 ] . Further, the map determines the point [A, B] = [a1(0) , b2(0)] E JlI'l (C) up to switching the two coordinates. In this way we identify
[A, B]
f---l
[AXY, XY
+ By 2 ],
where the symmetric group on two letters 5 2 acts on JlI'l by interchanging the coordinates . (It is an exercise to show that JlI'l /52 is isomorphic to JlI'l.)
4.7 Automorphisms and Twists As we have repeatedly seen, from a dynamical perspective the geometric properties ofa rational map (z) and its conjugates 1 = 1- 1 1 are the same, since conjugation by 1 E PGL 2 (C) is simply an invertible change of variables. However, matters become more complicated if we restrict the coefficients of 1 to lie in a field that is not algebraically closed. This leads to a notion of conjugation equivalence relative to a particular field, as in the following definitions.
Definition. Two rational maps (z) , 'ljJ(z ) E K (z) are PGL 2-equivalent if there is a linear fractional transformation 1 E PGL 2 (k ) such that 'ljJ = I . More generally, we say that and 1/J are PGL 2 (K) -equivalent if there is a linear fractional transformation 1 E PGL 2 (K ) such that 1/J = I. When PGL 2 is clear from context, we refer more simply to ic-equivalence and K -equivalence. We leave as an exercise (Exercise 4.31) the proof that these are equivalence relations . We denote the set of rational maps that are k -equivalent to by
and similarly the set of rational maps that are K -equivalent by
Remark 4.63. In some sense we already have a notation (
lu(f-l ) for some I E A.
The cohomology set H1(r, A) is defined to be the collection of l-cocycles r --; A modulo the equivalence relation that two l-cocycles gl and 92 are cohomologous if the map gl"lg2 is a l-coboundary. Remark 4.78. If the group A is abelian, then H1(r, A) is itself an abelian group, and in this situation it is possible to define cohomology groups Hn(r , A) for all n ~ O. For the general theory of group (and Galois) cohomology, with many important applications to class field theory, Diophantine equations, and arithmetic geometry, see for example [97, 396, 410]. A slight elaboration of the proof of Proposition 4.77 shows that if Au t (X) is abelian, then there is a well-defined one-to-one map
i y u(i y1)) , (4.47) where iy : Y --; X is a K-isomorphism; see Exercise 4.42. In some cases, for example when X is an algebraic variety, the map (4.47) is an isomorphism.
Twist (X /K ) ---. HI (Gal(K/K ), Aut (X)) ,
[Y] K
f--->
(u
f->
4.9. Twists of RationalMaps
203
4.9 Twists of Rational Maps According to Proposition 4.77, every twist corresponds to a l-cocycle, and a twist is trivial if and only if its l-cocycle is a l-eoboundary. The natural question that arises is whether every l-cocycle comes from a twist. It turns out that the answer depends on the category from which the objects are being chosen. For example, in the category of algebraic varieties, every l-cocycle does come from a twist. We will not prove this general result, but in the next section we treat the case of twists of pI . In this section we describe what happens for rational maps.
Definition. We define an action of the Galois group Gal(K I K) on the group of linear fractional transformations PGL2(K) in the natural way, thus
a(f) = a (az + b) = a(a)z + a(b) cz + d a(c)z + a(d) az + b for f = - - E PGL 2(K) and a cz+d
-
E Gal(K
I K).
The automorphism group Aut( (z) E K (z), so O"(¢» = cPo
We have proven that ¢>f = 0"(¢> f) for all 0" E Gal( K j K ), which shows that ¢> f (z) E K (z) and thus completes the proof that (b) implies (a). In order to prove the cohomological description of Twist (¢>j K ), we first use Proposition 4.77 , which says that there is a natural embedding of Twist (¢>j K ) into the cohomology set Hi (Gal (K j K ), Aut (¢») (cf. also Remark 4.78). Then the (Gal (K j K ), Aut (¢») comes equivalence of (a) and (b) tells us that an element of from an element of T wis t (¢> j J() if and only if it becomes trivial when mapped to (G al (Kj K ), PGL 2 (K)). 0
u:
u:
Remark 4.80. A formal argument with commutative diagrams shows that
where Br(K ) is the Brauer group of K. The Brauer group plays an important role in class field theory and many other areas of number theory and arithmetic geometry. For example, Br(iQp) = iQjZ, so Br(iQp)[2] has only two elements, and the same is true for finite extensions of iQp. The Brauer group of a number field is more complicated.
Example 4.81. Let K be a field of characteristic not dividing n, and let ¢>(z ) E K(z) be a rational map whose automorphism group is Aut (¢» where we recall that JLn C isomorphism
K *jK*n
= {( z : (
E JLn},
K denotes the set of nth roots of unity. Then there is an
~Twist ( ¢>jK ) ,
(4.49)
4.9. Twists of Rational Maps
205
Note that by assumption we have ¢(z) = (-l¢((Z) for all ( E J.t n . In particular, the function Z-l¢(Z) is invariant under the substitution z ----+ (z, so it has the form ¢(z) = z'lj;(zn) for some 'lj;(z) E K(z) (Exercise 4.37). Thus the twist of ¢(z) given in (4.49) is equal to z'lj;(bz n), so it is indeed in K(z). Since we are assuming that the automorphism group of ¢ is isomorphic to J.t n , not merely as an abstract group, but also in terms of the way in which Gal( R / K) acts on Aut(¢) and on J.t n , a standard result in Galois cohomology? says that there is an isomorphism
b I-------t
(CTf---+ 0-( V'b))
V'b
.
(4.50)
This allows us to identify Twist (¢ / K) with a subset of K*/ K* n. In order to show that they are isomorphic, Theorem 4.79 says that we must show that every cocycle in (4.50) becomes a coboundary when we consider it as a cocycle with values in PGL 2(R). But this is clear, since with our identification of J.t n with Aut(¢), the cocycle in (4.50) is equal to
z
V'b
with j(z) =
-
E PGL 2 (K ).
Note that this example generalizes Example 4.75, which dealt with the case n = 2.
Example 4.82. Let ¢(z) E K(z) be a rational map with automorphism group Aut( ¢) = {z, Z-l}. (See Exercise 4.35.) The Galois group Gal(R/ K) acts trivially on Aut( ¢), so we have
The isomorphism is given explicitly by associating to any b E K* / K* 2 the cocycle
0-
I-------t
{
z
if 0-( Vb) =
z-l
if 0-( Vb) =
Vb, -Vb,
(4.51)
To ease notation, we let (3 = Vb and let 9 be the cocycle described by (4.51). Then Theorem 4.79 says that 9 comes from a twist of ¢ if and only if there is some j E PGL 2(R) satisfying ga = jo-(1-1). Using thefact that ga(z) E {z, Z-l }, we are looking for an j E PGL 2(R) satisfying 9This statement is equivalent to H 1 (Gal( K / K), K*) = 0, which is a version of Hilbert's Theorem 90. Using this and taking Galois invariants of the Kummer sequence
1 ------+ J-t n ------+ K yields the cohomology long exact sequence
*
Q:t---+Q:n
-------+
K
*
------+
1
206
4. Families of Dynamical Systems
if a({3) = (3, f (Z) a(f)(z ) = { l/f(z ) ifa({3) = - (3. It is not hard to construct such an
f , for example
f( z ) = {3z + 1 - {3z + 1 Note that det( ! {3 ~) = 2{3
:I
0, so
f
is in PGL 2 (k ). Hence every 9 gives a twist
of ¢>, and indeed we can write the twist associated to b = using f ,
{32
E K */ K * 2 explicitly
1 {3¢> ( ~) + {3 ¢> (z) = _J... ( {3z+1 ) + 1 . If'
-{3z+1
For example, let Md(z) = zd be the £ilh-power map. Then a judicious use of the b) binomial theorem yields a formula for the b-twist Md of Md'
Mdb )(z) =
I: (2~)bkz2k /I: k
k
(2k
~ 1)bkz2k+
1 .
In particular, the first few b-twists of M d are
1 + bz
2
2 M (b)(Z) _ 1 + 3bz 3 3 - 3z + bz '
M 2(b) ( Z )
_ -
4.10
Fields of Definition and the Field of Moduli
2z
'
If ¢>(z) is a rational map whose coefficients lie in a field K , then it is interesting to study the orbits of points whose coord inates 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 ) = Z3 + 1 is clearly in Q(z) and the map ¢>(z) = Z3 + 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
= r 1 (¢>(f (z ))) = - i¢>(iz) = -i( -iz 3 + i) = _ z3 + 1 E Q(z). map ¢>(z) = z3 + i is really a Q(z) map that has been altered by
¢>1 (z)
Thus the injudicious change of variables.
an
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' / K be an extension field. We say that K' is afield 0/definition/or ¢> 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 of variables, ¢> 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( K j K) act on K (z) in the natural way by applying (J E Gal (K j K) to the coefficients of ¢ E K (z),
+ adzd) (J(ao) + (J(a1)z + + bdzd = (J(b o) + (J(bdz +
() (a o + a1Z + (J ¢ = (J bo + b1z +
+ (J(ad)zd + (J(bd)zd .
Then Galois theory (and Hilbert's Theorem 90) tell us that
¢
E
K' = (fixed field of{ (J
K' (z) for the field
E Gal(K j K)
: (J( ¢) = ¢} ).
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/ E K'(z). Then (J(¢f) = ¢f for all (J E Gal(KjK'),so
f-1¢f = ¢f = (J(¢f) = (J(r1¢J) = (J(J-1)(J(¢)(J(J) = (J(J)-1(J(¢)(J(J). Solving for (J( ¢) yields
(J(¢) = (J(J)f-1¢f(J(J-1) = (J(J(J-1)f1¢(J(J(J-1)) = ¢fa(J-l). Thus (J(¢) is equal to ¢ conjugated by the map f(J(J-1) E PGL2(K), so in particular (J( ¢) and ¢ are equivalent. We have proven that
K ' is a field Of) ( definition for ¢
===}
(for every (J E ~al(K j K') there exists ) a ga E PGL 2 (K) such that (J (¢) = ¢gu .
(4.52)
Turning this around, we start with an arbitrary rational map ¢(z) E K(z) and study the automorphisms (J E Gal( K j K) for which d ¢) is equivalent to ¢.
Definition. Let rjJ(z) E K(z) be a rational map. We associate to rjJ a subgroup G¢ of Gal(K j K) and a field K¢ defined by G¢ = {(J E Gal(KjK): d¢) = ¢gu for some ga E PGL 2(Kn,
K¢ = fixed field ofC¢ = {o
E
K: (J(O:) = 0: for all (J
E
G¢}.
The field K ¢ 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 = ¢ f. If we consider as usual the equivalence class
r:'
consisting of all maps that are conjugate to a particular map ¢, then C ¢ is the subgroup of Gal(K j K) consisting of elements that map the set [¢J to itself. Equivalently, if ¢ has degree d, let (¢) be the image of ¢ in the moduli space Md = Rat., j PSL 2 that we defined and studied in Section 4.4. The space Md 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
Webeginby provingsome elementarypropertiesofFOM 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 K (z ). (a) The set Gel> is a subgroup ofGal(K / K ). (b) Let K' be afield of definition for ¢. Then K eI> FOM ~ FOD.
~
K'. Informally, we say that
Proof The proof of this proposition is simply a matter of unsorting definitions. Thus let (J,T E Gel> . Then
Thus ((JT) (¢) is equivalentto ¢, so (JT E Gel>. Similarly,
which shows that (J-l(¢) is equivalent to ¢, and hence that (J-l E Gel>. This proves that Gel> is a subgroup of Gal(K/ K ). Next let K' be a field of definition for ¢. Under this assumption, we proved earlier (4.52) that for every (J E Gal(K/ K' ) there is a g(7 E PGL 2 (K ) such that (J (¢) = ¢9" . In other words, (J(¢) is equivalent to ¢, so (J E Gel> = Gal(K / K eI» by definition. This provesthat Gal(K/ K' ) c Gal(K/ KeI», and henceby Galoistheory, that KeI> c K'. (Weare also using the fact that K eI> is a finite extension of K .) 0 It follows from Proposition 4.84 that the smallestpossiblefield of definition for ¢ is the field of moduli of ¢, but it is not clear whether the field of moduli is always a field of definition. The following example shows that we can have FOM i=- FOD.
Example 4.85. Let
¢(Z) =i (: ~ ~ r Clearly Q(i) is a field of definition for ¢. Let (J be complex conjugation, so Gal(Q(i)/Q) = {I, (J}, and let g(z) = -1/ z. Then we obtain 1 ¢(-I/z)
=i
1
i(-I / Z-I)3 -I / z + 1
(- 1/z _ 1)3= - i (z + 1)3= (J (¢)(z). - I/z + I
z - I
This shows that (J E Gel> . so Gel> = {I , (J} and K eI> = Q. In other words,Q is the field of moduli of ¢. Now suppose that Q is actually a field of definition for ¢. This means that we can find some f E PGL 2 (Q) such that ¢! E Q(z). In particular, letting (J denote complex conjugation, we have
4.10. Fields of Definition and the Field of Moduli
209
It is not hard to verify (see Exercise 4.39) that Aut(¢) = 1, so we deduce that f = ga(f). Using the fact that g(z) = -1/ z, this equation says that
az+b cz +d where we use an overscore to denote complex conjugation. The fact that these two linear fractional transformations are equal means that there is a A E Q* such that
a=-AC,
b=-Ad,
c=Aa,
d=Ab.
Multiplying the first and third equations gives
This is a contradiction, since A -I- 0 and we cannot have a = C = O. Hence Q is not a field of definition for ¢, so we have an example of a map with FOM -I- FOD. In order to investigate more closely the question of when the field of moduli of a rational map ¢ is a field of definition, we make the simplifying assumption that Aut(¢) = 1.
(4.53)
Replacing K by K p, we may assume that K is the field of moduli of ¢. This means that for every a E Gal(K/ K) there exists a ga E PGL 2(K) satisfying
Note that the assumption (4.53) that Aut(¢) = 1 implies that ga is uniquely determined by a. The next proposition describes some of the properties of the map a ~ gao Proposition 4.86. Let ¢ E K (z) be a rational map of degree d 2: 2 with field of moduli K and satisfying Aut(¢) = 1, andfor each a E Gal( K / K) write a( ¢) = ¢9rr as above. (a) The map
is a l-cocycle, i.e., it satisfies for all a, T E Gal(K/ K). (b) K is a field ofdefinition for ¢ if and only if there is an f E PGL 2(K) such that
if g is a l-coboundary,
for all a E Gal(K/ K).
i.e., if and only
210
4. Families of Dynamical Systems
Proof (a) Let a, r E Gal (K j K ). Then ¢9u r
= a (r (¢ )) = a (¢9r ) = a (¢t
(9r ) = ¢9uU(9r) .
Since Au t (¢ ) = 1, we conclude that gUT = gua (gT)' (b) Suppose first that K is a field of definition for ¢ . This means that there is an f E PGL2(K ) such that ¢f E K (z ). Hence for any a E Gal(K j K ) we have ¢f
= a (¢ f ) = a ( ¢ t(f ) = ¢9u u (f) .
Again using the fact that Au t (¢ ) = 1, we find that f = gua(J), and hence gu =
fa (J-1 ). Conversely, suppose that there is an 1 E PGL2(K ) such that gu Then a( ¢f) = a( ¢t(f) = ¢9uu (f ) = ¢f.
= la(J-1 ).
This is true for every a E Gal( K j K), so ¢f E K (z). Hence K is a field of definition for ¢ . 0 The criterion for FOM = FOD given in Proposition 4.86 may seem complicated, but it represents a tremendous simplification. lfwe try to use the definition ofFOD directly, we need to search for an 1 E PGL2 (K ) that makes the coefficients of ¢f lie in K. The substitution ¢f (z) =
(az+b) - b cz+d , - c¢ (a z+b) + a cz+d d¢
even for a rational function ¢ (z) of small degree , has coefficients that are very complicated expressions in the quantities a, b, c, d. It is thus difficult to determine whether there is some choice of a, b, c, d that makes the coefficients lie in K. On the other hand, the cocycle--eoboundary criterion FOM
= FOD
{:::=:}
gu has the form la (r 1)
in Proposition 4.86 involves only linear functions, i.e., elements of PGL2(K), so it is often considerably easier to apply. The 1-cocycles that arise in the FOD = FOM question have the form g : Gal (K j K ) -+
PGL2(K ).
The group PGL2 (K ) is the automorphism group of the projective line p1 , so these cocycles should be associated to twists of p l. We saw in Section 4.8 that every twist gives rise to a 1-cocycle , but in general it is a delicate question to determine whether every 1-cocycle comes from a twist. It turns out that this is true for algebraic varieties , but since we do not need the most general result , we are content to construct the twists of p 1 that are needed to answer our question about FOD = FOM . (See also Example 4.87.)
211
4.10. Fields of Definition and the Field of Moduli Proposition 4.87. Let
g: Gal(K/K)
--+
PGL 2 (K )
be a 1-cocycle, and assume that g has the property that there is a finite extension L / K such that grr = 1 for all IJ E Gal( K / L). 10 Then there is an algebraic curve C defined over K and an isomorphism i : C -+ ]p>1 defined over K such that the 1cocycle Gal(K/K) --+ PGL 2 (K ), IJ f--------' iIJ(i- I ) , is equal to the 1-cocycle g. Hence C is a twist of]p>1 / K, and
C is the trivial twist of]p>1 / K
if and only if
g is a I-co boundary.
Proof We construct the curve C by describing its field of rational functions. Note that the field of rational functions for the curve ]p>1 is the field K (z) and that the Galois group Gal( K / 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(K/ K). We define this twisted action by letting Gal(K/ K) act on K as usual, but setting
IJ(w) = g;;I(W). In other words, if gl7(z) = (az + b)/(cz + d), then
IJ(w) = g;;I(W) =
dw -b , -cw+a
and for any rational function
we have
IJ(F)(w) = IJ(ao) + IJ(at}g;;l(w) IJ(bo) + IJ(bt}g;;I(W)
+ IJ(a2)g;;l(w)2 + + IJ(b2)g;;I(W)2 +
+ IJ(ad)g;;l(w)d. + IJ(bd)g;;l(w)d
It is clear that IJ(F + G) = IJ(F) + IJ( G) and that IJ(FG) = IJ(F)IJ(G). However, to be an action, we must also have (IJr)(F) = IJ(r(F)). We use the cocycle relation to verify this condition,
IJ(r(w)) = IJ(g;lw) = IJ(g;I)a(w) = IJ(g;l)g;;I(W) = g;;;(w) = (IJr)(w). We now look at the subfield of K (w) that is fixed by the twisted action,
K = {F E K(w) : IJ(F) = F for all IJ E Gal(K/ K)}. We prove that the field K has the following properties: lOOne says that 9 is a continuous l-cocyc1e for the profinite topology on Gal( K / K) and the discrete topology on PGLz(K).
212
4. Families of Dynamical Systems
(i) K n K
= K.
(ii) K has transcendence degree lover K.
(iii) KK = K (z). To verify (i), let a E K n ic. Then a E K, so the action of Gal(K j 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 L j K be a finite Galois extensionwith the property that gT = 1 for every T E Gal(K j L). This implies that the action of an element <J E Gal(K j K ) on W depends only on the image of <J in the quotient group Gal(L jK ) = Gal(KjK)jGal(K jL) ,
since if r E Gal(KjL), then
(<JT)(W) = <J (gT(W)) = <J (w). It follows that the polynomial P (T ) =
II
II
(T - ..\(w )) =
>'EGa l(L/ K )
(T - g>: l (W)) E K (w )[T]
(4.54)
>'EG al (L/ K )
is well-defined. Further,we claim that the coefficients of P (T ) are in K. To see this, for any <J E Gal( K j K ) we note that
<J(P )(T ) =
II
(T - <J..\(w)) = P (T ),
>'E Gal(L/K )
since the effect of replacing ..\ with <J..\ is simply to permute the order of the factors in the product. Wehavenow constructeda polynomial P(T) E K[TJ, and we observethat W is a root of P(T). Hence by definition, the extension K(w)j K is an algebraic extension. Since the element W is transcendental over K, this proves that K has transcendence degree at least lover K. On the other hand, K is contained in the field K (w), and it is clear that K (w)j K has transcendence degree lover K . Therefore K has transcendence degree exactly lover K , which proves (ii). Finally, consider the splitting field I:- over K of the polynomial P (T). As already noted, we have wE I:-. The definition of P (T ) showsthat there is a naturalsurjection Gal(L j K ) ~ Gal(l:- j q , which implies in particularthat I:- = LK. Hence w E L K, which proves that K (z) c KK . This gives (iii), since the other inclusion is true from the definitionof K. We now have the tools needed to complete the proof of Proposition 4.87, but we pause briefly for an example illustratingthe general construction.
213
4.10. Fields of Definition and the Field of Moduli
Example 4.88. Define a map g by the rule
z
ga(z)
=
{
-
II Z
ifcr(i)=i, ·f (.) . 1 cr Z --to
It is easy to check that g is a l-cocyc1e, and indeed it is the l-cocycle described in Example 4.85. Fix an embedding ofQl into C and let p E Gal(Ql/Q) denote complex conjugation. The twisted action on Ql(w) is given by
p(a) =
a
for a E Ql
p(w) = -l/w.
and
The field K. 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) give us elements in K;
This yields one interesting element in K; namely u = w - w- I , and we observe that the quantity v = i( w + w- I ) gives another element in K. It is not hard to show that u and v generate K,
x=
K(u, v) = K ( w -
~,i ( w + ~) ) .
(See Exercise 4.45.) Of course, u and v are not independent, since
u2
+ v2
= (
W _
~
) 2
+
(i ( + ~ ) ) w
2
= -4.
The field K. is the function field of the curve C : u 2 + v 2 = -4. Notice that C is defined over Q, and C is Ql-isomorphic to ]pI, but C is not Q-isomorphic to ]pI, since C(Q) = 0. From our general theory, the fact that C is a nontrivial twist of ]pIIQ is equivalent to the fact (proven by a direct calculation in Example 4.85) that the l-cocyc1e g is not a I-coboundary. Resuming the proof of Proposition 4.87, we have constructed a field K that is the function field of a curve C I K, and we have an isomorphism
KlC = K(w) ~ K(z),
W
f------+
Z,
that induces a K-isomorphism i : C ---+ ]pl. In other words, the functions w on C and z on]pI are related by the formula w = z 0 i. The curve C is a twist of ]pII K, and its associated cocyc1e is given by a ---+ icr(i- I). We compute
214
4. Familiesof Dynamical Systems
z 0 a (i ) = a(z 0 i) = a(w)
since a (z ) = z, since w
= g;;I (W)
= z 0 i,
by definition of the twisted action on
dw - b - cw +a dz 0 i - b - cz 0 i + a - 1 . = Z 0 ga 02 .
az
since w =
+b
= --d
where ga(z )
cz+
R (w),
-
E PGL 2(K ),
z 0 i,
s:'
I Thus a (i) = 0 i, which proves that ga = ia(i- ) is the l-cocycle associated to the R -isomorphism i : C --> pl . This completes the proof that the algebraic curve C is a twist of pI / K whose associated l-cocycle is g. Finally, Proposition 4.77 tells us that C is the trivial twist if and only if its associated l-cocycle is a l-coboundary. 0
Returning to the question ofFOD = FOM, let ¢(z) E R( z) be a rational function with field of moduli K and trivial automorphism group. We have constructed a
I-cocycle g : Gal(R/ K)
----t
PGL 2 (R)
that is associated to ¢ (Proposition 4.86) and a twist C'" of p I/ K that is associated to the l-cocycle 9 (Proposition 4.87). We have also proven the following chain of equivalences: K is a field of definition for ¢ {::=}
9 is a l-coboundary
(Proposition 4.86),
{::=}
C'" is K -isomorphic to p I
(Proposition 4.87).
It remains to find a way of determining whether a twist of pi is K -isomorphic to pl . 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 O.
Theorem 4.89. (Riemann-Roch Theorem) Let C / K be a smooth projective curve ofgenus 9 defined over K. (a) There is a divisor on C ofdegree 2g - 2 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 ) ~ 29 - 1. Then there is a function
div(J) + D
~
f
E K (C) satisfying
o.
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 2 on C .
215
4.10. Fields of Definition and the Field of Moduli
(b) Let D E Div(C) be a divisor defined over K and satisfying deg(D) 2 1. Then there is afunction f E K(C) with div(f) + D 2 o. 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 [410, 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 Krational divisor of degree 2. 0
Proposition 4.91. Let C be a twist of pI / K. The following are equivalent: (a) C is the trivial twist of pI / K, i.e., Cis K -isomorphic to pl. (b) C(K) is nonempty, i.e., C has a point with coordinates in K. (c) There is a divisor D = 2:= n;(Pi) on pI (K) ofodd degree such that D is defined over K, i.e.i for all a E Gal(K/K) we have 2:=ni(a(Pi)) = 2:=ni(Pi) as a formal sum ofpoints. Proof If C is the trivial twist of pI / K, then there is a K -isomorphism j : C --t pl. In particular, j : C (K) --t pI (K) is a bijection, so C (K) is certainly nonempty. This proves that (a) implies (b). Next, it is clear that (b) implies (c), since if P E C(K), then the divisor D = (P) has odd degree (one is an odd number!) and is clearly defined over K. Finally, suppose that the degree n = deg(D) = 2:= ni of D = 2:= ni(Pi) is odd and that D is defined over K. We also use the fact that C and pI are K -isomorphic, so in particular C is a curve of genus zero. It follows from the Riemann-Roch theorem (Corollary 4.90(a)) that there is a K-rational divisor on C having degree -2, say
Consider the divisor
E=D
n-1
n-1
+ -2- D' = (Pd + (P2) + ... + (Pn) - -2-((Qd + (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 'l/J on C such that 'l/J is defined over K and 'l/J has degree 1. In other words, 'l/J is a map 'l/J : C --t pI of degree 1 defined over K, and hence C is K-isomorphic to pl. This shows that (c) implies (a) and completes the proof of the theorem. 0 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 K (z) be a rational map ofdegree d 2 2. Then the field ofmoduli of ¢ is a field ofdefinition for ¢ in the following two situations:
4. Families of Dynamical Systems
216 (a) ¢(z) has even degree. (b) ¢( 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 g : Gal( K / K) --+ PGL 2 (K) be the l-cocycle associated to ¢ (Proposition 4.86) and let C be the twist of pI / K associated to the l-cocycle g (Proposition 4.87). This means that there is a K -isomorphism 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 l-coboundary
(Proposition 4.86),
{=::}
C is K -isomorphic to pI
(Proposition 4.87),
{=::}
C(K) is not empty
(Proposition 4.91),
(There is a divisor on C(K) Of) odd degree and defined over K
(Proposition 4.91).
{=::}
We are going to produce divisors and points on the curve C using the map --+ C defined by the composition
'ljJ : C
We begin by checking that the map 'ljJ = i- 1¢i is defined over K. To verify this, we let a E Gal(K/ K) and compute
a('ljJ)
= a(i- 1¢i) = a(C 1)a(¢)a(i) = a(i- 1)¢g"a(i) 1)a(i) = a(i- 1)g;;I¢gaa(i) = a(i- 1)(ia(C 1))-I¢ia(i= C 1¢i = 'ljJ.
Hence e : C --+ C is defined over K. We also note that ¢ and 'ljJ have the same degree, since i is an isomorphism. (a) Let D'lj; be the divisor of fixed points of'ljJ, that is, the collection of fixed points of'ljJ counted with appropriate multiplicities. In the language of algebraic geometry, D'lj; is the pullback by the diagonal map
Pf------t(P,P), of the graph {(x,'ljJ(x)): x E C} 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, i.e., it is fixed by the Galois group. Thus D'lj; is a divisor of degree d + 1 on C
4.10. Fields of Definition and the Field of Moduli
217
and D'ljJ is defined over K. By assumption, d is even, so D'ljJ has odd degree. Hence by the chain of equivalences derived earlier, the field K is a field of definition for ¢. (b) The map ¢ : pI ----t pI 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 e p ( ¢) = d. The map i : C<jJ ----t pI is an isomorphism, so the point
is a totally ramified fixed point of 7/J. Suppose first that P is the only fixed point of ¢ in pI (K) with ramification index d. Then Q is the only fixed point of 7/J in C<jJ (K) with ramification index d. However, for any 0" E Gal(K/ K) and any point R E C<jJ(K) we have
since 7/J is defined over K. In particular, taking R = Q to be the given fixed point of ramification index d, we see that 7/J(O"(Q)) = O"(Q) and d = eQ(7/J) = ef7(Q)(7/J), so 0" ( Q) is also a fixed point of 7/J of ramification index d. Hence 0" ( Q) = Q, and since this holds for every 0" E Gal(K/ K), we conclude that Q E C<jJ(K). Hence C<jJ(K) is not empty, so we again conclude that K is a field of definition for ¢. We are left to consider the case that ¢ has a second fixed point of ramification index d, say ¢( PI) = P' and e pi (¢) = d. (Note that the Riemann-Hurwitz formula, Theorem 1.1, precludes more than two such points.) Choose some linear fractional transformation h E PGL 2(K) satisfying h(oo) = P and h(O) = Pl. Then ¢h satisfies
SO
¢h must have the form ¢h (z)
= czd
for some c E K*.
But any rational map of this form has a nontrivial automorphism group. Indeed, its automorphism group is a dihedral group of order 2(d - 1) generated by the maps Z
f----+
(z
and
Z
f----+
a/z,
where (d-I = 1 and a d-I = 1/c. In any case, we have ruled out this case by our assumption that Aut( ¢) = 1, which completes the proof of Theorem 4.92. D
Remark 4.93. The distinction between the field of moduli and fields of definition is important in the study of abelian varieties; see for example the work of Shimura [398]. More recently, the FOM = FOD question has been investigated for the collection of of covering maps ¢ : X ----t B up to automorphism of the base B. In particular, this is much studied for B = pI (cf. Grothendieck's "dessins d'enfant"). If also X = B = pI, then one studies the set of rational maps ¢(z) E K (z) under the left composition equivalence relation ¢ rv f¢ for f E PGL 2(K). This bears
4. Families of Dynamical Systems
218
considerable resemblance to the material in this section, where we use instead the equivalence relation ¢ rv j -l ¢j, but there are significant differences. For example, Couveignes [111] shows that using the relation equivalence ¢ rv j ¢, there are polynomials in Q[z] with FOM =I FaD, in direct contrast to Theorem 4.92. For further results, see for example work of Debes and Douai [117, 118, 119] and Debes and Harbater [120].
4.11
Minimal Resultants and Minimal Models
Let P (X ,Y ) = l: aij X iy j E K[X, Y] be a polynomial and let p be a prime of K . We define the order of P at p to be ord, (P)
= min ord, (aij ). t,)
Notice that ord, (P) = 0 if and only if all of the coefficients of ¢ are p-integral and at least one coefficient is a p-unit. In Section 2.4 we defined the resultant of a rational map ¢ (with respect to p) by writing ¢ = [F, G] using homogeneous polynomials F, G E K [X , Y] satisfying min {ord, (F), ord, (G)} = aand setting Res p (¢) = Res(F, G).
The resultant Res, (¢) is well-defined up to multiplication by the 2lf h power of a p-adic unit, so in particular ord, (R esp (¢)) depends only on ¢ and p. We also recall that ¢ has good reduction at p if and only if its resultant is a p-adic unit.
Example 4.94. Even if ¢ has bad reduction at p, it may be possible to change coordinates and achieve good reduction. In other words, there may be some j E PG L2 (K) such that Res(¢/ ) is a p-unit, For example, the map ¢(z ) = z + p2z - 1 has bad reduction at p, since
However, ifwelet j(z) = pz, then ¢I (z) = z + z-l, which has good reduction at p. Our aim in this section is to study this phenomenon. In particular, we study the extent to which we can eliminate, or at least ameliorate, bad reduction in ¢ by conjugating ¢ with a linear fractional transformation in PGL2 (K ). Definition. Let ¢ = [F, G] be a rational map given by homogeneous polynomials F, G E K[X, Y], let j E PGL2 (K ), and choose a matrix A=
(~ ~)
E
GL2 (K )
representing f. We define polynomials FA,GA E K[X ,Y ] by the formulas
4.11. Minimal Resultants and Minimal Models
219
FA(X , Y ) = e5F(aX + (3Y" X + e5Y ) - (3G(a X + (3Y, , X + e5Y) , G A(X , Y) = - ,F(aX + (3Y, , X + e5Y ) + aG(aX + (3Y, , X + e5Y).
(4.55) (4.56)
It is often convenient to write this in matrix notation as
[~~] where
Aadj = (
1- 10 ¢ 0 I
=
Aadj
[~ :~] ,
!-y -!) is the adjoint matrix to A. Note that the conjugate ¢t
=
of ¢ by I is equal to
Proposition 4.95. Let ¢ = [F, G] be a rational map ofdegree d described by homogeneous polynomials F, G E K[X,Y] and let p be a prime ideal. (a) The valuation ofthe minimal resultant of ¢ is given by the formula ordp(Res p (¢)) = ord p(Res( F , G)) - 2dmin{ord p (F), ord p(G)}.
(4.57)
Note that there is no requirement that the coefficients ofF and G be p-integral or that some coefficient be a p-adic unit. (b) Let A E GL 2 (K ). Then with FA and G A defined by (4.55) and (4.56),
ord, (Res(FA, G A)) = ord, (Res(F, G)) + (d2
+ d) ord, (de t A ),
min{ordp (FA ),ordp (G A )} ~ min{ ordp (F), ordp (G )}
+ (d+ l )ord p (A ),
where ord, (A) denot es the minimum ofthe order ofthe coordinates ofthe matrix A. (c) In particular, ifU E GL2 (Rp ), then
ord p( Res(Fu, G u ))
= ordp(Res(F, G )) ,
min{ ordp (Fu ), ord p( Gu )} = min{ ord, (F) , ord, (G) }. Proof (a) Choose a constant c E K * satisfying
ord, (c) = min{ ord; (F), ord p( G)}. Then
where we have made use of the homogeneity property of the resultant (Proposition 2.13(d)). This gives the desired result (4.57) . (b) An elementary calculation (see Exercise 2.7(c)) shows that 2+d
Res(FA,G A) = (det A)d
Res(F,G) ,
so taking ord; gives the first part of (b). For the second part , we observe that every coefficient of FA and G A is a sum of terms of the form
4. Families of Dynamical Systems
220
coefficient ) x ( homogeneous polynomial of ) ( of For G degree d + 1 in Z[a, (3, 'I, 8] . Hence min{ ord, (FA), ord, (G A)} 2: min{ ord, (F), ord, (G)}
(c) The assumption that U
+ (d + 1) ord, (A).
E GL 2 (R p ) is equivalent to the two conditions
ordp(U) 2: 0
and
ord, (det U)
= 0,
i.e., the coefficients of U are p-adic integers and the determinant is a p-adic unit. Applying (b) with A = U gives
ord, (Res(Fu, Gu») = ord, (Res(F, G»), min{ ordp(Fu), ordp(Gu)} 2: min{ ordp(F), ordp(G)}.
(4.58)
This almost completes (c). We next apply (b) to the polynomials Fu and Gu and to the matrix A = U- 1 E GL 2(R p ) . Using the fact that (Fu )u-1 = F and similarly for G, we find from (b) that min{ ord, (F), ord; (G)} = min{ ord, ((Fu )u-1), ord, (( Gu )u-1)}
2: min{ ord, (Fu ), ord, (Gu )}. This gives the opposite inequality to (4.58), which completes the proof of (c).
0
Definition. Let K be a number field and let ¢(z) E K(z) be a rational map. For each prime p of K, define a nonnegative integer by Ep(¢) =
min !EPGL 2(K)
ord, Res p(¢!).
In other words, Ep (¢) is the exponent of the power of p dividing the resultant of the conjugate ¢! that is closest to having good reduction at p. Then the (global) minimal resultant of ¢ is the integral ideal (4.59) We say that (F, G) is a minimal model for ¢ at p if ¢ = [F, G], the coefficients of F and G are p-integral and
ord, Res(F, G) = ord, 9t.p. (See Section 6.3.5 for the analogous definition of mimimal models of an elliptic curve.) Remark 4.96. The product (4.59) defining 9t.p makes sense, since Ep(¢) = 0 for all but finitely primes p. To see this, write ¢ = [F, G] for any F and G having coefficients in the ring of integers of K. Then there are only finitely many primes p with ord; Res( F, G) > 0, and it is clear that Ep(¢) = 0 for all other primes.
4.11. Minimal Resultants and Minimal Models
221
The minimal resultant of a rational map is clearly invariant under PGL 2(K)conjugation. It measures the extent to which the conjugates of cj; have bad reduction, so provides a convenient measure of the arithmetic complexity ofthe conjugacy class of cj;. A coarser way to measure arithmetic complexity is simply to take the product of the primes with bad reduction, which we denote by (4.60)
It is clear that !)1¢ divides 9l¢, and it is tempting to conjecture an inequality in the opposite direction that would be a dynamical analogue of Szpiro's conjecture [205, FJ.2] for elliptic curves.
Conjecture 4.97. Let KjQ be a number field and d ~ 2. There is a constant c = c(K, d) such that for all rational maps cj;(z) E K (z) ofdegree d,
N K / Q 9l¢ < (N K / Q !)1¢t The minimal resultant gives one way to measure the arithmetic complexity of a rational map, but note that there are infinitely many PGL 2(K)-inequivalent rational maps of a given degree whose minimal resultants are the same. For example, the minimal resultants of the polynomials
are the same as u ranges over all units in the ring of integers of K. The moduli space Md provides an alternative way to measure the arithmetic complexity of the conjugacy class of a rational map. If we fix a projective embedding M d '---7 lPN, then cj; E K(z) determines a point (cj;) E Md(K) and we can take the height of the corresponding point in JP'N (K). However, this way of measuring arithmetic complexity is also not entirely satisfactory, since twists of a rational map give the same point in Md, yet are arithmetically quite different. Note that the same situation arises in the theory of elliptic curves, where curves with the same jinvariant need not be arithmetically identical. This suggests combining the primes of bad reduction with the height coming from moduli space. We do this and formulate a dynamical version of a conjecture of Lang (cf. [202], [254, page 92], or [410, VIII.9.9]). Recall that the canonical height ofa point P satisfies h¢(P) = 0 if and only if Pis preperiodic for cj; (Theorem 3.22). The following conjecture says that the height of nonpreperiodic points grows as cj; becomes more arithmetically complicated.
Conjecture4.98. Fix an embedding ofthe moduli space Md in projective space and let h M d denote the associated heightfunction. Let K be a numberfield and d ~ 2 an integer. Then there is a positive constant c = c(K, d) such that for all rational maps
cj; E K(z) ofdegree d and all wandering (i.e.. non-preperiodic) points P E JP'I (K),
222
4. Families of Dynamical Systems
In the theory of elliptic curves , the notion of global miminal Weierstrass equation is extremely useful; see the discussion in Section 6.3.5 and [410, VIII, §8] for further details. We briefly discuss a dynamical analogue. Definition. Let K be a number field and let ¢(z) E K (z ) be a rational map . Then ¢ has a global minimal model if there is a linear fractional transformation f E PGL 2 (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 integers of K and
ord; (Res( F,G)) = ord; (rytc/»
for every prime p.
In other words, the pair (F, G ) is simultaneously a minimal model for
= Res(F, G) u~G '
This is an equality of ideals, so by unique factorization of ideals we conclude that COcF ,cG = OF,G. Hence the ideals 0 cF ,cG and OF,G differ by a principal ideal. Next suppose that d is even. Then by a similar calculation we find that c
Hence ideal.
2d
Res(F, G)U~F,cG
2) (C OcF ,cG
=
OF,G ,
= Res(cF, cG) O~F,cG = 9'-tcl> = Res(F, G)oi,G' so
OcF ,cG
and
OF,G
differ by the square of a principal 0
Definition. Let K be a number field, let ¢(z) E K (z) be a rational map of degree d, and write ¢ = [F,G] with polynomials F and G as usual. If d is odd, we write o.cI> / K for the ideal class of OF ,G in the ideal class group of K. If d is even, we write aN K for the image of OF,G in the group of fractional ideals modulo squares of principal ideals . In both cases, by analogy with the theory of minimal equations of elliptic curves (cf. [410, VIII §8]), we call acl> / K the Weierstrass class of ¢ over thefield K. By Proposition 4.99 the Weierstrass class acl> / K depends only on ¢, independent of the chosen lift [F,G].
The triviality of the Weierstrass class gives a necessary condition for the existence of a global minimal model. Proposition 4.100. Let K be a number field and ¢( z) E K (z ) a rational map of degree d ~ 2. If ¢ has a global minimal model over K. then its Weierstrass class acl>/ K is trivial.
Proof Replacing ¢ by ¢f for an appropriate choice of f E PGL 2(K), we may assume that ¢ = [F, G] with polynomials F and G having coefficients in the ring of integers of K and satisfying for every prime p.
It follows from the defining equation (4.61) of OF,G that for every prime p, Hence OF,G = (1), so its image in the ideal class group (if d is odd) or in the group of ideals modulo squares of principal ideals (if d is even) is also trivial. 0
Exercises
224
Exercises Section 4.1. Dynatomic Polynomials 4.1. Let¢(z) = z + liz, or inhomogeneous form, ¢([X, YJ) = [X 2 + y 2 , X Y J. (a) Compute the first few dynatomic polynomials iI>;,n (X, Y) for ¢, say for n = 1,2,3,4. (b) Prove that for all n 2:: 2, the dynatomic polynomial iI>;,n satisfies iI>;,n(±X, ±Y) = 2 2 iI>;,n(X, Y). Deduce that iI>;,n(X, Y) E Z[X , y ]. (c) Prove that the field 1Ql2,¢ generated by the points of exact period 2 is the field IQl ( A). (d) Prove that the field 1Ql3,¢ generated by the points of exact period 3 is an 53 extension oflQl. (Hint. Show that the roots of iI>¢,3 (1, VW,) E Z[w] generate a cyclic cubic extension oflQl.) (e) Prove that iI>;,4(X, Y) factors into a polynomial of degree 4 and a polynomial of degree 8. (See Exercise 4.40 for a more general result.) Describe the fields generated by the roots of each factor.
4.2. Prove the following elementary properties of the Mobius function. (See (4.2) on page 148 for the definition of the Mobius function.) (a) Let n 2:: 1 be an integer. Then ifn
= 1,
ifn 2:: 2. (b) Let g(n) be a function whose domain is the positive integers, and define a new function f(n) by f(n) = Ldln g(d). Prove that
g(n)
=
L
f(d)J1
G) .
din
This is called the Mobius inversion formula. (c) Prove that the nth cyclotomic polynomial ~(Zk
_ l)~(n/k)
kin is indeed a polynomial. (Hint. Use (b) and the fact Zk - 1 has distinct complex roots.)
4.3. Let l/d(n) = deg(iI>;,n) be the number offormal n-periodic points ofa rational map of degree d, counted with multiplicity (see Remark 4.3). (a) Prove that l/d(l) = d + 1 and
z:= J1 (iD dk
for n 2:: 2. kin (b) Make a table of values ofl/d(n) for some small values ofn and d. (c) Prove that l/d(n) =
formally as power series. For what range of x
> 0 do the series converge?
225
Exercises I Type of Point
I Char K I a;'(n) values
Multiplier
a;'(n) = 0 for all n
Wandering
a;'(n) > o for exactly one n
Periodic
not root of unity
Periodic
root of unity
0
Periodic
root of unity
p>O
a;'(n) > ofor exactly two n's a;' (n) > 0 for n = ip", exactly two t's and all k ::0: 0
Table4.4: Values of ap(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 + ze'ljJ(z) with e ::0: 2 and'ljJ(O) =1= 0, where e = ao(¢, 1) in the notation of Theorem 4.5. (a) Prove that
¢n(z)
=
Z + nze'ljJ(z)
(b) Assume further that K has characteristic p
+ O(z2e-l).
> O. Prove that for all k ::0: 1.
4.5. Verify that the description of the values of a;'(n) given in Table 4.4 is correct. 4.6. Let ¢( z) be a rational function, let n ::0: 1, let p be a prime with p t n, and let P E pl. (a) Prove that
a;'(n, ¢P) = a;'(n, ¢) + a;' (np, ¢). (b) Deduce that the nth dynatomic polynomial for ¢P factors as
(c) More generally, ifgcd(n,k) = 1, prove that
*nk,¢ -- Il(*n,¢J. )p(kfj) . jlk
4.7.
* Let ¢(z) = Zd
+ ad_lzd-1 + ... + a2z 2 + alZ + ao
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). (a) Prove that there is a (unique) monic polynomial
whose nth power satisfies
m(p) has formal period n. Points satisfying ::",n (P) = 0 are called preperiodic points with formal preperiod (m, n). (b) ** Prove (or disprove) that ::",n(X, Y) is a polynomial.
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 polynomialon,¢(x) by
On,¢(xt=
II
(x-A¢(a))=
4>n,C a)=o
and for min with m
II
(x-(¢>n)'(a)),
(4.65)
4>n,Ca)=O
< n we set ~n,m
= Res(Cn/m(X),Om,¢(X)),
where Ck(X) is the k th cyclotomic polynomial. We now specialize to ¢>c(z) write on(e,x) = on,¢c (x) and ~n,m (c) to indicate the dependence on e.
= z2 + e and
228
Exercises
(a) Prove that bn(c,x) E Z[c,x] and that ~n,m(c) E Z[c]. Then use Exercise 4.7(c) to explain the powers that appear in Table 4.3. (b) Prove that 8n (c, x) = T deg , ot>;'bn(C, 2nx) is in Z[c, x]. (This eliminates many powers of 2 from the coefficients of bn (c,x).) Compute
8n (c,x)
forn = 1,2,3,4. (c)
** Prove that there is a monic polynomial Wn,m(X)
E Z[x] such that
~n,m(c) = Wn ,m(4c),
and deduce that the set {c E
Q: ~n,m(C) = 0 for some m < n with min}
is a set of bounded height. Conclude that the set of bifurcation points in the Mandelbrot set M is a set of bounded height. (Hint. The formula ~n,m(C) = wn ,m(4c) bounds the denominator of c, then mimic the proof of Proposition 4.22.) (d) Prove that ~n,l (c) and ~2n,2 (c) are irreducible in lQi[c] for all n 2 1. (e) ** Prove the following conjecture of Morton-Vivaldi [314]: The polynomial ~n,m( c) is irreducible in lQi[c] for all min with m < n.
4.13. Let cPe(Z) = Z2 + c and write ~ (c, z) for the nth dynatomic polynomial of cPe. Continuing with the notation from Exercise 4.12, we let bn(c,x) E Z[c,x] be the polynomial defined by (4.65) and 8n (c, x) = T degot>;'bn(C, 2n x) the normalized polynomial in Exercise 4.l2(b). (a) Recall that the dynatomic curve Y1 (n) C A2 attached to the polynomial
Z(n) such that the composition
is the map F. (c) Prove that the map Yo(n) ---> Z (n) in (b) has degree 1, so the equation 8n (y, x) = 0 gives a (possibly singular) model for Yo(n). (d) Prove that Z(l) and Z(2) are nonsingular and that Z(3) and Z(4) are singular. Resolve the singularities of Z (3) and Z (4) and check directly that Yo(3) and Yo (4) are curves of genus O.
229
Exercises
4.14. Let ¢c(z) = Z2 + c and let ;'",n be the generalized dynatomic polynomial defined in Exercise 4.11. (a) Use a computer algebra system to compute the first few generalized dynatomic polynomials ;'",n for ¢c, say for (m , n ) in the list
{ (I, 1), (2, 1), (3, 1), (4, 1), (1,2 ), (2, 2), (3,2) , (4, 2), (1,3), (2, 3), (1, 4) } . (b) From your list, it should be apparent that many of the leading terms of ip;;' , 1 and ;'",2 coincide . Prove that for all m
~ 1.
(Note that this is a special property of the generalized dynatomic polynomi als for ¢c .) 4.15. Let c E C and let ¢c(z ) holomorphic function
= z2 + c. Prove
that there is a number R c
'l/Jc : {z E C : Izl > R c }
----t
>
0 and a
C
satisfying log I'l/Jc (z) I rv log
and
[z]
as
(Hint. Show that there is a consistent way to choose square roots so that
Izi ...... 00 . 'l/Jc can be defined as
limn ~ oo 2\!¢6' (Z) .) 4.16. Consider the doubl ing map
D : Q/ 'l. ----t Q/ 'l.,
D(t ) = 2t mod Z.
Fix t E Q/Z and let m ~ 0 and n ~ 1 be the smallest integers such that the denom inator of t divides 2m (2n - 1). (a) Prove that t is periodi c for D if m = 0 and that t is strictly preperiodic for D if m ~ 1. (b) Prove that m ~ 0 and n ~ 1 are the smallest positive integers such that t can be written as a (not necessarily reduced ) fraction of the form
(c) If t is preperiodic for D , prove that the preperiod of t is equal to m and that D'" (t ) is periodic with exact period n. (d) For t as above, we say that t is of type (m , n) for the doubling map D . For a given pair of positive integers (m, n) , how many t E Q/ 'l. are of type (m, n)? The map D is used in the analytic characterization of Misiurewicz points; see Theorem 4.25. However, note that (d) does not count the exaet number of Misiurewicz points of type (m, n) , because distinct t may give the same Misiur ewicz point. 4.17. ** Let ¢c(z ) (of c) by
(c) =
F
m,n
IT ( kin
= z2 + c as usual, and for integers ¢;;,+k(O) - ¢;;'( O)
¢~ - I+ k ( O ) _ ¢~ - I (O )
) J1o(n/ k) =
m , n ~ 1, define a rational function
IT (-I,m- l+k(O) + -I,m- l (O))J1o( n/ k). kin 'Pc
'Pc
Then set
Gmn(C) ,
= {Fm,n(C)
ifm ,!-1 (mod n ) orm = 1, Fm,n(C)/F1,n(c) ifm=l(mod n)andm=l=-l.
230
Exercises
(a) Prove that Gm,n(e) is a polynomial in e. More precisely, prove that Gm,n(e) is in :E[e]. (b) If n ~ 2, prove that the roots of O ,«,« are the Misiurewiczpoints of type (m , n). (c) Provethat G,« ,« (e) is irreduciblein lQ>[e). 4.18. Let V be a variety and suppose that the points of V algebraically parameterize a family of quadratic polynomials 'Ij; together with a marked point Aof formal period n. Theorem4.11 says that there is a unique morphism TJ : V -+ YI (n) satisfying TJ(?)
= ('Ij;p (z), A(P ))
E Formal(n).
Note that by construction, each point I = (e, Q;) E YI ( n) is identified with a quadratic polynomial
]p~
1"'1
V
1"'1 4>
]ph (n) ----->
]pI
YI (n )
/ -1 0 )..
----->
]p~
1"'1 YI (n)
1"'1 I-'
----->
]pI
YI (n)
(Hint . During the proof of Theorem 4.11 we defineda linear fractional transformation f p( z). Provethat fp is uniquely determinedby P . Deducethat the map P f-+ fp is a morphism.)
4.19. In Example 4.14 we gave an explicit description of the quotient map YI(2) Performa similar analysis and describe the map YI (3) -+ Yo(3).
-+
Yo(2).
4.20. Weprovedthat the dynatomicmodularcurves X I (1), X I (2), and X I (3) are isomorphic to ]pl . This exerciseasks you to investigate X I (n) for other small values of n. (a) Prove that X I ( 4) is a curve of genus 2 and X 0 ( 4) is a curve of genus l. (b) Prove that Xo (4) (IQ» is finite, and Yo (4) (IQ» is the empty set. (c) Prove that X I(5) has genus 14 and X o(5) has genus 2. (d) * Prove that X o(5)(IQ» is finite, and Yo (5) (IQ» is the empty set. (e) Computethe genera of Xl (6) and X o(6 ). (f) ** Find all rational points in Yo(6)(IQ» . Section 4.3. The Space Ra t d of Rational Functions
4.21. Let
11 > 1}. We leave the details to the reader. 0
It is now a simple matter to use Theorem 5.26 to prove a nonarchimedean version of Monte1's theorem for rational functions. Theorem 5.27. (Nonarchimedean Montel Theorem, Hsia [208]) Let
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 5.31. 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 zd or z:", in which case its Julia set is empty.) Corollary 5.32. Let ¢ : lP'I (K) assume that J (¢) -I- 0. (a) J(¢) has empty interior. (b) Let P E J(¢) and let
---+
lP'I (K) be a rational map of degree d 2: 2, and
O;(P) =
U¢-n(p) n2:0
be the backward orbit ofP. The Julia set J( ¢) is equal to the closure of in pI (K). (c) J( ¢) is a perfect set, i.e..for every point P E J( ¢), the closure ofJ( ¢) contains P. (d) J (¢) is an uncountable set.
0; (P) O ¢n(u) omits at most one point, and Proposition 5.29(a) says that ifit 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 .:J(¢) by assumption. This proves that P E Un>O ¢n(u), or equivalently, that there is some n 2': 0 such that Un ¢-n(p) i- 0. This proves that every open set U that intersects .:J(¢) nontrivially also intersects (P) nontrivially. Hence .:J (¢) is contained in the closure the backward orbit
0;
ofO;(P). (c) Let Po E .:J(¢ ). We claim that the backward orbit (Po) must contain a nonperiodic point. To see this, suppose instead that (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 PI E (Po). The point PI is in .:J(¢ ), since .:J(¢) is completely invariant, so (b) tells us that
0;
0;
0;
Po E closure of On the other hand, Po is not in
0; (Pd.
0; (PI), since otherwise PI would be periodic. Hence
Po E closure of (.:J(¢) ,,{P}). (d) The Baire category theorem [387, §5.l,5.2] implies that a nonempty perfect subset of]p'1 (K) is uncountable. 0
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 Cpo
Lemma 5.33. Let ¢1(Z) and ¢z(z) be power series that converge on D(a, r), and suppose that ¢1 (D(a, r)) n ¢z (D(a, r)) = 0. Then inf
p(¢I(Z), ¢z(z)) > O.
zED(a,r)
Proof Let
M1 =
syP zED(a,r)
1¢I(Z)1
and
Mz =
sup
l¢z(z)l.
zED(a,r)
The maximum modulus principle (Theorem 5.l3(a)) says that there are points ZI,Zz E Di«, r) such that ¢1(zd = M 1 and ¢z(zz) = M z. In particular, M 1 and M z are finite, since ¢1 and ¢z are power series that converge on D(a, r).
269
5.7. Periodic Points and the JuliaSet Let M
= max{M1 , M2 , I}.
Then for any
On the other hand, the function (PI so Theorem 5.13(b) tells us that
-
Z
E
Dto; r) we have
¢2 does not vanish on D( a, r) by assumption,
for all z E D(a, r). Hence
o 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 {O, I}.
Lemma 5.34. Let A, B c C p be bounded sets that are at a positive distance from one another. In other words, there are constants ~, 0 > 0 such that
sup lal ::; "'EA
~,
sup 1,81
0, depending only on ~ and 0, such that
p(L",,(3(z), L"",(3' (z')) ::; C· max{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 ]P'l(C p ) . Remark 5.35. Although Lemma 5.34 appears somewhat technical, it is not saying anything mysterious. The linear fractional transformation L",,(3 is determined by the three conditions L",,(3(O) = a,
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))} PElF" (iC p )
to measure the closeness oftwo maps. Thus Lemma 5.34 is equivalent to the assertion that the map (a,,8, z) ~ L",,(3(z)
is Lipschitz.
= (,8 -
a)z
+a
270
5. Dynamics over Local Fields: Bad Reduction
ProofofLemma 5.34. Toease notation, for x , y E C p we write
Ix, yl = maxj ]»], Iyl}· We also assume (without loss of generality) that fj. any 0:, 0:' E A and (3, (3' E B we have
~
1 and 8 :::; 1. Then for
10: - 0:'1 = p(o: , 0:') , 10:, 11, 10:', 11:::; fj. 2p (0:,o:'),
(5.17)
1(3 - (3' 1= p((3, (3' ) , 1(3,1 1,1(3' ,1 1:::; fj. 2p((3 ,(3' ). Let 0:, 0:' E A, let(3, (3' E B, andlet z, z' E of L o ,{3 and the chordal metric, we have
]P'I(Cp ) . Directly from
the definitions
, 1(((3' - o:')z ' + 0:') - (((3 - o:) z + 0:) I p(L o ,{3 (z ), L o ' ,{3' (z )) = max{ 1((3 _ o:)z + 0:1, I} . max{I((3' - o:') z' + 0:'1, I} . (5.18)
Assuming for the moment that z i:- 00 and z' and estimate it using the triangle inequality:
i:-
00,
we multiply out the numerator
1(((3' - 0:') z' + 0:' ) - (((3 - 0:) Z + 0:) 1 = 1(3' z' - (3z - 0:' Z '
+ o:z + 0:' -
0: /
= 1(3' (z' - z) + ((3' - (3 )z - o:' (z' - z ) + (0: - o:') z
+ (0:' -
0:)1
:::; max{I(3' - (31· lzl, 10:' - o: l· lz, 11, Iz' - zl ' lo:' ,(3' I} :::; max{fj.2p( (3, (3') ·lzl, fj.2p( 0:,o:') ·Iz, 11 , Iz' - zl·lo:' , (3'I} from (5.17),
:::; max{fj.2p(I3,I3' ) ' lzl, fj.2p (0:, o:') ' Iz, 11 , fj. ·I z' - zl} from (5.16), :::; max{fj. 2p(I3,I3') 'lz l, fj. 2p (0:, o:') 'Iz, 11, fj.p (z, z' ) · Iz , II · Jz' , I I} definition of p, :::; fj.2 . max{p(l3, (3'), p(o:, 0:'), p(z , z' )} · Iz, 11·l z' , 11. Substituting this into (5.18) and doinga little bit of algebrayields
max{p(l3, 13'), p(o:, 0:' ), p(z, z') } Iz' , 11 < fj. 2 . _~-,--,-:I---, z '---,1-,--1--,----""7 max{ I(13 - o:)z + 0:1 , I} max{ 1(13' - o:')z'
+ 0:' 1, I} .
(5.19)
We are left to show that the righthand side is bounded in terms of 8 and fj.. By symmetry, it suffices to boundthe first fraction. Weconsidertwo cases. First, if Izi :::; fj. / 8, then we havethe trivial estimate Iz,11 K have disjoint images, there is at least one point a omitted by both of them. Making a linear change of variables z f---+ z - a, we may assume that cPl and cP2 omit the value O. Then Theorem 5.13(b) tells us that
for all z E D(a, r). We are going to apply Lemma 5.34 to the disjoint bounded sets andcP2(D(a,r)). Thus for each point w E D(a,r),ifwelet
cPl (D(a, r))
5. Dynamics over Local Fields: Bad Reduction
272
then Lemma 5.34 says that there is a constant C such that
p(Lw(z), Lu(z')) ::; C max{p( l (Cp ) .
Proposition 5.45. Let V open and Vc10sed be, respectively, the collections ofstandard open and closed disks in jp>l (Cp ) as defined above. (a) Let D 1 , D 2 E Vc1osed. Then one ofthe following is true: (i) D 1 n D 2
= 0.
(ii) D 1 U D 2 = jp>l(Cp ) .
(iii) D 1
~
D2.
(iv) D 2
~
D 1.
(b) Let U C jp>l (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 Vc10sed U V open.
Proof (a) If D 1 U D 2 = jp>l(Cp ) , we are done. Otherwise, choose any point in the complement of D 1 U D 2 and use a linear fractional transformation to move that point to 00. This reduces us to the case that neither D 1 nor D 2 contains 00, so they have the form D 1 = D(al,rd and D 2 = D(a2,r2). If D 1 and D 2 are disjoint, we are done, so we may assume that there is a point
and switching D 1 and D 2 ifnecessary, we may also assume that rl ::::: r2. Let f3 E D 1 . Then
so f3 E D 2 • Hence D 1 ~ D 2 . (b) If V = jp>l(Cp ) , we are done, so we assume that V -=I- jp>l(Cp ) . Using a linear fractional transformation to move a point not in V to 00, we are reduced to the case that 00 ~ V.
279
5.8. Nonarchimedean Wandering Domains
Let D 1 , • .• , D n E Dclo sed be any path of disks contained in V . Each pair of adjacent disks (D i , Di+d 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 D, is itself a closed disk, i.e., the union is in D closed. 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 ~ 0 : D(a, r ) c V }. Note that R > 0, since a E V and V is open. If R done, so we assume that R < 00 . We claim that
= 00, then V = C p
D(a, R ) ~ V ~ D(a, R ).
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) ~ V tell us that s :s; R, and then the fact that b E D(a, s) tells us that bE D(a, R ). This gives the other inclusion . But we get even more. If there is even a single b E V satisfying Ib - al = R, then s = Rand D(a,R) = D(b, s ) ~ V, so we find that V = D(a,R) E D closed. On the other hand, if Ib- al < R for every b E V, then V c D ( a, R), so (5.23) tells us that V = D (a, R) E D open • 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 ajinite extension ofp-adicjields and let ¢(z ) E K(z) be a rational fun ction ofdegree d ~ 2. Proposition 5.20(c) tells us that F( ¢) =1= 0, so changing variables if necessary, we may assum e that 00 E F( ¢). Then the following are equivalent : (a) There are no critical points in :J( ¢). (b) For every jinite extension L/ K there exists an integer m ~ 1 such that for all ex E :J(¢) n L. In other words, ¢ffi is strictly exp anding on :J (¢)
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 N -(k + 1)M , so k
+ 2 > N 1M. We have thus found integers k and r
satisfying
first be-
5.8. Nonarchimedean Wandering Domains
283
(¢N)'(o:) = 'ljJ~o(O:O)1/( (o:d'ljJ~2(O:Z)" ·'ljJ~k(O:k)· (¢T)'(O:k+l) with k
> N / M - 2 and r < M.
Hence 1(¢N)'(o:)1
= 1'ljJ~o(O:O)'ljJ~l (o:d'ljJ~2(O:Z)" ·'ljJ~k(O:k)I·I(¢T)'(O:k+dl ,
v
I
2 Zk from (5.26)
2: 2N / M -
Z
. J-LM
>1
since k
'-----v---"
2
r
M from (5.27)
> N/M - 2 and r < M,
from the choice (5.28) of N.
This shows that 1(¢N),(o:)1 > 1 for all N > MZlogz(J-L- 1 ) and all which completes the proof of Theorem 5.46.
0:
E :J(¢)
n L, 0
5.8.3 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. Let ¢( z) E C p (z) be a rational map defined over a nonarchimedean field, and let
'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
'DC (¢) ------; 'DC (¢ ),
U
(disk component of F (¢) containing ¢( U)). (5.29) We say that U E 'DC (¢) is periodic, preperiodic, or wandering according to the behavior of U under iteration of the map (5.29). I----t
Example 5.49. Let p be an odd prime and let
z ¢(z)=z -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,
:J(¢)
c D(O, 1) U D(I, 1),
and that :J (¢) contains the repelling fixed points 0 and 1. For 0: E C p , let B(o:) denote the disk component of 0: 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 O. On the other hand, D(-I, 1) is in F(¢), since it is disjoint from D(O, 1) U D(I, 1). Hence B( -1) = D( -1,1).
284
5. Dynamics over Local Fields: Bad Reduction
Now consider the image point ¢ (-1 ) disk component
= 2p-1
and the image of the associated
¢ (B( - l) ) = ¢ (D( - l , 1)) = D (2p-l ,p). The disk D(2p-l,p) is contained in F (¢), but it is certainly not the largest disk around 2p-1 contained in F (¢ ).Indeed,
2p-1 E pI (Cp ) " D(O, 1) c F (¢). It is not hard to check that pI (rCp ) containing ¢ (B ( -1 )).
d rules out the existence of wild critical points.
> d,
Remark 5.54. If a recurrent critical point P is in the Fatou set F (¢), then one can show that P is in fact per iodic (Exerc ise 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 C p and some k :::: 2. Thus if Q is sufficiently close to P, then
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 JP'l(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 C p 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 ofQp. 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 5.51, 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 afinite extension ofp-adic fields, 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 contains an algebraic point. (Note that Qp is dense in C p.) Replacing K by a finite extension and changing coordinates, we may assume that 00 E F( ¢), and indeed we may even assume that .:J(¢) c D(O, 1). Equivalently, we may assume that the disk component of 00 contains P! (C p ) -, D(O, 1). We suppose that U C F( ¢) is a wandering disk component of F( ¢) and derive 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 00. In particular, ¢n(u) C D(O, 1) for all n 2: O. Again taking a finite extension of K if necessary, we can find a point 000 E K and a radius ro such that D(ao,ro) cU. At this stage we fix the field K and we use Theorem 5.46 to find an integer m such that I(¢m)'l > 1 on .:J(¢) n K. Replacing ¢ by ¢m, it suffices to consider the case that WI> Ion .:J(¢) n K. For each n 2: 0, the image ¢n(D(ao,ro)) is a (closed) disk centered at the point an = ¢n(ao), say
In particular, ¢(fJ(a n, rn)) yields
= D(an+l' rn+d, so applying Proposition 5.16(b)
286
5. Dynamics over Local Fields: Bad Reduction
rn l¢/(an) 1< rnH .
(5.30)
We also know that the disks D (an, r n) are disjoint, since D(an , r n) is contained in ¢n(u) , and further, each disk D (a n ,r n ) contains a point of D(O , l )nK. It follows that the radii must satisfy (5.31) lim r « = 0, n- oo
since the set D(O ,I) 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 n with the property that i.e., since r n of points
---->
0, we must have rn+l
< r n infinitely often. Consider the infinite set
{an: 1¢'(a n)1 < 1,
n
= 1,2,3, ... }.
It is contained in the compact set D(O, 1) n K, so it contains an accumulation point E K. The continuity of ¢' implies that 1¢'((3) 1:::; 1, which shows that (3 E F(¢),
(3
since we used Theorem 5.46 to ensure that WI > Ion .J (¢ ). Let V be the disk component of F (¢) containing (3. Then by construction V contains infinitely many of the iterates an = ¢n(ao). Since the radii of ¢n(u) and ¢n( v) shrink to 0 as n ----> 00, 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 0 wandering, contradicting our original assumption.
5.8.4 Wandering Disk Domains Exist in c, 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, Sullivan'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 Cp that have wandering disk domains. Theorem 5.56. (Benedetto [59]) For c E C p, let ¢c(z) be thepolynomial
¢c(z) = (1 - c)zpH
+ cz".
Then there exists a valuea E C p such that: (1) .J(¢a) =I- 0, (2) F(¢a) has a wandering disk domain, (3) F( ¢a) contains every criticalpoint of¢a. Proof See [59] for a proof of this specific theorem, and see [63, 62, 373, 378, 380] for generalizations and related results. 0
287
5.9. Green Functions and Local Heights
5.9 Green Functions and Local Heights The canonical height h the Green function of .
(5.35)
5.9. Green Functions and Local Heights
289
(b) The Greenfunction is the uniquefunction A; (K) twoproperties: Q(1)(x,y)) = dQ(x,y) Q(x, y) = log] (x, y) Ilv
+ 0(1)
----+
IR having the following
for all (x, y) E A;(K).
(5.36)
for all (x,y) E A;(K).
(5.37)
(c) Ifv is nonarchimedean and iIl satisfies IliIlll v = 1 and IRes(F, G) Iv = 1, i.e.,
the map ¢ = [F, G] : lP'1 Q(x,y)
=
----+
if
lP'1 has good reduction at v, then
logll(x,y)llv
for all (x,y) E A;(K).
(The converseis also true. See Exercise 5.27.) (d) Forall (x, y) E A; (K) and all c E K*, the Greenfunction Q has thefollowing
homogeneity properties: Q(cx, cy) = Q(x, y)
+ log Icl v.
1 Qc(x, y) = Q(x, y) + -d-log Jelv. -1
(5.38) (5.39)
(e) The Greenfunction Q : A; (K)
----+ IR is continuous. (In fact, Q is Holder continuous, but this is more difficultto prove. See Exercise 5.28.)
Proof We consider the two functions and Proposition 5.57(a) tells us that they satisfy loglliIl(x, y) Ilv
= dlogll (x, y) Ilv + 0(1)
for all (x, y) E A;(K).
(5.40)
This is exactly the situation needed to apply Theorem 3.20, from which we conclude that the limit (5.35) exists and satisfies (5.36) and (5.37). Further, Theorem 3.20 says that Q is the unique function satisfying (5.36) and (5.37). This completes the proof of (a) and (b). (c) The assumptions 111>llv = 1 and IRes(F,G)lv = 1 combine with Proposition 5.57(b) to tell us that II iIl(x, y) Ilv Hence by induction we obtain
= II (x, y) II~
for all points (x, y) E
A; (K).
Then the definition of Q immediately gives Q(x,y) = logll(x,y)llv' which proves (c). dn1>n(x,y). (d) The map iIl n is homogeneous of degree d", so iIln(cx,cy) = c Substituting this into the definition of Q gives (5.38). Similarly, ifwe let iIl c (x, y) = ciIl(x, y), then homogeneity and an easy induction argument show that
290
5. Dynamics over Local Fields: Bad Reduction
Hence
1 9c (x,y ) = fl.-+ limOO dn
10gllcp~ (x,y) 11 v
1(
= J~~ dn
d
n-l)
log llcpn( x , y)llv + d=1 log [c] , 1
= 9 (x,y) + d _ 1 log Iclv·
(e) Let 9n(x,y) = d- n 10g llcpll(x , y) Ilv' Then for any fixed value of n, the function 9 n : A; (K ) ----t ~ is continuous, since it is the composition of the continuous maps cpn : A;(K) ----t A;(K) and log II . Ilv : A;(K) ----t R Further, the telescoping sum argument used in Theorem 3.20 to prove the existence of limn->oo 9n(x, y) implies that for all n ~ 1 and all (x, y) E A;(K), where C = C(CP , v) depends only on the 0 (1 ) constant in (5.40). (In the inequality (3.15) on page 98, switch the roles ofm and n and then let m ----t 00.) Hence the sequence of continuous functions 9n(x,y) converges uniform ly to 9 (x, y), so the limiting function 9 is also continuous. (See also Exercise 3.14.) 0 The Green function allows us to decompose the canonical height he/> into a sum of local terms. Theorem 5.59. Let K be a number fi eld. let ¢ : JP>1 ----t JP>1 be a rational function of degree d ~ 2 defined over K , and let cp be a fixed lift of ¢. For each absolute value v E M K , let 9,v be the associated Green function as described in Proposition 5.58, where we now include v in the notation so as to distinguish between different absolute values. Then the (global) canonical height decomposes as a sum
L
he/>(P) =
n v9 ,v(x,y)
for all P
= [x ,y] E JP>l(K).
vEMK
Proof. Let
TJ(x ,y) =
L
n v9 ,v(x,y)
for (x, y) E A; (K ),
v EM K
so a priori the function TJ is a function on A;(K) . However, applying the Green function homogeneity property (5.38) from Proposition 5.58(d) and the product formula (Proposition 3.3), we see that
TJ (CX,cy)=
L v EM K
n v (9 ,v(x,y) + log Iclv) =
L vE MK
n v9 ,v(x,y) = TJ (x,y),
5.9. Green Functions and Local Heights
291
sc n gives a well-defined function on pi(K). Next we use the transformation property (5.36) from Proposition 5.58(b) to compute
".,(¢(P)) =
2:
n v9 ,v( (P )) =
vEMK
2:
n vd9 ,v(P ) = d".,(P).
vEM K
Similarly, we use the normalization property (5.37) from Proposition 5.58(b) to estimate
".,(P) =
2:
n v9 ,v(P )
vEMK
where h(P) is the usual height of the point P E pi (K) and the Ov(1) are the bounded functions appearing in (5.37). We further observe that Proposition 5.58(c) says that we may take Ov(1) = 0 for all but finitely many v E M K . More precisely, we may take Ov(1) = 0 for all v satisfying
(i) v E M~,
(ii) 11llv
= 1,
and
(iii) IRes( oo r, exists and is called the
Remark 5.65. A point
radius of~a ,r . We claim that the radiu s of a Type-IV point is strictly positive. To see this, suppose that ~a, r has radiu s O. Then the sequence of points a 1, a2, . . . is a Cauchy sequence in Cp, so it converges to a point a E Cpo Let 6i =
j nf
[z -
al
zED(ai,r;)
be the distance from a to the i th disk. Notice that 0 ~ 61 ~ 62 ~ 63 ~ ... , since the disks form a decreasing sequence. On the other hand, 6i ~ la - ai I -7 0 as i -7 00. Hence 6i = 0 for all i, so a E b i«; ri ) for all i and the intersection is nonempty, contradicting the assumption that ~a,r is a Type-IV point. 5.10.1.2
The Berkovich Disk as a Set of Seminorms
The description of the Berkovich disk DB as a union of points of Types I, II, III, and IV is very concrete, but it can be awkward to apply, since one must deal with four different kinds of points ." There is an alternative description of DB as a collection of seminorms on the ring C p [z] that is sometimes easier to use and that also naturally generalizes to other rings and other spaces. Definition. Fix R
II . IIRon Cp[z] is defined by
> O. The Gauss norm
The maximum modulus principle (Theorem 5.13(a)) tells us that the Gauss norm is equal to the sup norm,
IlfliR=
~up
If(z)l ·
zE D (O,R )
Definition. A (nontrivial)
II . IIR-bounded seminorm on Cp[z] is a nonconstant map
with the following properties: 1.
If I 2: 0 for all f
E
Cp[z].
2. Ifgl = If I·Igl for all I, g E Cp[z] . 40ne can unify the four types of points by defining all of them as equivalence classes of nested sequences of closed disks. See Exercise 5.40.
5.10. Dynamics on Berkovich Space
297
3. If + gl :S max{l f l, Ig l} for all I. s E Cp[z]. 4. If I :S Il fli R for all f E Cp[z]. (This is the boundedness condition.) Thus a seminorm has all of the properties of an absolute value except that there may be nonzero elements f E Cp[z] with If I = o.
Definition. The (closed unit) Berkovich disk fjB is the set fjB
=
{ II . IiI-bounded seminorms on Cp[z]}.
More generally, the closed Berkovich disk ofradius R is the set
fj~
= {II . Ilwbounded seminorms on Cp[z ]}.
The definition of fjB as a set of bounded seminorms is quite unintuitive, but its utility becomes clear when one examines Table 5.2 and sees how each of the four types of Berkovich points naturally defines a seminorm on Cp[z]. A fundamental theorem of Berkovich, whose proof we omit, says that every bounded seminorm on Cp[z ], and more generally on certain power series rings containing Cp[z], comes from one of the four types of Berkovich points. (See [64, page 18] or [26, Proposition 1.1].) Notice that the sem inorm I . 10 1 corresponding to the Berkovich point ~0, 1 is exactl y the Gauss norm Iflo ,1 = Ilf ll; , so following Baker and Rumely, we call ~0,1 the Gauss point.
I Point I Type-I
~a , O
Types-II & III
~a ,r
Type-IV
~a ,r
Seminorm
Ifl a,o = If( a)1 Ifla ,r = sup If(z)1 zE D(a,r) Ifl a,r = lim If lan,rn n~ oo
Table 5.2: Seminorms on 1 The open and closed branches in 1/. 0
11\0 ,1
-
and
r
< 1} U {~oo .o} .
A\"oo are defined using the natural
identification
1/. 0
.--.. 11\00 '
For example, a basic open neighborhood of the Gauss point ~0 ,1 is obtained by removing from jp'13 a finite number of closed branches, some of which may be in the vertical branch A\"oo at infinity.
5.10. Dynamics on Berkovich Space
303
Extra branch "at infinity"
Figure 5.5: The Berkovich projective line
jp'B.
Remark 5.70. There is a natural embedding of AP into jp'B. However, the inversion map I (z) = 1/z used to glue together the two pieces of jp'B can cause notational confusion regarding the "radius" of points of Types II and III. For example, the point ~p-2 ,p-3 in the branch /r..:a of jp'B would be denoted by ~p-l ,p when viewed as a point in A B . Thus it might be wiser to denote the points in /r..:a using some
alternative notation, for example
i.; but we will not do so.
Remark 5.71. A useful alternative construction oflp>B mimics the construction of the scheme jp'i as the set of homogeneous prime ideals. It starts with the set of bounded seminorms on the two-variable polynomial ring Cp[x, y] that extend the usual absolute value on C p and that do not vanish on the maximal ideal (x, y). Two seminorms I 11 and I . 12 are considered equivalent if there is a constant c > 0 such that I/iI = cdegfl/b for all homogeneous IE Cp[X, y].
Then jp'B is the set of equivalence classes of such seminorms. For details of this construction, see [66]. Let P E jp'I (C p ) . To create a seminorm from P, choose homogeneous coordinates P = [a, b] and set III = I/(a, b) I. Notice that a different choice of homogeneous coordinates for P gives an equivalent seminorm. This embeds jp'I (C p ) into jp'B.
5.10.2.3
Properties of AB and jp'B
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
Theorem 5.72. (a) The Berkovich disks D~ are compact, Hausdorff, and uniquely path connected. 5 (b) The Berkovich affine line A B is locally compact, Hausdorff, and uniquely path connected. (c) The Berkovich projective line JlDB is compact, Hausdorff, and uniquely path connected. D
Proof See [26] and [64].
Remark 5.73. As noted earlier, the Berkovich affine line A B contains a copy of Al(C p) = Cp, since each a E C p gives an associated Type-I point ~a,O in D~ provided R :::: [c]. Similarly, the Berkovich projective line JlDB contains a copy of the classical projective line JlDl (C p) via the map
a
f------+
{
~a ,o
E
.0(0,1)
iflal::; 1,
~a,O E
1r..:O ~oo,o E 1r..:O
if1
= d - Mel>
and
5.10. Dynamics on Berkovich Space
307
(Recall that a probability measure is a nonnegative measure of total mass 1.) We call fJ the canonical measure associated to ¢, since the property ¢* fJ = d . fJ resembles the analogous property h (¢( P)) = d . h (P) of the canonical height. The reader should be aware that other common names for fJ in the literature include Brolin measure, Lyubich measure, and invariant measure. For the construction and applications of fJ, see Baker and Rumely [26, Theorem 7.14], [29], ChambertLoir [98], Thuillier [434], and Favre and Rivera-Letelier [168], as well as [360].
Theorem 5.78. Let ¢(z) E Cp(z) be a rational map ofdegree at least 2. The support ofthe canonical measure fJ is equal to the Julia set :JB (¢). In particular, the Berkovich Julia set :JB (¢) is not empty.
Proof This theorem is an amalgamation ofresults due to Baker, Rumely, and RiveraLetelier. We refer the reader to [26, Section 7.5] for the construction ofthe canonical measure and to [26, Theorem 7.18], [27, Theorems 8.9 and A.7], and [381] for the proof that fJ is supported exactly on :JB (¢). The last part of the theorem is then 0 clear, since the empty set cannot provide the support for a nontrivial measure.
Example 5.79. Let ¢(z) E Cp(z) be a rational map of degree at least 2 and suppose that ¢ has good reduction. We know (Theorem 2.17) that the classical Julia set :J(¢) C jp'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., fJ(U)
=1
if ~O,l E U
and
fJ(U)
=0
if ~O,I ~ U.
Thus :JB(¢) = {~O, I }, 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 ¢f of ¢ have bad reduction for every f E PGL 2(Cp ) , then :J(¢) is a perfect set, and hence uncountable. (See Theorem 5.82.)
5.10.3.3
The Map ¢(z) = z2 on Berkovich Space
To conclude our brief foray into Berkovich space, we illustrate Berkovich dynamics by studying the simplest possible map, namely ¢( z) = z2. For any a E C p, we expand
¢(z) - ¢(a) = z2 - a2 = 2a(z - a) + (z - a)2. Assuming henceforth that p 2: 3 and using our convention that all points in jp'B have radius satisfying r ::; 1, we find that with
s = max{12alr, r 2 } = r- maxj ]«], r}.
(5.46)
This explicitly gives the action of ¢ on points of Types I, II, and III in jjB, 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 lal < 1 and r < 1. Then IcPn (a) I ----+ 0, so the radius of cPn (~a ,r) also goes to O. Thus [o]
. (a) Prove that there is a disk D (P,r ) centered at P such that ¢>- I (D (P,r ») consists of d disjoint open sets
Show that the Vi are disks. (b) Let ¢>i : Vi ---+ D(P, r) be the restriction of ¢> to Vi for each 1 :::; i :::; d. Prove that ¢>i is bijective. (c) Possibly after reducing r , prove that the inverse maps ¢>; I : D (?, r) ---+ Vi are given by convergent power series. (If 00 E ¢> -1 (P) , either change coordinates or instead use a convergent Laurent series.) 5.6. (a) Let K be an algebraically closed field that is complete with respect to a (nonarchimedean) absolute value and let cP : IF'1(K) ---+ IP'1(K) be a rational map. Prove that ¢> is an open map, i.e., the image of an open set is an open set. (b) Consider the map ¢> : IF'1(Q3) ---+ IF'1 (Q3) defined by ¢>( z ) = Z2 . Prove that ¢> is not an open map by showing that the image of the open disk D(O , 1) in IF'1 (Q3) is not open. Thus rational maps over complete, but not algebraicall y closed, fields need not be open.
Exercises
313
5.7. Let ¢(z) E Cp[z] be analytic on D(a, r) and assume that ¢(a) -I- o. Prove that l/¢(z) is analytic on some closed disk centered at a. More precisely, prove that l/¢(z) is analytic on D(a, t) for any t < rl¢(a)I/II¢II.
Section 5.3. Newton Polygons and the Maximum Modulus Principle 5.8. This exercise generalizes Proposition 5.16(b) by asking you to prove a general Cauchy estimate for p-adic analytic functions. Let ¢(z) E Cp[z] be a power series that converges on oi«, r) and let ¢(D(a, r)) = D(¢(a), s). Prove that for all n (Note that as n increases, the estimate becomes worse, since
;::>:
In!1
1. ---+
0 as n
---+
00.)
Section 5.4. The Nonarchimedean Julia and Fatou Sets 5.9. Prove the implications between uniform Lipschitz, uniform continuity, and equicontinuity stated in Section 5.4. More generally, give a definition of what it means for a family of functions to be equi-Lipschitz and prove the following implications:
I uniformly Lipschitz I
===}
I equi-Lipschitz at every point I
I uniformly continuous I ===} I equicontinuous at every point I 5.10. This exercise develops the abstract theory of residues (on pl) in order to prove Theorem 1.14 in arbitrary characteristic. Let ¢( z) E K (z) and Q E K. The function ¢( z) can be written as a partial Laurent series
a-N ¢ (z ) = (Z-Q )N
a-N+l a-l ( ) + (Z-Q )N-l +...+ -(--) Z-Q + 'IjJ z
and we define the residue of ¢ at
Q
with 'IjJ(z) E K[z - Q],
to be
Res(¢(z)dz) z =o:
=
a-l.
The residue of ¢ at 00 is defined by using the substitution z = l/w; thus Res (¢(z)dz) Z=CXl
= Res(-¢(w- 1)w- 2dw). w=o
(a) Compute all of the nonzero residues of each of the following functions. In each case, check that the sum of the residues is O. 2 Z2 2 2· (i) ¢t(z) = Z : 2 (ii) ¢2(Z) = 4 z - z z - z 5 z(b) If ¢( z) has a simple pole at Q, prove that the residue of ¢ at Q is the value of the function (z - Q) ¢( z) evaluated at z = Q. (c) More generally, for any integer n ;::>: 0, let 'ljJn (z) be the function
2
~
3 2:
n
1 d ( 1/Jn () Z = n! dz n (z - Q)n+l) ¢(z).
Exercises
314
Here we are taking formal derivatives in K(z). Note that in characteristic p, one must be careful to "cancel" the n! before setting p = O. (If you have not seen this kind of computation before, rewrite the expression for 'l/Jn (z) to make it clear that it makes sense in any characteristic.) Suppose that ¢(z) has a pole of exact order n 2: 1 at 0:, i.e., writing ¢(z) = F(z)/G(z) as a ratio of polynomials with no common factors, we have n = ordz=a (G(z)). Prove that Res(¢(z) dz) = 'l/Jn-l(O:). z=o:
(d) Prove that
{p E jp'l(K): Res(¢(z)dz) -# O} P is a finite set. (e) Assume that K is algebraically closed and prove the Cauchy residue formula
L
~s(¢(z)dz)=O.
PElI'l(K)
(This is a hard exercise. Try it first under the assumption that ¢(z) has simple poles and ¢( 00) -# 00, which suffices for (f).) (f) Use (e) to prove that Theorem 1.14 is true in arbitrary characteristic, and use it to deduce that Corollary 5.19 and Proposition 5.20(c) are true. Hence Fatou sets are nonempty for all algebraically closed nonarchimedean fields. 5.11. (a) Let p 2: 3 be a prime and let eE Cp satisfy lei> 1. Prove that the function ¢(z) = z2 + c has exactly one nonrepelling periodic point, namely the totally ramified fixed point at infinity. (b) * Suppose that ¢(z) E Cp(z) has an indifferent periodic point, i.e., a periodic point P whose multiplier satisfies lAp(¢) I = 1. Prove that ¢(z) has infinitely many indifferent periodic points. (This result is due to Rivera-Letelier [372].) (c)
** Do there exist rational maps ¢(z) E Cp(z) that have at least two, but only finitely many, nonrepelling periodic points? (From (b), the nonrepelling periodic points would have to be attracting.)
Section 5.5. The Dynamics of (Z2 - z)/p 5.12. Let S be a finite set, let SM be the space of sequences on S with the symbolic dynamics metric, and let L : SM ----> SM be the left shift map. (See Section 5.5.1.) (a) Let 0:, f3 E SM and suppose that limn~oo p( Ln(o:),L n (f3 )) = O. Prove that there exists an m such that Lm(o:) = L m (f3 ). (b) Prove that the periodic points of L are dense in SM. (c) Prove that there is an element 'Y E SM with the property that the orbit
Ch("() = {C("() : n E N} is dense in SM. One says that L is topologically transitive on SM. (Hint. Create 'Y by first listing all possible blocks oflength I, then all possible blocks oflength 2, and so on.) (d) Let 0: E SM. Prove that the backward orbit Vi (0:)
= Un 2:0 L-n(o:) is dense in SM.
Exercises
315
(e) More precisely, prove that the backward orbit of any following sense: For all (3 E SM and all k 2': 0,
0:
E SM is equidistributed in the
1 (#S)k· (In fact, prove that the limit stabilizes as soon as n. > k.) 5.13. Let ¢(z) = (Z2 - z)/p with p 2': 3. Use Exercise 5.12 and the identification provided by Proposition 5.24 to prove that the backward orbit of a point a E :J(¢) is equidistributed in the following sense: For all b E :J(¢) and all radii r = p-\
5.14. Let p 2': 3 and for any c E C p , let ¢c(z) = (Z2 - cz)/p. (a) If c E Z;, prove that the statement of Corollary 5.25 is true for ¢c. (b) More generally, prove an analogous statement if c is a unit in a finite extension of Zp. (c) What happens if c E pZp? 5.15. Let d
2':
2 be an integer, let p
¢(z)
> d be a prime, and consider the dynamics of the map
= _z(,---z_-_l,--,)(_z_----'2)'---._.·---,-(z_-_d+----,-1) p
over C p . Prove that :J (¢) can be described using symbolic dynamics and use this identification to prove thefollowing generalization of Corollary 5.25: :J(¢) c IQip, the set Per n ( ¢) consists ofthe fixed point at CXJ and dn repelling points, the Julia set :J(¢) is the closure ofthe repelling periodic points, and ¢ is topologically transitive on :J (¢). Section 5.6. A Nonarchimedean Version of Montel's Theorem 5.16. Complete the proof of Theorem 5.26 (see page 265) by writing down the details in the case that 0: = O. 5.17. Let be a collection of power series that converge on D( a, r), and suppose that there is a point 0: E K such that 0: ~ ¢( D(a, r)) for all ¢ E . We proved in Theorem 5.26 that there is a constant C = C ( 0:, a, r) such that
p(¢(z), ¢(w)) S Cp(z, w)
for all ¢ E and all z, wE D(a, r).
Find an explicit value for the Lipschitz constant C in terms of 0:, a, and r, 5.18. Let be a collection of rational, or more generally meromorphic, functions
D(a,r)
-->
jp'l(K),
and suppose that 0:, (3 E jp'l (K) do notlie in ¢ (D(a, r)) for all ¢ E . Our proofofthe nonarchimedean Montel theorem (Theorem 5.27) shows that there is a constant C = C( 0:, (3, a, r) such that
p(¢(z), ¢(w)) S Cp(z, w)
for all ¢ E and all z, wE D(a, r).
Find an explicit value of C in terms of 0:, (3, a, and r.
Exercises
316 Section 5.7. Periodic Points and the Julia Set
5.19. As in the statement of Lemma 5.34, let A, B C C p be bounded sets for which there are constants 0 < 8 < 1 ::; ~ such that sup lal ::;~,
sup 1)31 ::;~,
"EA
(3EB
For each (a,)3) E A x B, let L",(3(z) showed that
p(L",(3(z),L"',(3I(Z'))
jp'l be a rational map of degree at least 2 and let P E jp'l (iC p ) be a recurrent critical point that is in the Fatou set F( ¢). Prove that P is a periodic point.
318
Exercises
Section 5.9. Green Functions and Local Heights 5.27. Let K be a field with a nonarchimedean absolute value v, let ¢ : pI -+ pI be a morphism of degree d 2:: 2, let 1> = (F, G) : A? -+ A? be a lift of ¢ satisfying 111> II v = 1, and let Y4> be the associated Green function. If Res(F, G) v =I- 1, or equivalently if ¢ has
I
I
bad reduction at ¢, prove that there exists a point (x, y) E A; (K) such that
This is the converse to Proposition 5.58(c). 5.28. Let K be a field with an absolute value v, let ¢ : pI gree d 2:: 2, and let 1> : A 2 -+ A2 be a lift of ¢. (a) Prove that the map
g(x, y)
= logll1>(x, y) Ilv -
-+
pI be a morphism of de-
dlogll (x, y) Ilv
induces a well-defined function 9 : pI (K) -+ R (b) Prove that 9 is Lipschitz, i.e., prove that there is a constant C
= C( ¢) such that
Ig(p)-g(Q)1 ~Cpv(P,Q) forallP,QEpl(K). (c) If v is nonarchimedean, prove that 9 is locally constant. More precisely, if the lift 1> is chosen to satisfy 111>llv = 1, prove that
g(P)
= g(Q)
forall P,Q E pl(K) with Pv(P,Q)
(d) Define a modified Green function
< IRes (1)) Iv'
94> by
Prove that 94> is a Holder continuous function on pI (K). In other words, prove that Q4> is well-defined on pI (K) and that there are positive constants C and (P) is given by the telescoping series 'L':=o d- n9 (¢n (P) ). For small n, say n < N, estimate the difference I9 (¢n(P)) - 9 (¢n (Q)) I using (b) and the fact that ¢ is Lipschitz, and for large n use an elementary bound. Then make an appropriate choice for N.) 5.29. The definition of the canonical height h¢(P) as the limit of d:" h( ¢n(p)) is not practical for numerical calculations, even for P E pI (Q), because one would need to compute the exact value of points ¢n (P) whose coordinates have O( dn) digits. A better method to compute h¢(P) is as the sum of the Green functions Y4>,v(P) as described in Theorem 5.59. Let ¢ : pI -+ pI be a morphism of degree d 2:: 2 and fix a lift 1> of ¢. If v E MK is a nonarchimedean absolute value such that 111>llv = 1 and IRes(1))Iv = 1, then Proposition 5.58(c) says that Y4>,v(x,y) = logll(x,y)llv. This covers all but finitely many absolute values, so it remains to devise an efficient method to compute Y4>,v (x, y) in the remaining cases.
Exercises
319
Let P = [x, y] E jp'1(K) and use the algorithm described in Figure 5.6 to define a sequence of triples with iu, Xi, Yi E K. This exercise asks you to prove that N iterations ofthe algorithm gives the value of 9, v (x, y) to within O(d- N ) . (a) Prove that IUilv -:; 1 for all i 2: O. Hence the computation of quantities such as F(ui, 1) in the algorithm do not involve excessively large numbers. (b) Prove that for all i 2: 1 we have
(c) As in Exercise 5.28, we let
g(X, Y)
= logll(X,Y)llv
- dlogll(X, Y)t
The homogeneity of shows that 9 is a well-defined function on jp'1(K), and Proposition 5.57 says that 9 is bounded. Prove that
9(x,y)
= logll(x,Y)llv +
:f= ~ig(i-1(X,y)). i=l
(d) Prove that N
9(x,y)
= logll(x,Y)llv +
L ~i
logmax{lxilv, IYilv} + O(d-
N
(5.48)
),
i=l
where the big-O constant depends only on . Deduce that the algorithm described in Figure 5.6 computes 9 (x, y) to within O(d- N ) . (e) Find an explicit value for the big-O constant in (5.48) in terms ofthe quantities d, 11llv,
I
and Res( 1 (K).
Section 5.10. Berkovich Space and Dynamics 5.33. Let ¢(z) E Cp[z] be a polynomial and let a E D(O, 1). Write ¢(z) as a polynomial in z - a, say
Prove that
¢(D(a,r))
=
D(¢(a),s),
where the radius s of the image disk is given by s=r.max{lcII, IC2Ir, IC31r2, ... , ICdl r d}.
(Hint. We already proved that ¢ (D(a, r)) is a disk; see Proposition 5.16. Now use the maximum 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 . Ibe a nonconstant seminorm. Prove that I I = 0 and 111 = 1. (b) Suppose that I . I is an II . Ilwbounded seminorm. Let f(z) = Cbe a constant polynomial. Prove that if I = [c] is the usual absolute value on Cpo (c) Suppose that we replace property (3) of bounded seminorm with the usual triangle inequality If + gl < If I + Igl· 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 II 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 TypesII, 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) = zd has no Type-IVfixed points in AB. Does ¢(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) =
f(D(a,r))
= {Z-I: Iz-al
II z. Prove that
::;r} =D(a- I,rllaI 2 ) .
5.38. Let d
¢(z) =
Lcd
E
Cp[z]
be a polynomial with ICil ::; 1 and ICdl = 1.
i=O
The polynomial ¢ induces a map ¢ : DB --+ DB on the Berkovich disk. (a) Prove that the Gauss point ~O,l is fixed by ¢.
(5.49)
322
Exercises
(b) Let a E Cp be a fixed point of c/> : Cp --+ Cpo Prove that ~a,O is fixed by c/>.
°
I
(c) Let a E Cp be a neutral fixed point, i.e., Ic/>'(a) = 1, and let ~ r ~ 1. Prove that ~a,r is fixed by c/>. (In other words, if a is a neutral fixed point, then the entire line segment La,o C DB is fixed by c/>.) (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?
5.39. Let c/>(z) = L: CiZi E Cp[z] be a polynomial as in (5.49). Prove directly that the Julia set of c/> is .JB(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,
n
D(ai, r.) is nonempty and a is a point in the (a) Let r = lim rio If the intersection intersection, prove that Ifla,r = Ifla,r. Thus every point in DB, 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,l) ::::J D(a1, ri ) ::::J D(a2, r2) ::::J ... D(O,l)::::J D(b1,Sl)::::J D(b2,S2)::::J""
,
are defined to be equivalent if the following two conditions are true: • For every i • For every i
21 thereexistsaj 21 suchthatD(bj,sj) 21 there exists aj 2 1 such that D(aj,rj)
C C
D(ai,ri)' D(bi,Si)'
Prove that two sequences of disks are equivalent if and only if their seminorms I . la,r and I . Ib,. are equal.
5.41. Let ~a,r E DB be a Type-IV point. We defined the "line segment" running from ~a,r to the Gauss point ~0,1 to be the set 00
La ,r
= {~a,r} U U{~ai,t
:
r,
~ t ~ ri-d·
i=l
(Note that ro
= 1 by definition.) Let r = limi--->oo rio Prove that the map h: [r,l]
--->
La,r,
h(t) =
{~a,r
~ai,t
~ft =
r,
if r, ~ t ~ ri-1,
is a homeomorphism, where [r,l] C R has the usual topology and DB has the Gel'fond topology. Thus La,r is indeed a line segment.
323
Exercises
5.42. There is a natural metric on the Berkovich disk DB coming from the tree structure. We first identify the interval [0,1] with each of the line segments
La,o = {~a,r :
°.,:;
r .,:;
-B
I} ED,
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 1I;(~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 1I;(~a,r, ~b,s) = Ir - 81 is simply the distance between them on that line segment. (b) In general, prove that
1I;(~a,r,~b,s)
= max{lr - 81, 21b - al- r - 8}.
(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 DB : II;(~, ~o,I) < ~}
contains no Type-I points. Prove that every neighborhood of ~O,l in the Gel'fond topology contains infinitely many Type-I points. Deduce that the path metric II; does not define the Gel'fond topology, and indeed that II; is not even continuous in the Gel'fond topology! In Baker and Rumely's terminology, the set DB with the metric II; is called the small mode/. There is also a big model, in which the edges are reparameterized so that if bt«, r) C D(b, 8), then the distance from ~a,r to ~b,s is !log(r/ 8) 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.
I.
5.43. For points of Type I, II, and III in DB, the Hsia kernel is defined to be
(For points of Type IV, one takes the appropriate limit.) The Hsia kernel is used in studying potential theory on DB; see [26,209]. (a) Prove that 8(~a,r, ~a,r) = r. (b) Prove that 8(~a,o, ~b,O) = la - bl. Thus 8 extends the usual norm on !Cp , where we identify the unit disk in !Cp with the Type-I points in DB. (c) Let ~c,t be the point where the line segments La,r and Lb,s first meet. Prove that 8(~a,r, ~b,s)
= t.
(d) Prove that
8(~,C).,:; max{8(~,e'),8(e',(')} 5.44. There are natural inclusions -
-B
D(O, 1) '---t D , 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 metric on D(O, 1) and A 1 (!C p ) and the p-adic chordal metric on jp'l (!C p ) .
324
Exercises
5.45. Let ¢ be a rational map of degree at least 2, let P E
jp'B, and
let U C
jp'B be
a neighbor-
hood of P with the property that
¢(U)
~
U
and
n
¢n(U) = {P}.
n;:>O
Prove that P is a Type-I point and that, considered as a point in jp'l(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 polynom ials, and Lattes maps, whose complex dynamics were briefly discussed in Section 1.6. The underlying structure that they possess comes from an algebrai c group , namel y the multiplicative group for the power maps and Chebyshev polynomial s and elliptic curves for the Lattes maps. Althou gh such maps are special in many ways, they yet provide important examples , testing grounds, and boundary condition s for general results in dynamic s. In this chapter we investigate some of the algebraic and arithmetic propertie s of the rational maps associated to algebraic groups.
6.1
Power Maps 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 compositi on,
A more intrinsic description of the maps associated to the monic monomial s M d is that they are endomorphisms of the multiplicati ve group ,
More preci sely, there is an isomorphi sm of rings, 325
6. Dynamics Associated to Algebraic Groups
326
The fact that the M d are endomorphisms of the multiplicative group G m makes it quite easy to describe their preperiodic points.
Proposition 6.1. Let d E Z with map Md(Z) = zd. Then
PrePer(Md) = (Gm)tors
Idl 2:
2 and let M d
= {( E Gm : C
=
:
pI
-+
1for some n 2:
p I be the power
1} =
U J.Ln , n~ 1
where recall that J.Ln denotes the group ofnIh roots ofunity. Proof We proved this long ago for any abelian group G and homomorphism z t--+ zd with d 2: 2; see Proposition 0.3. The prooffor d :::;; - 2 is similar and left to the reader. 0 The iterates of Md are given by
so the periodic points of M d are also easy to characterize,
Proposition 6.2. Let Idl 2: 2 and let ( E Per~* (Md ) be a point of exact period n ;::: 2. Then the multiplier of Md at ( is given by
Proof Using M'd(z) = zd" , 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 d" 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 00. 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 a fi eld and let .ft!Id(z ) = zd be the d lh -power map for some Idl 2: 2. Further, if K has finit e characteristic p, assume that p f d. (a) The set ofrational maps that commute with Md(z) is given by
{¢(z ) E K (z ) : ¢ 0 Md = Md 0 ¢ } = {cz e : c E J.Ld-1 and e E
Z}.
6.1. PowerMaps and the Multiplicative Group
327
(b) The automorphism group ofM d is
Aut (Md) = {az : a E ltd-I} U {bz- I : bE Itd-d , where we recall that the automorphism group Aut (¢) ofany rational map ¢ is the set of f E PGL 2 (K ) satisfying ¢f = ¢. In particular, Aut (Md) is a dihedral group oforder 2n , where n is the number of (d - 1)51roots ofunity in K*. Proof (a) It is clear by a direct computation that the indicated maps cz" commute with Md(z ), so it suffices to prove that they are the only commuting maps. Suppose that ¢(z) E K (z ) commutes with Md(Z) , so
¢(zd) = ¢(z) d. Let ( E K be a primitive cfh root of unity. (This is where we use the assumption that p f d if K has positive characteristic p.) Then ¢((z)d = ¢(z)d, so for some cfh root of unity ( k .
¢( (z) = (k ¢ ( z)
Consider the function 'l/J (z) = z- k¢(z). It satisfies
'l/J((z)
= ((z)-k¢((z) = (( z)- k(k ¢(z) = z-k¢(z) = 'l/J(z) .
Hence 'l/J is a function of zd. In other words, there is a rational function 'l/JI(z) E K (z) such that 'l/J(z) = 'l/JI (zd), and thus ¢ (z) = zk'l/JI (zd). n More generally, for any n 2: 1 we have ¢ 0 j\;f :l = M:l 0 ¢ , so ¢( zd ) = n ¢(z) d . The above argument then yields an integer 0 :S kn < dn and a rational function 'l/Jn(z) E K (z) such that n ¢(z) = zkn'l/Jn(Zd ). We write 'l/Jn(z) = zj An(Z) for some integer j and some An(Z) E K (z) satisfying An(O ) =f. 0 and An(O ) =f. 00. Then deg(¢ (z))
n
= deg(zkn'l/Jn(zd
))
n n n = deg(zkn+j d An(zd )) ;::: deg(An( Zd
) )
= d" deg(An(z)).
Hence deg(A n) = 0 for sufficiently large n, which proves that ¢(z) has the form ¢(z) = cz" for some c E K* and some e E Z. With this information in hand, it remains only to observe that ¢(zd ) = ¢(z)d if and only if c = cd, i.e., if and only if either c = 0 or c E ltd- I . (b) By definition, Aut(¢ ) is the set of rational maps of degree 1 that commute with ¢ . It follows from (a) that Aut (¢ ) is the set of all CZ± I with c E ltd-I ' In particular, if we let fa(z) = az and g(z ) = Z- I , then
Aut (lVfd ) = {fa : a E Itd-d U {fa 0 9 : a E
ltd-I} '
and the dihedral nature of the group law is evident from the identities f a 0 f b = f ab ,
fa
0
9= 9
0
f a-I.
o
6. Dynamics Associated to Algebraic Groups
328
Example 6.4. We can use the map Md(z) = zd to illustrate the construction of dynamical units in Section 3.11. First we use Theorem 3.66, which says that if 0: has exact order n and gcd(i - j , n ) = 1, then 0:
di
- 0:
dj
d
0:
Taking j
. IS
- 0:
.
a umt.
= 0 and gcd( i , n) = 1, this implies that
:d-l di_ l
1
~1
is a unit for all primitive (dn - 1)st roots of unity
0:.
These are examples of classical cyclotomic units. Similarly, let m and n be positive integers with m n and n m . Then Theorem 3.68 says that if 0: is a primitive (dm - L)" root of unity and if (3 is a primitive (dn - 1)st root of unity, then 0: - (3 is a unit. These are again classical examples of cyclotomic units.
t
t
Example 6.5. Recall that a nontrivial twist of a rational map ¢( z) E K (z ) is a rational map 'ljJ(z ) E K (z) such that 'ljJ(z ) is PGL 2 (K )-conjugate to ¢( z), but 'ljJ(z ) is not PGL 2 (K )-conjugate to ¢(z ). See Section 4.9 for the general theory. Since the power maps Md(z ) = zd have a large automorphism group, they tend to have many twists. For example , the twists associated to the subgroup of Aut (¢ ) are given by (cf. Example 4.81)
K *j (K *)d- l
---+
{cz : c E J.Ld-l}
Twist (Md),
There are also some rather complicated-looking twists associated to the subgroup { Z , Z -l} C
Aut (Md ) .
Each bE K *j (K *)2 leads to a twist
O O. (c) Td(- w ) = (-l)dTd(w ). Thus Td is an oddfunction ifd is odd and it is an even function ifd is even. (d) The Chebyshev polynomials satisfy the recurrence relation
(e) For all d ;::: 1, the dth Chebyshev polynomial is given by the explicitformula
'~ "
T d ( w) =
( -1 )
d (d -k k)wd-2k.
kd _ k
(6.5)
0'5,k'5,d j2 Proof Suppose first that there do exist polynomials Td(w) satisfying (6.3). Then To(z + Z- I) = ZO+ z - o = 2, T 1 (z + Z- I ) = Z + Z- I, T 2(z
+ Z- I) = z2 + z-2 = (z + Z- I )2 -
2,
so we see that To(w) = 2, T 1 (w) = w, and T2(w) = w 2 - 2 are uniquely determined monic polynomials with integer coefficients. Still assuming that the set of Chebyshev polynomials exists, we next observe that they satisfy
(z
+ Z- I )T d+1 (z + Z- I) -
Td(Z + z - l )
= (z + z - l )(zd+l + z - d- l)
_
(zd + z -d)
+ z - d- 2 T d +2 ( Z + z - I ).
= z d+l =
Putting z + Z- 1 = w, this mean s that T d+2(W) = wTd+l(w) - Td(W). Hence if the Chebyshev polynomials exist, they are unique, because they are completely determined by the recurrence
330
6. Dynamics Associated to Algebraic Groups
To(w) =2,
T1(w)=w,
ford~O.
Td+2(W)=wTd+l(W)-Td(W)
and
(6.6) And from this recurrence we see immediately by induction that Td(w) is a monic polynomial of degree d in Z[w]. We now tum this argument around and use the recurrence (6.6) to define a sequence of polynomials Td(w). We claim that Td(w) then satisfies (6.3). This is clear for To and T 1, so we assume that (6.3) is true for To,T 1, ... ,Td+l and use (6.6) with w = z + z-l to compute
Td+ 2(Z + Z-l) = (z + Z-l )Td+l (z + z-l) - Td(Z + Z-l) = (z + Z-l)(zd+l + z-d-l) _ (zd + z-d) = zd+2 + z-d-2. Hence (6.3) is true for Td+2, so by induction we have for all d
~
O.
This proves that the recurrence defines polynomials satisfying (6.3), so Chebyshev polynomials of every degree exist. We have now shown that Chebyshev polynomials exist, are unique, and have the properties in (a) and (d). Next we make repeated use of (6.3) to compute
Td{Te(z+Z-l))
=
Td(ze+Z-e)
=
(ze)d+(ze)-d
=
zde+z-de
=
Tde(z+Z-l).
Hence Td{Te(w)) = Tde(W), which proves (b). To prove (c), we replace z by -z in (6.3) to obtain
Td(-(Z+Z-l)) =Td( -z+(-Z)-l) = (_z)d
+ (_z)-d = (_l)d(zd + z-d) = (-l)dTd(z + Z-l).
Therefore Td(-w) = (-l)dTd(w), which is (c). Next we prove the explicit summation formula (6.5) given in (e). Substituting d = 1 and d = 2 into the formula yields the correct values T 1 ( w) = wand T2(w) = w2 - 2. We now assume that the formula is correct up to Td+ 1 and use the recurrence (6.4) to check it for d + 2. Thus
Td+2(W)
=
wTd+l(w) - Td(w)
=w
""'
r: O~k~(d+l)/2
d+l-k
(d+l-k) wd+l- 2k k
d_(d k
'r:
(_l)k d+l
""' (_l)k_ s: d-k
O~k~d/2
""'
(_l)k d+l
s:
d+l-k
O~k~(d+l)/2
_
""' L...i
(_l)k-l
1~k~d/2+l
(d+l-k)wd+2-2k k
d (d-k+l)wd-2k+2 d-k+l k-l
6.2. Chebyshev Polynomials
T2 = W 2 T3 = w
3
T4 = w
4
T5 = w 5
331
-
2
-
3w
-
4w 2 + 2
-
5w3 + 5w
+ 9w2 - 2 T7 = w 7 - 7w 5 + l4w 3 - 7w T s = w S - 8w6 + 20w4 - l6w 2 + 2 T 9 = w 9 - 9w 7 + 27w5 - 30w3 + 9w T lO = w lO - lOws + 35w6 - 50w4 + 25w2 T ll = Wll - llw 9 + 44w 7 - 77w 5 + 55w3 T6 = w 6
T 12 =
W
6w 4
-
12
-
l2w
10
+ 54w
S
-
112w
6
+ l05w
2
llw 4
-
36w2 + 2
Table 6.1: The first few Chebyshev polynomials.
~ ~
(d + l) (d + 1 - k) + d (d + 1 - k) k
(_l)k
k- 1
d+l-k
O:s;k:S;d/2+l
wd+2-2k '
where we use the standard convention that (;:,) = 0 if n < m or if m < O. A simple algebraic calculation that we leave for the reader (Exercise 6.4) shows that
(d + l)
(d+ k1 - k) + d (d +k _1 -1 k)
=
d+l-k
d + 2 (d + 2 - k) . d+2-k k
Hence
T
() =
d+2 W
~ ~
O:s;k:S;(d+2)/2 which completes the proof of (e).
(_l)k
d
+2 d+2_ k
(d + k2 - k) w d+2-2k , o
Remark 6.7. As mentioned in Section 1.6.2, the classical normalization for the Chebyshev polynomials is
or equivalently,
332
6. Dynamics Associated to Algebraic Groups
Td ( cos B) =
cos(dB)
for all B E R
The two normalizations are related by the simple formula i; (w) = ~ Td(2w ). We have chosen the alternative normalizationbecause it has better arithmeticproperties. In particular, the map Td : pI -> pI has good reduction at all primes. The classical normalization i; has bad reduction at 2. As with the power maps, it is not difficult to describe the periodic points of the Chebyshev polynomials and to compute their multipliers. We state the result and leave the computationas an exercise.
Proposition 6.8. Let T d(w) be the dth Chebyshev polynomial for some d 2: 2. (a) Thefixedpoints ofTd in AI(C) are
27rj ) d+ 1} { ( 27rj ) { 2 cos ( d + 1 : 0 S; j S; -2- U 2 cos d _ 1 : 0 < j
d - 1} . < -2-
(b) The multipliers ofTd at itsfixed points are given by
(2 (}:jl)) =-d AT (2 (d2: d
ATd
cos
d
cos
j
1) )
=
d+l
forO
< j < -2-'
forO
< j < -2-'
d-l
ATd(±2) = d2 . (Note that -2 E Fix(Td) if and only if d is odd.) In general, the periodic points and multipliers ofTd can be derived from the above formulas using T d = Tdn andPern(Td) = Fix(Td).
o
Proof See Exercise 6.5.
We now prove an analogue of Proposition6.3 for Chebyshevpolynomials.
Theorem 6.9. Let K be afield and let Td(W) be the dth Chebyshev polynomial for some d 2: 2. Further, if K has finite characteristic p, assume that p f d. (a) The automorphism group ofTd is given by
Aut(T = d)
{I
J.L2
if d is even, if d is odd.
(b) Assume that K does not have characteristic 2. Let ¢( w) E K (w) be a rational map that commutes with Td(w), i.e., ¢(Td(w)) = Td(¢(w)). Then ¢(w)
= ±Te(w)
The minus sign is allowed stronger result.)
for some e 2: 1.
if and only if d
is odd. (See Theorem 6.79 for a
6.2. Chebyshev Polynomials
333
Proof (a) The assertion that Aut(Td ) C JL2 is an immediate consequence of (b), since (b) implies that any f E Aut(Td) satisfies f (w) = ±T1 (w) = ±w. However, since the proof of (b) is somewhat intricate, we give a direct and elementary proof of (a). Suppose that f E PGL 2 (K ) satisfies Tl = Td . The polynomial Td has a unique totally ramified fixed point at 00 (cf. Exercise 6.8), and Tl similarly has a unique totally ramified fixed point at f- 1(00 ), so the equality Tl = Td tells us that f- I (00) = 00. Hence f (w) = aw + b is an affine transformation. (The same argument applies to any polynomial not of the form awd .) Proposition 6.6(c) says that Td(w) satisfies Td(- w) = (- I )dTd(w), so in particular, T d ( w) = wd + (terms of degree at most d - 2). (6.7) The identity Tl (w)
= Td (w) with f(w) = aw + b can be written as
We evaluate both sides using (6.7) and look at the top degree terms. This gives
adwd + dad- Ibw d- 1 + (terms of degree at most d - 2) = awd
+ (terms of degree at most d -
2).
Hence and By assumption, d¥-O in the field K , so we conclude that ad- 1 = 1 and b = O. In order to pin down the value of a, we use the explicit formula for Td(w) given in Proposition 6.6(e). In fact, we need only the top two terms,
Td(W ) = wd - dwd- 2 + (terms of degree at most d - 4). By assumption we have Td(aw) = aTd(W ), so
Hence ad = a and ad- 2 = a, where we again use the assumption that d¥-O in the field K. It follows that a2 = 1. Further, a = -1 is possible only if (_I)d = -1, so when d is odd. This completes the proof that Aut(Td) is trivial if d is even and is equal to JL2 if d is odd. (b) It is easy to verify that the Chebyshev polynomial Td(W ) cannot be conjugated to a polynomial of the form cw d (Exercise 6.8). It follows that any rational map commuting with Td(w) is necessarily a polynomial, a fact whose proofwe defer until later in this chapter; see Theorem 6.80. We now describe a proof due to Bertram [69] that the only polynomials commuting with T d are ±Te . We begin with two lemmas. The first characterizes the Chebyshev polynomials as the solutions of a nonlinear differential equation, and the second explains how to exploit such equations.
6. Dynamics Associated to Algebraic Groups
334
Lemma 6.10. Assume that K does not have characteristic 2. Let d 2 1 and let F (w) be a polynomialsolution to the differentialequation (6.8)
Then F (w) = ±Td(W). Proof We first check that ±Td(w) are solutions. We differentiate the functional equation (6.2) defining the Chebyshev polynomials to obtain the identity T~ ( z
+ Z- I )(1 -
Z- 2) = dzd-I - dz-d- I,
and then solve for T~,
Putting w
= z + Z-I
as usual and noting that w 2
-
4 = (z d
Z- 1) 2 ,
we compute
_d) 2
(4 - w2)T~(wf = (4 - (z + Z- 1)2)d2 ( Zz ~ ZZ_ l = _d2(zd _ z- d)2 = d2(4_ (zd + z - d)2)
2(4- Td(w f) ·
= d
This proves that ±Td(W) are solutions to (6.8). Next suppose that F (w) is any polynomial solution to (6.8). If F' (w) is identically 0, then (6.8) implies that F (w) = ± 2 = ±To(w), so we are done. We may thus assume that F' (w) f:. O. We differentiate both sides of (6.8) and divide by 2F' (w) to obtain (6.9) (4 - w 2)F"(w) - wF' (w) + d2F (w) = O.
In particular, Td(w) is a solution to (6.9). Suppose now that F is any polynomial of degree k that is a solution to (6.9). We write F (w) = aw k + bw k - I +... with a f:. 0 and substitute into (6.9). The leading term is
so we must have k = d. In other words, we have shown that every nonzero polynomial solution of (6.9) has degree d. But F(w) - aTd(W) is a polynomial of degree strictly less than d that satisfies (6.9); hence F (w) = aTd(w). Finally, substituting w = 2 into (6.8) yields F (2) = ±2, while we know that Td (2) = Td(1 + 1- 1) = I d + I d = 2. Hence F (w) = ±Td(W) , which completes the proof of Lemma 6.10. 0
Lemma6.11. Let A(w) be a polynomial ofdegree r 2 1 and suppose that F (w) is a polynomial ofdegree d 2 2 satisfying
A (w)F' (wr = dr A(F(w)).
(6.10)
335
6.2. Chebyshev Polynomials
Suppose further that G (w) is a polynomial ofdegree e 2: 0 that commutes with F , i.e., F (G(w )) = G(F(w)). Then
A(w)G'(wY = eT A(G(w)). Proof Consider the polynomial
B (w) = A(w)G'(wY - e"A (G(w)). We assume that B (w) i- 0 and derive a contradiction, which will prove the desired result. First we observe that the leading coefficients of A(w )G' (w r and e"A (G(w)) cancel , so (strict inequality). degB < r e Next we use the various definitions and given relations to compute
dTB(F(w)) = dTA(F(w))G'(F(w)f - dT eTA(G(F(w))) = dTA(F(w))G'(F(w)f - dTe TA (F(G(w)))
definition of B, using F oG
= G 0 F,
= A(w )F' (wYG'(F (w)f - e"A(G(w)) F'(G(w)f
using (6.10) twice,
= A(w)( G 0 F )'(wy - e"A( G(w)) F' (G(w)f
chain rule,
G)'(wy - eTA(G(w)) F'( G(w)f = A(w )F' (G(w)f G' (wY - eTA (G(w))F'(G(w)f
using F o G
= A (w)(F
0
= G o F,
chain rule,
= F'(G (w)f [A(w)G'(wr - e"A (G(w))] = F'( G(w)f B (w )
definition of B.
Taking degrees of both sides gives
(deg B)(deg F) = r( deg F - l)(degG) + (deg B) , and the assumption that deg F
2: 2 means that we can solve for
deg B = r (deg G) = reo This contradicts the earlier strict inequality deg B < reoHence B must be the zero polynomial, which completes the proof of Lemma 6.11. 0 We now resume the proof of Theorem 6.9(b). Let ¢(w) be as in the statement (c) with ¢(w) a polynomial and let e = deg(¢). Lemma 6.10 tells us that
(4 - w2) T~(w)2 = d2(4 - Td(w)2). Hence we can apply Lemma 6.11 with A(w) mials ¢ and Td to deduce that
= 4 - w2 and the commuting polyno-
Then another application of Lemma 6.10 implies that ¢( w) = ±Te (w).
0
336
6. Dynamics Associated to Algebraic Groups
Using Theorem 6.9, it is easy to describe all of the twists of the Chebyshev polynomials. Corollary 6.12. Continuing with the notation and assumptions from Theorem 6.9, if d is even, then Td has no nontrivial K I K -twists, and if d is odd, then each a E K * y ields a twist 1 Td,a(W) = ..;aTd( yaw) . Two such twists Td,a and Td,b are K -conjugate
if and only if alb
is a square in K *.
Proof We use the description of Aut (¢ ) from (a). If d is even, then the automorphism group Aut( ¢ ) is trivial, so Proposition 4.73 says that ¢ has no nontrivial twists. For odd d we have Aut (¢) = {±z}, so the desired result follows from Example 4.81 (see also Example 4.75) . D Remark 6.13. Over C, there is a short proof that Aut(Td ) C /L2 using the fact (Exercise 1.31) that the Julia set of Td is J(Td) = [-2,2]. Then the assumption that Tl = Td implies that f maps the interval [- 2, 2] to itself. Since f is bijective on pI (C), it follows in particular that f permutes the endpoints of the interval [-2,2] . Hence f (2) = ±2 and f ( -2) = =f2. Writing f as f (w ) = aw + b, this gives two equations to solve for a and b, yielding b = 0 and a = ±1. Note that this proof does not carry over to characteristic p, since, for example , working over IFp we have Aut(Tp ) = PGL 2 (lFp ) ; see Exercise 6.10.
6.3 A Primer on Elliptic Curves The remainder ofthis chapter is devoted to rational maps associated to elliptic curves . In this section we give some basic definitions and review, without proof, some of the properties of elliptic curves that will be needed later. The reader should also review the summary of elliptic curves over C given in Section 1.6.3. For further reading on elliptic curves and for the proofs omitted in this section, see for example [96, 198, 248,250,254,257,410,412,420].
6.3.1 Elliptic Curves and Weierstrass Equations Definition. An elliptic curve E over a field K (of characteristic different from 2 and 3) is described by a Weierstrass equation, which is an equation of the form 2 E : y = x 3 + ax
+b
(6.11)
with a, b E K and 4a3 + 27b2 =1= O. Of course, we really mean that E is the projective curve obtained by homogenizing equation (6.11), so E has one extra point "at infinity," which we denote by O . If K has characteristic 2 or 3, then equations of the form (6.11) are insufficient, and indeed they are always singular in characteristic 2, so one uses the generalized Weierstrass equation E: y2
+ al XY + a3Y =
x
3
+ a2x2 + a4X + a6.
(6.12)
6.3. A Primer on Elliptic Curves
337
Remark 6.14. Let E / K be an elliptic curve defined over a field K. When we write E, we mean the geometric points of E, i.e., the points in E(K) for some chosen algebraic closure of E. Ifwe want to refer to points defined over K, we always explicitly write E(K), and similarly we write E(L) for the points defined over some extension field L of K. More intrinsically, an elliptic curve is a pair (E, 0) consisting of a smooth algebraic curve E of genus 1 and a point 0 E E. For convenience we often call E an elliptic curve, with the understanding that there is a specified point o. We say that E is defined over afield K if the curve E is given by equations with K -coefficients and the point 0 is in E(K). Using the Riemann-Roch theorem, one can prove that every elliptic curve E / K can be embedded in J!D2 by a cubic equation of the form (6.12) with 0 mapping to the point at infinity. (See [410, III §3].) Then, if the characteristic of K is neither 2 nor 3, we can complete the square on the left and the cube on the right to obtain the simpler Weierstrass equation (6.11). In order to simplify our discussion, we will generally make this assumption. The discriminant t::.(E) and j-invariant j(E) of the elliptic curve E given by (6.11) are defined by the formulas
Proposition 6.15. (a) Let a, b E K and let E be the curve given by the Weierstrass equation (6.11). Then E is nonsingular; and thus is an elliptic curve, if and only
if t::.(E) =J O. (b) Two elliptic curves E and E' are isomorphic over Kifand only ifj(E) = j(E'). More precisely, E and E' are isomorphic if and only if there is a u E K* such that a' = u 4a and b' = u 6b
o
Proof See [410, III §1].
6.3.2 Geometry and the Group Law There is a natural group structure on the points of E that may be described as follows. Let L be any line in J!D2. Then counted with appropriate multiplicities, the cubic curve E and the line L intersect at three points, say
EnL = {P,Q,R}, where P, Q, and R need not be distinct. The group law on E is determined by the requirement that the sum of the points P, Q, R be equal to 0,
P+Q+R= O. The point 0 serves as the identity element of the group. The inverse of a point P, which we denote by - P, is the third point on the intersection of E with the line through P and O.
338
6. Dynamics Associated to Algebraic Groups
Theorem 6.16. Let E / K be an elliptic curve defined over a field K. (a) The addition law described above gives E = E(K) the structure ofan abelian group. (b) The group law is algebraic, in the sense that the addition and inversion maps, E xE
(P,Q)>-+P+Q I
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). Proof See [410, m §§2,3].
D
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 Z
E :y = x
3
+ ax + b,
and let PI = (Xl, Yl) and Pz = (xz, yz) be points on E. If Xl = Xz and Yl = -Yz, then PI + Pz = 0. Otherwise, define quantities
A = yz - Yl Xz - Xl'
Xz -
A = 3XI + a 2Yl '
l/
=
-xi
Xl
+ aXl + 2b
--=------2Yl '
Then Y = AX + v is the line through PI and P z , or tangent to E sum of PI and Pz is given by
if Xl =
Xz·
if PI = P z, and the
As a special case, the duplication formulafor P = (x, y) is x([2]P)
=
4 Z z x - 2ax - 8bx a 4x 3 4ax 4b
+
+
+
D
Proof See [410, m.2.3]
6.3.3 Divisors and Divisor Classes Definition. A divisor on E is a formal sum of points D
=
L np(P), PEE
339
6.3. A Primeron EllipticCurves
with np E Z and all but finitely many np = o. The set of divisors under addition forms the divisor group Div(E). The degree of a divisor d is deg(D)
=
L
n.p:
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)
----t
E,
L np(P)
f------t
PEE
L [np](P). PEE
(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(J) =
L ordp(J)(P), PEE
where ordp(J) is the order of zero of fat P if f(P) = 0, and ordp(J) is negative the order of the pole of fat P if f(P) = 00. A divisor of the form div(J) is called a principal divisor. The principal divisors form a subgroup ofDiv(E), and the quotient group is the Picard group Pic( E). Within Pic( E) is the important subgroup Pico (E) generated by divisors of degree o. 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 o. (b) A divisor D E Div(E) is principal if and only sum(D) = O. (c) The summation map induces a group isomorphism sum: Pico(E)
----t
if both
deg(D)
E.
Proof See [410, III.3A and 111.3.5].
6.3.4
o and
D
Isogenies, Endomorphisms, and Automorphisms
Definition. An isogeny between two elliptic curves E 1 and E 2 is a surjective morphism 'ljJ : E 1 ----+ E 2 satisfying 'ljJ( 0) = O. (Note that any nonconstant morphism E 1 ----+ E 2 is automatically finite and surjective.) The curves E 1 and E 2 are said to be isogenous if there is an isogeny between them. Remark 6.19. Every nonconstant morphism 'ljJ : E 1 ----+ E 2 is the composition of an isogeny and a translation (cf. [410, IIIA.7]). To see this, let ¢(P) = 'ljJ(P) - 'ljJ(0). Then the map ¢ : E ----+ E is a morphism, and ¢( 0) = 0, so ¢ is an isogeny. Hence
'ljJ(P) = ¢(P) +'ljJ(0) is the composition of an isogeny and a translation.
340
6. Dynamics Associated to Algebraic Groups
Remark 6.20. We observe that an isogeny is unramified at all points. This follows from the general Riemann-Hurwitz formula (Theorem 1.5) applied to the map
'IjJ : E 1 --+ E 2 , 2g(E 1) - 2 = (deg 'IjJ)(2g(E 2 )
-
2) +
I: (ep ('IjJ ) -
1).
P EEl
The elliptic curves E 1 and E 2 both have genus 1, and the ramification indices satisfy ep ('IjJ ) ::::: 1, so it follows that every ep('IjJ) is equal to 1, so 'IjJ is unramified. Theorem 6.21. An isogeny tb : E 1 --+ E 2 is a homomorphism ofgroups, i.e.,
'IjJ (P
+ Q) = 'IjJ (P ) + 'IjJ(Q)
for all P,Q E E 1(K) .
o
Proof See [410, 111.4.8]
The degree of an isogeny 'IjJ : E 1 --+ E 2 is the number of points in the inverse image 'IjJ - 1(Q) for any point Q E E 2 . This number is independent of the point Q, since, as noted earlier, 'IjJ is an unramified map. It is clear that if deg( 'IjJ ) > 1, then 'IjJ is not invertible, since it is not one-to-one. However, there does exist a dual isogeny that provides a kind of "inverse" for 'IjJ. Theorem 6.22. Let 1/J : E 1
isogeny ,¢ : E 2
--+
'¢('IjJ (P )) = [d] P
--+ E 2 be an isogeny ofdegree d. Then thereis a unique E 1 , called the dual isogeny of 'IjJ, with the propertythat
and
'IjJ( '¢(Q )) = [d] Q
for all PEEl andQ E E 2 .
o
Proof See [410, III §6].
Definition. Let E be an elliptic curve. The endomorphism ring of E, which is denoted by End (E), is the set ofisogenies from E to itself with addition and multiplication given by the rules
(In order to make End(E) into a ring, we also include the constant map that sends every point to 0.) The automorphism group of E, denoted by Aut(E), is the set of endomorphisms that have inverses. Equivalently, Aut(E) = End(E)* is the group of units in the ring End(E). Every integer m gives a multipli cation-by-in morphism in End(E). For m this is defined in the natural way as
>0
m tenus
[m] : E
----+
E,
[m] (P) = 'p + P ; . . . + P.
For m < 0 we set [m](P) = - [-m ](P), and of course [O](P) = O. This gives an embedding of Z into End (E ), and for most elliptic curves (in characteristic 0), there are no other endomorphisms.
6.3. A Primer on Elliptic Curves
341
Definition. An elliptic curve E is said to have complex multiplication if End (E) is strictl y larger than Z . The phrase "complex multipli cation" is often abbreviated byCM. Example 6.23. The elliptic curve E : y2 = x 3 + x has Clvl, since the endomorphism .ljJ : E
----1
E,
'ljJ( x, y) = (-x ,iy),
is not in Z . An easy way to verify this assertion is to note that
'ljJ2(X, y) = (x , -y) = - (x , y) , so tjJ2 = [-1]. This gives an embedding of the Gaussian integers Z [i] into End (E ) via the association m + ni f---+ [m] + [n] 0 'ljJ , and in fact it is not hard to show that End(E ) 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 = O. These are the curves
E a' .. Y2 -_ EI: : y2 =
+ ax , 3 x + b, X
3
j (E~ ) = 1728,
End(E~ )
= Z[i],
Aut (E~ ) = JL4 '
j (EI: ) = 0,
End (EI: )
= Z[p],
Au t (EI: ) = JL6 '
Here p = (- 1+A )/2denote s a cube root of unity and JL n is the group of nIh roots of unity. Of course , all of the E~ are isomorphic over an algebrai cally 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 / K be an ellip tic curve. Then the endomorp hism ring ofE is one ofthe following three kinds ofrings : (a) End(E) = Z. (b) End (E) is an order in a quadratic imaginaryfi eld F. This means that End (E ) is a subring of F and satisfies End (E ) ® iQl = F. In particular, End (E ) is a subring offinite index in the ring ofintegers ofF. (c) End(E) is a maximal order in a quaternion algebra. (This case can occur only if E is defined over a finite fi eld.)
o
Proof See [410, III §9]. The automorphisms of an elliptic curve are very easy to descr ibe.
Proposition 6.26. Let K be a fi eld whose characteristic is not equal to 2 or 3 and let E / K be an elliptic curve. Then
Au t (E)
=
{
JL2
if j (E ) # 0 and j (E ) #
JL4
ifj (E)
= 1728,
JL6
i/j (E )
=
O.
1728,
342
6. Dynamics Associated to Algebraic Groups
o
Proof See [410, III §1O].
Remark 6.27. It is easy to make the description of Aut(E ) in Proposition 6.26 completel y explicit. Assuming that E is given by a Weierstrass equation (6.11) as usual, for an appropriate choice of n there is an isomorphism
[.J: JL n
---+
Au t (E),
(6.13)
Here we take n = 4 if j(E) = 1728, we take n = 6 if j( E) = 0, and we take = 2 otherwise. Of course, for n = 2 and n = 4 the formula simplifies somewhat to (x, ~y) and (x, ~ - ly), respect ively.
n
6.3.5
Minimal Equations and Reduction Modulo p
Let K be a local field with ring of integers R, maximal ideal p, and residue field k RIp . As in Section 2.3, we write for the reduction of x modulo p.
x
=
Definition. Let ElK be an elliptic curve defined over a local field K. A minimal Weierstrass equation / or E is a Weierstrass equation whose discriminant 6.(E ) has minimal valuation subject to the condition that the coefficients of the Weierstrass equation are all in R.
Example 6.28. If k does not have characteristic 2 or 3, then a Weierstrass equation (6.14) for E is minimal if and only if
a,bE R
and
min{30rd p (a ),2 0rd p (b)}
< 12.
In general, if the residue field k does not have characteristic 2 or 3, then any Weierstrass equation (6.14) can be transformed into a minimal equation by a substitution of the form (x, y) I-t (u 2x , u 3 y) for an appropriate u E K *. If k has characteristic 2 or 3, 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. Definition. Fix a minimal Weierstrass equation for ElK. Then we can reduce the coefficients of E to obtain a (possibly singular) curve E lk. We say that E has good reduction if E is nonsingular, which is equivalent to the condition that 6.(E) E R* . In any case, we obtain a reduction modulo p map on points ,
E (K )
---+
E(k),
Pt-------->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, VU.2.1].
o
6.3. A Primer on Elliptic Curves
343
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 Weierstrass 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 elliptic 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 4.11.
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 'l/J E End(E) we write E['l/J] = Ker('l/J) = {p E E : 'l/J(P) = O}. Of particular importance is the kernel of the multiplication-by-m map,
E[m] = {P E E: [m]P = O}. The group E[m] is called the m-torsion subgroup ofE. The union of all E[m] is the torsion subgroup ofE,
E tors
=
U E[m].
m~l
Theorem 6.31. Let E / K be an elliptic curve and assume that either K has characteristic 0 or else that K has characteristic p > 0 and p f m. Then as an abstract group, E[m] = Z/mZ x Z/mZ.
In other words, E [m] is the product oftwo cyclic groups oforder m.
o
Proof See [410, III.6.4].
The next result gives conditions that ensure that the reduction modulo p map respects the m-torsion points. It may be compared with Theorem 2.21, 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 2: 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
E(K)[m]
-+
E(k)
is injective. In other words, distinct m-torsion points have distinct reductions modulo p.
344
6. Dynamics Associated to Algebraic Groups D
Proof See [410, VII.3.1].
Let E j K be an elliptic curve defined over the field K . Then the points in E [m ] are algebraic over K, so their coordinates generate algebraic extensions of K. An immediate corollary of the preceding theorem limits the possible ramification of these extensions .
Corollary 6.33. Let K be a local fie ld whose residue fi eld has characteristic p. let E j K be an elliptic curve with good reduction, and let m 2 1 be an integer with p m . Then the field K (E [m]) obtained by adjoining to K the coordinates ofthe m-torsion points ofE is unramified over K .
t
ProofSketch. Let K' = K (E [m ]), 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 ' j K ) is in the inertia group . Then CY fixes everything modulo p', so in particular,
CY(P) == P
(mo d p')
for all P E E[m].
(6.15)
But from Theorem 6.32, the reduction map E[m] --+ E (k' ) is injective, so (6.15) implies that CY(P ) = P for all P E E [m ]. The points in E [m ] generate K' j K, so CY fixes K' . Hence Gal (K' j K ) has trivial inertia group , so K ' j K is unramified . (For further details, see [410, VIlA I].) D
Remark 6.34. The coordinates of the points in E[m] are algebraic over K , so the absolute Galois group G K = Gal(K jK ) acts on E [m ] compatibly with the group structure. In this way we obtain a representation
In order to create a characteristic-O representation , we fix a prime f. and combine all of the f.-power torsion to form the Tate module
(Here Ze denotes the ring of f.-adic integers.) Then the f.-adic representation of E is the homomorphism
These representations are of fundamental importance in the study of the arithmetic properties of elliptic curves .
6.3.7
The Invariant Differential
Definition. Let E : y 2 = x 3 + ax + b be an elliptic curve given by a Weierstrass equation . The invariant differential on E (associated to the given Weierstrass equation) is the differential I-form
dx
WE
dy
= -2y = 3x 2 + a .
6.3. A Primer on Elliptic Curves
345
The next result explains why the invariant differential is so named and shows that it linearizes the group law in a useful way. Theorem 6.35. Let E be an elliptic curve given by a Weierstrass equation and let WE be the associated invariant differential on E. (a) For any given point Q E E, let TQ : E ----t E be the translation-by-Q map defined by TQ (P) = P + Q. The differentialform WE is translation-invariant in the sense that for every Q E E. TQ(WE) = WE
(b) The differential form WE is holomorphic at every point of E. (c) Up to multiplication by a nonzero constant, WE is the only holomorphic translation-invariant 1-form on E. (d) For every m E Z the differential form WE satisfies [m]*WE
= mWE·
o
Proof See [410, III §5].
The invariant differential can also be used to fix an embedding of the endomorphism ring of E into Co Of course, this is of interest only when E has Clvl, since there is only one way to embed Z into E corresponding to a E R, we observe that deg(a)
=#
Ker
(elL ~ elL) = (L : O'L).
If a E Z, then it is clear that (L : a L ) = a 2 , since L = Z 2 as an abstract group. Suppose now that a (j. Z. Then continuing with the earlier notation, the transformation formulas (6.18) imply that the index of eel. in L is (L : a L) = ad - be. On the other hand , the product a& is the constant term in the minimal equation (6.19) for a 0 over Q, hence also equal to ad - be. The theory of complex multiplication uses elliptic curves to describe the abelian extensions of a quadratic imaginary field F in a manner analogous to the description of abelian extensions of Q using torsion points in G m , i.e., using roots of unity. For a complete introduction to the theory of complex multiplication, see, for example, [257, Part II], [399 , Chapter 5], or [412, Chapter II]. We now describe the tiny piece of the theory that will be needed in Section 6.6 in order to prove Theorem 6.62.
6. Dynamics Associated to Algebraic Groups
350
Let F be a quadratic imaginary field, let RF be the ring of integers of F, and let IF be the group of fractional ideals of F. If we fix an embedding FcC, then each fractional ideal a E IF is a lattice a C C; hence it determines an elliptic curve En whose complex points are (6.20) We observe that En has complex multiplication by RF, since any a E R F has the property that ou C a, hence it induces a holomorphic map
[a] : Cia
1----+
CI a,
Z 1----+
az,
which in turn yields an isogeny [a] : En ----t En. In fact, since R F is the maximal order in F, we have End(En) = R F . We denote by £e£(R F ) the set
£e£(R F ) = {isomorphism classes of elliptic curves E with End(E) ~ R F
} .
We thus have a natural map
a 1----+ (isomorphism class of En).
(6.21)
We also observe that if we multiply a by a principal ideal, then the isomorphism class of En does not change, since for any e E K* there is an obvious isomorphism
Cia ~ C/ea,
Z
1----+
ez.
Hence the map (6.21) induces a natural map from the ideal class group CF to elliptic curves with complex multiplication by RF,
= IF I F*
Proposition 6.40. Let F be a quadratic imaginary field with ring of integers RF and ideal class group CF, and let h F = #C F be the class number ofF. Then with notation as above. the natural map
is a bijection. In particular, there are exactly h F isomorphism classes of elliptic curves whose endomorphism ring is RF.
Proof See [412, 11.1.2].
D
6.4 General Properties of Lattes Maps Chebyshev polynomials arise by restricting the power map z" to the quotient of JlDl by the finite group of automorphisms {z, z-l }. As already briefly described in Section 1.6.3, quotients of elliptic curves lead similarly to rational maps called Lattes maps. In this section we define and discuss general properties of these Lattes maps. For an excellent introduction to Lattes maps over C, including historical remarks and proofs of their basic geometric and analytic properties, see [300].
351
6.4. GeneralProperties of Lattes Maps
Definition. A rational map ¢ : pI ~ pI of degree d 2: 2 is called a Lattes map if there are an elliptic curve E, a morphism 7/J : E ~ E, and a finite separable' covering 7r : E ~ pI such that the following diagram is commutative:
(6.22)
pI
..«: pl.
Example 6.41. Let E : y2 = x 3 + ax + b be an elliptic curve. Then the classical formula for x(2P) (Proposition 6.17) and the isomorphism x : E / {±l} ~ pI yield the Lattes map
2 2 2ax - 8bx + a 4x 3 + 4ax + 4b Here 7/J is the duplication map 7/J(P) = [2]P, and the projection 7r is given by 7r(P) = 7r(x, y) = x. Example 6.42. Let E be the elliptic curve E : y2 = x 3 + ax with j (E) = 1728 and again let 7/J(P) = [2]P be the doubling map. Ifwe take 7r(x, y) = x, then we are in ¢(x)
= x(2P) = x
4
-
the b = 0 case of Example 6.41, and we obtain the Lattes map
¢(x)
x(2P)
=
=
(x 2 _ a)2 4 (2 )' X x +a
However, for this curve we may instead take 7r (x, y) map ¢1. We find a formula for ¢1 using the relation 2
¢l(X) =¢(JX) =
((x_a)2)2 4)X(x+a)
= x 2 . This gives a new Lattes
(x - a)4 16x(x + a)2'
Note that the map 7r(x, y) = x 2 corresponds to taking the quotient of E by its automorphism group Aut(E) ~ 1L4 via the association described in Remark 6.27.
Example 6.43. In a similar manner, the doubling map on the elliptic curve
E: y2
= x3 + 1
with j(E) = 0 and Aut(E) = 1L6 gives various Lattes maps corresponding to taking the quotient of E by the different subgroups of 1L6' Explicitly, the Lattes maps corresponding, respectively, to 1L2' 1L3' and 1L6 are
We leave the verification of these formulas to the reader; see Exercise 6.12. 1The assumption that 1r is separable is relevant only when one is working over a field of characteristic p, in which case it is equivalent to the assumption that 1r does not factor through the p-power Frobenius map.
352
6. Dynamics Associated to Algebraic Groups
We begin with an elementary, but useful, characterization of the preperiodic points of a Lattes map (cf. Proposition 1.42). Proposition 6.44. Let ¢ be a Lattes map associated to an elliptic curve E. Then PrePer(¢)
= Jr(Etors).
Proof Let ( E pi and let PEE be any point satisfying Jr(P) orbits of ( and P. Thus
= (. We consider the
Jr(01jJ(P)) = Jr ((~n(p): n ~ a}) = {Jr~n(p): n ~ o} =
{¢nJr(p): n ~ o} = {¢n((): n ~ o}
=
Oq,(().
The map Jr is finite, so this shows that 01jJ(P) is finite if and only if Oq,(() is finite. Hence PrePer(¢) = Jr(PrePer(~)), and it is left to prove that PrePer( ~) = Etors. We observe that the map ~ : E ~ E has the form ~(P) = ~o(P) + T for some ~o E End(E) and some point TEE. (See [410, III.4.7].) We are going to prove Proposition 6.44 in the case that ~ = ~o E End(E), i.e., assuming that T = O. For the general case, which requires knowing that the point T is a point of finite order, see Exercise 6.14. Suppose first that P E Etors, 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 nEnd(E)).
In other words, there is an endomorphism f3 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 Etors C PrePer( ~). Next suppose that P E PrePer(~), say ~i(p) = ~j(P) for some i > j. We rewrite this as (~i - ~j) (P) Theorem 6.22 to obtain
=0
-----
and apply the dual isogeny ~i - ~j described in
[deg(~i - ~j)](P)
= O.
We know that ~i -I ~j, since deg(~) = deg(¢) ~ 2 and i > j, so ~i - ~j has positive degree. This proves that P E E tors, which gives the other inclusion 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.
6.4. General Properties of Lattes Maps
353
Definition. Let n
from the commutativity of(6.22),
1r
= CritVal1>n U
n . This holds for all n 2: 1, which gives the inclusion 00
CritVal1r
:2
U ) = PostCrit1> . n=O
In order to prove the opposite inclusion , suppose that there exists a point Po E E satisfying Po E CritPt" and 7f( Po) t/: PostCrit1> . (6.23) Consider any point Q E 7j; - I (PO) . Then Q is a critical point of 7f'lj;, since 'Ij; is unramified and 7f is ramified at 7j; (Q) by assumption. But 7f'lj; = (() of ¢ at ( can be computed using the differential form dt via the equation
¢*( dt) dt
I t =(
=
d¢ (t ) dt
I t=(
= ¢'(() = A¢(( ).
(6.31)
360
6. Dynamics Associated to Algebraic Groups
Using the relation ¢ 0 x = x
x* ( ¢*(dt)) dt
'l/J and formula (6.30), we compute
0
= x*¢*( dt) = x*(dt)
(¢ox)*(dt) x*(dt )
= (x 0 'l/J )*(dt ) = 'l/J*x*(dt ) = 'l/J*(dx) = m y o 'l/J x*( dt)
x*(dt)
dx
(6.32)
y .
We lift ( to a point PEE satisfying x( P) = ( and evaluate both sides of (6.32) at P to obtain
¢*(dt) I = m (y 0 'l/J ) (P) . dt t=( Y
(6.33)
Equating (6.31) and (6.33) gives the useful formula
y o'l/J ) (P) A,p(() = m ( -y-
for ( = x(P ) E Fix(¢) with (
i
00 .
Assume first that [2]P i 0 , which ensures that y(P) i 0 and y(P ) i we can directly evaluate the fraction in (6.34) and conclude that
(6.34)
00.
Then
A (() = m (y 0 'l/J )(P) ¢ y(P) ' We are assuming that ( = x(P ) is a fixed point of ¢, so
x(P) = ¢(x( P)) = x('l/J(P)). Thus 'l/J(P) = ±P, and hence y('l/J(P )) = ± y(P), which proves that A,p(( ) = ± m. More precisely, A (() = m if mP = P and A¢(( ) = - m if mP = -Po To summarize, we have proven that if ( ~ x (E[2]), then
( E Fix(¢)
A (() = { ¢
m -m
if ( E x(E [m -1]) , if( E x (E [m + 1]).
(6.35)
Next suppose that y(P ) = 0, so [2 ]P = 0 , but P i O. We also have [m]P = ±P from the assumption that ( = x(P) E Fix(¢), so m is odd and 'l/J fixes P. The functions y and y 0 'l/J both vanish at P , so we can use l'Hopital's rule to compute
(y ~ 1jJ) (P) = (d(Yd~ 'l/J )) (P) , assuming that the righthand side has a finite value. (Note that this formula is valid algebraically, since what we are really doing is looking at the linear terms in the local expansions of y and y 0 'l/J at P.) We now use the chain rule to compute
6.5. Flexible Lattes Maps
d('lj;*y) dy
361
d('lj;*y) 'lj;*(3x2 + a) 3x2 + a 3x2 + a . dy 'lj;*(3x2 + a) 2+a). 2+a 3x ='lj;* ( dy ). 'lj;*(3x 2 2 3x + a 3x + a dy 2 'lj;*(w) 'lj;*(3x + a) -- . from (6.28), w 3x 2 + a 'lj;*(3x2 + a) =m from (6.29), 3x 2 + a 3(x o'lj;)2+ a - m --'-----,--'---3x 2 + a .
We evaluate both sides at P = ((, 0). Note that the quantity 3(2 + a is nonzero, since otherwise P would be a singular point of E. Also, x('lj;(P)) = x(P) = (. Hence
(
y 0 'lj;) (P) = d('lj;*y) (P) = m 3(x 0 'lj;(p))2 + a = m 3(2 y dy 3x(P)2 + a 3(2
+a +a
= m.
Substituting this value into (6.34) yields the desired result .\1' (() = m 2 . It remains to deal with the case P = O. There are several ways to do this case. First, we could perform an explicit calculation using local coordinates around O. Second, since we are missing only one multiplier, we could use Theorem 1.14, although this would require knowing a priori that>. i- 1. Third, at least for odd m, we could observe that 'lj;(P) = [m]P looks the same locally around each of the points in E[2], and we already computed>' = m 2 for the nonzero points in E[2]. We leave as an exercise for the reader (Exercise 6.20) to complete the proof using whichever argument he or she prefers. Finally, to compute the multiplier of a periodic point ( E Per., (¢), we apply the results that we have just derived to the fixed points of the Lattes map ¢n satisfying ¢n 0 X = X 0 [m n ]. 0 Remark 6.53. Let ¢ : ]lD1 ----+ ]lD1 be a flexible Lattes map associated to 'lj;(P) = [m]P as in Proposition 6.52. If we work over C, then the multiplier of every periodic point ( E Per(¢) satisfies
for some e
= e( () 2':
1.
Hence ( is repelling, so Per(¢) is contained in the Julia set (cf. Exercise 1.27). Choosing a lattice L and a complex uniformization C/ L ----+ E(C) as described in Section 1.6.3, it is clear that E(C)tors is dense in E(C). Therefore Per(¢) = x(E(C)tors) is dense in ]lD1(C) and is contained in .:J(¢). Further, the Julia set is closed. This proves that .:J(¢) = ]lD1(C) and F(¢) = 0, which is Lanes's Theorem 1.43 discussed in Section 1.6.3. On the other hand, if we work over a p-adic field such as Qp or C p , then every periodic point ( E Per(¢) satisfies for some e = e(() 2': 1.
362
6. Dynamics Associated to Algebraic Groups
Thus every periodic point is nonrepelling, so Per (¢ ) c F (¢) from Proposition 5.20. Further, ifplm, then I\ t> (() I = Iml e < 1, so in this case every point in Per (¢ ) is attracting.
Remark 6.54. Recall that the multiplier spectrum of a rational map ¢ : pI -; pI of degree d is the map that associates to each integer n ~ 1 the set
where we treat Per n (¢ ) as the set of dn + 1 (not necessarily distinct) fixed points of ¢n. Two maps with the same multiplier spectrum are called isospectral. (See Section 4.5, page 187.) Proposition 6.52(b) shows that flexible Lattes maps of degree m 2 are isospectral , since their multiplier spectrum depends only on m . A deep theorem of McMullen [294, §2] (Theorem 4.53) says that these are the only isospectral rational maps that vary in a continuous family. Not surprisingly, good reduction of Lattes maps is closely related to good reduction of the associated elliptic curve.
Proposition 6.55. Let K be a local fi eld, let R be the ring of integers of K, and be a flexible Lottes map of degree m 2 associated to an elliptic let ¢ : curve E / K . Suppose that E has good reduction and that m E R*. Then there exists an f E PGL 2 (K ) such that ¢i has good reduction.
Pk -; lPk
Proof Since E has good reduction , we can find a Weierstrass equation for E with coefficients in R and discriminant in R*. We then use Proposition 6.51 to replace ¢ by ¢i so that it fits into a Lattes diagram (6.26). In other words, the Lattes projection map 7f is the x-coordinate function on a minimal Weierstrass equation for E. For any n ~ 1 we can find polynomials Fn(X) ,Gn(X ) E R [X] such that
x([nJ P) = Fn{x(P)) .
c; (x(p))
This is easily proven by writing out the first few polynomials explicitly and then computing the subsequent ones by a recurrence formula. The recurrence also shows that the leading terms of Fn and G n are 2 n G nX=nX ()
and
2
1 -+ ... .
(See [96, page 133], [410, Exercise 111.3.7], or Exercise 6.23.) We note that the roots of Gn(X ) are the x-coordinates of the n -torsion points , and one can check that Gn(X ) factors as
Gn(X ) = n 2
II
(X - x(P) ).
P EE [n] , P ol-V
We are assuming that 'l/J(P ) = [mJ(P ) + T for some fixed integer m and some T E E[2J. For simplicity we prove here the case T = 0 and leave the general case as an exercise. Then
6.5. Flexible Lattes Maps
363
¢(x(P)) = x(1jJ(P)) = x ([m](P)) , We fix an auxiliary prime £ satisfying £ f m and £ E R* and consider the polynomial
H(X) =
II
(Fm(X) - x(Q)Gm(X)).
QEE[R],Qi-O
Notice that H(X) is a monic polynomial, since Fm(X) is monic and deg(Fm ) > deg(G m ) . We also note that all of the x(Q) are integral over R, since they are roots ofGR(X) and £ E R*. It follows that H(X) E R[X]. Further, Proposition 2.13(b) (see also Exercise 2.6) tells us that the resultant of H(X) and Gm(X) is
II II
Res(H(X), Gm(X)) = ±m2degH
H(()
Gm()=O
= ±m2degH
Fm(( / 2- 1
Gm()=O
=
Res(Fm(X),Gm(X))
R2~1
.
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 coordinates of the points of order m£ to K, let R' be the ring of integers of K', let 1" be the maximal ideal in R', and let k' = R' /1" be the residue field of R'. The extension 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.
Claim 6.56. The roots of H(X) are given by {roots of H(X)}
= x(E[m£] "E[m])
C
R'.
ProofofClaim. The roots of H(X) are the solutions to
for some Q E E[£]. Writing a root of H(X) as x(P) for some PEE, this means that
x([m]P)
=
Fm(x(P)) Gm(x(P))
=
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)) i- 0, so
e2
H(x(P)) = Fm(x(P)) -
I
i- O.
This gives the inclusion {roots of H(X)}
c x (E[m£] "E[m]).
The other inclusion is clear from the definition of H(X), since
P
E
E[m£] "E[m]
=?
[m]P E E[£]
jp'l, for which only the iterates of rjJ commute with rjJ. The inclusion Z c End(E) leads to the existence of Heeke 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 D to non-Lattes maps on jp'l. Corollary 6.64. For all integers n curves E / K we have
#(
>
1, all number fields K/Q, and all elliptic
U E(L)tors)::; B(n[K : Q])3,
(6.41 )
[L:K]:Sn
where B(D) is the constant appearing in Theorem 6.63. Proof To ease notation, we let D = [K : Q]. Every field L appearing in the union in (6.41) satisfies [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 E(L)tors C
[L:K]:Sn
U
E[b].
l:Sb:SB(nD)
Then using #E[b] = b2 yields
#(
U E(L)tors)::; L [L:K]:'On
l:'Ob:'OB(nD)
#E[b] =
L l:'Ob:'OB(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 Lanes map ¢, which may be surprising at first glance. However, it is easily explained by the fact that Lanes 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 that for all numberfields K /Q ofdegree D and all Lottes maps ¢ : pI -+ pI defined over K we have #PrePer(¢,pI(K))::; C(D). Proof Without loss of generality we fix a reduced Lanes diagram (6.37) for ¢. Then Proposition 6.26 says that the projection map 7r : E -+ pI has degree at most 6, and indeed if j (E) i- 0 and j (E) i- 1728, then deg( 7r) = 2. Proposition 6.44 tells us that PrePer(¢,pI)
= 7r(Etors),
so the fact that deg( 7r) ::; 6 yields PrePer(¢,pI(K)) C
U
7r(E(L)tors).
(6.42)
[L:K]::;6
Corollary 6.64 says that the set on the righthand side of (6.42) has size bounded D solely in terms of D, hence the same is true of # PrePer(¢, pl (K)). Example 6.66. The rational map x4
¢a,b(X) =
2ax 2 - 8bx + a 2 4x3 + 4ax + 4b
-
is the Lattes map associated to multiplication-by-2 on the elliptic curve
Ea,b : y2 = x 3 + ax + b. The j-invariant and discriminant of Ea,b are given by the usual formulas
j(Ea ) = 1728 4a 3
4a 3 + 27b2
and
Theorem 6.65 tells us that # PrePer( ¢a,b, pI (K)) is bounded solely in terms of the degree d = [K : Q]. In general, the best known bounds are exponential in d, 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 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
6.7. Uniform Bounds for Lattes Maps
371
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[3J), a field of degree at most 48 over K.) Then a result of Hindry-Silverman [204] impliesa bound slightly strongerthan
#E(K )tors ~ 22 1 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 : QJ) for all numberfields K impliesa bound of the form
U E(L)tors) < B(n[K : QJ) 3.
#(
[L:K) :=; n
Hence as in the proof of Theorem6.65 we have
U
PrePer(¢,lP'l(K)) C
7r(E(L)tors) ~ c(D log D)3
[L :K ]:=; 6
o
for an absolute constantc.
Theorem 6.65 proves uniformity for rational preperiodic points of Lattes maps. In the other direction,recall that we proved(Theorem3.43) that the orbitsof rational wandering points contain only finitely many integersexcept in a few preciselyspecified 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 uniformityestimates for the number of integerpoints lying in Lattes orbits. For simplicity we state results over Q and Z, but we note that an appropriately formulated versionis true for rings of S-integers in numberfields. Definition. Let E /Q be an elliptic curve. Recall that a global minimal Weierstrass equation for E is a Weierstrass equationthat is simultaneously minimalat all primes (Remark 6.30). We define a quasiminimal Weierstrass equation for E /Q to be an equation of the form E : y2
= x 3 + ax + b,
a,b E Z,
(6.43)
such that 14a3 + 27b2 j is as small as possible. Equivalently, the equation (6.43) is quasiminimal ifthere are no primes p such that p4ja and p6lb. Remark 6.68. Givenan arbitrary Weierstrass equation
E : y 2 = x3
+ ax + b,
a.b E Z,
it is easy to create a quasiminimal equation. Simply let u be the largest integer such that U 12 divides gcd(a3 , b2 ) , and then
6. Dynamics Associated to Algebraic Groups
372
is a quasiminimal equation for E /Q. Elliptic curves over Q have global minimal Weierstrass equations, see Remark 6.30, and it is not hard to show that a quasiminimal equation is minimal at every prime p ::::: 5, and that it is almost minimal at 2 and 3; see Exercise 6.26. The following theorem is a conditional resolution of a conjecture of Lang [254, page 140].
Theorem 6.69. (Hindry-Silverman) Let E /Q be an elliptic curve given by a quasiminimal Weierstrass equation and let E(Z) be the set of points in E(Q) having integer coordinates. Also let v(E) be the number ofprimes dividing the denominator ofthe j-invariant of E. (a) There is an absolute constant C such that for any subgroup r c E(Q),
# (r n E(Z)) < cv(E)+rankr. (b)
If the "ABC conjecture" is true.' then there is an absolute constant C such that for any subgroup r c E(Q), # (r n E(Z)) < c
rankr.
Proof The proof of (a) is given in [407] and the proof of (b) is in [202].
D
We can use Theorem 6.69 to prove a uniform bound for integer points in orbits of flexible Lattes maps (cf. Conjecture 3.47).
Theorem 6.70. Let E / Q be an elliptic curve given by a Weierstrass equation with integer coefficients, let m ::::: 2 be an integer, and let ¢(z) E Q(z) be the Lattes map satisfying for all PEE. ¢(x(P)) = x([m]P) Assume further that ¢ is affine minimal in the sense that
Res(¢) =
min
Res(¢!).
(6.44)
!EPGL2(iQ) !(z)=az+b
(See page 112 for the definition ofthe resultant Res(¢) ofa rational map.) Thus the assumption (6.44) says that we cannot reduce the resultant of ¢ by conjugation by an affine linear transformation j(z) = az + b. Let ( E Q and consider the orbit O¢(() of( by ¢. (a) There is an absolute constant C such that
where v( E) is the number ofprimes dividing the denominator ofthe j-invariant
ofE. 2The ABC conjecture of Masser and Oesterle says that if A, B, C integers satisfying A + B = C, then C «€ TIplABC pH€.
> 0 are pairwise relatively prime
6.7. Uniform Bounds for Lattes Maps
(b)
373
If the ABC conjecture
is true, then the number of integer po ints in 0 4> (() is bounded by an absolute constant independent ofE and (.
Proof Write the given Weierstrass equation for E as with a , b « Z . We begin by showing that the minimality assumption (6.44) implies that there are no primes p with p21a and p3lb. The rational function ¢ (x ) = F (x )/G (x ) is associated to the rnultiplication-by-rn 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
¢ (x ) = x
4
2 2 2ax - 8bx + a 4x 3 + 4ax + 4b
-
Hence ifp2 a andp3 lb, then conjugating ¢ (x ) by f (x ) = px yields 1
¢ f (x ) =p_1F(a,b;px ) G(a, b;px )
= F (p- 2a ,p- 3b;X) . G(p- 2a, p- 3b;x )
The assumption that p2ja and p31b implies that these polynomials have integer coefficients, and then homogeneity yields
Res(¢f ) = Res(F(p- 2a , p- 3b;x), G(p-2 a , p- 3b;x )) 2
2_1
2
2
= p_ m (m
) Res(F(a ,b;x) ,G(a,b ;x))
= p_m (m _ 1) Res (¢ ). This contradicts (6.44), so we have proven that there are no primes p satisfying p 2 1a andp 3 lb. We would like to apply Theorem 6.69 to the rank-I subgroup generated by a point P = ((, TJ) of E lying above (. Unfortunately, although ( E Q, there is no reason that TJ 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 = (3 + a( + b is rational. We write
J
then we factor with u 1 squarefree, and consider the elliptic curve
6. Dynamics Associated to Algebraic Groups
374
(6.45) (In the terminology of Section 4.7, E' is a twist of E; cf. Example 4.71.) Notice that the point P{ = (UIXI,uiv l) 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 or Since
UI
ordp(b) < 3.
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-l subgroup generated by P' to conclude that
#{ n 2:
1 : [n]P' E E'(Z)} :S
ClI(E')+I.
(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 iQ! (JUl). This isomorphism, which we denote by F, is given explicitly by
E : y2 = x 3 + ax (x,y)
+b
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 and
P~ = [n]P{ = (x~, y~).
Since the isomorphism F respects multiplication by n, we have
In particular, since P{ E E' (iQ!), itfollows that[n]P{ = P~ E E'(iQ!), so F maps the multiples of PI, which, note, are not in E(iQ!), to points in E'(iQ!). Further, if X n E Z, then it is clear from the definition of F that x~ = UIX n E Z, and hence y~ is also in Z, since we just showed that y~ is in iQ! and the equation of E' shows that y~ is the square root of an integer. To summarize, we have proven that Xn
E
Z
=}
P~ =
F(Pn ) E E'(Z).
(6.47)
6.8. Affine Morphisms and Commuting Families
375
= x([mk]p ) = Xmk, and hence
By con struction we have qi (()
#(()(p)nz) = #{ k ?O : Xmk EZ } < # { k ? 0 : P;"k E E'(Z)}
from (6.47),
:s: #{n ? 0 : P~ E E' (Z )} < cv (E' )+l
from (6.46),
= cv (E )+l
since E and E' are isomorphic.
Further, if the ABC conjecture is true, then we may replace the upper bound by C.
o Remark 6.71. 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 # (() (() n Z ) arbitrarily large by replacing ¢(z) with B¢(B- l z) . This conjugation has the effect of multiplying every point in the orbit by B, hence allow s 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 ko such that
¢k(() E Z
for
O :S: k
:s: k o
and
¢k(( ) ¢:c Z
for
k
> k o.
This reflects the more general fact that if x( [n] P) E Z and if r ln, then x( [r]P) E Z ([4 10, Exerc ise 9.12]). Note that no such cutoff statement holds for general rational maps that are not non-Lattes map s.
6.8
Affine Morphisms, Algebraic Groups, and Commuting Families of Rational Maps
Power maps and Chebyshev maps are attached to endomorphisms of the multiplicative group lG m and its quotient lG m I {z = z - l }, and similarly Lanes maps are attached to maps of quotients of elliptic curves. In this section we put these constructions 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 Ie be a connected commutative algebraic group of dimension g. Then its universal cover is e 9 and every affine morphism 'ljJ : G --+ G lifts to an affine map 9 --+ 9 , i.e., there are a matrix A and vector a such that the following diagram commutes:
e
e9
z ....... A z + a )
e9
1
1
G
G
e
376
6. Dynamics Associated to Algebraic Groups
Example 6.74. Every affine morphism of the multiplicative group G m has the form 'ljJ(z) = az d for some nonzero a and some dE Z. More generally, for any commutative group G, any a E G, and any dE Z there is an affine morphism 'ljJ(z) = az", Notice that it is easy to compute the iterates of this map,
Proposition 6.75. Let ib : G ----7 G be an affine morphism ofan algebraic group G, so 'ljJ has theform 'ljJ(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 'ljJ. (b) Let a and a be as in (a). Then the iterates of i] have the form
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 'ljJ(z) = aa(z). Evaluating at the identity element e E G yields 'ljJ(e) = aa(e) = a, so a is uniquely determined by 'ljJ. Then a(z) = a-I'ljJ(z) is also uniquely determined by 'ljJ. This proves (a). The proof of (b) is an easy induction, using the commutativity of G and 0 the fact that a is a homomorphism.
Definition. A self-morphism of an algebraic variety ¢ : V ----7 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 I' c Aut (G), and a morphism G If ----7 V that identifies G If with a Zariski dense open subset of V (possibly all of V) such that the following diagram is commutative: G
7jJ ------7
1 cit
1
1
------7
1
V
G
ott
(6.48)
1 1
q, ------7
V
Example 6.76. Examples of dynamically affine rational maps ¢ : pI ----7 pI include the power maps ¢( z) = z" with G = Gm and I' = {I}, the Chebyshev polynomials Tn(z) with G = G m and I' = {z, Z-I}, and Lattes maps with G an elliptic curve E and f 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 I' c Aut( G) be the associated quantities fitting into the commutative diagram (6.48). (a) For every ~ E I' there exists a unique E I' with the property that 'ljJo~ = o'ljJ. (b) Write'ljJ(z) = a . a(z) with a E G and a E End(G) as in Proposition 6.75. Then ~(a) = afor every ~ E f.
e
e
6.8. Affine Morphisms and Commuting Families
377
(c) Assume that #f ::::: 2 and that C is simple. (An algebraic group is simple if its only connected algebraic subgroups are {I} and C.) Then a E C tars, i.e., the translation used to define 'Ij; is translation by a point offinite order. Proof (a) The uniqueness is clear, since if'lj; 0 ~ = 6 o'lj; = 6 0 sb, then 6 = 6 because the finite map 'Ij; : C ----+ C is surjective. We now prove the existence. The commutativity of (6.48) tells us that for all z E C and all ~ E I',
(n
0
'Ij; 0 O(z) = (¢ 0 n
0
~)(z) =
(¢ 0 n)(z) = (n 0 'Ij;)(z).
Thus ('Ij;o~)(z) and 'Ij;(z) have the same imagefor the projection map n : C ----+ C If, so there is an automorphism E f satisfying
e
'Ij;(~(z))
= (('Ij;(z)).
e,
We claim that the automorphism which a priori might depend on both ~ and z, is in fact independent of z. To see this we fix ~ and write 'Ij; (~( z)) = ~~ ( 'Ij; (z)) to indicate the possible dependence of on z. In this way we obtain a map (of sets)
e
C
------t
I',
Since I' is finite, there exists some C E f such that ~~ = C for a Zariski dense subset of z E C. (Note that a variety cannot be a finite union of Zariski closed proper subsets.) It follows that 'Ij; 0 ~ is equal to Co 'Ij; on a Zariski dense subset of C, and hence they are equal on all of C. (b) From (a) we see that there is a permutation off defined by the rule T :
I'
------t
I',
Evaluating both sides of'lj; 0 ~ = T(~) o'lj; at the identity element 1 E C and using the fact that ~(1) = 1 and 'Ij;(1) = a· a(l) = a, we find that
a = 'Ij;(~(1)) = T(O('Ij;(l)) = T(O(a). But T is a permutation of I', so as ~ runs over I', so does T (0. Hence a is fixed by every element off. (c) From (b) and the assumption that #f ::::: 2, there exists a nontrivial ~ E I' with ~(a) = a. It follows that a is in the kernel of the endomorphism
C
------t
C,
The kernel is not all of C, since ~ is not the identity map, so the simplicity of C tells 0 us that the kernel is a finite subgroup of C. Hence a has finite order. Remark 6.78. In this book we are primarily interested in dynamically affine maps of pI, but higher-dimensional analogues, especially of Lanes maps, have also been studied. See for example [68, 134, 145,439].
6. Dynamics Associated to Algebraic Groups
378
The commutativity of (6.48) implies that deg(¢) = deg(7jJ). It follows that all dynamically affine maps for the additive group G a have degree 1, since every affine morphism ofG a has the form 7jJ(z) = az + b. Hence nonlinear dynamically affine maps on pI are attached to either the multiplicative group G m 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.' More precisely, the multiplicative group has endomorphism ring End(G m ) = 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 ¢ 0 7jJ = 7jJ 0 ¢. Then one ofthe following two conditions is true: (a) There are integers m, n 2: 1 such that ¢n = ib": (b) Both ¢ and 7jJ are dynamically affine maps, hence they are either power maps, Chebyshev polynomials, or Lottes maps. In all cases, the commuting maps ¢ and 7jJ satisfy
F(¢)
= F(7jJ),
:J(¢) = :J(7jJ),
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 7jJ. 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 Exercise 1.15. D
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 polynomial 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 ¢(z) E K[z] be a polynomial ofdegree d 2: 2, and let 7jJ(z) E K(z) be a nonconstant rational map. We assume that both ¢ and 7jJ are separable, i.e., neither ofthe derivatives ¢' (z) and 7jJ' (z) is identically O. Suppose further that ¢ and 7jJ commute under composition, ¢ 0 7jJ = 7jJ 0 ¢. Then one of the following is true: (a) 7jJ(z) E K[z], i.e., 7jJ is also apolynomial. 3Evenin characteristicp, most elliptic curves have commutative endomorphism ring. However, there are a finitenumberof ellipticcurveswhoseendomorphism ring is a maximalorder in a quatemionalgebra. These supersingular curves are all definedover IFp2. See [410, V §3].
379
6.8. Affine Morphisms and Commuting Families
(b) After simultaneous conjugation by an affine map f (z) z + (3. the polynomial ¢(z) has the form ¢(z) = azd and the rational map 'ljJ(z) has the form 'ljJ(z) = bz" for some r < O.
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 00 is a totally ramified fixed point of ¢. Suppose that 'ljJ(z) is not a polynomial. This means that we can find a point a E 'ljJ-l(00) with a=/=- 00. We use the commutativity of ¢ and 'ljJ to compute ea(¢n
0
'ljJ)
= ea('ljJ)
n-l rr eqhj;(a)(¢) = ea('ljJ)eoo(¢t = ea('ljJ)dn i=O
(6.49)
II ea('ljJ Hence
0
¢n) =
Cg
e¢i(a)(¢)) e¢n(a) ('ljJ).
nrr-l e¢i(a)(¢) __ i=O
d
ea('ljJ) > _1_ e¢n (a) ('ljJ) - deg 'ljJ
for all n :::: 1.
(6.50)
Every ramification index e¢i(a) (¢) is an integer between 1 and d, so letting n --+ 00, we see that e¢i(a)(¢) = d for all sufficiently large i. On the other hand, ¢ is a polynomial and a =/=- 00, so ¢i (a) =/=- 00 for all i. Hence there is at least one point (3 =/=- 00 with e{3 (¢) = d. The Riemann-Hurwitz formula then implies that (3 and 00 are the only two points at which ¢ is ramified. It follows that ¢i (a) = (3 for all sufficiently large i, which implies that ¢((3) = (3. In other words, (3 and 00 are both totally ramified fixed points of ¢, i.e., (6.51 )
and
and ¢ has no other ramification points. In particular, since by construction we have ¢i(a) = (3 for some i, it follows that a = (3. But a =/=-00 was an arbitrary point in 'ljJ-l(00), so we have also proven that 'ljJ-l(00) C {(3, oo}. Next let, E 'ljJ-l((3). We use the fact that ¢i'ljJ(r) = (3 and ee (¢) = d to repeat the calculation (6.49) with a replaced by v. This again leads to the inequality (6.50), but with, in place of a, and hence to the conclusion that ¢ is totally ramified at some iterate ¢i (,). It follows that ¢i (,) E {(3, 00} for some i, and hence from (6.51) that, E {(3,00}. We have now proven that
'ljJ-l({(3,OO}) C {(3,00}. Thus 'ljJ is totally ramified at (3 and 00, and since 'ljJ((3) =/=- 00 by assumption, the map 'ljJ must switch (3 and 00. Since we also know that (3 and 00 are totally ramified fixed points of ¢, it follows that ¢ and 'ljJ have the form
¢(z) = (3 + a(z - (3)d Then conjugation by f(z)
and
'ljJ(z) = (3 + b(z - (3y
= z + (3 puts them into the desired
for some r form.
< O. D
380
Exercises
Exercises Section 6.1. Power Maps and the Multiplicative Group
6.1. (a) (b) (c)
Let K be a field of positivecharacteristic p. Let Mp(z) = z", Prove that the automorphism group of Mp over K equals PGL 2(lFp ) . More generally, if q is a power of p, prove that Aut(Mq ) = PGL2(lFq). Again let q be a power of p, and let d be an integerwith p f d. Describe Aut(Mq d ) .
6.2. Let K be an algebraically closed field, let d E Z, and let a E K*. Further, if K has positive characteristic p, assume that p f d. Describe all rational functions ¢(z) E K (z) that commutewith az d under composition. 6.3. ** Let Md(z) = zd be a power map for some Idl 2: 2, and if K has positivecharacteristic p, assume that p f d. Example6.5 describestwo types of twists of Md(Z). The first type has the form ¢a(Z) = az d and the second type 'l/Jb(Z) is given by the complicatedformula (6.1). Does Md(z) have any other twists? If so, describe all of the twists of Md(z). Section 6.2. Chebyshev Polynomials
6.4. Completethe proof of Proposition6.6(e) by verifyingthe identity 1 [( d+ 1) (d + 1 - k) + d(d + 1 - k)] = d+ 2 (d + 2 - k) . d+l-k k k-l d+2-k k
6.5. Let Td(w) be the dth Chebyshevpolynomial for some d 2: 2. (a) Prove that the fixed points ofTd( w) are as describedin Proposition6.8(a). (b) Prove that the multipliers of Td(W) at its fixed points are as described in Proposition 6.8(b). (Hint. For (a) use the trigonometric identity cos(A) _ cos(B) = sin ( B ; A) sin ( B ; A) , and for (b) differentiate Td(z
+ z ~ 1) = Zd + Z-d /
Zd - Z-d
-1
Td(z
to obtain the identity
+Z )=d z _
Z-1 .)
6.6. Proposition 6.8 describes the multipliers of the fixed points of the Chebyshev polynomial Td(w). Prove directly that the multipliers satisfy the summation formula described in Theorem 1.14,
L
(EFix(Td)
= 1.
1 1-
ATd (()
6.7. Let K be a field of characteristic p 2: 3, let n 2: 1 be an integer with p f n, and let I-t n C K* be the nth roots of unity. There is no "cosine function" for tile field K, but we can define a set of cosine values by Cos.,
=
a + a2 {
We also let 2 Cos; = {2( : ( E Cosn } . Let d 2: 2 be an integer with p f d(d2 mial.
-
1
: a E I-t n
}
.
1) and let Td(w) be the a;h Chebyshevpolyno-
381
Exercises (a) Prove that Fix(Td)
= 2 COSd+l U2 COSd-1 .
Also compute the intersection 2 COSd+1 n2 COSd-l. (b) Prove that the multipliers of Td at its fixed points are given by if ( E 2 COSd+1 and ( if ( E 2 COSd-1 and (
f ±2, f ±2,
if( = ±2. (c) Give a similar description of the fixed points and their multipliers in the case that d == ±1 (modp). 6.8. We stated during the proof of Theorem 6.9 that for d ~ 2, the Chebyshev polynomial Td(W) is not equivalent to a monomial, i.e., no conjugate (i-I 0 Td 0 f)(w) has the form cu/". Prove this assertion. 6.9. Prove that the (formal) derivatives of the Chebyshev polynomials satisfy the following identities:
+ dwTd(W) = 2dTd-dw). (4 - w2)T~f(W) - wT~(w) + d2Td(W) = O.
(a) (4 - w2)T~(w)
(b)
6.10. Let K be a field of positive characteristic p. (a) Prove that the pth Chebyshev polynomial Tp(w) is equal to w P in K[w]. q) (b) In general, if q is a power ofp, prove that Tqd(W) = Td(wF = Td(w for all d ~ 1. (c) Again letting q be a power of p, deduce that Aut(Tq) = PGL2 (IF q). (Cf. Exercise 6.1.) 6.11. Let K be a field of characteristic 2 and let d ~ 1 be an odd integer. Prove that wT~ (w) = Td(w). What is the derivative T~ (w) if d is an even integer?
Section 6.4. tattes Maps - General Properties 6.12. Let E be the elliptic curve E : y2 = x 3 + 1 with j(E) = 0, so Aut(E) = 1-t6 is cyclic of order 6. Let 1/J(P) = [2]P be the doubling map. (a) Let 7f : E --> E / 1-t2 ~ pl. Prove that we can take 7f( X, y) = x and that the Lattes map corresponding to 1/J is
=
tih(z)
Z(Z3 - 8b) 4(z3 + b) .
(b) Let 7f : E --> E / 1-t3 ~ pl. Prove that we can take corresponding to 1/J is A,
(
)
=
Z4
,+,2 Z
7f (
+ 18bz2 8z 3
x, y) = y and that the Lattes map
2 27b
(c) Let 7f : E --> E/l-t6 ~ pl. Prove that we can take 7f(x,y) = x 3 and that the Lattes map corresponding to 1/J is
z(z - 8b)3 cP3(Z)
= 64(z + b)3
Compute the conjugate cP3 (z - b) + b of cP3 (z), compare it to cP2 (z), and explain. 6.13. Let cP be a Lattes map. Prove that there does not exist a linear fractional transformation f E PGL 2 such that the conjugate cP f is a polynomial. (Cf. Exercise 6.8.)
382
Exercises
6.14. Complete the proof of Proposition 6.44 in the general case that 'l/J(P ) = a( P ) + T with a E End (E) and T EE not necessarily equal to O . However, you may assume that T E Etars, 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 Jr : E --.... p I 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 tt ( - (P + Po)) = tt (P + Po) for all PEE. (c) Conclude that after choosing a new identity element for E, the map Jr is even, i.e., satisfies Jr (P ) = Jr ( - P ) for all PEE. 6.17. Fix an elliptic curve E and and a degree-2 map Jr : E --.... p I satisfying Jr (P) = Jr(- P ). For any integer m and any point T E E [2], let ¢m,T : pI --.... p I be the flexible Lattes map associated to the map 'l/J(P ) = [m](P ) + T as in the commutative diagram (6.22). (a) Prove that ¢m,TO¢m' ,T' = ¢mm',mT'+T . In particular, the maps ¢m,O commute under composition. (b) Prove that cP':n.,T is either cPmn,T or cPmn, O. More precisely, if m is odd and n is even, prove that ¢':n. ,T = ¢m n,O, and prove in all other cases that ¢':n.,T = ¢mn,T . (Of course, ifT = 0, the cases are all the same.) (c) It follows from (a) that the collection of maps { ¢m,T : m ~ 1, T E E[2]} is closed under composition. Prove that cP I,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 ¢ : pI --.... pI be a flexible Lattes map associated to 'l/J (P ) point T E E[2] is not necessarily equal to O. (a) Prove that the set of fixed points of ¢ is given by Fix(¢)
= x([m-
= [m]P + T, where the
lr I(T)) u x([m + l r I(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. I7 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 Lanes map. Using these values, verify directly that the formula
from Theorem 1.14 is true for flexible Lattes maps.
Exercises
383
6.20. Complete the proof of Proposition 6.52(b) by computing the multiplier A¢(00) at the fixed point 00 = x(O). (Hint. Move 0 to (0,0) using the change of variables z = x/y and w = l/y. Then write the invariant differential in terms of z and wand mimic the proof in the text.) 6.21. Let K be an algebraically closed field and let ¢ and ¢' be flexible Lattes maps defined over K that are associated, respectively, to elliptic curves E and E'. Suppose that ¢ and ¢' are PGL 2(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 (6.12) to define E.) 6.22. We proved Proposition 6.55 in the case that 'IjJ( P) = [m](P). (a) Prove Proposition 6.55 for general flexible Lattes maps, i.e., Lattes maps associated to maps oftheform 'IjJ(P) = [m](P) + Twith 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 ¢f has bad reduction for every f E PGL 2(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 = x
3
+ ax + b.
Let m ~ 1 be an integer and write x ([m] p) as a quotient of polynomials
_ Fm(x(P)) x ( [mJP) c.; ( x(P) ).
(6.52)
(a) Prove that Fm and G,« can be taken to be polynomials in x, a, and b. More precisely, prove that there are polynomials Fs«, G,« E Z[a, b, x] satisfying (6.52) and that they are uniquely determined by the requirement that F m be monic in the variable x. (b) Prove that deg(Fm ) = m 2 and deg(G m ) = m 2 - 1 and that their leading terms m2 + ... and Gm(x) = m 2xm2- 1 + .... are Fm(x) = x (c) If m is odd, prove that there is a polynomial 'ljJm(X) E Z[a, b,x] such that Gm(x) = 'ljJm(X)2. Similarly, ifm is even, prove that there is a polynomial 'ljJm(X, y) E Z[a, b,x, y] such that G m (x) = 'ljJm (x, y?, where in the computation we replace y2 by x 3 + ax + b. The polynomial 'ljJm is called the m" division polynomial for E, since its roots are the nontrivial points of order m. (d) Prove that Fm and G m satisfy 2
m Fm(a,b;x) Fm(t 2 a,t 3 b;tx)=t
and
2 G m (2 ) t a,t 3 b;tx ) =t m _ 1 G m ( a,bjx.
Thus Fm and G m are homogeneous if x, a, and b are respectively assigned weights 2, 4, and 6. (e) Let !::>.(E) = -16(4a 3 + 27b2 ) . Prove that the resultant of F m and G m with respect to the variable x is given by
384
Exercises
Section 6.6. Elliptic Curves and Rigid t.attes Maps 6.24. This exercise extends Proposition 6.61. Let ¢ : jp'l --> jp'l be a Lattes map and fix a reduced Lattes diagram (6.37) for ¢. Write 'IjJ(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) : [0: - ~](P) = -T}.
(6.53)
~Er
(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 I": [~]P =
pl.
In particular, if P is not a critical point, then there is a unique ~ E I' that fixes P. (c) Assume that T = O. Let 1r(P) E Fix(¢) and choose some automorphism ~ E f such that 'IjJ(P) = [WP). Compute the multiplier of ¢ at 1r(P) as in the following table (we have given you the first four values):
ifep(1r) = 1, iff
= JL2 and ep(1r) = 2,
iff = JL3 and ep(1r) = 3, iff = JL4 and ep(1r) = 2, iff = JL4 and ep(1r) = 4, iff
= 1-L6 and ep(1r) = 2,
iff = 1-L6 and ep(1r) = 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 [vi- D d] down to Z. In other words, there are integers u and v such that
u
2
+ D d v 2 = d.
Then with notation as in the statement of Theorem 6.62, prove that for every is a constant CE such that
E
> 0 there
for all d.
(Hint. Use elliptic curves with CM by the ring Z [vi- Dd] and Lattes maps associated to the endomorphism (b) Prove that for every E
[u + vvl- D d ]
and follow the proof of Theorem 6.62.)
> 0 there is a constant C~ > 0 such that max S(d - u 2 )
2
C~dl-E
for all d.
O:O;u, and
¢ fit together into the commutative diagram
Example 7.2. The map
induces the rational map
and has indeterminacy locus Z (¢)
7.1.2
= {[O,0, I]} consisting of a single point.
Affine Automorphisms
Of particular interest are affine morphisms that admit an inverse.
Definition. An affine morphism ¢ : AN -; AN is an automorphism if it has an inverse morphism. In other words, ¢ is an affine automorphism if there is an affine morphism ¢- I : A N -; AN such that
Somewhat surprisingly, ¢ and ¢-I need not have the same degree , nor does deg(¢n ) have to equal (deg¢) n .
Example 7.3. Consider the map ¢(x, y ) = (x, y + x 2 ) . It has degree 2 and is an automorphi sm, since it has the inverse ¢- I (x , y ) = (x, y - x 2 ) . The composition ¢2 IS
¢2(x l y)
= ¢(x , y + x 2 ) = (x, Y + 2x 2 ) ,
so deg(¢2) = 2 = deg(¢). More generally, ¢n( x, y) = (x ,y + nx 2 ) has degree 2, so the degree of ¢n does not grow. This contrasts sharply with what happens for morphism s of jp'N.
Example 7.4. Let a E K * and let f (y ) E K [y] be a polynomial of degree d
~
2.
The map
¢(x, y) = (y , ax
+ f(y )),
is called a Henan map. It is an automorphism of A 2 , since one easily checks that it has an inverse ¢ - I given by
7.1. Dynamics of Rational Maps on Projective Space
391
Henon maps, especially those with deg(f) = 2, have been extensively studied since Henon [200] introduced them as examples of maps R'' -7 ]R2 having strange attractors. There are many open questions regarding the real and complex dynamics of Henon maps; see, for example, [132, §2.9] or [211], as well as [212,413] for a compactification of the Henon map. The rational maps J!D2 -7 J!D2 induced by ¢ and ¢-l are
¢([Xo, Xl, X 2]) = [xg, Xg- lX 2, aXg-lXl + f(XO,X2)], ¢-l ([X o, Xl, X 2 ]) = [xg, a-I xg- l X 2 - a-I f(x o, Xd, xg- l Xd, where we write f(u, v) = ud f (v/ u) for the homogenization of f. It is easy to see that the loci of indeterminacy of ¢ and ¢-l are
Z(¢)
= {[a, 1, oj}
and
Z(¢-l) = {[O,O, 1J}.
In particular, the locus of indeterminacy of ¢ is disjoint from the locus of indeterminacy of ¢ -1. Maps with this property are called regular; see Section 7.1.3. Example 7.5. Consider the very simple Henon map
¢(x, y) = (y, -x + y2). The extension ¢ = [xg, X OX2 , - XOXI + Xi] of ¢ to J!D2 has degree 2, but it is not a morphism, since it is not defined at the point [0, 1,0]. And just as in Example 7.2, there is no height estimate of the form h (¢( P)) = 2h(P) + 0(1) for ¢. We can see this by noting that
¢-( [b, a, b] )
=
[b 2,b2,-ab + b2] = [b, b, -a + b],
so if a,b, E Z with gcd(a, b) = 1 and b > a > 0, then [b, a, bJ and ¢([b,a, bJ) have the same height. Hence for every E > 0 even the weaker statement
h(¢(P)) ::::: (1 + E)h(P)
+ 0(1)
for all P = (x, y) E A 2(Q)
is false. It turns out that ¢ has only finitely many Q-rational periodic points (Theorem 7.19), but the proof does not follow directly from a simple height argument. Example 7.6. More generally, if ¢ : AN -7 AN is an affine automorphism, then it is not possible to have simultaneous estimates of the form
h(¢(P)) ::::: (1
+ E)h(P) + 0(1),
h(¢-l(p)) ::::: (1 + E)h(P)
+ 0(1),
(7.2)
for some E > 0 and all P E AN (K). To see this, suppose that (7.2) were true. Then we would have for all P E AN (K),
392
7. Dynamics in Dimension Greater Than One
Thus h(P ) would be bounded, leading to the untenable conclusion that AN (K ) is finite. So it is too much to require that both ¢(P) and ¢- l(p ) have heights larger than the height of P. However, as we shall see, it is often possible to show that some combination of h( ¢(P )) and h( ¢-I (P )) is large, which is then sufficient to prove that P er ( ¢) is a set of bounded height. We conclude this section with two useful geometric lemmas . The first relates the locus of indeterminacy of an affine automorphism and its inverse, and the second characterizes when the degree of a composition is smaller than the product of the degrees .
Lemma 7.7. Let ¢ : AN ----; AN be an affine automorphism ofdegree at least 2 and denote the hyperplane at infinity by Hi, = {X o = O} = IF'N = (Xo,Fl ,Fz, ... , FN)
and
cI>
-1
e
-
-
-
= (XO ,Gl ,GZ, .. . , G N )
be the lifts of ([> and ([>- 1, respectively. The fact that ¢ and ¢ - l are inverses of one another implies that there is a homogeneous polynomial f of degree de - I with the property that
xge, so we see that f = xge- Thus (xge, xge- l Xl , xge- l X l , .. . ,xge- l X N ), 1
But the first coordinate of the composition is (cI>- 1 0 cI» (Xo, . . , , X N ) =
.
or equivalently, -
d
-
-
Gj(XO' F l , · .. , F N )
= X od e-l x,
for aliI 5:. j 5:. N.
Now let P = [0, Xl, .. " XN] E Hi, -, Z(¢), so ([>(P) with at least one Fi(P) # O. From (7.3) we see that
=
(7.3)
[0, F\ (P), ... , FN(P )]
Hence
so ([>- 1 is not defined at ([>( P) . Therefore ([>(P) E Z (¢ - l ).
o
Lemma 7.8. Let ¢ : AN ----; AN and 1fJ : AN ----; AN be affine morphisms, and let H o = {X o = O} = IF'N -, AN be the usual hyperplane at infinity. Then
deg(1fJ 0 ¢) < deg(1fJ) deg(¢) if and only if ([> (H o " Z (¢ )) C Z (1fJ ).
7.1. Dynamics of Rational Maps on Projective Space
393
Proof Let d = deg( 1», let e = deg( 'ljJ), and let
' D' and 4>' D" are linearly equivalent. (c) Using (a) and (b), the map
4>' : {D' E Div(V' ) : 4>(V) ~ ID'I } - - - t Div(V) extends uniquely to a homomorphism
4>' : Pic(V' )
Pic(V).
---t
Example 7.26. A prime divisor W of ]P' N is the zero set of an irreducible homogeneous polynomial F E K [X o, ... , XN]. Wedefinethe degree ofW to be the degree of the polynomial F and extend this to obtain a homomorphism deg(l: n w W) = l: n wdeg(W ).
w
w
It is not hard to see that a divisor on ]P'N is principal if and only if it has degree 0, so the degree map gives an isomorphism
deg : p ic(]P'N)
~
) Z.
Any hyperplane He ]P'N is a generator ofPic(]P'N ). Example 7.27. A prime divisor oflP'N X ]P'M is the zero set of an irreducible bihomogeneous polynomial F E K[ X o, .. . , X N , Yo, ... , Y M]. We say that F and W have bidegree (d, e) if F satisfies
F (oXo, . .. , aXN ,,GYO, • •• ,,GYM) = ad,GeF (Xo, .. . , X N,Yo, ··· ,Y M). The bidegree map can be extended linearly to give an isomorphism bideg : Pic(lP'N x ]P'Af) ~ Z x Z. Let PI : lP'N x ]P'M ----t ]P'N and P2 : ]P'N x ]P'Af ----t ]P'Af be the two projections and let HI be a hyperplane in lP'N and H 2 a hyperplane in ]P'Af. Then p ic(lP'N x lP'M) is generated by the divisors piHl = H I
X
lP'M
and
P'2H2 = lP'N
X
H 2.
406
7. Dynamics in Dimension Greater Than One
7.2.2 Ample Divisors and Effective Divisors Definition. A divisor D = LnwW is said to be effective (or positive) ifnw ~ 0 for all W. We write D ~ 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 ,
L
D=
nw W -
w with
L
(-nw)W.
W with nwO
Definition. Let D E Div(V). Associated to D is the finite-dimensional K-vector space £(D) = {J E K(V) : (I) + D ~ o} U {O}. We write £(D ) = dim £(D) for the dimension of £(D). Let D E Div(V) be a divisor with £(D) ~ 1. We choose a basis for £(D ) and use it to define a rational map ¢D =
[ft ,... ,fi (D)] : V
---+
fl ,··· , f i (D )
p i(D)-I.
The map ¢D is well-defined up to a linear change of coordinates on p i (D)-I, i.e., up to composition by an element ofPGLi(D)(K). Further, if D and D' are linearly equivalent, then ¢ D and ¢ D' differ by a change of coordinates. Conversely, let i : V fil'N be a morphism (or even a rational map) and let H C p N be a hyperplane with i (V ) H . Then i is equal to the composition of ¢i' H with a change of coordinates and a projection.
rt
Definition. A divisor D E Div(V) is very ample if the map ¢D : V -4 fil'i (D)-l is an embedding, i.e., an isomorphism onto its image. A divisor D is ample if some multiple nD with n ~ 1 is very ample . Ampleness and very ampleness are properties of the divisor class of D . Notice that ifi : V fil'N is an embedding and H E Div(fil'N) is a hyperplane, then ¢* H is a very ample divisor on V . Example 7.28. Let p! HI and P2H2 be the generators of Pic(fil'N x pM) described in Example 7.27. Then p!H1 + P2H2 is a very ample divisor on pN x fil'M. The associated embedding is called the Segre embedding. It is given explicitly by the formula fil'N X pAJ ---+ fil'N M+N+M ([Xo, . .. , X N ], [Yo , ... ,YM
])
f----+
[X oYO, X OY1 , .. . , X ) 'j , .. . , X NYM
Now let V be a subvariety ofpN x pM, say ¢ : V
¢*(prHI
+ p;H2) = (P I 0
is a very ample divisor on V.
¢)* HI
pN
X
+ (P2 0
pM . Then
¢)* H 2
].
7.3. The WeilHeight Machine
407
7.3 The Weil Height Machine The theory of height functions that we developed in Sections 3.1-3.5 provides a powerful tool for studying the arithmetic of morphisms ¢ : ]p'N - t ]p'N of projective space. For example, if ¢ has degree d ~ 2 and is defined over a number field K, then the fundamental estimate (Theorem 3.11)
h(¢(P )) = d· h(P ) + 0 (1)
for all P E ]p'N (Q)
(7.14)
and the fact (Theorem 3.7) that there are only finitely many points in ]p'N (K) of bounded height lead immediately to a proof of Northcott's theorem (Theorem 3.12) stating that ¢ has only finitely many K -rational preperiodic points. Recall that the height h(P ) of a point P E ]p'N(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 '---'> ]p'N and define hv (P) to equal h(¢(P )) . Unfortunately, different projective embeddings yield different height functions . But letting H denote a hyperplane in ]p'N , one can show that if the divisors ¢* Hand 1jJ* H are linearly equivalent, then the height functions attached to ¢ : V '---'> ]p'N and 1jJ : V '---'> ]p'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)
-t
JR},
D
f--t
hV,D,
with the following properties: (a) (Normalization) Let H C ]p'N be a hyperplane and let h : ]p'N (Q) - t JR be the absolute logarithmic height fun ction on projective space defined in Section 3.1. Then for all P E ]p'N (Q). (b) (Functoriality) Let ¢ : V - t V' be a morphism ofnonsingular varieties defined over Qand let DE Div (V' ). Then
hV,q,"D(P ) = hV1,D(¢(P )) + 0 (1)
for all P E V(Q).
408
7. Dynamics in Dimension GreaterThan One
(c) (Additi vity) 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,o(P) = hV,E(P ) + 0 (1 )
for all P E V (Q).
(e) (Pos itivity) Let D E Div (V ) be an effective divisor. Then
bv.o 'P) ;::: 0 (1)
for all P E V(Q) " Base(D ).
That is, hV,D is bounded below fo r all points not in the base locus of D . (f) (Finiteness) Let D E Div (V ) be ample. Thenfor all constants A and B , the set
{p E V(Q) : [Q(P) : Q] ::; A is finite. In particular, finite extension, then
if V
and hV,D(P)
< B}
is defin ed over a number fi eld K and
{p E V( L)
if L/ K
is a
: hV,D (P ) ::; B}
is a fin ite 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 assume functoriality for projective embeddings V '-+ jp'N.) Proof See [76, Chapter 2], [205, Theorem B.3.2], or [256, Chapter 4].
0
Remark 7.30. All ofthe 0 (1) constants appearing in the Weil height machine (Theorem 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 precisely, 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 Nullstellensatz effecti ve, so they tend to be rather large. Remark 7.31. Many of the properties of the Weil height machine may be succinctly summarized by the statement that there is a unique homomorphism
hV such that if ¢ : V
:
· V P 1c( ) -----+
'-+ jp'N
{functions V( K) -7 R} {bounded functions V (K) -7 R}
-;-,-------'-.,..---:-....,---- -,------'--:'-::-:-::=::---=-----=-o-
is a projective embedding, then bv,« H = h + 0(1).
7.3. The Weil HeightMachine
409
Example 7.32. Let ¢ : jp'N - t jp'N be a morphism of degree d and let H E Div(jp'N ) be a hyperplane. Then ¢* H '" dH, so Theorem 7.29 allows us to compute
hpN,H(¢(P )) = hpN, 'H (P ) +0( 1) = hpN,dH (P ) + 0( 1) = dhpN ,H(P)+ O(l). This formula is Theorem 3.11. Example 7.33. Let V be a subvariety of jp'N x jp'M, say ¢ : V
]P'2,
PI(X,y)
= x,
P2(X,y) = y.
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
Pll (a) = {(a ,y) E 1P'2 x p 2 : L(a,y) = Q(a ,y) =
o}
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 1P'2 . And similarly, P2 is a map of degree 2. In general, a degree-2 map between varieties induces an involution on the domain given by switch ing the two sheet s of the cover. In our situation the maps PI and P2 induce involutions LI and L2 on SA ,B . Explicitly, if P = [a, b ] E SA ,B , then LI (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 S A ,B isfying and PI 0 LI = PI P2 0 L2 =P2·
----7
S A,B sat-
We note that LI and L2 are rational maps on S, i.e., they are given by rational functions . To see why this is true, obse rve that band b ' are the intersection point s in p2 of the line and the conic
L(a,y) = 0
and
Q(a ,y) =
o.
Thus each of band 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 LI and L2. Example 7.36. We illustrate the involut ions on S A ,B using the example
L (x , y )
= XoYo + X I YI + X2Y2,
Q (x , y )
= x~Y6 + 4X~YOYI
+ 7X~YIY2 + 3XOXIY6 + 3XOX IYOYI + XOXIY2 + XIYo + + 4 X2IYIY2 - XOX2YI2 + 5XOX2YOY2 - 4XIX2yi - 4XIX2YOY2 - 2X~YOYI + 3x~y~. 2
- x 6yi
2 2
2 2 2 X I YI
The point P = ([1 ,0,0], [0,7,1]) is in S(Q). In order to compute tute the value x = [1,0,0] into Land Q and solve for y. Thus
L([I , 0, 0], y ) = Yo = 0 and Q([I , 0, 0], y ) = Y6 + 4YOY I -
LI
(P), we substi-
yi + 7YIY2 = 0,
so the solutions are y = [0, YI , Y2 ], where YI and Y2 are the roots of the polynomial -y~ + 7YIY2 = O. One solution is YI = 7, which gives the original point P , and the other solution is YI = 0, which gives LI(P ) = ([1,0,0],[0,0, 1]). Next we compute L2(P ), To do this, we substitute y = [0, 7, 1] into Land Q to obtain
412
7. Dynamics in Dimension Greater Than One L(x, [0, 7,1]) = 7XI + X2 = 0, Q(x, [0,7,1]) = XOXI - 49xoX2 + 126xi - 196xI X2
Substituting X2
=
+ 3x~ = o.
-7XI into the second equation gives
Q([XO' Xl, -7XI], [0,7,1]) = 344xoXI + 1645xi =
o.
The solution Xl = 0 gives back the original point P. The other solution is [Xo, Xl ] [1645, -344] , and then setting X2 = - 7XI = 2408 gives ~2 (P)
= ~2 ([1, 0, 0], [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 ~l (~2(P)) are already integers with 12 to 13 digits.
Remark 7.37. The surface S described by (7.15) is an example of a K3 surface. Formally, a K3 surface is a surface S of Kodaira dimension 0 with the property that HI (S, Os ) = O. However, all of the information that we will need is contained in the explicit equations (7.15) defining S. The reader desiring more information about the geometric properties ofK3 surfaces might consult [40, 44, 178,298]. 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 W8 x W35 . However, many of the surfaces are isomorphic. For example, we can use elements ofPGL 3 to change variables in each of two factors oflP'2. This reduces the dimension of the parameter space by 16, since PGL 3 has dimension 8. Further, the surface SA ,B really depends only on the ideal generated 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
'-v-" A Ep8
+ '-v-" 35 B EP35
-
8
'-v-" P G L3
-
8
'-v-" P GL 3
-
9
'-v-"
= 18.
AI
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 surfaces SA,B .
Definition. There are several linear, quadratic, and quartic forms that come up naturally when one is working with the surface S A,B. We define linear and quadratic forms by setting
7.4. Dynamicson Surfaces with Noncommuting Involutions
Lj(x)
= the coefficient
413
ofYj in L(x, y),
L¥(y) = the coefficient of Xi in L(x,y), Q'h£(x) = the coefficient ofYkY£ in Q(x, y), Q¥j(y) = the coefficient of XiXj in Q(x, y).
(7.16)
This notation allows us to write the bilinear form L and the biquadratic form
L(x, y) = Q(x,y)
=
2
2
j=O
i=O
L Lj(x)Yj = L L¥(y)Xi, L Q'h£(X)YkY£ L =
0~k~£~2
Q as
Q¥j(y)XiXj.
0~i~j~2
Then for each triple of distinct indices i, j, k E {O, 1, 2} we define quartic forms
G% = (Lj)2Qfi -
Lf LjQfj + (Ln 2Qjj'
G% = (Lj)2Q¥i - L¥LjQ¥j + (Ln 2Qjj' HfJ
=
H0 =
(7.17)
+ (Lk) 2Qij, 2L¥LjQ%k - L¥L%Qjk - LjL%Q¥k + (L%)2Q¥j' 2Lf LjQ%k -
Lf L'hQjk -
Lj L%Qfk
For some choices of A and B, there may be points on the surface SA,B at which ~l or ~2 is not well-defined. The next proposition, which provides a criterion for checking whether ~l 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 ~l is defined at P unless G~(a)
(b) The involution
= Gf(a) = ~2
G~(a)
= Hg1 (a) = Hg2(a) = Hf2(a)
=
O.
is defined at P unless
Proof By symmetry, it is enough to prove (a). The map ~l is defined at P = [a, b] if and only if the fiber PI 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 lP'2. 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 ~l is a morphism' except in the following two situations: I 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) = O.
7. Dynamics in Dimension Greater ThanOne
414
• L( a, y) is identically O. • L( a, y)
= 0 is a line that is contained in the set where Q(a, y) = O.
With the notation defined by (7.16) and (7.17), we use the bihomogeneity of Q to write
and then we eliminate the variable Yo by substituting LoYo = L - LfYl - L'2Y2. After some algebra, we obtain an identity of the form
+ Hf2YIY2 + GfY~ + L{ QooL + (LOQOl - 2LfQoo)Yl + (LOQ02 - 2L'!jQoO)Y2}'
(LO)2Q = G'!jyi
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 Yl or Y2, as congruences in the polynomial ring Z[A i j , Bijk£, Xi, Yj]. Thus
+ Gf(x)y~ Lf(x)2Q(X, y) = G2(x)Y5 + Hg2(x)YOY2 + Go(x)y~ L2(X)2Q(x,y) = Gf(x)Y5 + Hg1(x)YOYl + Go(x)yi LO(X)2Q(X, y) = G'!j(x)yi + Hf2(X)YIY2
(mod L(x,y)), (7.18) (mod L(x, y)), (7.19) (mod L(x,y)). (7.20)
Suppose first that L( a, y) is identically O. Substituting x = a into (7.18), we find that the quadratic form
is identically O. Hence G2(a) = H 12(a) = G1(a) = O. Similarly, substituting x = a into (7.19) and (7.20) shows that all of the other values G'k(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 O. Then the assumption that Ll 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 L1(a) = L'2 (a) = 0, then the definition (7.17) of Go shows that Go(a) = O. And if L1(a) and L2(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 G1(a) = G2(a) = O. Next we evaluate (7.18), (7.19), and (7.20) at x = a and use the fact that we now know that Go(a) = G1(a) = G2(a) = O. This yields
415
7.4. Dynamics on Surfaces with Noncommuting Involutions
°
Hf2(a)YIY2 = H~2 (a) yOYz = H~l (a)YoYI = for all y = [Yo,YI , yz] satisfying L(a, y ) = 0. (7.21 ) Wewill provethat H 12(a) = 0; the others are done similarly. Ifthere is a point on the line L(a ,y) = with YIY2 i- 0, then (7,21) immediately implies that H 1z(a) = 0. So we are reduced to the cases in which the line L(a , y ) = is either YI = or yz = 0. If it is the line YI = 0, then L(a , y ) = CYI for some constant c i- 0, so Lo(a) = L2(a) = 0, and similarly ifi t is the line yz = 0, then Lo(a) = Ll(a) = 0. In either case, the definition (7.17) of H 1z yields
°
Hfz(a) =
°
°
2Lf (a)L~ (a)Qoo (a)
- Lf (a)Lo (a)Q~o (a) - L2(a)Lo(a)Qfo(a) + Lo(a)2Qfz(a) = 0. 0 Example 7.40, We illustrate Proposition 7.39 using the surface described in Example 7.36. The polynomials G'k and Htj for this exampleare givenin Table7,1, Proposition 7.39 says that [1 is defined at P = [a, b] provided that at least one of the six polynomials Go, Gf , G2, HOI' Hoz , H 12 does not vanish at a. For convenience we say that a point a E ]P'2 is degenerate if
Our first observation is that and Hence there are no degeneratepoints with ao = 0. (Weassume 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, aI, az] , Weuse a tilde to indicate the dehomogenization Xo = 1 of the G'k and Htj polynomials. So for example,
Now suppose that a = (aI, a2) is a degenerate point. Then Xl = al is a common root of the polynomials
Hence if we take resultants with respect to the X2 variable, then Xl = a l is a root of both of the polynomials
R I (X2) = Res X 1 (Gf(X I, X2) , ii~I (Xl , X2)) , R 2(xz) = Res X 1 (G2(X I , X2 ) ,ii~I (X I , X2 )) ' Explicitly, these polynomials are
416
7. Dynamics in Dimension GreaterThan One
y 3 + YOY2 2 2 Go - - 2YOY13 + 4Y1Y2 + 4YOY1Y22 + 5Y12Y22 + 4Y1Y23 y 2 2 2 4 2 2 2 7 3 G 1 = - 2Yo3 Y1 + YOY1Y2 - YOY2 + YOY1Y2 - Y1Y2 + Y1 Y2
3 c: = Yo4- 3YOY1 + 4YOY13 2
y H 01
4+4YOY1Y2 + 73 Y1 Y1Y2 - YOY1Y22 2
2 2 3 = - 4YOY1 + 4YOY12Y2 + Y1Y2 + 7YO2Y22 + 4YOY1Y22 + Y24
Hl;02 = 4YOY1 3 - Y14 + 2YoY2 3 + YoY1Y 2 2 2 + 6YOY1Y2 + 8YOY1Y22 - Y1Y23 tt:12 = - 4YOY1 2 2 3 - 6YOY1Y2 2 2 3 + 14Y1Y2 2 2 + YOY13 - 7YoY2 + 8YOY1Y2 - 2Y1Y2 - YOY23
Table 7.1: The polynomials R 1 (xd
G~
and H i} for the surface in Example 7.36.
= 4X~2 + 112x P + 11 60x ~ o + 5112 xr + 7052x~
R 2(x d =
- 227 1xi
+ 1 8573x~
+ 2 160xr + 16053x f - 7304xr + 1045x I - 49x I, 4x~6 + 80X ~ 5 + 600X~4 + 2064 x ~ 3 + 2548x }2 - 36 16x P - 1 421 6x ~ o - 10892xr + 98 56x~ + 2170 8xi + 1 5648x~ + 1000xf - 13986xf - 10462xr - 3 124xI - 412xI - 18,
and the assumption that R 1 and R 2 have a common root implies that their resultant must vanish. However, when we comput e it, we find that
Res(R 1 , R 2 )
=
198929 . . . 3830147072 ~ 1.99 . 1087 .
Hence t 1 is defined at every point of SA ,B unless the characteristic of K divides this (large) nonzero integer Res(R 1 , R 2 ) . We can use other resultants to reduce the list of possible bad characteristics. For example, let R O(X2) = Res X 1 (Go ,HOl )' Then tl is everywhere defined unless both Res(flo , R 2 ) and Res(R l , R 2 ) vanish. We compute gcd (Res( flo, R 2 ) , Res( R 1 , R 2 ) ) = 439853213743020234882809856 = 217 . 36 . 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 appearing in (7.22). We sketch a similar calculation for L2. Let
= ResY1 ( Gg (Y1, Y2), iIlf1 (Y1, Y2) ) , T 1(Y2) = Res Y1 (Gi (Y1, Y2), iIlf1 (Y1' Y2)), T 2(Y2) = Res Y1 (G~ (Y1, Y2), iIlf1 (Y1' Y2))' To (Y2)
Then Res(To, T 2 ) ~ 2.57.10 97 and Res(T1 , T 2 ) ~ 2.75.10 114 , and
gcd (Res (To, T 2 ) , Res(T1 , T 2 ) ) = 216 ·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 involution 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 S A,B are morphisms. Proposition 7.41. There is a proper Zariski closed set Z C if(A, B) ~ Z, then the involutions
lP'8 X lP'35
such that
are morphisms. Proof According to Proposition 7.39, the involution provided that the system of equations
L1
is defined on all of SA,B
(7.23) has no solutions in lP'2. A general result from elimination theory (see [198, I.5.7A]) says that there are polynomials h, ... , I- in the coefficients of
such that the equations (7.23) have a solution if and only if h = ... = ir = O. The coefficients of Go, ... ,H'f2 are themselves polynomials in the coefficients of L(x, y) and Q(x, y), so we can write each ii as a polynomial in the variables A and B. There are then two possibilities: • The set of ii consists only of the zero polynomial, and hence every surface S A,B has some point at which L1 is not defined. • The set of ii contains at least one nonzero polynomial defined on SA,B provided that h(A, B) # o.
h, and then L1 is well-
7. Dynamics in Dimension GreaterThan 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 Proposition 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. 0 Remark 7.42. See Exercise 7.28 for a less computational proof of Proposition 7.41.
7.4.2 Divisors and Involutions on SA ,B In this section we study how the involutions LI and L2 act on divisors on S A,B. Later we use this information to study how iterates of the involutions act on points. This prompts the following definitions.
Definition. Let LI and L2 be the involutions of the surface S A,B defined by (7.15). 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 ofP 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 ofJP'2 x JP'2 is isomorphic to ,;£2 via
(By abuse of notation, we write nlHI + n2H2 for its divisor class.) We note that nlHI + n2H2 is a very ample divisor on JP'2 x JP'2 if and only ifboth nl and n2 are positive. Next we define two divisors D I , D 2 E Div(SA,B) using the two projections of SA,B to JP'2, and Wehler [448] has proven that the Picard group of a general surface SA,B satisfies PiC(SA,B) ~ ,;£2 and that D I and D 2 are generators.? but for our purposes it will not matter ifPic(SA,B) is larger than ,;£2, we will simply use the part ofPic(SA,B) generated by D I and D 2 . We now compute the action of LI and L2 on D I and D 2 • 2What this means is that the set of coefficients (A, B) E lP'8 X lP'35 for which Pic(SA,B) is strictly larger than ;Z2 forms a countable union of proper Zariski closed subsets of lP'8 x lP'35. This implies that most (A, B) in lP'8(C) x lP'35(C) have Picard group ;Z2, but it does not directly imply that there are any such values in lP'8(1Qi) x lP'35(1Qi), since IQi is countable.
7.4. Dynamics on Surfaces with Noncommuting Involutions
419
Proposition 7.43. Let D l = P~ Hand D 2 = p'2H. The involutions ~l and ~2 act on the subspace ofPic(SA,B) generated by D l and D 2 according to the following rules: ~~Dl ~iD2
= D l, = 4D l - D 2,
~'2Dl
=
-o, + 4D 2,
(7.24)
~2D2 = D 2·
(7.25)
Proof The involution ~l switches the sheets of the projection Pl, so it is clear that Pl 0 ~l = Pl. This allows us to compute
This proves the first formula in (7.24). Next we observe that for any P E SA,B, the two points in the set PIl (Pl (p)) are P and ~l(P), Thus if we start with a divisor on SA,B, use Pl to push it down to ]P'2, and then use Pl to pull it back to SA,B, we get back the original divisor plus its translation by ~l. In other words,
Using this formula with D
= D 2 allows us to compute (7.26)
The divisor Pl*P'2H on]P'2 is linearly equivalent to some multiple of H. For simplicity, let H be the line Y2 = O. Then P'2H is the curve in ]P'2 x ]P'2 (lying on SA,B) given by the equations
Y2 =
o.
We solve the linear equation L = 0 to express Yo and Yl as linear functions of [xo, Xl, X2], and then substituting into the quadratic equation Q = 0 yields a homogeneous equation of degree 4 in xo, Xl, X2. SO when we use Ph to push P'2H down to ]P'2, we get a curve of degree 4 in ]P'2. Hence
where this is an equality in Pic(]P'2). Substituting into (7.26) yields the second formula 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 ~l and ~2 are ~i = 1 and ~§ = 1. In other words, A is isomorphic to the free product of the groups of order 2 generated by ~l and ~2. An alternative description is that A is isomorphic to an infinite dihedral group. See Exercise 7.31.
420
7. Dynamics in Dimension Greater Than One
7.4.3 Height Functions on SA,B In this section we use the WeiI height machine to translate the divisor relations and transformation formulas from Proposition 7.43 into relations among height functions. We recall from Example 7.33 that the height functions associated to the divisors D 1 and D 2 are given at a point P = [x, y] E SA,B by h D 1 (P) h D 2(P)
= =
Proposition 7.45. Assume that
hpI H(P) hp~H(P) SA,B
= =
hH(P1P) hH(P2P)
= =
h(x), (7.27)
h(y).
is defined over a numberfield K. Let
a=2+V3 and define functions h +, h - : S A,B (K) ----; lR by the formulas h+([x,yJ)
=
-h(x)
+ ah(y)
and
h-([x,yJ) = ah(x) - h(y).
Then h+ and h: transform according to the following rules: h+
0 [1
= ah- + 0(1),
t.: 0
[1
= a- 1h+ + 0(1),
h+
0 [2
= a- 1h- + 0(1),
h-
[2
= ah+ + 0(1).
0
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 twodimensional real vector space V with basis elements D l and D 2 , where we view V as a subspace of Pic(SA,B) @ R Then the formulas (7.24) and (7.25) in Proposition 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
=
(10-14)
and
*
[2
=
(-10)
4 1 .
We now look for a new basis {E 1 , E 2 } for V with the property that change the basis. More precisely, we ask that
[i and [2 inter-
for some constants a, b, c, d. This problem can be solved directly, but it is easier to observe that [2[i E 1 = adE l and [2[i E 2 = bcE2. Thus E 1 and E 2 must be eigenvectors for the linear transformation [2[i, whose matrix is
* *
[2[1
=
(10-14) (-1410) = (15-4 -14) .
It is easy to check that 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 [2*E 1
= a -lE2,
(7.28)
It is then natural to define height functions h+ and ti: corresponding to the divisors E 1 = -D 1 + aD 2 and E 2 = aD 1 - D 2 , since the divisor relations (7.28) and the Weil height machine should then yield corresponding relations for the height functions.
ProofofProposition 7.45. Having given the motivation, we commence the proof of Proposition 7.45, which is a formal calculation using the additivity and functoriality 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) = h D1([x, vl)
and
h(y) = h D2([x, y])
from (7.27). We compute h+
0 [1
+ ahD 2 0 [1 = -h'~Dl + ah'~D2 + 0(1) = -h D1 + ah4D,-D2 + 0(1) = (-1 + 4a)h D1 - ah D2 + 0(1) 2hD = a 1 - ah D2 + 0(1) = oh" + 0(1). = -h D1 0
[1
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+
0 [2
=
-h D1 0
[2
+ ah D2 0
[2
by definition of h - ,
= -h'~Dl + ahL~D2 = -h- D1+4D2 + ah D2
from Theorem 7.29(b),
+ (-4 + a)h D2 = h D1 - a- 1h D2 + 0(1) = a- 1h- + 0(1).
from Theorem 7.29(c),
= h D1
from Proposition 7.43, since a 2
=
4a - 1,
This proves the transformation formulas for h +. The proof for h - is similar and is left for the reader. 0 We can use Proposition 7.45 and the general theory of canonical heights (Theorem 3.20) to construct two heights on SA,B that are canonical with respect to both [1 and [2. Proposition 7.47. Let functions
SA,B
be defined over a number field K. There exist unique
422
7. Dynamics in Dimension Greater Than One and
satisfying both the normalization conditions
i: =
and
obo, - h D 2 + 0(1)
(7.29)
and the canonical transformation formulas
Proof Let ¢ = ~1
~1
h+O~l=ah-,
h-
h+
i: O~2 =
0 ~2
~2 = a -1
0
h- ,
0
= a-1h+, (7.30)
ah+.
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
h+ 0 ¢ = h+ 0 ~1
0
~2 + 0(1)
= ah-
0
~2 + 0(1)
= a 2h+ + 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
h-
0 ¢-1
= h"
0
~2
0
~1
¢-1
= ~2
+ 0(1) = ah+
Applying Theorem 3.20 to the functions tion h- satisfying
¢-1
and
~1, we find that
0
0
~1
+ 0(1) = a 2h- + 0(1).
and h: , we find that there is a func-
h-
= tc:
+ 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+ 0 ~1 = h+ 0 ~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+ O~l = h+ o¢n 0 ~1 = h+ O~l o¢-n = ak- o¢-n +0(1) = an+lk- + 0(1). Divide both sides by an and let n ----t 00 to obtain the desired result h+ 0 ~1 = ak-. This proves the first of the transformation formulas (7.30). The others are proven similarly.
423
7.4. Dynamicson Surfaceswith Noncommuting Involutions Finally, in order to prove uniqueness, suppose that isfying (7.29) and (7.30). Then
g+
and
g-
are functions sat-
and similarly g- 0 ¢J-1 = a 2 g- . Hence g+ and g- have the same canonical properties as 11,+ and 11,-, so the uniqueness assertion in Theorem 3.20 tells us that g+ = 11,+ andg- = 11,-. 0
Remark 7.48. In practice, it is infeasible to compute the canonical heights 11,+ and 11,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 11,+ and 11,- as sums oflocal heights that may then be computed using rapidly convergent series (cf. Exercise 5.29). See [89] for details.
7.4.4
Properties and Applications of Canonical Heights
The next proposition describes various useful properties of the canonical height functions 11,+ and 11,- 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 3.12 on preperiodic points ofmorphisms on lP'N and of Theorem 7.19 on periodic points of regular affine automorphisms.
Proposition 7.49. Let SA,B be defined over a numberfield K, let 11,+ and 11,- be the canonical height/unctions constructed in Proposition 7.47, and let
(a) The set
{p
E
SA,B(K) : h(P) < c}
is finite. (N.B. This is not true if we replace see Exercise 7.35.) (b) Let P E SA,B(K). Then
h+(P)
=0
-¢==}
h-(P)
=0
-¢==}
11, by either 0/the heights 11,+ and 11,-,
h(P)
=0
-¢==}
P has finite A-orbit.
(c) There are only finitely many points PESA,B (K) with finite A-orbit.
Proof (a) Using the properties of 11,+ and
11,-, we find that
11,=11,++11,= (-h D 1 + ah D 2 )
+ (ah D 1 - h D 2 ) + 0(1) = (a -1)(h D 1 + hD 2 ) + 0(1).
As noted earlier, the heights h D 1 and h D 2 are given by
by definition of 11" from Proposition 7.47,
424
7. Dynamics in Dimension Greater Than One
hD 1 ([x, vl) = h(x )
and
where h(x ) and h(y ) are the standard heights ofx and yin jp'2 . Hence
it ([x, y]) = (0 - l )(h(x ) + h(y )) + 0 (1 ), so if it( [x, vl) is bounded, then both h(x) and h(y ) are bounded. (Note that 0 satisfies 0 > 1, which is crucial for the argument to work.) This completes the proof of (a), since Theorem 3.7 tells us that lP2( K) contains only finitely many points of bounded height. (b) Since it = it+ + it- and both it+ and it- are nonnegative , it is clear that
Suppose next that it+(P) = O. Let ¢ = ~l
0
~2 as usual. Then
it(¢n(p)) = it+(¢n(p)) +it-(¢n(p)) = 0 2nit+(P ) + 0- 2nit- (P ) =
0 -
2nit- (P ).
The righthand side is bounded (indeed, it goes to 0) as n --t 00, so we see that { ¢n(p ) : n ~ O} is a set of bounded it-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 )) + A- (¢- n(p )) = 0 - 2nh+(P) + 0 2nh- (P ) =
0
2nit- (P ).
The lefthand side is bounded , since P is periodic for ¢, so letting n --t 00 implies that h- (P) = o. This proves that it + (P) = 0 implies it- (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 it(¢n(p )) is bounded, since it takes on only finitely many values. Letting n --t 00 in (7.31) and using the fact that 0 > 1, we deduce that h+(P) = o. Finally, suppose that h(P) = O. Then it+(P) = it- (P) = 0, so (7.31) 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
o
so A (P ) is also finite.
The canonical height functions on S A ,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 7.21 for an analogou s (conditional) estimate for regular affine automorphisms of ]P'N . Proposition 7.50. Let S A ,B be defin ed over a numberfield K. andf oranypoint P [x, y] E S A ,B( K), let h(P ) be the height fun ction
AIso let let
0:
=
=
2 + V3 as usual. Fix a point Q E SA ,B (K) with infinite A-orbit and JL (Q )
= #{ 7jJ E A
: 7jJ(Q)
= Q}
be the order ofthe stabilizer ofQ. Then 2
#{ P
E A (Q ) : h(P ) ~ B } =
JL
1 logo ( (Q)
A
B
)
A
h+(Q )h-(Q)
+ 0 (1) as B
-+ 00,
where the 0 (1) constant is independent ofboth Band 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
-+ 00,
where the 0 (1) constant depends only on u. Proof We start by writing the real number log., ( vr;ra) as the sum of an integer and a fractional part,
log.,
If
= m
+r
with m E Z and
Irl
~ ~.
(The reason that we do this is because the function au X + bu : " has a minimum at x = logu (vr;ra).) Then replacing n by n + tti in the expression au" + bu- n yields Hence
It thus suffices to prove that if c, dE 1R are both between u -
1 2 /
and
1 2, U /
then
7. Dynamics in Dimension Greater ThanOne
426
as t ~
We note that if n
--+ 00 .
(7.32)
0, then
and similarly if n ::::: 0, then
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: cu" +du- n
:::::
#{n E Z: logu (cu n +du- n ) ::::: logu(t)} =#{n E Z: Inl + 0(1) ::::: logu(t)}
t} =
= 210gu(t)
+ 0(1).
This is the desired inequality (7.32), which completes the proof of Lemma 7.51.
D
ProofofProposition 7.50. We do the case that JL (Q) = 1 and leave the similar case JL(Q) = 2 to the reader. (It is easy to check that JL (Q) ::::: 2; see Exercise 7.29.) Let 4> = ~ I 0 ~2 . Every element of A is given uniquely as an alternating composition of ~I 's and ~2 's, so
A OUT
A splits up as a disjoint union = {¢n : n E
Z} U {¢n 0
~I :
n E Z}.
assumption that JL (Q) = 1 then implies that the A-orbit of Q is a disjoint union (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+ 04> = o:h+ and h- 0 ¢ = 0:- 1 h-. This allows us to compute
#{P
E
O¢(Q) : h(P) ::::: B}
B} #{ n E Z : h+(¢nQ) + h-(¢nQ) ::::: B} #{ n E Z : 0:2nh+(Q) + 0:- 2nh- (Q) ::::: B}
= #{ n = =
E
Z : h(¢nQ) :::::
2
1 = -100'
2
bet
(
B ) h+(Q)h-(Q )
+ 0 (1)
since p,(Q) = 1, definition of h, from Proposition 7.47, from Lemma 7.51.
Further, if we replace Q with ~ I (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 O(Q) : h(P) < B} + #{ P E O(tlQ) : h(P)
:::; B}
~ log. (j,+(~~_(Q)) + 0(1) Finally, in order to replace the canonical height that
h with the naive height h, we note
h = h+ + h- = (ah D1 - hD2) + (-h D1 + ah D2) + 0(1) = (a - 1)(hD1 + hDJ + 0(1) = (a - l)h + 0(1). Thus
#{ P E A(Q) : h(P)
:::; B} =
#{ P E A(Q) : h(P)
:::; (a - l)B + 0(1)},
and replacing B with (a - l)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
¢(x,y,z) = (ax+ by2+(cx 2+dz)2, ey+(ax+ by2)2, dz+cx 2).
(7.34)
(a) Prove that ¢ is invertible if and only if ade =1= 0. (b) Prove that ¢ is a regular automorphism if and only if abcde =1= 0. (c) Clearly (0,0,0) is a fixed point of ¢. Let b = -1, d = 1, and e = 1 - t 3 . Prove that (0, t, t) is also a fixed point of tb. Hence there are infinitely many maps ¢ E Q[x, y, z] of the form (7.34) such that Fix( ¢) n A,.3(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 - t 3 .) (e) Suppose that ¢ E lR[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(lR) = {(O, 0, On. In particular, ¢ has only one rational fixed point. 7.2. Let ¢ : A3
-->
A3 be the map
¢(x,y,z) = (x 2z,xy,yz). (a) Calculate the indeterminacy locus of ¢. (b) What are the values of · . f I im III
(x,y,z)EZ 3 h(x,y,z)~oo
h(¢(x,y,z») h(x, y, z)
and
.
lim sup (x,y,z)EZ3 h(x,y,z)~oo
h(¢(x,y,z») h(x, y, z)
?
428
Exercises
(c) Same question as (b), but with the points (x , y, z ) E Z3 restricted to satisfy x yz
i= o.
7.3. Let ¢ : AN ---> AN be a regular affine automorphism and let n ~ 1. Prove that P er n (¢ ) is a discrete subset of AN (C), and that counted with appropriate multipl icities,
(This is Theorem 7.10(c). Hint. Rewrite ¢Nn( p) = Pas ¢l2 n (P) = ,p-(N- l 2)n(p ), show that the homogenizations of ¢l2 n and ¢(N- (2 )n have the same degree, and use Bezout 's theorem to count the number of solut ions.) 3 2 ---> A be the map ¢( x , y, z) = (y, Z, x ) . (a) Find an explicit expression for ¢n(x, y, z) . (There may be more than one case.)
7.4. Let ¢ : A3
(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 n>odnT n and prove that it is in (jJl(T) . (d) Prove that PrePer (¢ ) C { P E A3 (Q) : h(P ) (e) Let 8
= dyndeg(¢ ) and P
= a}.
E A 3 (Q). 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
.
# {n ~ a: h(¢n(p )) < T}
~ oo
~T
lim
7.5. Let ¢ : 1P'3 --->
1
= - -. ~8
Jr3 be the rational map
(a) Prove that ¢ is a birational map, i.e., find a rational map 'IjJ so that ¢ the identity map at all points where they are defined. (b) Compute Z (¢) and Z (¢ - 1). Where do they intersect?
0
'IjJ and 'IjJ
0
¢ are
(c) * Let dn = deg (¢ n ). Prove that the generating function L n>odnT n is not in Q(T ). The map in this exercise and the map in the previous exercise are examples of monomial maps, see (199]. 7.6. Let u , al , .. . , a N , bl , .. . , bN E
Q with u i= o. We proved in Lemma 7.17 that
Prove that this inequality need not be true if u =
a.
Exercises
429
7.7. This exercise generalizes Theorem 7.15. Let 1,. .. , t : AN phisms with the property that
-+
AN be affine automor-
Let d; = deg(i) for 1 ~ i ~ t. Prove that there is a constant C = C (rP l , .. . , t) so that for all P E AN (Q), I I I
d h( I (P)) l
+ d2h(2(P )) + ... + dt h( t(P )) ?
h( P) - C.
7.8. Let < /JI , 2 : A2 -+ A 2 be the maps and (a) Prove that rP1 and rP2 are jointly regular. (b) Let a a positive integer and P = (0, a) E A2 (1Q). Prove that
This proves that the lower bound in Theorem 7.15 cannot be improved in general for jointly regular affine morphisms . (c)
** Can the lower bound in Theorem 7.15 be improved for jointly regular affine automorphisms , i.e., if we add the requirement that rP1 and 2 be invertible, although not necessaril y inverses of one another?
7.9. Let rP : AN -+ AN be an affine automorphism (not necessarily regular) and let d l deg and d2 = deg rP- 1. Prove that
. {h( (P )) h( - l (P ))} > h(P.)
mill
d1
'
da
- d 1d2
+
0 (1) "orallP E
"
~N (If1,).
""
"'>e
Exercises on Integrability and Reversibility The notions of integrability and reversibility play an important role in classical real and complex dynamics. Their algebraic analogues lead to dynamical systems with interesting arithmetic properties, which we explore in Exercises 7.10-7.14. Definition. An affine automorphism : AN -+ AN is said to be algebraically reversible if there is linear transformation 9 E G LN satisfying
det (g) = -1 ,
and
The terminology is meant to reflect the idea that conjugation by the involution 9 has the effect of reversing the flow of the map .
,2
7.10. Assume that rP is reversible, say 9 = - 1. Let , = rP 0 g. Prove that is the identity map. Thus 9 and , are both involutions, so a reversible map can always be written as a composition = , 0 9 of two, generally noncommuting, involutions.
430 7.11. Let a
Exercises
i- 0, let f (y) be a polynomial of degree d ~ 2, and let ¢: A2 -----t A?, ¢(x,y) = (y,ax + f(y))
be the associated Henon map. Suppose that ¢ is reversible. Prove that ¢ and its reversing involution 9 E GL 2 have one of the fol1owing forms: (a) a = 1, g(x, y) = (y, x). (b) a = 1, g(x, y) = (-y, -x), f satisfies f( -y) = f(y)· (c) a=-I, g(x,y) = (-y,-x), f satisfies f( -y) = - f(y)· 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 ¢ : AN -> AN, we let
Cp (¢)
= number of distinct orbits of ¢ in A2 (IFp).
For each of the fol1owing Henon maps, compute C p (¢) 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 C p (¢) / p. (Notice that Exercise 7.11 says that the maps in (a) and (c) are reversible, while the map in (b) is not reversible.) Definition. Let ¢ : AN -> AN 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 ¢-I : AN -> AN such that ¢ 0 ¢-I is the identity map wherever it is defined. The map ¢ is said to be algebraically integrable if there is a nonconstant rational function I : AN -> Al satisfying I 0 ¢ = I. 7.13. Let ¢: A 2
->
A 2 be the rational map
¢(x,y)
=
(y,-X-
Y::ll).
(a) Prove that ¢ is a rational automorphism. (b) Let ¢ and ¢-1 be the extensions of ¢ and ¢ -1 to maps jp'2 -> jp'2. Compute Z (¢ ) and Z ( ¢ -1), the sets of pointes) where ¢ and ¢ -1 are not defined, and verify that
Z(¢) n Z(¢-I) = 0. (c) Prove that ¢ is integrable by the function I(x, y) = x 2y + xy2 + x 2 + y2 + X + y. In other words, verify that! 0 ¢( 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 -> jp'2 (using dehomogenized coordinates on A 2) by
e)
by2 + dy+ ¢(x, y) = ( y, -x 2 b ay + y + C
.
Prove that ¢ is integrable by the function
I(x, y) = ax 2y2 + b(x 2y + xy2)
+ c(x 2 + y2) + dxy + e(x + y).
Exercises
431
7.15. Assume that ¢ is integrable by the function I. For each c E K, the set I(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 ofjp'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 7.16.
Theorem 7.52. ([172, Proposition 4.2]) Let ¢ : jp'N ---+ jp'N be a morphism ofdegree d ::::: 2 and let V C jp'N be a completely invariant hypersurjace, i.e., ¢-l(V) = V = ¢(V). Then V has at most N + 1 irreducible components. 7.16. Let ¢ = [¢o, ... , ¢ N 1be a morphism ¢ : jp'N ---+ jp'N of degree d given by homogeneous polynomials ¢i E qXo , ... , XN]. We say that such a map is a polynomial map if its last coordinate function is equal to x'fv. Equivalently, ¢ is a polynomial map if the inverse image of the hyperplane H = {X N = O} 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 ¢ : jp'N ---+ jp'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 ¢ : jp'N ---+ jp'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 7.17-7.22 describe Kawaguchi's construction [230] of canonical heights for regular affine automorphisms assuming the validity ofConjecture 7.18, which is presently known only in dimension 2 [230, 413]. Let ¢ : AN ---+ AN be a regular affine automorphism of degree at least 2 defined over Q and let
d1
= deg(¢)
and
We assume that Conjecture 7.18 is true, i.e., we assume that there is a constant C = C( ¢) ::::: 0 such that for all PEA N (Q),
Assumption:
11 h(¢(P))
+ 12 h(¢-1(P))::::: (1+
1dJh(P)-C. dl
(7.35)
For any point P E AN (Q), Kawaguchi defines canonical heightfunctions by the formulas
h+(P) = lim sup n-too
d~h(¢n(p)),
h-(P) = lim sup
1
h(P)
n-tCX)
= h+(P) + h-(P).
d~h(¢-n(p)),
(7.36)
2
(7.37)
7.17. Assuming that (7.35) is true, prove that the canonical height functions h +, h- , and h defined by (7.36) and (7.37) have the following properties: (a) h+(P) :::; h(P)
(b) h(P)
+ 0(1)
and
h-(P):::; h(P)
+ 0(1) < h(P) < 2h(P) + 0(1).
+ 0(1).
432
Exercises
(c) h+(P) 2 0
and
(d) h+(P) = 0
~
h-(P)20
and
h-(P) = 0
h(P) 20.
~
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 following transformation formulas:
h+(¢(P))
= d1h+(P),
i: (¢ -l(p)) = d2h- (P),
11 h(¢(P)) + 12 h(¢-l(p)) =
(1 + d1~J h(P).
(7.38) (7.39)
7.19. Suppose that and are two functions satisfying (7.39), and suppose further that
hi = hI! + 0(1). Prove that hi
= hI!.
7.20. Let ¢ : AN ~ AN be a regular affine automorphism satisfying (7.35) and let PEA N (Q) be a wandering point for ¢, i.e., P is not a periodic point. Prove that
Hence the constant 1 larger constant.
+
dl
1d2
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) = ¢,
(_1_ + _1_) log d1
log d2
10gT- (IOgh+(P) log d1
+ IOgh-(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 ~ AN 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
A o = 0,
dlA i d2Ei
Eo = 0,
-
(1 + d ld2)A i - l
+ d2Ai - 2 = 0, (1 + d ld2)E i - l + d lE i - 2 = 0,
and use a telescoping sum argument.) 7.23. ** Let ¢ : AN ~ AN be an automorphism defined over Qand denote the dynamical degree of ¢ by 8(¢) = lim deg(¢n)l/n. n~oo
We associate to ¢ the number
Remark 7.16 tells us that S (¢) satisfies S(¢)
< 2.
If ¢ is regular, then 8(¢) = deg(¢) and 8(¢-1) = deg(¢-l). If in addition ¢ satisfies assumption (7.35), then Exercises 7.18 and 7.20 imply that 1
S(¢)
= 1 + 8(¢)8(¢-1)
(a) Do there exist automorphisms ¢ : A N ~ A N of degree at least 2 satisfying S(¢) = I? What if we require that ¢ be algebraically stable? (See Remark 7.13.) (b) Do there exist automorphisms ¢ : A N ~ AN of degree at least 2 satisfying S (¢) = 2? (c) What are the possible values of S (¢) for automorphisms of AN? (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 ¢ : ]P'N (K) ~ ]P'N (K) be a morphism. Prove that ¢ is an open map, i.e., the image of an open set is an open set. Section 7.4. Dynamics on Surfaces with Involutions
7.25. Let G'k and H;*j be the quartic forms defined by (7.17). Prove that the following algorithm computes Ll and L2. (a) Let [x, y] E SA,B and write Ll([x, vl) = [x, y']. Then
[YoGo(x), -Yo Hal (x) - ylGO(x), -YOH02(X) - y2GO(x)] if Yo ::j= 0, v' =
{
[-ylHal (x) - Yo Gf(x), ylGf(x), -Y1H12(X) - y2Gf(x)]
ifYl ::j= 0,
[-y2Ho2(x) - YoG~ (x), ~Y2Hf2(X) ~ Y1G~(X),Y2G~(X)]
ifY2 ::j= 0.
(b) Let [x, y] E
x' =
{
SA,B
and write Ll([x,
vl)
= [x', y]. Then
[xoG!j(y), -xoHgl(y) - xlG!j(y), -x OH02(Y) - X2G!j(y)]
ifxo ::j= 0,
[-xlHgl(y) - xoGi(y),X1GJ!(y), -x lH12(y) - x2GiCy)]
if Xl ::j= 0,
[-x 2Hg2(y) - xoG§(y), -x 2Hf2(y) - xlG~(y), X2G~(y)]
ifx2 ::j= 0.
Exercises
434
7.26. The K3 surface given in Example 7.36 contains the following 12 points of small height:
PI
= ([0,1,1],[1,1,-1]),
P2 = ([1,0,0], [0,0,1]), P3
= ([0,1,0],[0,0,1]),
= ([0,0,1],[0,1,0]),
Pg = ([8,6,9],[-6,5,2]),
P6 = ([0,0,1], [1,0,0]),
P lO = ([1,0, -1]' [9,1,9]),
P5
P7
= ([3,1,3],[-3,3,2]),
P4 = ([1,0, -1], [0,1,0]), Ps = ([1,0,0], [0,7,1]),
Pll
= ([3,8,11],[1,1,-1]),
P12 = ([12,1, -20], [2,-4, 1]).
(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 Ix; I :s; 100, substitute into Land 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, 14521485737273461} find an example of a surface SA,S defined over IFp such that [1 and [2 are defined at every point of SA,S (JFp ) . (These examples can be used to complete the proof of Proposition 7.41.) 7.28. This exercise sketches a noncomputational proof of Proposition 7.41 using more advanced methods from algebraic geometry. (a) Let Sand 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 -> Sand 'ljJ' : T -> S' so that ¢ 0 'ljJ = 'ljJ' [198, v'5.5]. Do this so that 'ljJ is a minimal number of blowups and let E be an exceptional 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· K s' :s; E· Ks = -1, which contradicts the fact that K s ' = 0, since S' is a K3 surface.) (b) Prove that there is a proper Zariski closed set Z C lP's X lP'35 such that for all (A, bf B) ¢: Z, the surface SA,S is nonsingular. (Hint. Elimination theory says that the set of (A, bfB) E lP's X lP'35 such that SA,S is singular is a Zariski closed set. Thus it suffices to find a single (A, bfB) for which SA,bj B is nonsingular.)
(c) Combine (a) and (b) to prove Proposition 7.41. 7.29. Let P E SA,S with infinite A-orbit. Prove that the A-stabilizer of P,
{'ljJ
E
A : 'ljJ(P) = P},
has order either 1 or 2. 7.30. This exercise describes intersections on the surface SA,S. For the basics of intersection theory on surfaces, see, for example, [198, V §1]. (a) Let D: = piH and D 2 = P2H be the usual divisors in Pic(SA,S). Prove that
and
435
Exercises
(b) Let 0: = 2 + J3 and define divisors E+ and E- in PiC(SA,B) 0lR by the formulas
E+
= -D l + o:D2
= o.I),
E-
and
Prove that ~iE± = o:±l E~ and ~2E± = o:~l E~. (c) Prove that E+ . E+ = E- . E- = 0 and E+ . E-
=
- D2 .
120:.
7.31. (a) Prove that under composition, the involutions ~l, ~2 E Aut(SA,B) satisfy no relations other than ~i = ~~ = 1. Thus A is the free product of the subgroups generated by ~l and ~2. (Hint. Use Exercise 7.30. Apply a composition of ~l'S and ~2'S to E+ + Eand intersect with E+ .) (b) Show that A is isomorphic to the infinite (discrete) dihedral group
V
Vt
= oo
{t
2
j
:
i, j E Z}
= 1 and ts = rlt}
via the map
s (t
f---+ f---+
~l~2) .
~l
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,B fixed by some nontrivial element 'lj; E A and consider the intersection of C with the divisors E+ and Edefined in Exercise 7.30.) 7.33. Let 1>
= ~l
0 ~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,B) be the divisor D and 0: = 2 + V3 as usual. Prove that
= D l + D 2 = piH + P2H.
Let 1>
=
for all P E SA,B(K) and all n
~l
0
~2
?: o.
(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 j,,+ and j,,- be the canonical height functions constructed in Proposition 7.47. Assuming that SA,B(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 j" is replaced by either j"+ or j"- . 7.36. Let SA,B be defined over a number field K, let j,,+ and j,,- be the canonical height functions constructed in Proposition 7.47, and let j" = j,,+ + i:'. Fix a point Q E SA,B(K). (a) Prove that the product j,,+(P)j,,-(P) is the same for every point P E A(Q). This product measures, in a certain sense, the arithmetic complexity of the A-orbit of Q. Notice how j"+ (Q) (Q) naturally appears in Proposition 7.50 counting points of bounded height in the A-orbit of Q.
t.-
Exercises
436 (b) Prove that
(Here
0:
= 2 + .j3 as usual.)
7.37. Let SA,B be defined over a number field K and let Q E SA,B (in be a point whose Aorbit A(Q) is infinite. Further, let it = it+ + it-, and define a heightzetafunction for the Aorbit of Q by the series
Z(A(Q), s)
L _, h(Q)s
1_.
=
PEA(Q)
(a) (b) (c) (d)
Prove that the series defining Z(A(Q), s) converges on the half-plane Real(s) > O. 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.
7.38. Let SA,B be defined over a number field K and let P E SA,B(K) be a point whose Aorbit A(P) is Galois-invariant, i.e., if Q E A(P) and o E Gal(K/ K), then er(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 {I, 2} such that pj(1jJ(P)) E 1P'2(K). 7.39. Let V C jp'N biquadratic form,
X
jp'N be a variety given by the vanishing of N - 1 bilinear forms and one
LI(x,y) = ... = LN-I(X,y) = Q(x,y) = 0, and let PI : V ---+ jp'N and P2 : V ---+ jp'N be the usual projection maps PI (x, y) = x and P2(X,y) = y. (a) Prove that PI and P2 are generically 2-to-1, so they induce involutions LI : V ---+ V and L2 : V ---+ V. In other words, there are rational maps LI and L2 such that LI and L~ are the identity map wherever they are defined. (b) If N 2: 3, prove that LI and L2 are not morphisms. 7.40. Let a E K*. The Markoffequation
u, : x 2 + l + Z2 = axyz defines an affine surface in A 3 . (a) Prove that there are involutions
L12, L13,
LI2(X,y,Z)
and L23 of M a defined by the formulas
=
(x,y,axy-z),
LI3(X, y, z) = (x, axz - y, z), L23(X, y, z)
= (ayz -
x, y, z).
Explain how these involutions correspond to natural double covers M a by projection maps A3 ---+ A2 .
---+
A 2 induced
437
Exercises (b) Prove that the involutions
~ij
Ma =
do not extend to morphisms on the projective variety {x
2
w
+ y 2 w + z2 w =
axyz}
C p3,
and determine the points at which they fail to be defined. (c) Find a birational map p2 ---+ 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 ofW.) In particular, this implies that Ma(Q) contains many points. (d) Prove that every point in M 3(Z) with positive coordinates can be obtained by starting 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 M 3 (Z) with x :s; y :s; z. Let
N(T) =
# {normalized Markoff triples (x, y, z) with z :s; T}.
Prove that there are positive constants C1 and C2 such that (7.41 ) More precisely, prove that there is a constant c such that
N(T) (g)
= c(logT)2 + O((1ogT)(loglogT)2).
(7.42)
** Let (Xl, Y1, Zl) and (X2, Y2, Z2) be normalized Markoff triples. Prove that if Zl = Z2, then also Xl = 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 Be C p1 x p1 X p1 are described by the vanishing of a trihomogeneous polynomial of degree 2,
Q(x,y, z) =
L
CijkimnXiXjYkYiZmZn
=
o.
O.¢ (a) set of points of period n, 18 Per n(¢) Per~*(¢) set of points of exact period n, 18 ~; space of differential one-forms, 19 the Mande1brot set, 26 M multiplicative group, 29 Gm the d th Chebyshev polynomial, 29 Td Ga the additive group, 30 addition on an elliptic curve, 31 EB Lattes map associated to multiplication by d, 32 ¢E,d Lattes map associated to an endomorphism u, 32 ¢E,u complex uniformization of an elliptic curve E, 33 7/JE Weierstrass !:J function, 34 !:J ~(¢, a) residue fixed-point index of ¢ at a, 38 the nth dynatomic polynomial, 39 q>~ (z) v-adic chordal metric on jp'l, 45 pv p reduction of the point P modulo a prime, 48 ¢ reduction of a rational map ¢ modulo a prime, 52 the resultant of A and B, 53 Res(A,B) Res(¢) resultant of a rational map, 56 ",(P1 , P2, P3 , P4) cross-ratio of P 1 , P2, P3 , P4, 71 K(¢) the filled Julia set of ¢, 74 (multiplicative) height of a rational point, 82 H(P) MQ set of standard absolute values on Q, 82 the usual absolute value on JR, 82 I . 100
¢o
445
446 ordp(a)
I . Ip MK M 'j(
MfJ< RK
Rs nv HK (P) H(P)
k K(P)
VI V(I) I(V)
IPlv IfIv c5 v (m)
hK h 0(1) h
Av ). .v
Res(c,i» Per ~· (cP) K n. Gn.
Wreath(H, S )
G~.
K n. J-t
C(P/K)
c5p J-tp S
'P J-t
e;«;
, 149 that nth dynatomic polynomial of ¢, 149 dehomogenized period and dynatomic polynomials, 149 the set of points of period n for ¢, 150 the set of points offormal period n for ¢, 150 the set of points of primitive period n for ¢, 150 number of points of formal period n for a map of degree d, 150 order of the period polynomial at the point P , 151 order of the dynatomic polynomial at the point P , 151
List of Notation
¢c(z) ~(c,z)
YI(n) XI(n) Formal(n)
Yo(n) Xo(n)
e
Fa(X, Y) [a,b] Ratd jp'~ Md
(-) Q[Ratd]PGL2 SL 2 PSL 2 Rat d Rat dS
Md
M ds
F¢,n,G¢,n
A n (¢) A~(¢)
a;n)(¢) ;;n)(¢) O"d,N O"d,N
A(¢) (T
M2
[¢] [¢]K Aut(¢) Twist(¢/ K) Twist(X/K)
g" HI(r,A) Gf Kf ordp(P) FA,GA
Ep(¢) 9l¢ lJ1¢ Up,G
a¢/K
8n ,¢(x )
447 the quadratic polynomial Z2 + C, ISS the nth dynatomic polynomial for Z2 + c, 157 (affine) dynatomic modular curve for z2 + c, 157 (projective) dynatomic modular curve for Z2 + c, 157 PGL 2-classes of quadratic with point offormal period n, 158 the quotient of YI (n) by ¢, 161 the quotient of Xl (n) by ¢, 161 conformal isomorphism to the complement of Mandelbrot set, 167 the homogeneous polynomial aoX d + alXd-Iy + ... + adyd, 169 the point lao, ... .o«, bo, . . . ,b d] in jp'2d+ 1, 169 the set of rational maps of degree d, 169 the projective line over V, 171 the moduli space of conjugacy classes of maps of degree d, 174 map from Ratd to Md, 174 the ring of PGL 2-invariant functions on Ratz., 175 the special linear group, 175 the projective special linear group, 175 set of stable rational maps, 178 set of semistable rational maps, 178 stable completion of Md, 178 semistable completion of Md, 178 coordinate functions of the nth iterate of ¢, 181 the n-multiplier spectrum of ¢, 182 the formal n-multiplier spectrum of ¢, 182 symmetric polynomial of n-periodic multipliers of ¢, 183 symmetric polynomial of formal n-periodic multipliers of ¢, 183 map of Md using a;n) with n :::; N, 187
;;n)
map of Md using with n < N, 187 the multiplier spectrum of ¢, 187 map Rat, ---> A 2 inducing an isomorphism M 2 ~ A 2 , 188 the completion M 2 = M 2s of M2, 194 set of rational maps K -equivalent to ¢, 195 set of rational maps K -equivalent to ¢, 195 the automorphism group of ¢, 196 the set of twists of the rational map ¢, 197 the set of twists of the object X, 199 the l-cocycle associated to a twist, 201 cohomology set (group), 202 subgroup of Gal( K / K) associated to ¢, 207 field of moduli of ¢, 207 minimum ord, of the coefficients of the polynomial P, 218 new coordinate functions for conjugate of [F, G], 218 exponent of the minimal resultant of ¢, 220 the (global) minimal resultant of ¢, 220 product of the primes of bad reduction for ¢, 221 ideal connecting minimal resultant to resultant of a model, 222 the Weierstrass class of ¢ over K, 223 polynomial whose roots are multipliers of ¢, 225
448
List of Notation
Fm,n(X, Y) :'n,n(X, Y) (S¢)(z) BiCritd M~iCrit
Cp R=RK R=R'K 9J1
D(a,r) D(a,r) Sea, r) D(Q, r) D(Q,r) ord., (¢)
11¢llr,a
v(z)
F(¢) 3(¢) F(¢,K) 3(¢,K) A = A(¢)
u.: N
sN
D Dc10sed
D open De(¢) 71"
A:
II(x,Y)llv
9q,(x,y) )..,E
DB
([x, y])
~a,a ~a,r ~a,r
IlfliR
DB
-B
DR
1l'R La,r La,r
U(j,B) V(j,B) ffi.a,r
A:,T
generalized period polynomial, 227 generalized dynatomic polynomial, 227 the Schwarzian derivative of ¢, 231 the space ofbicritical rational maps, 233 the moduli space ofbicritical rational maps, 233 the completion of the algebraic closure ofQp, 239 the ring of integers of a local field K, 242 the unit group of the ring of integers of a local field K, 242 the maximal ideal of the ring of integers ofa local field K, 242 open disk of radius r at a, 242 closed disk of radius r at a, 242 circle of radius r at a, 242 open disk of radius r in jp'l(K), 243 closed disk of radius r in jp'l (K), 243 order of vanishing of a holomorphic function, 244 norm of ¢ relative to the disk D(a, r), 247 valuation -logp 14 249 the Fatou set of ¢, 255 the Julia set of ¢, 255 the K -valued points of the Fatou set of ¢, 255 the K -valued points of the Julia set of ¢, 255 set of Il.0ints with bou~ded ¢n (z) 258 disks D(O, lip) and D(l, lip), 258 the set of natural numbers, 258 set of sequences chosen from the set S, 258 set of "disks" in a topological space, 277 standard collection of closed disks in jp'l (rep), 277 standard collection of open disks in jp'l (Cp ) , 278 set of disk components of the Fatou set, 283 projection map A2 -+ jp'l, 287 the affine plane with (0, 0) removed, 287 sup norm of x and y, 288 Green function attached to : A 2 -+ A2 , 288 local canonical height associated to ¢ and E, 291 the unit Berkovich disk, 295 Type-I point in Berkovich disk DB, 295 Type-II or III point in Berkovich disk DB, 295 Type-IV point in Berkovich disk DB, 295 Gauss norm of f E Cp[z], 296 the closed unit Berkovich disk, 297 the closed Berkovich disk of radius R, 297 the Tate algebra, 297 line segment in Berkovich disk, 298 line segment in Berkovich disk for Type-IV point, 298 open set in Gel'fond topology on Berkovich disk, 300 open set in Gel'fond topology on Berkovich disk, 300 closed branch of Berkovich disk rooted at (a, r), 300 open branch of Berkovich disk rooted at (a, r), 300
I
I,
List of Notation ~O,R B
Gauss point of Berkovich disk of radius R, 301 the Berkovich affine line, 301 the Berkovich projective line, 302 annulus in Berkovich space, 302 branch at infinity of Berkovich projective line, 302 radius of the image disk ¢(D(a, r)), 305 R(¢, a, r) :JB (¢) the Berkovich Julia set of ¢, 306 ;:B(¢) the Berkovich Fatou set of tb, 306 canonical measure on Berkovich space associated to ¢, 306 J1¢ Resz=o (¢(z) dz) residueof¢(z) atz = a, 313 modified Green function, 318 9 local canonical height associated to a divisor, 320 ).,¢,D divisor of the rational function g, 320 Dg path metric on the Berkovich disk, 323 I\;(~a,r, ~b,s) the Hsia kernel on the Berkovich disk, 323 8(~a,r, ~b,s) the power map zd, 325 Md Gm multiplicative group, 325 the group of nth roots of unity, 326 ILn the dth Chebyshev polynomial, 329 Td point at infinity on an elliptic curve, 336 the set of points of E defined over the field K, 337 E(K) Div(E) the group of divisors on E, 339 deg(D) the degree of the divisor D, 339 sum the summation map on the divisor group of E, 339 div(f) the divisor associated to the rational function f, 339 ordp(f) the order of the zero (pole) of fat P, 339 Pic (E) the Picard group (group of principal divisors on E), 339 cJ; the dual isogeny of 1/J, 340 End(E) the endomorphism ring of E, 340 Aut(E) the automorphism group of E, 340 E the reduction of E modulo p, 342 kernel of the endomorphism 1/J, 343 E[1/J] kernel of multiplication-by-m map, 343 E[m] E tors torsion subgroup of the elliptic curve E, 343 the Tate module of E, 344 Te(E) the fl-adic representation on an elliptic curve, 344 PE,e the invariant differential on E, 344 WE group of fractional ideals of F, 350 IF elliptic curves with CM by RF, 350 E££(RF) ideal class group of F, 350 CF class number of F, 350 hF CritPt¢ set of critical points of ¢, 353 CritVal¢ set of critical values of ¢, 353 PostCrit¢ forward orbit of critical values of ¢, 353 E(1,) the set of points in E(Q) with integer coordinates, 372 I/(E) number of primes dividing the denominator of j(E), 372 P homogenization of the polynomial F, 389 Z(¢) locus of indeterminacy of the rational map ¢, 389
A IP'B Ann B
ffi.::'
o
449
450
List of Notation
lift of the rational map ¢ to A N + 1 , 389 the divisor group of V, 403 local ring of V at W, 403 order of vanishing of I along W, 403 ordw(J) the Picard group of V, 403 Pic(V) linear equivalence of divisors, 404 D 1 '" D 2 ¢*D pullback of the divisor D, 404 pushforward of the divisor D, 404 I*D ¢* pullback homomorphism on Picard groups, 405 degree map on Div(jp'N) and Pic(jp'N), 405 deg base locus of the divisor D, 406 Base(D) vector space associated to the divisor D, 406 L(D) dimension of the vector space L(D), 406 £(D) rational map associated to the divisor D, 406 ¢D height on the variety V associated to the divisor D, 407 hV,D the K3 surface in jp'2 x jp'2 determined by A and B, 410 SA,B projections of a K3 surface to jp'2, 410 Pl,P2 involutions on a K3 surface SA,B, 411 Ll, L2 linear forms associated to a K3 surface, 412 Lj,Lr quadratic forms associated to a K3 surface, 412 Q'k£,Qrj G~, Htj, H~ quartic forms associated to a K3 surface, 413 subgroup of Aut(SA,B) generated by Ll and L2, 418 A the A-orbit of P on the K3 surface SA,B, 418 A(P) divisors on the K3 surface SA,B, 418 D 1,D2 h+,hheight functions on SA,B, 420 it+,itcanonical heights on the K3 surface SA,B, 421 the sum it+ + it- on the K3 surface SA,B, 423 it the order of the stabilizer of a point on a K3 surface, 425 Jl(Q) number of distinct IFp-orbits of ¢, 430 Cp (¢ ) it+,it-,it canonical heights on AN for a regular affine automorphism, 431 E+,Eeigendivisors in Pic(SA,B) @ JR, 435 subspace ofPic(Sc) generated by D 1 , D 2 , and D 3 , 438 D Div(V) Ov,w
ct.
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s.-s.
Index composition of elementary maps and Henon maps, 402 composition of involutions, 429 degree, 394 degree of inverse, 394, 395 disjoint indeterminacy locus, 394 dynamical degree, 433 fixed point, 427 height inequality,399, 431 height of periodic points, 402 Henon map, 390 indeterminacy locus, 389, 392 dimension of, 394 disjoint, 394 of inverse, 392 of iterate, 394 inversemay have different degree, 390 iterate may have wrong degree, 390 number of orbits over a finite field, 430 periodic points of, 394, 400 regular, 391, 394, 427 periodic points of, 428 reversible Henon map, 430 affine coordinate ring of M 2 , 189 of Md, 176, 179 as 2-scheme, 186, 189 183 contains a
ABC conjecture, 372, 373 abelian group preperiodic point equals torsion point, 2, 41, 326 torsion subgroup, 2 abelian variety, 409, 442 Bogomolov conjecture, 129 field of definition, 217 field of moduli, 217 Manin-Murnford conjecture, 127 Neron local height, 104 Raynaud's theorem, 127 absolute height, 85 logarithmic, 93 absolute value, 43 archimedean, 82, 83 completion of a field at an, 83 effect of polynomial map on, 288 equivalent, 44 extension formula, 83, 84 local degree, 83 nonarchimedean, 44, 82, 83, 242 of a point, 90, 288 of a polynomial, 91, 288 on function fields, 44 on IQ, 82, 312 p-adic, 44, 82 product formula, 83 standard set of (MK), 83 abstract dynamical system, I realizable sequence, 6 addition on an elliptic curve, 31 additive group, 30 additivity of height, 408, 421 affine automorphism, 390, 427 algebraically reversible, 429 algebraically stable, 396, 433 canonical height, 431
in),
;'i
n
) , 183,232 contains of Rat.i, 169, 174 affine line, Berkovich, 301 affine minimal model of a rational map, 112,372,385 affine minimal rational map, 372 affine morphism, 375, 388 algebraically stable, 396,433 degree of composition, 392
473
474 affine morphism (continued) dynamical degree, 397,428 finite quotient, 376 homogenization, 389 indeterminacy locus, 389, 392, 427 iterates, 376 jointly regular, 397, 429 lift of, 389 uniquely determines endomorphism and translation, 376 affine plane absolute value, 288 lift of rational map, 287 projection to projective line, 287 with origin removed, 287 affine space absolute value, 90 height, 397 projection to projective space, 287 algebraic closure, 85 algebraic dynamics, 6 algebraic entropy, 397 algebraic family, 159,230 field of definition, 159 induces map to dynatomic curve, 159 oflP'l,s, 171 of morphisms, 171 of quadratic maps, 194 algebraic geometry, 402 algebraic group, 325 commutative, 375 conjugation is action on, 173 finite quotient, 376 iterates of affine morphism, 376 simple, 377 universal cover, 375 algebraic point in Julia set, 40 algebraic set, 90 attached to an ideal, 90 ideal of, 90 algebraic variety, 89,147,159 dynamically affine, 376 quotient by finite group, 161 quotient by infinite group, 161, 174 algebraically closed field, 26, 242 C p is an, 239 algebraically integrable map, 430
Index algebraically reversible affine automorphism, 429 algebraically stable morphism, 396, 433 degree, 396 indeterminacy locus, 396 ample divisor, 406 bounded height is finite set, 408 analytic function, see holomorphic function annulus, 302 archimedean absolute value, 83 arithmetic complexity height in Md, 221 minimal discriminant, 221 of orbit on K3 surface, 435 arithmetic function, 6 Arzela-Ascoli Theorem, 25 attracting basin of 00, 74,138,140,141 attracting periodic point, 19,47,326,362 accumlates on a critical point, 78 basin of attraction, 310 every periodic point is an, 78 immediate basin of attraction, 310 in Berkovich space, 310, 324 in Fatou set, 22, 40 limit is repelling, 274 of Type I, 310 reduction is critical, 78 automorphism action on critical point, 234 affine, 390, 427 affine regular, 394 algebraically reversible, 429 canonical height, 431 finitely many periodic points, 435 height inequality, 399,431 height of periodic points, 402 induced by degree 2 map, 411, 436, 437 is subgroup ofPGL2, 234 nondegenerate, 226 of order two, 96, 197,235 rational,430 automorphism group, 340, 341 containing (nZ, 205, 234 contains 1-L2' 235 cyclic, 197, 204 dihedral, 217, 234, 327 extension of cocycle to PGL2, 203
Index automorphism group (continued) involution, 410 is finite, I 96 list of possible , 197 no twists if trivial, 198 noncommuting involution s, 4 10, 437 of Chebyshev polynom ial, 332, 336 in characteristic p, 381 of conjugate, 196, 234 of elliptic curve, 365 of monomial map, 327 in characteristic p, 380 of power map, 327 in characteristic p, 380 of projective line, 10 of z d ,234 order two, 198, 200 points stabilized by, 425, 434 polynomial map with nontrivial, 234 rational map, 196 symmetric group , 197 trivial for most maps, 199, 234 {Z±l }, 205, 234 Backward orbit dense for shift map, 314 dense in Berkovich Julia set, 3 11 dense in Julia set, 267 equidistributed, 315 equidistributed for shift map, 3 15 of rational map , 109, 142 bad reduction becomes good reduction, 218 Berkovich Julia set of map with, 307,311 complex dynamics has, 294 composition of maps, 77 composition of maps can be good , 77 Green function of map with, 294, 318 of if>(P) is not ~ ( p) , 77 product of prime s of, 22 1 Baire category theorem , 268 Baker 's theorem , 107, 155 Banach algebra, 298 Baragar 's theorem, 443
475 base locus, 406, 408 basin of attraction, 74,138,140,141 ,310 immediate, 3 10, 311 Benedetto's theorem, 284- 286 Berkovich affine line, 301 as ringed space, 304 contained in Berkovich projective line, 303 contains C p , 302, 304, 323 Gel'fond topology restricts to Cp topology, 304, 323 has no Gauss point, 302 is Hausdorff, 304 is locally compact, 304 is tree, 302 is uniquely path connected, 304 no Type-IV fixed points , 321 polynomi al map, 304 Berkovich disk, 295 annulu s, 302 base of open sets, 300 big model, 323 branch, 300 closed branch, 300 closed of radius R, 297, 30 I closed unit , 297 conta ins unit disk in C p , 295, 301, 323 dynamic s of z 2 , 307 Gauss point, 297, 30 I Gel'fond topology restricts to C p topology, 301, 323 Gel' fond topology, 300, 322 good reduction polynomial map, 321, 322 Hsia kernel, 323 is compact, 295, 300, 304 is connected, 295 is Hausdorff, 300, 304 is metric space, 295 is path-connected , 300 is set of bounded seminorms, 297 is set of seminonn s, 296 is uniquel y path connected, 304 Julia set of good reduction map, 322 line segment from Gauss point to Type-IV point , 322 line segment in, 298 neutral fixed point, 322
Index
476 Berkovich disk (continued) of radius R, 297,301 open branch, 300 potential theory, 323 radius of Type-IV point is positive, 296 seminorm associated to each type of point, 297 small model, 323 topology on, 299 union of annulus and open branch , 302 visualiz ing, 298 Berkovich Julia set backward orbit dense in, 311 equals closure of repelling periodic points, 311 has empty interior, 311 is connected or infinitely many components, 311 is perfect set, 307, 311 is uncountable, 307, 311 Berkovich project ive line, 301, 302 as homogeneous two-variable seminorms, 303, 305 as ringed space, 304 attracting fixed point, 324 basin of attraction, 310 branch at infinity, 302 canonical measure, 306 support of, 307 contains Berkovich affine line, 303 contains ]pI (iC p ) , 304, 323 domain of quasiperiodicity, 310, 311 dynamics of Z2, 307 Fatou set, 306, 311 contains classical, 306 Gel'fond topology restricts to iC p topology, 304, 323 immediate basin of attraction , 310, 311 is compact, 304 is Hausdorff, 304 is uniquel y path connected, 304 Julia set, 306 contains classical, 306 equals closure of repelling periodic points, 311
is nonempty, 307 of bad reduction map, 307, 311 of good reduction map, 307, 311, 322 Montel theorem , 310, 311 rational map, 305 recurrent point, 310 Berkovich space, 294 attracting fixed point, 324 attracting periodic point, 310 basin of attraction, 310 canonical measure , 6, 306 support of, 307 connectivity of, 276 different centers yield same point, 295 domain of quasiperiodicity, 310, 311 dynamics of Z2 , 307 dynamics on, 304 Fatou set, 306, 311 fixed Gauss point, 321 Gauss point , 297 Gel'fond topology, 322 good reduction polynomial map, 321, 322 Hsia kernel, 323 immediate basin of attraction, 310, 311 invariant measure, 104 is set of seminorms, 296 Julia set, 306 equals closure of repelling periodic points, 311 is nonempty, 307 of bad reduction map, 307, 311 of good reduction map, 307, 311, 322 Laplacian oflocal canonical height,
104 line segment from Gauss point to Type-IV point, 322 Montel theorem, 310, 311 neutral fixed point, 322 path metric, 306, 323 potential theory, 323 radius of a point , 296 radius of Type-IV point is positive, 296
Index Berkovich space (continued) recurrent point, 310 repelling periodic point , 310 rigid analytic space, 298 seminorm associated to each type of point, 297 topology on, 299 Type-I-IV points , 295 Bezivin 's theorem , 274, 275 Bezout 's theorem, 428 bicritical rational map, 233 bidegree, 405, 410 bifurcation point, 165 bifurcation polynomial, 165 is a power, 165,226,227 big model of Berkovich disk, 323 big-Oh notation, 93 bihomogeneous polynomial, 405, 410 binomial theorem, 206, 261 birational map on Markoff variety, 437 Bogomolov conjecture, 129 dynamical, 129 Borel probability measure , 127 bounded height finitely many points of, 86, 407 is finite set, 408 Misiurewicz points have, 167 bounded seminorm automatically nonarchimedean, 321 properties, 321 set of is Berkovich disk, 297 bounded seminorm on Cp[z], 296 branch at infinity in jp'B, 302 branch of Berkovich disk, 300 Brauer group, 204 Brauer-Siegel theorem, 368 Brolin measure, 307 bulb, 166 Canonical divisor, 214 canonical height, 97, 99, 100,287 algorithm to compute, 99, 318 associated to an eigendivisor class, 104 commuting maps have same, 137 conjugation invariance, 137 finitely many points of bounded , 423
477 infinitely many points of bounded, 435 is sum of Green functions, 290, 318 Lehmer's conjecture, 101, 138 local, 102, 291, 320 local for a polynomial map, 140, 141 lower bound for, 101, 137, 138,221 normalization conditions, 422 normalized local, 141 of point in Fatou or Julia set, 138 on abelian variety, 409 on elliptic curve, 409 on K3 surface, 421, 423, 426, 435, 436 with three involutions, 438 product h+ k, 435 quadratic map , 137, 138 regular affine automorphism, 431 sum of local canonical heights, 103, 293 Tate construction, 97 transformation formulas , 422, 432 uniqueness, 98 zero iff finite orbit, 423, 431 zero iff preperiodic point , 99 zeta function , 436 canonical measure, 127,306 on Berkovich space, 306 support is Julia set, 307 supported at Gauss point, 307 cardiod, 166 Cauchy estimate for norm of derivative, 252,313 Cauchy residue theorem, 20, 248, 314 Cauchy sequence, 98, 240, 296 chain rule, 18, 360 calculate multiplier with, 19,47 change of variables, II chaos, 3, 22 Chebyshev polynomial, 29, 30, 95, 329 algebraic properties, 41, 329 alternati ve normalization, 30, 331 automorphism group, 332, 336 in characteristic p, 381 binomial coefficient identity, 380 commutativity, 41, 329, 332, 378 defining equat ion, 30 derivative identity, 381
478 Chebyshev polynomial (continued) derivative in characteristic 2, 381 differential equation characterizing, 334 dynamical properties, 41 even, 41, 329 explicit formula, 41, 329 fixed point, 332, 380 in characteristic p, 381 in characteristic 2, 381 in characteristic p, 380, 381 is monic of degree d, 329 Julia set, 30, 41, 336 multiplierof fixedpoint, 332, 380 in characteristic p, 381 multipliersummationformula, 380 not conjugateto monomial, 381 odd, 41, 329 periodic point, 41 recurrencerelation, 41, 329 twist, 336 uniform boundedness of periodic points, 137 value of T 2 , T 3 , T4 , 41 chordal metric, 102 complex, II , 144 dehomogenizedform, I I distance between closed disks, 268 effect of linear fractional transformation, 76 equals distance on Riemann sphere, 11 ,35 invariant maps for, 35 inversion is isometry, 35 Lipschitzproperty satisfied, II , 24, 36,56 on inverseimage of rational map, 115,144 related to Euclideandistance, 119 relation to cross-ratio,72 resultant measuresexpansion, 56, 76 v-adic, 45, 46,144,243 value is between zero and one, II chordal sup norm, 269 circle, 242 is open and closed, 243 class field theory, 202, 204 class number, 350
Index of quadratic imaginaryfield, 368 closed branch of Berkovich disk, 300 closed disk, 242 closed under addition, 243 distancebetween, 268 equivalent nested sequences, 322 Gel'fond topology restricts to iCp topology, 30I, 323 image of D(O, 1) by PGL 2 , 317 image under inversion, 321 in IP'! , 243 is open, 243 is ring of integers, 243 nested sequence, 295, 322 radius of image by polynomial, 321 rational radius, 243, 278, 295 standard collection, 277 CM, see complexmultiplication coarse moduli space, 160 coboundary, 201 , 202, 209 trivial when extended, 203 cocycle, 20I, 202 cohomologous, 202 determines field of moduli, 209 PGL 2 gives twist of IP!, 211 cocycle corresponds to twist?, 203 Cohn's theorem, 443 cohomologygroup, 202 maps to Brauer group, 204 twists map to, 202, 236 cohomologyset, 202 commutative algebraic group, 375 universal cover, 375 commuting maps, 8 Chebyshev polynomial, 332 Fatou set, 378 have same canonicalheight, 137 have same preperiodicpoints, 38 higher-dimensional, 378 Julia set, 378 Lattes, 382 monomial, 325, 326, 380 polynomial, 378 power map, 326, 380 commutingrational map, 378 complete field, 240 circle, 242 closed disk, 242 iCp, 242
Index complete field (continued) extension, 241 holomorphic function, 244 open disk, 242 power series converges iff an -> 0, 244 projective space compact iff K locally compact, 243 completely invariant set, 17,258,266 boundary of Julia set, 23 closed,267 component at infinity, 40 contains Berkovich Julia set, 311 Fatou set, 23, 40, 266 finite, 16, 266 Julia set, 23, 40, 266 completion of a field, 83 valued,241 complex chordal metric, II, 144 complex multiplication, 32, 341, 348 by Z[p], 341 by Z[iJ, 341 degree of an endomorphism, 349 fractional ideal gives curve with, 350 ideal class group, 350 is quadratic imaginary, 349 complex orbifold, 176 complex projective line, see projective line complex torus, 33 fundamental domain, 33,127 conformal map of Mandelbrot set complement, 167 conjugacy class in space of rational maps, 174 conjugation, 11 canonical height is invariant, 137 linear, 11, 173 new coordinate functions, 218 connectivity Berkovich space, 276 disk,276 Julia set, 26, 165 Mandelbrot set, 167 rigid analytic, 276 continuity, 22 equi-, 254, 264, 265, 271, 306, 313 uniform, 39, 254, 313
479 continuous function, 10, 253 on metric space, 22 convex hull, 249 coordinate ring, see affine coordinate ring counting lemma, 425 C p,239 bounded seminorm on on Cp[z], 296 Cauchy sequence, 296 compact Julia set example, 275 is algebraically closed, 239, 242 is complete, 242 is contained in Berkovich affine line, 302 is totally disconnected, 239, 276, 294 not locally compact, 239, 294 projective space is bounded, 243 projective space not compact, 243 unit disk contained in Berkovich disk, 295, 301, 323 valuation, 249 value group is Q, 294 critical point, 12,284,353 bounded orbit, 26, 165 distinct modulo p, 237 image under automorphism, 234 Julia set with no, 285 number of for rational map, 14,37 number offor separable map, 37 of Lattes projection, 384 orbit for quadratic map, 26, 165 periodic, 38 postcritical set, 280 ramification index, 12,36 rational map with two, 233 reduction is if point is attracting, 78 repelling, 166 strictly preperiodic, 26, 166 tamely ramified, 37 totally ramified, 12 unbounded orbit, 26 whose reduction is periodic, 78 wildly ramified, 37 critical value, 353, 382 distinct modulo p, 237 cross-ratio, 71, 132,348 j-invariant in terms of, 348 permutation invariant, 79
480 cross-ratio (continued) relation to chordal metric, 72 cubes, sum of two, 141 curve cubic, 30 function with given divisor, 214 genus, 15,37 of genus at least two, 107 quotient by finite group, 161 Riemann-Roch theorem, 214 twist, 200, 211, 215, 236 cuspidal cubic map, 193 cycle of n-periodic points, 150, 226 cyclic automorphism group, 204 cyclic group, 197 cyclotomic extension, maximal, 95 cyclotomic field, 69, 123 dynamical analog of, 123 Frobenius, 130 Galois group, 129 cyclotomic polynomial, 39,148,224 resultant with multiplier polynomial, 226 cyclotomic unit, 69, 72, 133,328 action of Galois, 133 Degenerate point for involution on K3 surface, 415 degenerate quadratic map, 194 degree map on divisor class group, 405 map on divisor group, 405 map on function fields, 44 map on Picard group, 405 of algebraically stable morphism, 396 of composition of affine morphisms, 392 of divisor, 339, 405 of endomorphism, 349 of family of morphisms, 171 of flexible Lattes map, 357 of inverse of affine automorphism, 390 of isogeny, 340 of iterate of affine automorphism, 390 of Lattes map, 34 of morphism, 389
Index of polynomial, 389 of rational map, 10,89,388,389 two map induces involution, 411 universal map of given, 171 degree 2 map induces involution, 436, 437 derivative formal, 12,37,313 norm of, 252, 313 of Chebyshev polynomial, 381 in characteristic 2, 381 dessins d' enfant, 217 diagonal map, 150,216,226 differential equation characterizes Chebyshev polynomial, 334 differential form used to define multiplier, 19 dihedral group, 197,217,234,327 infinite, 419, 435 dimension of family of K3 surfaces, 412, 438 Diophantine approximation, 104 Baker's theorem, 107 Dirichlet's theorem, 104, 141 effective version, 107 Roth's theorem, 104, 108 Diophantine equation, 4, 104, 202 Pell, 106 Thue's theorem, 105 Dirac measure, 128 Dirichlet's theorem on Diophantine approximation, 104, 141 disconnected Julia set, 26 discrete dynamical system, 1 discrete periodic point, 428 discrete valuation, 44 discriminant, 132, 134, 146,368 elliptic curve, 30, 337, 343, 370, 385 minimal, 221, 342 of dynatomic polynomial, 226, 227 disk,242 Berkovich, 295 c1osed,242 closed under addition, 243 connected, 317 distance between, 268 image of unit disk by PG L2, 317 image under inversion, 321
Index disk (continued) in pI , 243 in topological space, 277, 317 nested sequence, 295 open, 242 polynomial sends to disk, 312 power series sends to disk, 252, 305 radius of image by polynomial, 32 1 rational radius , 243, 248, 252, 278, 295 Siegel,28 standard collection, 277 sup norm of function on, 247 disk component, 277, 317 image may not be disk component, 283 is maximal disk connected set, 317 is open, 317 map on induced by rational map, 283 no wandering, 284 nonarchimedean has simple form, 278
of (z2 - z)/p, 283 open set is disjoint union of, 277, 317 periodic, 283 preperiodic, 283 same as path-connected component over C, 277 set of, 283 wandering, 283 disk connected set, 276, 317 maximal is disk component, 317 disk domain, no wandering, 284, 285 distance between closed disks, 268 between images via linear fractional transformation, 269, 316 logarithmic, 102 division polynomial, 362, 383 resultant, 383 divisor, 338,403 action of involution, 418, 419, 438 ample, 406 associated to a rational function , 339 associated vector space L(D ), 406 base locus, 406,408
481 degree of, 339, 405 effective, 406 positivity of height, 408 group of principal, 339 intersection on K3 surface, 434 is difference of effective, 406 linear equivalence, 404 linearly equivalent give same height, 408 local ring, 403 of a rational function , 320, 403 of degree zero, 405 on K3 surface, 418 positive, see effective divisor principal , 320, 403, 405 principal iff degree and sum are zero, 339, 409 pullback,404 pullback of linearly equivalent, 405 pushforward, 404 rational map associated to, 406 Riemann-Roch theorem , 214 support , 403 very ample, 406 divisor class ample, 406 base locus, 406 very ample, 406 divisor class group, 403 degree map, 405 exact sequenc e, 404 height map on, 408 pullback homomorphism, 405 divisor group, 339, 403 degree map, 339,405 summation map, 339 domain of quasi periodicity, 310, 311 domain, no wandering, 276, 284, 285 Douady-Hubbard theorem, 167 double cover, 410 of elliptic curve, 347 Drinfeld module , 6 dual isogeny, 340, 352 duplication map elliptic curve , 32, 338, 351, 370 on Q/Z, 168,229 dynamical Bogomolov conjecture, 129 dynamical degree, 397, 428, 433 dynamical Galois equidistribution, 128
482
Index
dynamical Lehmer conjecture, 101, 138 dynamical Manin-Mumford conjecture, 127 dynamical system, 1 abstract, 1 algebraic, 6 discrete, 1 integrable, 429, 430 iteration of arithmetic functions, 6 on Drinfeld modules, 6 on finite field, 2, 5 on finite set, 1 on foliated spaces, 6 on function field, 5 on group, 2 on metric space, 3 on topological group, 8 on topological space, 3 reversible, 429 symbolic, 258,314 dynamical unit, 69 action of Galois, 133 for power map, 328 generic monic polynomial, 146 generic rational map, 146 polynomial map, 71, 132 quadratic map, 134, 145, 146 rational map, 71, 72, 79, 132, 133 dynamically affine morphism, 376 dynatomic curve, 156, 157, 161 action of Z2 + con, 161 genus, 164, 230 is coarse moduli space, 160 is fine moduli space, 230 is irreducible, 164 is modular curve, 158 map X l(2) -+ X o(2), 162 map X l(3) -+ Xo(3), 230 points solve moduli problem, 159 quadratic polynomial, 156 quotient, 161
Xo(4), Xo(5), X o(6), 230 Xl (1), Xl (2), Xl (3) are rational, 157
X l (4), X l(5),Xl (6), 230 dynatomic divisor, 150 dynatomic field, 123 action of Galois, 124
Galois group is subgroup of wreath product, 125 quadratic polynomial, 123 ramification, 129, 131 unit, 129 dynatomic polynomial, 39, 148, 149, 181 associated multiplier polynomial, 225,227 dehomogenized, 149 discriminant, 226, 227 for Z2 + c, 156 generalized, 227, 229 homogeneous of degree lJd(n), 181 is a polynomial, 151,226 of z + 1/z, 224 of z2 + bz, 182 order at P (ap(n)), 151,225,226 quadratic map, 39 reducible, 235 resultant of two, 165,226,227 dynatomic O-cycle, 150,226 is positive, 226 Effective divisor, 406 positivity of height, 408 eigenvalue, 421 involution, 438 elementary symmetric polynomial, 87, 180 elimination theory, 417 elliptic curve, 30, 336,409 addition law, 31 automorphism group, 340, 341 explicit description, 342 canonical height, 409 complex multiplication, 32, 341 complex uniformization, 33,127 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
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, 351, 370 endomorphism ring, 31, 340, 341, 378 field generated by torsion, 344, 363 flexible Lattes map, 355 fundamental domain, 33,127 geometric points, 337 global minimal Weierstrass equation, 343, 371 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, 127 Hecke correspondence, 369 height on, 409 ideal class group maps to eM, 350 identity element, 337 integer point, 372 integral j-invariant, 370 invariant differential, 344, 359, 367 inverse of a point, 337 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, 351 preperiodic points, 41, 352 level curve of integrable map, 430 L-series, 409 minimal Weierstrass equation, 222, 342 modified group law, 382
483 morphism is isogeny and translation, 339 m-torsion subgroup, 343 multiplication map, 31, 112, 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, 371 principal divisor, 339 has degree zero, 339 quasiminimal Weierstrass equation, 371,385 quotient by automorphism group, 346,365 rational point, 337 reduction modulo p, 342 reduction oftorsion 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, 127 torsion point, 41 torsion subgroup, 32, 343, 352 twist, 198,341,374 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, 163 endomorphism ring elliptic curve, 340, 341, 378 kernel, 343 multiplicative group, 325, 378 noncommutative, 378
484
Index
endomorphism ring (continued) normalizedembeddingin C, 345 order in quadratic imaginaryfield, 341 order in quatemion algebra, 341 entropy, 6 algebraic,397 equicontinuity, 3, 22, 254, 264, 265, 271, 313 Fatou set is maximal open, 22 is not open condition,40, 254 on a metric space, 22 topological, 306 equidistribution, 6 dynamicalGalois, 128 Galois of a sequence, 128 of preperiodic points, 128 of small height points, 128 torsion on elliptic curve, 127 equi-Lipschitz,313 equivalence relation from group action, 234 Eremenko's theorem, 378 ergodic theory, 6 essential singularity, 23 etale map, 353 Euclidean distance versus chordal metric, 119
Eulertotientfunction, 137, 164, 182 iterationof, 6 exact period, 1, 18,38,150 conditionfor larger formal period, 151, 165,226 rational map has points of different, 154, 155, 196 exceptional set, 16, 17,266 exercise hard ( textbf*),7 unsolved (**), 7 extensionformula, 83, 84 Faltings' theorem, 107,442 family of maps, 171, 194 Fatou domain, 240 Fatou set, 22, 255 all oflP'l, 26, 59, 239 Berkovich,306, 311 canonicalheight of points in, 138
classicalcontained in Berkovich, 306 classification of connected components, 28 complementof Julia set, 22 componentat infinityis completely invariant, 40 containsattracting periodic points, 40 contains nonrepelling periodic points, 256 empty, 26, 35, 361 Herman ring, 28 is completelyinvariant, 23, 40, 266 is nonempty, 256, 285, 314 is open, 22 Julia set is boundary, 25 local canonicalheight of points in, 141 no wanderingdisk domains, 284, 285 no wanderingdomains, 28 nonarchimedean, 254 number of connectedcomponents, 27 of an iterate, 24, 255 of commutingmaps, 378 of (z2 - z)/p, 283 of Zd, 22 paraboliccomponent, 28 polynomialcannot have Herman ring, 28 recurrentcritical point is periodic, 284,317 set of disk components, 283 Siegeldisk, 28 Fermat's little theorem, 2, 252, 275 Fermat's sum of two squares theorem, 200,236 field complete, 240 completion, 241 extensionof complete,241 finite, 2 of characteristic p, 37 separableextension,37 valued,44, 240 field of definition, 206 abelian variety, 217
Index field of definition (continued) cocycle is coboundary, 209 contains field of moduli, 208 covering map, 217 intersection of all is field of moduli, 236 not equal to field of moduli, 208, 213,236 of algebraic family, 159 of morphism, 89 of point, 86, 122, 145 of quotient variety, 161 field of moduli, 177, 207 abelian variety, 217 cocycle, 209 contained in field of definition, 208 covering map, 217 equals field of definition for polynomials, 215 equals field of definition if degree is even,215 from image in Md, 207 is intersection of all fields of definition, 236 not equal to field of definition, 208, 213,236 of a point, 145 filled Julia set, 74, 140 Green function, 140 fine moduli space, 160, 172, 230 finite field dynamical system on, 2, 5 function field over, 5 finite invariant set, 16, 266 finite subgroup ofPGL 2 , 197 fixed point, 18 attracting in Berkovich space, 324 flexible Lattes map, 382 Galois conjugates, 180, 184 Galois group, 427 index summation formula, 20, 38, 255,314,380,382 Lattes map, 34 multiplier, 18,47, 180 at infinity, 38 neutral, 322 no Type-IV, 321 nonrepelling, 255, 314 in Fatou set, 256
485 of affine automorphism, 427 of Chebyshev polynomial, 332, 380 in characteristic p, 381 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, 384 summation formula, 20, 38, 255, 314,380,382 symmetric polynomial of multipliers, 180 totally ramified, 16, 17,37 flexible Lattes map, 187,355 composition of, 382 degree, 357 fixed points, 382, 384 good reduction, 362, 383 multiplier, 358, 366, 382, 384 spectrum, 356 multiplier at 00, 383 periodic points, 358, 382, 384 FOD, see field of definition foliated space, 6 FOM, see field of moduli formal derivative, 12,37,313 formal group, 67 formal multiplier spectrum, 182, 183 of Z2 + bz, 182 of z", 182, 183 formal period, 18,39, 150 algebraic family induces map to dynatomic curve, 159 algebraic family of points of given, 159 condition for smaller primitive period, 151, 165,226 moduli problem, 158 vd(n) points offormal period n, 150,224 formal power series, 5 periodic points over ring of, 78 forward orbit, I, 108, 280, 353 fractional ideal, 350 exact sequence, 404 height formula, 136
486 fractional linear transformation respects reduction, 50 free product of cyclic groups, 419, 435 Frobeniusmap, 351 cyclotomic field, 130 periodic points are attracting, 78 function analytic,244 equicontinuous family, 254, 264, 265,271,306,313 equi-Lipschitz family, 313 holomorphic, 244 meromorphic, 244 uniformly continuousfamily, 39, 254,313 unifonn1y Lipschitz family, 39, 254, 256,264,265,271,313,315 functionfield, 5 absolute values on, 44 dynamics, 5 functorialityof height, 407, 421 fundamental domain, 33,127 Galois cohomology, 205 Galois conjugate, measure supportedon, 128 Galois equidistributed sequence, 128 Galois equidistribution, 128 Galois group action on dynatomicfield, 124 action on PGL2, 203 action on projectivespace, 122 action on rational map, 207, 211 cyclotomicfield, 129 of dynatomicfield, 123, 125 offield generatedby periodic points, 122 of fixedpoint, 427 orbit of conjugatepoint, 137 profinitetopology, 211 subgroupassociated to rational map, 207 subgroup of wreath product, 125 Galois invariance of height, 85 Galois-invariant orbit on K3 surface,436 Galois theory, 122 Gauss norm, 296 associatedGauss point, 297 is sup norm, 296
Index on iCp[z], 296 on Tatealgebra, 298 Gauss point, 297 fixedby polynomial, 321 is Julia set of good reductionmap, 307,322 line segmentto Type-IV point, 322 of Berkovichdisk of radius R, 301 Gel'fond metric, 306 Gel'fond topology, 300, 322 base of open sets, 300 not inducedby path metric, 306, 323 restricts to iCp topology, 301, 304, 323 general linear group, 10 generalizeddynatomicpolynomial, 227 for quadraticmap, 229 generalizedWeierstrass equation,336 generatingfunctionfor degree of rational map, 397,428 generic fiber, 77 generic rational map dynamicalunit, 146 not very highly ramified, 231 genus, 15,37 of Xo(n), 164
of Xl (4), Xl (5), Xl (6),230 of Xl(n), 164 geometricinvarianttheory, 176 geometricpoint, 337 geometric quotient, 179 GL2,10 global minimal model, 222 impliestrivial Weierstrass class, 223 overQ,236 global minimal Weierstrass equation, 343,371 difference from quasiminimal, 385 global S-minimal model, 237 good reduction,43, 58, 218 abelian variety, 442 attractingperiodic points, 64 BerkovichJulia set of map with, 307,311,322 compositionof 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 18 Green function of map with, 289, 294,318 iff resultant is unit , 58, 2 18 implies reduction is homomorphism, 342 is everywhere nonexpanding, 59 Julia set is empty, 59, 239 local canonical height for, 103, 141 map on Berkovich disk, 321, 322 nonrepelling period ic points , 64 of ¢ (P ) is ¢(?), 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 periodi c points, 61 potential , 371 scheme-theoretic characterization, 59,61 ,77 separable, 64 twists with outside S, 237 v-adic distance between periodic points, 69 graph, ISO, 216, 226 Green function , 287 , 288 algorithm to compute, 318 complex, 140 defining properties, 289 homogeneity properties, 289 is continuous, 289 is Holder continuous, 318 modified, 318 of bad reduction map, 294, 318 of good reduction map, 289, 294, 318 relation to local canonical height , 291 sum is canonical height , 290, 3111 group, 2 action , 234 additi ve, 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 P GL2, 170 wreath product , 125
HI, 202, 236 Haar measure , 127 haifa loaf, 133 hard probl em ( textbf*),7 harmonic function , local canon ical height is, 140 Hausdorff space, Berkovich disk is, 300 Heeke correspondence, 369 height, 4, 81 absolute, 85 absolute logarithm ic, 93 additivity, 408 , 421 bounded for ample divisor is finite set, 408 canonical , 97, 99, 100,287,409 on K3 surface, 421, 426, 435, 436 effect of morphism of degree d, 90, 136,398,407,409 effect of rational map, 388 effectivity of constants, 408 elliptic curve , 409 effect of multiplication map, 409 equals one iffroot of unity, 88, 100 field extension formula , 84 finitely many points of bounded, 86, 407,423 for linearly equivalent divisors, 408 fractional ideal formula, 136 functoriality, 407, 421 Galois invariance of, 85
h([u, a, b]) :::; h([u, a]) + h([u, b]), 398,428 in affine space, 397 inequality for j ointly regular affine morphisms, 397, 429 inequality for nonregular affine automorphism, 429 inequality for regular affine automorph isms, 399,431 infinitely many points of bounded , 435
488 height (continued) is at least one, 84 is well-defined, 84 Lehmer's conjecture, 100, 10 I, 138 logarithmic, 93 lower bound for, 100, 101, 137, 138,221 machine , 407 map on Picard group , 408 Misiurewicz points have bounded , 167 number of points of bounded, 135, 136,441 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 , 136, 138 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 Weil, 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, 199,205, 207 Holder continuity of Green function , 318 holomorphic function, 244 coefficients uniquely determined, 312 distance between image disks, 268
Index dynamics of, 23 equicontinuous family, 254, 264, 313 equi-Lipschitz family, 313 family omitting one point, 264, 315 Fatou set, 24, 255 is continuous, 253 is open, 253 Julia set, 24, 255 Lipschitz bound , 248, 254, 272 maximum modulus principle, 250 Montel theorem, 264, 271, 315 Newton polygon, 249 norm, 247 norm of derivative, 252, 313 order of, 244 product is holomorphic, 312 reciprocal is holomorphic, 313 roots determined by Newton polygon, 249 sends disks to disks, 252, 305 uniformly continuous family, 254, 313 uniformly Lipschitz , 254, 264, 265, 271,313 ,315 zeros are isolated , 244 homogeneous coordinates, 10 homogeneous ideal, 89 homogeneous polynom ial, 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, 13,217 hyperbolic component of Mandelbrot set, 166 hyperbolic map, 317 nonarchimedean, 279, 285, 317 Icosohedral group, 197 ideal algebraic set attached to, 90 attached to an algebraic set, 90 Hilbert basis theorem , 90 homogeneous, 89 radical ,89 ideal class group, 350
Index ideal class group (continued) exact sequence, 404 map to eM elliptic curves, 350 identity element of an elliptic curve, 337 imaginary field, quadratic, 34 1 immediate basin of attraction, 310, 311 inclusion-exclusion, 148, 224 indeterminacy locus, 389, 427 dimension, 394 disjoint, 394 of involution on K3 surface, 413 of iterate, 394 indifferent periodic point, 19,47,326 one implies infinitely many, 314 induced rational map, 389 inertia group, 344 infinite dihedral group, 419, 435 integer point counting on Markoff variety, 437 cutoff for in orbits, 375 in orbit, 3, 108, 109, 112, 142, 143, 145 in orbit of z + l i z, 8 on lP'1 minus 3 points, 106, 108 on elliptic curve, 372 on Markoffvariety, 437 orbits with many, 110, III , 143, 375 uniform bound in orbits, 112, 372, 385 value of rational function, 143 integrable map, 429, 430 level curve is elliptic curve, 430 level set, 431 McMillan family, 443 QRT family,443 intersection theory, 434 invariantdifferential, 344, 359, 367 is holomorphic, 345 is translation-invariant, 345 transformation for [mJ, 345 invariant measure, 307 invariant set, 16 completely, 17, 266 contains Berkovich Julia set, 3 II Fatou and Julia sets are completely, 23 finite, 16, 266 inverse function theorem, 115, 144
489 over C p , 312 inverse of point on elliptic curve, 337 inversion maps disk to ], 321 involution, 410 action on divisor, 418, 419, 438 as linear transform on Picard group, 420 composition is reversible affine automorphism, 429 degenerate point, 415 eigenvalue of, 421, 438 formulas to compute, 433 indeterminacy locus, 413 induced by degree 2 map, 41 I, 436, 437 is morphism on most K3 surfaces, 417,434,438 noncommuting, 410 not a morphism, 436, 437 on Markoff variety, 436 quotient by, 410 relation between noncommuting, 419,435 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, 186, 188, 356, 362 iterate, coordinate functions of, 149,226 iteration commutes with conjugation, I 1 itinerary, 258 as map on sequence space, 260 left shift, 260 Jacobian variety, 369 j -invariant, 221, 337 in terms of cross-ratio, 348
490 jointly regular affine morphisms, 397, 429 height inequality, 397,429 Julia set, 22, 255, 284 algebraic points, 40 all ofIP!, 26, 35, 361 backward orbit is dense, 25 Berkovich, 306 backward orbit dense in, 311 for bad reduction map, 307, 311 for good reduction map, 307, 311,322 has empty interior, 311 is connected or infinitely many components, 311 boundary is completely invariant, 23 boundary of Fatou set, 25 canonical height of points in, 138 chaotic behavior, 22 Chebyshev polynomial, 30, 41 classical contained in Berkovich, 306 closure of repelling periodic points, 41 compact, 275 complement of Fatou set, 22 connected, 26, 165 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 Zp, 275 equals closure of repelling periodic points, 27,274, 311 filled, 74, 140 Green function for filled, 140 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
Index is nonempty, 25 is perfect set, 23, 25, 267, 307, 311 is uncountable, 267,307,311 local canonical height of points in, 141 no 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, 361 of(z2 - z)/p, 262, 283, 315 of Z2 - 2,40 of Zd, 22 of z ... (z - d + 1)/ p, 315 open set orbit omits at most one point, 266 orbit of open subset, 27 periodic points dense, 23, 263, 315 smallest closed completely invariant set, 267 strictly expanding map on, 279, 317 support of canonical measure is, 307 topologically transitive map, 263, 315 totally disconnected, 23, 26 with no critical points, 279 K3 surface, 410, 412 action of involution on divisors, 418,419,438 A orbit, 418 arithmetic complexity of orbit, 435 canonical height, 421, 423, 426, 435,436 infinitely many points of bounded,435 normalization conditions, 422 product it+ it-,435 transformation formulas, 422 zero iff finite orbit, 423 degenerate point for involution, 415 dimension offamily of, 412, 438 divisors on, 418 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 H 1(S,Os ) = 0,412 height functions on, 420 height transformation rules, 420 height zeta function , 436 homogeneous forms associated to, 412 ,433 in pi x jpl X jpl , 437 indeterminacy locus of involution, 413 intersection theory, 434 involution, 411 as linear transform on Picard group, 420 example, 411 formulas, 433 is morphism on most, 417 , 434, 438 Kodaira dimension, 412 nonsingular,438 number of points of bounded height, 425 orbit is Zariski dense , 435 Picard group of general , 418 projections to jp2 , 410 relation between noncommuting involutions, 419, 435 stabilizer of point on, 425, 434 with three involutions, 437, 438 Kawaguchi's theorem , 399, 431 Kodaira dimension, 412 Kronecker's theorem, 88, 100 Krylov-Bogolubov theorem, 127 Kummer sequence, 205 £-adic representation, 344 Lang's height lower bound conjecture, 101,221 Lang 's integer points conjecture, 112 Lanes diagram, 365 Lattes map, 32, 97, 186, 187, 351 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, 351 degree, 34 fixed points, 34, 366, 382, 384 flexible, 355 for multiplication-by- (l + i ), 32 for multiplic ation-b y-2, 32,351, 370 on j = 0 curve, 351, 381 on j = 1728 curve, 351 good reduction, 362, 383 higher-dimensional, 377 in characteristic p, 366 integral j -invariant, 370 isospectral family, 362 iterate, 382 multiplier, 34, 186, 358, 366, 382, 384 at 00, 383 of fixed point, 382 spectrum , 362 summation formula , 382 nonconjugate, 354 in characteristic 2, 383 to polynomial , 381 periodic points , 186, 358, 366, 382, 384 all attracting , 362 all nonrepelling, 362 expanding, 35 in Julia set, 35, 361 padically p-adically, 361 postcritical set, 353 postcritically finite, 353 preperiodic points, 32,41,352 projection to Elf, 365 reduced diagram, 365 rigid,364 set of is mono id, 382 torsion points map to preperiodic points, 41 translation fixed by I', 365 uniform bound for integral points in orbit, 372, 385 uniform boundedness of periodic point s, 137 lattice, 33
492 Laurent series, 245 coefficients uniquely determined by function, 312 of Schwarzian derivative, 232 product of, 312 least period , see exact period Lefschetz principle, 20, 365 left shift map, 259, 314 as itinerary map, 260 backward orbit dense , 314 backward orbit equidi stributed, 315 continuous, 259 is topologically transitive , 259 Lipschitz, 259 periodic points, 259 are dense, 314 properties of, 259 topologically transitive , 314 uniformly expanding, 259 Lehmer 's conjecture, 100, 138 dynamical, 101, 138 level curve of integrable map, 430 level set of an integrable map, 431 Lie group , 6 lift, 287 from IFp to Zp, 48 of affine morphism, 389 of rational map, 287,389 line bundle, metrized , 410 line segment from Gauss point to Type-IV point , 322 in Berkovich disk, 298 of Newton polygon, 249 linear conjugation, II , 173 commutes with iteration, II 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 , 269,316 effect on chordal metric , 76 image of unit disk, 317 Lipsch itz constant, 36 move three points , 36 on IP'N, 226 resultant of, 76
Index linear group general , 10 projective special, 175 special, 175 Lipschitz, II, 24, 36, 56 equi-, 313 holomorphic function, 248, 254, 272 shift map, 259 uniform family, 39, 254, 256, 264, 265,271,313,315 local canonical height, 102,291 associated to a divisor, 320 associated to an eigendivisor class, 104 computation of, 103 existence proof, 291 for a polynomial map, 103, 140, 141 good reduction case, 103, 141 Green function and, 291 is harmonic, 140 Laplacian is invariant measure, 104 normalized, 141 of point in Fatou or Julia set, 141 properties of, 102, 291, 320 sum to global canonical height, 103, 293 transformation law, 320 local degree , 83 local ring, 67, 79 at a divisor, 403 normalized valuation, 403 locally compact , 74, 239, 243, 304 is not, 239, 268, 294, 295 locally constant function , 318 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, 158 Lyubich measure, 307
c,
Index Mahler measure, 138 Mandelbrot set, 26, 165 analytic description of Misiurewicz points, 229 bulb,166 complement conformal to unit disk exterior, 167, 168 component of interior, 166 in disk ofradius 2, 166 is connected, 167 Misiurewicz point, 166 analytic description, 168 dense in, 166 in boundary, 166 root of hyperbolic component, 166 spider algorithm, 168 uniformization, 167, 168 Manes's theorem, 235 Manin-Mumford conjecture, 127 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, 187,367 M2,188 completion equals jp'2, 194 explicit formula for 0"1,0"2, 189 is a rational variety, 232 is a scheme over Z, 189 isomorphic to p,?, 188 locus of polynomial maps, 232 point with Aut(¢) = 53, 234 subvariety with Aut = IL2' 235
Md,174 affine coordinate ring, 176
493 coordinate ring contains 0"; n) and *(n) «: ,183,232 field of moduli, 207 generic map not very highly ramified, 231 has dimension 2d - 2, 176 height, 221, 237 is a rational variety?, 232 is a scheme over Z, 179, 186 is a unirational variety, 232 is a variety, 175 is affine, 176 is complex orbifold, 176 is connected, 176 is integral, 176 is nonsingular?, 193 M2 ~ p,2, 189 M3 embedded by O";n)?, 232 map from Ratd, 174, 175 multiplier 1 is Zariski closed subset, 230 not embedded by O";n), 186, 187, 356,367 rational points, 177 semistable completion, 178 stable completion, 178 subvariety with nontrivial automorphisms, 199, 234 twists have same image, 198, 221 M~,
178 has natural quotient property, 179 is a scheme over Z, 179 is geometric quotient, 179 is quasiprojective, 178 isomorphic to M~s iff d is even, 178
M~S,
178
is a scheme over Z, 179 is categorical quotient, 179 is projective, 178 isomorphic to M~ iff d is even, 178 measure, 6, 36 Brolin, 307 canonical on Berkovich space, 129, 306 Dirac, 128 Haar,127 invariant, 104, 127,307 Lyubich, 307
494 measure (continued) spherical, 36 support on Galois conjugates of point, 128 weak convergence, 128 Merel's theorem, 97, 369 meromorphic function, 244 coefficients uniquely determined, 312 dynamics of, 23 equicontinuous family, 254, 265, 313 equi-Lipschitz family, 313 family omitting two points, 265, 315 Fatou set, 24, 255 is continuous, 253 is open, 253 Julia set, 24, 255 Laurent series, 245 Montel theorem, 265, 315 order of, 244 pole, 244 product is meromorphic, 312 uniformly continuous family, 254, 313 uniformly Lipschitz, 254, 313 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,258,314 path in, 306, 323 SN is a, 258, 314 uniform continuity, 39 uniform Lipschitz, 39 metrized line bundle, 410 minimal discriminant, 221 minimal model, 220, 222 global S-, 237 minimal resultant, 220, 221, 237 exponent at p, 220 is PGL2(K)-invariant, 221
Index is resultant times power, 222 of conjugate, 219 valuation, 219 minimal Weierstrass equation, 222, 342 global, 343, 371, 385 quasi, 371, 385 Tate's algorithm, 342 Misiurewicz point, 166 analytic description, 168, 229 dense in Mandelbrot set, 166 has bounded height, 167 in boundary of Mandelbrot set, 166 minimal polynomial, 230 no neutral cycles, 166 of type (m, n), 166 repelling critical orbit, 166 spider algorithm, 168 subhyperbolic, 166 Mobius inversion formula, 148,224 Mobius JL function, 39, 148, 149, 164, 224 modified Green function, 318 modular curve dynatomic, 157, 158, 161 elliptic, 163, 369 Heeke correspondence, 369 modular Jacobian, 369 moduli space, 4, 26, 147 coarse, 160 degree of symmetric polynomial map, 187,232,367,384 fine, 160, 172,230 isospectral family, 188, 356 M2 S;' It?, 188 M 2 S;' jp'2, 194 map defined by symmetric polynomials of multipliers, 187,232,367,384 of rational maps of degree d, 174 scheme over Z, 179, 186, 189 moduli, field of, 207 Mobius transformation, see linear fractional transformation monoid of Lattes maps, 382 monomial map, 29, 95, 325, 428 automorphism group, 234, 327 in characteristic p, 380 commuting maps, 326, 380 dynamical unit, 328
Index monomialmap (continued) in characteristic p, 380 multiplierof periodic point, 326 periodic point, 326 preperiodicpoint, 326 twist, 206, 328, 380 monomial, number of given degree, 136 Montel theorem, 25 nonarchimedean, 264, 265, 271 , 281 ,311 ,315 ofHsia, 265, 271 of Rivera-Letelier, 311 on Berkovichspace, 310, 311 with movingtargets, 271 morphism, 89, 388 affine, 375, 388 algebraic family of, 171 bijective -I?- isomorphism, 193 Bogomolov conjecture, 129 canonicalheight associatedto, 99, 287 canonicalmeasure, 127 definedover K , 89 degree of, 389 degree of composition, 392 dynamical degree, 397, 428 dynamicallyaffine, 376 effect on absolute value, 288 effect on height, 90, 136,398, 407, 409 functoriality of height for, 407, 421 Green function, 288 induced on family, 171 invariant measure, 127 involution on most K3 surfaces is, 417,434,438 is open map, 433 jointly regular,397, 429 local canonicalheight, 102 over a variety, 171 pullback of divisor, 404 pullback of linearly equivalent divisors,405 pushforwardof divisor, 404 Morton-Silverman Conjecture, 96, 135, 368 for zd , 137 for Lattes map, 370 moving targets, 271
495 m-torsion subgroup of elliptic curve, 343 multiplication map by 2 on Q/Z, 168, 229 on elliptic curve, 31, 340, 383 effect on height, 409 kernel of, 343 multiplicative group, 29, 30 endomorphism, 29, 325 endomorphism ring, 378 preperiodicpoints, 326 quotient by {± I}, 29, 328 torsion, 326 multiplicative group scheme, 29 multiplier, 18, 19,47, 180 at infinity, 38 Lattes map, 383 chain rule used to calculate, 19,47 Chebyshev polynomial, 41 condition for unequal formal and primitiveperiods, 151,226 conjugacyinvariant, 19 differential one-formdefinition, 19 equal to one, 225, 230 flexible Lanes map, 186,358,382 Galois conjugates, 180, 184 integralover Q[Ratd], 180, 185 is PGL2 invariant, 18, 180, 185 minimal polynomial, 225, 227 not equal to one, 41 of a Lattes map, 34 of critical periodic point, 38 of degree two rational map, 190, 233 of fixed point of Chebyshev polynomial, 332, 380 in characteristic p, 381 of fixedpoint of Lattes map, 382 of Z2 + bz, 182 rigid Lattes map, 366, 384 root of unity, 151,226 symmetric polynomialof, 180, 183, 187,232,367 multiplierpolynomial, 225, 227 resultant with cyclotomic polynomial, 226 multiplierspectrum, 182, 187 flexible Lattes map, 356 formal, 182 isospectral family, 188,356,362
496 multiplierspectrum (continued) of Lattes map, 186 of Z2 + bz, 182 of zd, 182 Natural numbers,258 nth-dynatomic polynomial, 149, 181 homogeneous of degree vd(n), 181 Neron local height, 104 nested sequence of closed disks, 295, 322 associatedseminorm, 297, 322 equivalent, 322 neutral fixedpoint, 322 neutral periodic point, 19,47 Newtonpolygon, 248, 249, 252, 275, 276 determinesroots, 249 line segment,249 n-multiplier spectrum, 182 formal,182 of zd, 182 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 hyperbolicmap, 279, 285,317 nonarchimedean inversefunction theorem, 312 nonarchimedean chordal metric, see v-adic chordal metric noncommutative monoid of Lattes maps, 382 noncommuting involutions, 410 three, 437 nondegenerateautomorphism, 226 nonrepellingcycles,number of, 27 nonrepellingfixedpoint, 255, 314 nonrepellingperiodic point, 362 in Fatou set, 256 nonsingularelliptic curve, 337 nonsingularK3 surface,438
Index norm, 247,288 Gauss on Cp[z], 296 of a holomorphic function, 247 of derivative, 252, 313 semi-, 296 sup, 288 normal family, 25 normal forms lemma, 190,233 normalizedcoordinates, 49 normalizedMarkofftriple, 437 normalizedrational map, 51, 75 normalizedvaluation, 44, 249 Northcott's theorem, 94, 407, 423 false for rational maps, 136 notation, 7 n-periodpolynomial, 149, 181 homogeneous of degree d" + 1, 181 Nullstellensatz, 90, 92, 408 number field algebraic closure, 85 completionof, 83 extensionformula for absolute values, 83, 84 product formula, 83 ring of integers, 83 ring of S-integers, 83 numerical criterion for (semi)stability, 178 0(1),93 octahedralgroup, 197 one coboundary, 201, 202, 209 trivial when extended, 203 one cocycle,201, 202 corresponds to twist?, 203 determinesfield of moduli, 209 extensionfrom Aut(¢) to PGL 2 , 203 of twist, 201 open branch of Berkovichdisk, 300 open disk, 242 closed under addition, 243 image of D(O, 1) by PGL 2 , 317 in jp'1, 243 is closed, 243 is maximal ideal, 243 rational radius, 243, 278 standard collection, 278 open map, 10, 253, 312
Index open map (continued) morphism is, 433 rational map is, 10, 24 open set base offor Gel'fond topology, 300 disjoint union of disk components, 277,317 disk connected, 317 iterates omit at most one point, 266 orbit, I backward is dense in Julia set, 267 finite iff canonical height zero, 423, 431 Galois-invariant on K3 surface, 436 height zeta function, 436 integral points, 3, 108, 109, 142, 143 construction of many, 110, III, 143,375 cutoff for, 375 uniform bound for, 112,372,385 integrality estimate for points in, 112, 145 number over a finite field, 430 of critical points, 280 is bounded, 26, 165 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, 143 two sided, 432 under collection of rational maps, 143 Zariski dense on K3 surface, 435 orbit-counting function, 432 ord",,12 order in quadratic imaginary field, 341 in quatemion algebra, 341 of a holomorphic function, 244 of a meromorphic function, 244 ord p , 82 of a polynomial, 218 Ostrowski's theorem, 44,312 p, Weierstrass function, 34, 345 p-adic absolute value, 44
497 p-adic Cauchy residue formula, 314 p-adic hyperbolic map, 279, 285, 317 p-adic inverse function theorem, 312 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, 311 Julia set is, 267 period exact, 150 formal,150 of a point, I, 18 primitive, 150 period polynomial, 149, 181 dehomogenized, 149 for Z2 + c, 156 generalized,227 homogeneous of degree d" + 1, 181 multiple root, 230 order at P (ap(n)), 151,226 periodic disk component, 283 periodic domain, 28 periodic point, I algebraic family induces map to dynatomic curve, 159 algebraic family of with given formal period, 159 algebraic properties, 3 attracting, 19,47,326,362 for good separable reduction, 64 iff reduction is critical, 78 in Berkovich space, 310 in Fatou set, 22, 40 basin of attraction, 310 canonical height zero, 99, 431 Chebyshev polynomial, 41 closure contains Julia set, 273 closure of repelling equals Julia set, 274,311 condition for unequal formal and primitive periods, 151, 165, 226 critical, 38
498 periodic point (continued) criticalpoint with reduction a, 78 cycle of, 150,226 cyclotomic, 95 dense for shift map, 314 differential one-formdefinition of multiplier, 19 discrete, 428 dynamicalunit, 69, 71, 72, 79, 132, 133 dynamics on finite set, 7 dynatomicfield, 123 equation of, 122 equidistribution, 6 every is attracting, 78 exact period, 18,38,150 field generatedby, 122 finitelymany 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, 18,39, 150 moduli problem, 158 Galois theory offield generatedby, 122 generalizeddynatomicpolynomial, 227,229 generalizedperiod polynomial,227 immediatebasin of attraction, 310 index summationformula,20, 38, 255,314,380,382 indifferent, 19,47,326 irrationallyneutral, 19,47 isolated, 402 Julia set is closure of nonrepelling, 27 Lattes map, 186 least period, 18 multiplier, 18, 19,47 at infinity, 38 Lattes map, 35, 186 of critical, 38 power map, 326 nonrepelling, 362
Index for good reduction,64 in Fatou set, 256 number of, 39 given period, 122 nonrepellingcycles, 27 of commutingmaps, 8, 378 of power map, 326 of quadraticpolynomialmap, 96 of regular affineautomorphism, 394,428 of(z2 - z)/p,263 of Z2 + 1, 8 of Z2 - 1,8 of Zd + a over finite field, 8 ofz··· (z - d+ 1)/p, 315 on subvariety, 127 one indifferent implies infinitely many, 314 p-adic, 361 prime period, 18 primitive, 122 field generatedby, 123 primitiveperiod, 18, 150 product of multipliersin orbit, 19, 47 rational map can have infinitely many rational, 136 rational map has points of different periods, 154, 155, 196 rationallyneutral, 19,47 in hyperboliccomponent, 28 reduction of, 61, 62, 66, 78 regular affineautomorphism, 400 relativeabelian extension, 126 repelling, 19,47,326,361 in Berkovichspace, 310 in Julia set, 22, 40 limit of attracting, 274 repelling in Julia set, 256 residual degree, 310 rigid Lattes map, 366, 384 set of is Galois-invariant, 123 shift map, 259 summationformula, 20, 38, 255, 314,380,382 superattracting, 19,47 Type-I, 310 Type-II,31O Type-IV, 321
Index periodic point (continued) uniform boundedness conjecture, 96,135,368 forz d,137 for Lattes map, 370 v-adic distance, 69 periodic subvariety, 127 permutation group, 124 permutation polynomial, 5 Per n (c/» ,18 PGL2 ,1O action of Galois, 203 automorphism group is subgroup, 234 cocycle gives twist of jp>l , 211, 215 conjugation is algebraic group action, 173 equivalence, 195,233 extension of cocycle to Aut( c/», 203 finite subgroups, 197 invariant functions on Ratd, 175 is group variety, 170 is Rah, 170 map from PSL 2, 175,231 minimal resultant is invariant, 221 PGL2(K) equivalence, 195,233 quotient of Rat., by, 174 PGLn , 226 map from PSL n , 231 Picard group, 339,403 action of involution, 418, 419, 438 degree map, 405 eigenvalueof involution, 421, 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, 418 of jp>2 x jp>2 , 418 of jp>N x jp>AJ , 405, 406, 409 pullback homomorphism, 405 point absolute value, 90, 288 at infinity, 10, II
499 on elliptic curve, 336 field of definition, 86, 122 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, 91, 288 bidegree, 405, 410 bihomogeneous, 405,410 bounded seminonn on ring of, 296 cyclotomic, 148, 224 degree of, 389 dynatomic, 148, 149, 181 elementary symmetric, 87 Gauss norm, 296 height of roots, 136, 138 homogeneous, 10 homogeneous ideal, 89 local canonical height, 140 normalized, 51 number of monomials, 136 ords, 218 permutation, 5 resultant of two, 53 sup norm, 91, 288 trihomogeneous, 437 polynomial map, 17 cannot have Herman ring, 28 canonical measure on Berkovich space, 306 Chebyshev, 29, 95, 329 in characteristic p, 381 commuting, 378 complete invariance of component at infinity, 40 condition for unequal formal and primitive periods, 165 cyclotomic preperiodic points, 95 dynamical unit, 71, 132 of generic monic, 146 effect on absolute value, 288 filled Julia set, 74, 140 finitely many rational periodic points, 8 fixes Gauss point, 321 FOM = FOD, 215
Index
500 polynomial map (continued) good reduction, 32 i, 322 has primitive periodic points of different periods, 155 iff 0"1 (¢) = 2, 232 iterate is a, 17,37,431 iterates of open set omit at most one point, 266 Lattes map not conjugate to, 381 local canonical height , 103, 140, 141 monomial, 29, 95, 325 multipliers not equal to one, 41 nontrivial automorphism group , 234 number of nonrepelling periodic cycles, 27 on Berkovich space, 304 power, 95, 325 power map, 29,95,234,321,325 in characteristic p, 380 on Berkovich space, 321 radius of image disk, 321 sends disks to disks, 305, 312 wandering disk domains exist over C p , 286 polynomial type rational map, 143 positive divisor, see effective divisor positivity of height, 408 postcritical 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 , 312 converges iff an ---> 0, 244 distance between image disks, 268 family omitting one point, 264, 315
formal.f maximum modulus principle , 250 Montel theorem, 264, 27[ , 315 Newton polygon, 249 norm of derivative, 252, 3 [3 periodic points in ring of, 78 product of, 3 [2 sends disks to disks, 252, 305 Tate algebra , 297 preperiodic disk component, 283 preperiodic disk domain , 284, 285 preperiodic doma in, 28, 276 preperiodic point , I canonical height zero, 99 critical, 26, [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 Lanes map, 32, 352 of abelian group is torsion, 2, 41, 326 of commuting maps, 8, 38, 378 of multiplicative group , 326 of power map, 326 on subvariety, [2 7 reduction of, 61 Type-IV, 321 uniform boundedness conjecture, 96,135 ,368 for Zd , 137 for Lattes map, 370 preperiodic subvariety, 127, 128 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, 122
Index primitive periodic point (continued) dynatomi c field, 123 field generated by, 123 set of is Galois -invariant , 123 principal divisor, 320, 339, 403 has degree zero, 339 probability measure, 127 Borel, 127 canonical , 127 canonical on Berkovich space, 306 invariant, 127 produ ct formula , 83, 84, 290 profinite topology, 211 projection map, 287 ofK3 surface to jp'2, 410 projective genera l linear group , 10 projective line automorphism group , 10 Berkovich, 302 change of variables, II chordal metric , 11,35,45, 144, 243 construction of, 10 integer points on, 106, 108 inversion is isometry, 35 linear fractiona l transformation, 10 morphism of over a variety, 171 moving three points , 10, 36, 51 not locally compact over C p , 294 over a variety, 171 point at infinity, 10, II rational map, 10 Riemann sphere, 10 Riemann-Hurwitz formula , 13, 217 spherical measure , 36 twist, 200, 211, 236 twist trivial iffC(K) =1= 0,215 universa l map of degree d, I71 project ive space absolute value of a point , 90 action of Galois group, 122 compact iff K locally compact, 243 cross-ratio, 71, 79, 132,348 degree map on divisors, 405 field of definition of point , 86, 122 finitely many points of bounded height , 86, 407 height , 82, 84, 407 hyperplane generate s Picard group , 405
SOl monomial map, 428 morphism, 89, 388 is open map, 433 moving N + 2 points, 75 nondegenerate automorphism, 226 norma lized coordinates, 49 number of points of bounded height, 135, 136,441 probabil ity measure , 127 product of two, 405, 406 ,409,418 K3 surface in, 410 surface with involutions in, 436 projection map to, 287 rational map, 89, 388 Segre embedding, 406, 409 project ive special linear group, see PSL PSL, 175 map to P GL , 175,231 stable locus, 176 Puiseux series, 5 pullback divisor, 404 linearly equivalent , 405 push forward divisor, 404 Q RT family of integrab le maps , 443 quadratic formula, 261 quadratic imaginary field, 34 1 class number, 350, 368 fraction al ideals in, 350 ideal class group, 350 quadrati c map algebraic family induces map to dynatomic curve, 159 algebraic family of, 159, 194, 230 associat ed multiplier polyno mial, 227 automorphism group contai ns J.L2' 235 automorphism of order two, 96, 235 bifurcat ion point, 165 bound for periodic points , 96 canonical height, 137, 138 conjugation to Z2 + C, 156 degenerate, 194 dynamic al unit, 134, 145, 146 dynatomic curve, 156, 157, 161 dynatomic field, 123 ramification, 13I
502 quadratic map (continued) dynatomic polynomial, 39 explicit formula for al (¢), 181 generalized dynatomic polynomi al, 229 local canonical height, 141 Mandelbrot set, 26, 165 Misiurewicz point, 166 minimal polynomial, 230 orbit of critical point, 26, 165 parameter space has dimension three, 168 polynomial iff al (¢) = 2, 232 rational periodic points , 137 reducible dynatomic polynomial, 235 space of conjugacy classes has dimension one, 168 with preperiodic critical point , 166 quadratic twist, 198 rational map, 200 quasiminimal Weierstrass equation, 371 difference from global minimal, 385 quatemion algebra , 341, 378 quotient curve , 161 critical values , 353 dynatomic, 161 elliptic, 346 quotient variety, 410 by finite group, 161 by infinite group, 161, 174 categorical , 179 field of definition , 161 geometric, 179 R atd modulo PSL2 , 175 rational points on, 177 semistable, 179 stable, 179 Radical of an ideal, 89 radius of Berkovich disk, 297, 301 of Berkovich point , 296 of Type-TV point is positive , 296 ramification field generated by torsion on elliptic curve, 363 in dynatomic field, 129, 131
Index 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, 115, 144 large implies preperiodic, 37 of iterate , 117 of Lattes projection, 384 sum of over inverse image of a point, 15 ramification point, see critical point Ratl is PGL 2, 170 Rat2, 170, 174 normal forms lemma, 190,233 subvariety with Aut = JL2' 235 R atd,169 affine coordinate ring, 169, 174 n ) and coordinate ring contains *(n ) «; , 183, 232
ai
generic map not very highly ramified, 231 geometry of bounda ry, 170 in P SL 2-stablc locus, 176 is a rational variety, 232 is subset of IP'2d+l , 169 map induced by Schwarzian derivative, 232 map to M d, 174, 175 multipl ier 1 is Zariski closed subset, 230 PGL 2-invariant functions on, 175 quotient by PGL 2, 174 quotient by P SL 2 is a variety, 175 semistable rational maps, 178 stable rational maps, 178 subvariety with nontrivial automorphisms, 199,234 universal map over, 171 Rat d, 178, 179 Rat dS , 178, 179 rational automorphism, 430 rational function, 9 divisor, 320, 403 finitely many nonzero residues , 314 formal derivative, 12, 313 induces rational map, 10 integer values of, 106, 108
Index
503
rational function (continued) maximum modulus principle, 250 pole, 403 principal divisor, 320, 403 residue, 313 Schwarzian derivative, 231 space of associated to a divisor, 406 Taylor series, 12 zero, 403 rational map, 10,89,388 action of automorphism on critical point, 234 action of Galois, 207, 211 acts on X 1 (n), 161 affine minimal, 112,372,385 algebraic entropy, 397 algebraically integrable automorphism, 430 Arzela-Ascoli Theorem, 25 associated field extension, 207 associated Galois subgroup, 207 automorphism group, 196 contains J.Lz, 235 is finite, 196 of conjugate, 196, 234 order two, 235 53, 197
{z±1},205,234 automorphism of order two, 96, 197,198,200 az + biz, 96, 235 backward orbit, 109, 142 bad reduction, 77, 239 bicritical, 233 Bogomolov conjecture, 129 can have infinitely many rational periodic points, 136 canonical height, 99, 287 lower bound, 221 canonical measure, 127 on Berkovich space, 306 cocycle gives twist?, 203 coefficients well-defined up to homogeneity, 169 commuting, 378 have same canonical height, 137 have same preperiodic points, 38 with polynomial, 378 completely invariant set, 17, 266
contains Berkovich Julia set, 311 condition for unequal formal and primitive periods, 151, 165, 226 conjugation is algebraic group action, 173 coordinate functions of iterate, 149, 226 critical point, 12,284,353 distinct modulo p, 237 periodic, 38 critical value, 353, 382 cutoff for integral points in orbit, 375 cyclic automorphism group, 197, 204 defined at P, 89 degenerate quadratic, 194 degree, 10,89,388,389 of composition, 392 of regular, 394 degree generating function, 397, 428 determined by 2d + 2 coefficients, 169 domain of quasiperiodicity, 310, 311 dynamical degree, 397,428 dynamical unit, 71, 72, 79, 132, 133 of generic, 146 dynamically affine, 376 dynatomic field, 123 dynatomic polynomial, 39, 148, 149, 181 is a polynomial, 151,226 effect on absolute value, 288 effect on height, 90, 388, 398, 407, 409 equicontinuous, 22 equicontinuous family, 254, 265, 313 equi-Lipschitz family, 313 etale, 353 even degree implies FOM = FOD, 215 expanding on average, 36 family omitting two points, 265, 315 Fatou set, 22, 24, 255
504
Index
rational map (continued) field of definition , 206, 236 field of moduli, 177,207,236 not field of definition, 208, 213, 236 finitely many preperiodic points, 38 finitely many with bounded height and resultant, 237 fixed point , 149 formal n-multiplier spectrum, 182 Galois equidistribution, 128 generalized dynatomic polynomial, 227,229 generalized period polynomial, 227 generic not very highly ramified, 231 global minimal model, 222 over Q, 236 global minimal resultant, 220 global S-minimal model, 237 good reduction , 58, 59, 218, 239, 362,383 Berkovieh Julia set, 307, 3 II , 322 good reduction =} same degree , 58 graph,216 Green function , 288 has d + 1 fixed points, 149 has d" + 1 points of period n, 150 has vd(n) points of formal period
n,150,224 has nonrepelling fixed point , 255, 314 has primitive periodic points of different periods , 154, 155, 196 height of, 143 in Md, 221, 237 Herman ring, 28 homogenization, 389 hyperbolic, 279, 285, 317 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, 108, 109, 142, 143
integrality estimate for points in orbit, 112, 145 invariant measure , 127 inverse function theorem, 115, 144, 312 involution, 410, 436 is a morphism if.. . , 89, 388 is continuous, 10,24,253 is open, 10,24,253 ,312 isospectral family, 186, 188, 362 iterate is a polynomial map, 17, 37, 431 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, II , 173 minimal model, 220 minimal resultant , 237 is PGL2(K)-invariant, 221 of conjugate, 219 monomial,428 Montel theorem , 25, 265, 315 most have no automorphisms, 199, 234 most have no wildly critical point, 284 multiplier, 18 at infinity, 38 of fixed point, 18,47,180 of periodic point, 19,47 spectrum, 182, 187 summation formula, 20, 38, 255, 314,380,382 new coordinate functions of conjugate, 218 no critical Julia points, 279 no twists if trivial automorphism group , 198 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 , 190, 233 normalized form, 51, 75 n -period polynomial, 181 number of critical points, 14,37 number of nonrepelling periodic cycles, 27 number of periodic points , 39 of polynomial type , 143 omitting three points, 25 on Berkovich space, 305 orbit of collection of, 143 orbit with many integral points, 110, 111, 143,375 parabolic component, 28 period polynomial, 149 periodic point, 394 field generated by, 122 index summation formula, 20, 38,255,314,380,382 periodic subvariety, 127 PGL2-equivalence, 195,233 PGL 2 (K )-equivalence, 195,233 polynomial ifhas totally ramified fixed point, 17 postcritical set, 353 power map , 29, 95, 234, 321 , 325 in characteristic p, 380 preperiodic subvariety, 127, 128 primitive periodic point, 122 product of primes of bad reduction, 221 pullback of divisor, 404 linearly equivalent, 405 pushforward of divisor, 404 quadratic is polynomial iff (Tl (rfJ) = 2,232 quadratic twist, 200 ramification index, 12,36 of iterate, 117 ramification point , see crit ical point recurrent point, 284, 310 reducible dynatomic polynomial, 235 reduction modulo a prime, 52 repelling periodic point in Julia set, 256 resultant, 56, 58, 75, 112,218,372
505 of composition, 77 of conjugate, 76 of iterate, 184 Riemann-Hurwitz formula , 13,217 self similarity, 196 semistable, 179 separable, 37 separable reduction, 64 set of (Rats), 169 Siegel disk, 28 S-integer points in orbit , 143 space of conjugacy classes of, 174 stable, 179 strictly expanding on Julia set, 279, 317 symmetric polynomial of multipliers, 183, 187,232,367 Taylor series , 190 totally ramified fixed point, 37 totally ramified point, 12 twist, 197 same image in Md , 198,221 with good reduction outside S, 237 uniform bound for integral points in orbit , 112,372,385 uniformly continuous family, 254, 313 uniformly Lipschitz, 254,256,313 universal of degree d, 171 va luation of minimal resultant, 219 very highly ramified, 231, 284 wandering disk domains exist over C p , 285, 286 Weierstrass class, 223, 237 wildly critical point, 284 with Aut( rfJ) = 8 3,234 with ( n Z in Aut( rfJ), 205, 234 with same minimal discriminant, 221 with two crit ical points, 233 z + bi z, 197, 198 rational numbers, 7 absolute values on, 44 , 312 rational radius, 243 , 248 , 252, 278 , 295, 317 rational variety, 232 rationally neutral periodic point, 19, 47 in hyperbolic component, 28
506 Raynaud's theorem, 127 real fixed point, 427 realizable sequence, 6 recurrent point, 3,284, 310 critical in Fatou set is periodic, 284, 317 critical in 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, 61, 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, 391, 394, 427 algebraically stable, 396 canonical height, 431 zero iff finite orbit, 431 degree, 394 of inverse, 394, 395 height inequality, 399, 431 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 Ratd, 169, 174 repelling periodic point, 19,47,326,361 closure equals Julia set, 274, 311 dense in Julia set, 23, 263, 315 in Berkovich space, 310 in Julia set, 22, 40, 256 limit of attracting, 274 of Type II, 310 one implies infinitely many, 274 representation of Galois on elliptic curve torsion, 344 residual degree, 310 residue abstract, 313 at simple pole, 313 of a rational function, 313
Index rational function has finitely many nonzero, 314 residue fixed-point index, 38 residue formula, 314 resultant, 53, 107 chordal metric formula, 76 determinant formula, 54 homogeneity of, 54, 219 measures expansion, 56, 76 minimal, 219 nonvanishing defines Ratd, 169 normalized, 77 of a composition of maps, 77 of a linear fractional transformation, 76 of a rational map, 56, 58, 75, 112, 218,372 of conjugate, 76 of coordinate functions of iterate, 227 of division polynomial, 383 of dynatomic polynomial, 165 of iterate, 184 of quadratic polynomial, 75 product of differences of roots, 53, 75 properties of, 53 transformation formulas, 75, 173, 219 unit iff good reduction, 58, 218 with a linear polynomial, 75 reversibility, 429 reversible affine automorphism, 429 composition of involutions, 429 Henon map, 430 number of orbits over a finite field, 430 Riemann sphere, see projective line Riemann surface, 37 Riemann-Hurwitz formula, 37 V - E + F formula, 37 Riemann-Hurwitz formula, 13,65, 109, 117,217,340,347,379 application to finite invariant set, 16,266 for curves, 15,37 is local-global formula, 15 weak version, 15 Riemann-Roch theorem, 214, 337
Index rigid analytic connectivity,276 rigid analytic space, 298 rigid Lattes 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, 310 Fatou domain classification,240 strong Montel theorem, 311 root of hoiomorphic function determined by Newton polygon, 249 root of hyperbolic component of Mandelbrot set, 166 root of unity iff height one, 88, 100 multiplier, 151, 226 primitive, 148 Roth's theorem, 104, 106, 120 is ineffective, 107 over number fields, 108 Schanual's theorem, 135,441 scheme, 59, 303 good reduction, 59, 61, 77 moduli, 179, 186, 189 multiplicative group, 29 Schwarzian derivative, 231 O'd,N, 187,232 degree unknown for d = 3, 188 embeds M2, 188 Segre embedding, 406, 409 self similarity, 196 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,-IVpoints, 297,321 on c, [zJ, 296 properties, 321 set of bounded is Berkovich disk, 297
507 two variable homogeneous, 303, 305 semistable, 178 locus ofPSL 2 action on jp'2d+l, 178 numerical criterion, 178 quotient, 179 rational map, 179 separable field extension, 37 separable rational map, 37 number of critical points, 37 sequence space, 258, 314 backward orbit dense, 314 backward orbit equidistributed, 315 left shift, 259, 314 metric on, 258, 314 shift map periodic points dense, 314 shift map topologically transitive, 314 Shafarevich conjecture, 442 Sheshadri's theorem, 179 shift map, 259, 314 as itinerary map, 260 backward orbit dense, 314 backward orbit equidistributed, 315 continuous, 259 is topologically transitive, 259 Lipschitz, 259 periodic points, 259 periodic points dense, 314 properties of, 259 topologically transitive, 314 uniformly expanding, 259 Siegel disk, 28 Siegel's theorem integer points on jp'1 minus 3 points, 106,110 integrality of points on elliptic curves, 107 is ineffective, 107 over number fields, 108 (Ji(¢),180
explicit formula for (JI (¢), 181 is in Q[Md], 180 (J(n) 183
,
'
explicit formula for (JI, (J2, 189 for Lattes map, 186 in Q[Ul, (Jj for Rat2, 189 integral over Q[RatdJ, 185 is in Q[MdJ, 183
508
Index
in) (continued)
