Lecture Notes in Mathematics Edited by A. Oold and B. Eckmann
326 Alain Robert
Elliptic Curves Notes from Postgraduate Lectures Given in Lausanne 1971/72
Springer-Verlag Berlin Heidelberg New York Tokyo
Author Alain Robert Universite de Neuchitel, Institut de Mathematiques Chantemerle 20, 2000 Neuchitel, Switzerland
1st Edition 1973 2nd Corrected Printing 1986
Mathematics Subject Classification (1980): 12835, 12B37, 14G 10, 14H 15, 14H45, 32G 15 ISBN 3-540-06309-9 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-06309-9 Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright. All rights are reserved, whether the whole or Part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to -Verwertungsgesellschaft Wort", Munich. C by Springer-Verlag Serren Heidelberg 1973 Printed in Germany Printing and binding: Seltz Offsetdruck, Hemsbach/8ergstr.
2146/3140-543210
NOTATIONS AND
CON V E NT ION S
We have used the usual letters for the basic sets of numbers N (natural integers 0,1,2, ... ), ~
~
(ring of rational integers),
(field of rational numbers), R (field of real numbers), [ (field
of complex numbers), F q (finite field with q elements). As a rule, we denote by AX the multiplicative group of units (invertible elements) in a ring A. In formulas, the cypher 1 always represents the number one (except in log x ... so that in one occurence I have used log -1 to avoid ambiguities). Also ~(x) =e
2rrix
(normalized exponential).
In a theorem, I list properties under Latin letters a), b), ... keeping
i), ii), ... for equivalent properties, but the meaning is
always clear by the context. The following system has been adopted for cross-references. All theorems, propositions, corollaries, lemmas, remarks, definitions, formulas, errata, ... are numbered in one sequence. Such a cypher as (nL3.4) refers to the item (3.4) of chapter III, i.e. the fourth numbered in section 3. (This happens to be a lemma 3.J" From inside chapter III we would refer to (3.4) (in section 3, sometimes simply to lemma 3
this last system of numeration has not been used systema-
tically, but only when it can be more suggestive locally).
TABLE
OF
CONTENTS
I : COMPLEX ELLIPTIC CURVES
CHAPTER
1. Weierstrass theory
2
2. Theta functions (Jacobi)
19
3. Variation of the elliptic curve and modular fonns
35
4. Arithmetical properties of sane modular fonns
66
II : ELLIPTIC CURVES IN CHARACTERISTIC ZERO
CHAPTER
75
1. Algebraic varieties and curves
77
2. Plane cubic curves
98
3. Differential forms and elliptic integrals
124
4. Analytic p-adic functions
144
5. Tate's p-adic elliptic curves
160
CHAPTER
III : DIVISION POINTS
173
1. Division points in characteristic zero
174
2. t-adic representations
184
3. Integrality of singular invariants
197
4. Division points in characteristic p
215
CHAPTER
N: COMPLEMENTS
225
1. Hasse's invariant
226
2. Zeta function of an elliptic curve over a finite field
240
3. Reduction mod p of rational elliptic curves
246
REFERENCES
254
INDEX
262
I NT ROD UC T ION Elliptic curves are special cases of two theories, namely the theory of Riemann surfaces (or algebraic curves) and the theory of abelian varieties, so that any book concerned with these more general topics will cover elliptic curves as example. However, in a series of lectures, it seemed preferable to me to have a more limited scope and introduce students to both theories by giving them relevant theorems in their simplest case. I think that the recent recrudescence of popularity of elliptic curves amply justifies this point of view. I have not chosen the most concise style possible and sometimes have committed the tfcrime of lese-Bourbaki" by giving several proofs of one theorem, illustrating different methods or point of views. I shall not give here any idea of the topics covered by these notes, because each chapter has its own introduction for that purpose (prerequisites are also listed there). Let me just mention that I have omitted complex multiplication theory for lack of time (only integrality of singular invariants is proved in chapter III). In the short commented bibliography (given for each chapter at the end of the notes), I quote most of my sources and indicate some books and articles which should provide ample material for anyone looking for further reading. The origin of my interest in elliptic curves has to be traced to a series of lectures given by
~.
Demazure (Paris Orsay, oct.-dec.67)
on elliptic curves over [. Although the presentation I have adopted differs somewhat from his, I have been much influenced by the notes from these lectures (especially in the section on theta functions). I would also like to seize the opportunity of thanking here Y. Ihara,
VIII J.-P. Serre, G. Shimura for very helpful discussions, correspondence ••. Only at the end of the lectures did I learn through S. Lang that he had also written a book on elliptic curves. It seemed however that the material covered was sufficiently different to allow the publication of my notes, and I hope that they will still have some use. Finally it is a pleasure to thank the audience of the lectures whose interest stimulated me, my wife who gave me some hints on language and L.-O. Pochon who proof-read most of the notes, pointed out some mistakes and established an index. However, I take full responsability for remaining mistakes and would be grateful to anyone bothering to let me know about them !
September 1972
A. Robert Institut de Mathematiques Universite de Neuchatel CH-2000 NEUCHATEL
(Switzerland)
CHAPTER ONE COMPLEX ELLIPTIC CURVES
This chapter has as first aim the presentation of the classical theory of elliptic functions and curves, as first studied in the nineteenth century by Abel, Jacobi, Legendre, Weierstrass. In particular, the following mathematical objects will be shown to be equivalent i) Compact complex Lie group of dimension one. ii) Complex torus (tIL, with a lattice Le([ • iii) Riemann surface of genus one (with chosen base point). iv) Non-singular plane cubic of equation 2
Y =x
3
+
Ax
+
B
Here, we shall basically only assume that the reader is familiar with the theory of analytic functions of one complex variable. However, we shall mention briefly some "superior" interpretations which can be fully understood only with some knowledge of the basic definitions and properties of vector bundles and sheaves on Riemann surfaces.
- 1.2 -
1. Weiersnass' theory
A lattice in a (finite dimensional) real vector space V is by definition a subgroup L generated by a basis of V. Thus a lattice is isomorphic to a group ~d where d
dim(V) , and V/L is a compact group.
=
When we speak of lattice in a complex vector space, we always mean lattice for the real vector space of double dimension obtained by restricting the scalars from [ to R. Thus a lattice L in [ is a subgroup of the form (Jot /
L
=
Z w1 + Z ~1
wi th two complex numbers
lU· such that 1
is not real.
Wz.
Let L be a lattice in [ which we shall keep fixed. We say that a meromorphic function f on [ is an elliptic function (with respect to L, or L-elliptic) when it satisfies fez + tV)
=
fez)
for all GJE:L
(whenever one member is defined!). An elliptic function can be considered as a meromorphic function on the analytic space [/L. It is obvious that the set of elliptic functions (with respect to L) is a field with respect to pointwise addition and multiplication. It is nonetheless obvious that if f is elliptic, so is its derivative f' . Let frO be a non identically zero elliptic function. For any point a
~[
, we can use the Laurent expansion of f at a to define the
rational integer ord (f) (smallest index with non-zero Laurent coeffia
cient). This number is positive if f is regular at a, and strictly negative if f has a pole at a. By periodicity of f, we get ord (f) for any a
t.J E
orda+~(f)
L, hence we shall often consider a as an element of
[/L to speak of ord a (f). The formal sum of elements of [/L div(f) = ~orda(f).(a) , has only finitely many non-zero coefficients (because [/L is compact and the zeros and poles of f are isolated), hence can be considered as 2
- 1.3 -
element of Div([/L)
= ~[[/L]
(we consider this set as additive group,
without the convolution product). The first algebraic properties of elliptic functions are given by the foliowing (1.1) Theorem 1. Let f be a non-constant L-elliptic function, and div(f)
=
Ln. (a.) 1
.
1
Then we have: a) f is not holomorphic. b)
LRes a . (f) = 0
c)
Ln i =
d)
1
0
Ln.a. 1
in [/L
0
1
Proof: If an elliptic function f is holomorphic, it is bounded in a period parallelogram (this is a compact set), hence bounded in the whole complex plane by periodicity. It must then be constant by Liouville's "theorem, hence a). We choose now a basis 4)1 ' W
2
that f has no zero
of L and a E: [ such
or pole on the sides of the period parallelogram
Pa of vertices a, a +
~l
' a +
~2
' a +
~l
+ W2 · The integration of
f(z)dz on the (anticlockwise) oriented boundary dP 21Ci
.L Res a . (f) 1
=
a
of P
a
gives
0 because the opposite sides cancel their contributions
by periodicity of f. This proves b). Then c) follows from b) applied to the elliptic function f'lf • For d), we integrate the function zf'/f over the same boundary dP
a
. The contributions from parallel sides do
not cancel here, but e.g.
(Saa +
4Jof _
J
a +
4)1
+ W.z.
a + WI.
Since f admits the period wI ' (a
J
a
+ lUi
f'lf dz
)zf' (z)/f(z)dz f([a,a+~lJ)
a +
- 4)2
J a
4.11
f' I f d z •
is a closed curve C and
21Ci·Index(O;C)
Adding these two similar terms gives
3
Ln.a. 1
1
q.e.d.
- 1.4 -
We make a few comments on the meaning of this theorem. First, a) asserts that a non-constant elliptic function f has at least one pole. Moreover, if f and g are two elliptic functions with same divisor, the quotient fig has no pole, hence is a constant c : f = c.g . This shows that the divisor of an elliptic function determines the function up to a multiplicative constant. Then b) shows that an elliptic function f cannot have only one pole, this pole being simple. An elliptic function can have only one double pole with zero residue, or two simple poles with opposite residues. These are the simplest possibilities. Then c) brings out the fact that an elliptic function has as many zeros as poles. Thus if f is any non constant elliptic function and c any complex value, f and f - c having the same poles must have the same number of zeros. This explains that f takes all complex values the same number of times. This constant number of times f
(non-constan~
assumes every value may be called the valence (or the order) of f. For any divisor ~n.(a.) £Div(E), where we put E = ~/L to simplify the 1 1 notations, we define its degree as being the sum [n i • This gives a group homomorphism deg : Div(E)
~
~
With this definition, c) can be restated as deg(div(f))
=
0 for all
elliptic functions f ; O. Finally d) gives a further necessary condition on the divisor of an elliptic function (Abel's condition). We shall prove that the conditions c) and d) on a divisor of E are also sufficient to assure the existence of an elliptic function having precisely that divisor. We have to prove the existence of non-constant L-elliptic functions, and we shall follow Weierstrass' idea of constructing an elliptic function with only a double pole (on E), with zero residue. The construction is of transcendental nature. 4
- I.S -
For that purpose, we try an expression of the form L(z - w)-2, provided it converges. But for z fixed, and '4)/~ 00
,
(z - "') - 2
,-.J
tiJ-
2•
(1.2) Lemma. Let V be a (finite dimensional) real Euclidian vector space, and L a lattice in V. Then L IIwl- s converges absolutely for 0; tUE:L
Re(s)
>
dim(V).
Proof. We put d
= dim(V)
and we choose a basis
~l
' ... ,
~d
of L.
We observe first that it is sufficient to prove the convergence of the sum extended over the w=Ln.CJ. with positive n.~O, not all zero. We 1 1 1 group together those for which
Ln.1
n> 0 is a given strictly posi-
tive integer, and choose two constants A>O, B;>O such that 1
Aln ~ Iwl' Bin
The number of these w is asymptotically equal to the area of the simd plex in V with vertices n~. , hence is of the order of Cn - l (C~O). 1
(Actually the exact number can be computed, it is the binomial coefficient (n~~iz).) We find thus a contribution to the sum of the order of
nd-l/n s • The absolute convergence of this series occurs exactly when Re(s) - d
+
1
';>
1 , hence for Re(s»d as asserted.
Now, the series of general term (z
-~)
if we only subtract the asymptotic behavior (z - ~)-2 - w-
2
-2 does not converge, but
w- 2 ,
we get terms
which must be of the order c(z)~-3 (the incredulous
reader can check that by computing the difference of these
fractions~),
hence will give a convergent series. We define the Weierstrass' function (1. 3)
P(z)
=
p(z :
the sum being extended over
L)
=
z- Z
+
L:'{( z
- c.» - Z - w- Z
J
= L - {Of, which we indicate by a
w £ L'
prime in the summation (as a rule, we shall omit the prime for sums extended over all L, and keep it for sums over L'). This series is now absolutely convergent (for all z
L), and it is uniformly (and nor-
~
mally ) convergent when z stays in a bounded part of [ - L 5
- 1.6 -
p admits
It is not absolutely obvious that
L as lattice of periods,
but it is an even function, and we proceed as follows. The derivative of
~
can be computed by termwise differentiation
(1.4)
rl(Z)
=
=
fl(Z:L)
-Z['(z - 4»-3
(no convergence factors are needed here by the lemma, and they have disappeared automatically). This function is obviously L-periodic, hence L-elliptic, and odd. By integration, we get ~(z) + C'"
Since
~
is even,
=-
Z
gives
~/2
=
C~
(C~
a constant).
0 (we are in characteristic r2).
While we are at it, we give the Laurent expansion of
~
and
P'
at
the origin. By termwise differentiation, we see that ~
(2k)
(z)
=
~
(2k+l)! L-(z - ~)
-2k-2
so that the Taylor coefficients of the even function fez)
=
p(z) - z -2
are the f(Zk) (O)/(2k)!
=
(Zk+l) oG
.
where we have deflned G = G2k (L) = 2k get
,I -2k
+ Zk 2
(for k> 2). We eventually
L- lU
(0 < Izl < Min IwI)
(1. 5)
4J~L'
,
and by derivation (1.6)
~' (z)
-ZZ-3 +
~
(Zk+l)ZkG
k~l
+ z Zk Z
2k-l
It is easy to give explicitely the divisors of several elliptic
r
functions connected with being odd, it follows that -a
=a
~'
modL, i.e. 2a€L (and a
or its derivative. For instance,
p'
vanishes at the points a€([ such that
¢ L). A set of
points is 4Jl 12 , (J2 /2 , W3/2 wi th
W
3
l +
t.J
representatives of these 2 . Since
W
~'
has only a
triple pole on [/L, it can have only three zeros, and we must have ( 1 . 7)
d i v ( ~')
Similarly, the elliptic function
= -
3 ( 0) + (1-) + (~~) +
p- ~(a) 6
(for a
E
cY) ·
L) has only a
- 1.7 -
double pole on [/L, hence it has only the two zeros -a and a. At first,
p' (a)
this is true for a ;. -a mod L, but if Za eL, zero a of ~ -
p(a)
o shows
that the
is at least double, hence of order two, and
(1.8)
dive
(The two symmetric zeros of
r-
rea))
=
-Z(O)
+
(a)
+
(-a) ·
are not explicitely known.)
~
We are now able to derive the explicit structure of the field of all elliptic functions. (1.9) Theorem Z. The field of all L-elliptic functions is
([ (r
where
~
=X ,
and &2
=
60G
~'
'~')
(X) [y]/ (y2 - 4X 3 + &2 X + &3)
is identified with the image of Y in the quotient,
= 60l?w- 4 ,
4
';! ([
Ii
!!
=
&3
&~
-
140G
27&;
= 140I:'~-6
6
f
satisfy
0
Moreover, the subfield of eV61 elliptic functions is [( ~) . Proof. Let f be any elliptic function. We can write f as sum of an f = fl
even elliptic function and an odd elliptic function (indeed put fl(z) we can write f
=
=
£1
l(f(z) +
+
f(-z)) and f Z
p' (fZ/ P')
with fZ/
- f l )· Since
f
P'
+
fZ
r'
is odd,
even elliptic function.
This already shows that the field of all elliptic functions is a quadratic
extension of the subfield of even elliptic functions. We supVi = ord a . (f) if ai 1
pose now f is even. We let Y.
1.
~
-a i (mod L) and
= lord a (f) if Zai €,L. These are integers since f being even has i
a pole or a zero of even order at the points the product
&
=
TI(
r-
~. 1
. We consider then
tHai)))li extended over a set of representa-
tives of the classes {ai,-a i } with a i f/;. L . By construction, f and g have divisors with same coefficients, at least for all the points (a), where a
~
L, but the coefficient of (0) must also be the same in the
two divisors by condition c) of Theorem 1. It follows that f is a
p. To prove the P', we use the
constant multiple of g, hence a rational function in (quadratic) algebraic relation satisfied by 7
p and
- 1.8 -
Laurent expansions (l.S) and (1.6). Only the first few coefficients matter : z-2 + 3G Z2 4
f(Z)
pt (z) hence
= -2z- 3
+
SG Z4 6
+
6 (QCz )
+
6G z + 20G Z3 6 4
((}(zS)
+
4z -6 - 24G z -2 - 80G 6 + (D(z2) 4 z-6 + 9G z -2 + lSG + (!J(z2) ~(z) 3 6 4 -2 140G + CO(z2) -60G 4 z (z) 2 - 4 ~(Z)3 6
~' (z)-2
Thus,
P'
&3 + h(z)
-g2 P(z) where
,
h is L-e11iptic, holomorphic, and (f}(z2). This proves h =
To prove that the discriminant
does not vanish, we use a different
~
form of the algebraic relation between the divisor of
P'
o.
p and
r'.
by (1.7), we also know that of
all zeros of the cubic polynomial
4
Since we know
p,Z ,
p3 - g2 P - g3
hence we know
• They are the
w· for i = 1,Z,3, and are all double zeros. By comparing the e i = p.c~) leading terms, we get the equality 4(1' - el)(jJ - e 2 )([.J - e 3 ) (Note that this is also a special case of the explicit expression of p'2
any even elliptic function as rational function of By (1.8) we have ( 1. 1 0)
di v (
r-
e i) = - 2 (0) + 2 f~i. )
p as
derived aboveJ
( for i = 1, 2 , 3)
As the points Wi/Z are incongruent mod L , we conclude that these divisors are all distinct. In particular the functions
p-
e i must
all be distinct. This proves that the three complex numbers e i are
IT (e.-e.) r O. irj J
distinct, and the discriminant ~=
As is well known,
1
this discriminant can be expressed in function of the coefficients of the equation as g~ - 27g~ . Since the coefficient of cubic equation is 0, we also happen to see e
8
1
+
e
Z
+
p2
in the
e 3 = 0 , q.e.d.
- 1.9 -
Let Xoc[
2
2
be the curve of equation y = 4x
I claim that it is non-singular (when g~ - 27g~
3
r
- g2x - g3 .
0). Indeed, the
singular points would satisfy
1. (y2 ox
o o=
- 4x 3 + g2x + g3)
%;.(...)
2y
hence would be on the x-axis with x
= !(g2/l2)1 . But these points
3 2 are not on the curve because g2 - 27g 3 ' 0, as a short computation shows.
We can define a mapping (pez)
'P' (z.)).
fa :
Xo c 1E 2 by
IE - L -
f o (z)
=
It gives a bij ection between Eo and Xo where
Eo = E - {OJ (and E
=
f and P'
[/L as before). Indeed,
value exactly twice with
P(z) = P(-z),
assumes every complex separates z and -z
(all this mod L). This holomorphic map has a holomorphic inverse
f~l : Xo ~ Eo because its derivative z ~ (f'(z),~,,(z)) never vanishes (the only points z where simple zeros of
r',
hence
~'(z)
~"(Wi/ 2)
° are
the
~./2 1
which are
~ 0).
To go from the affine curve Xo to its natural "completion" (or "compactification") we embed [2 into the projective complex plane p2([) = [3 _ {oj /homoth., by (x,y) .....-.. (x,y,l) = (AX,AY, A ), and we
define XClP 2 (IE) by the homogeneous equation y2T = 4X3 - g2XT2 - g3T3 (I hope that there will be no confusion between the indeterminate X and the projective curve). Then we can extend fo to ep as follows.
(1.11) Corollary. Let
O)
it must be the only theta function of its
type (up to a multiplicative constant) by the theorem. It must in particular be proportional to ( Z • 11) ,
en z)
;: d' ?"-
);: 0
whenever
£ - £'
whenever
-
~
0
In particular if d is the divisor of a theta function 9, we see that div(9)
~
0
exactly when 9 is holomorphic. Although the divisor of
the zero function is not defined, we make the formal convention that div(O)
~
£ for all £ € Div(E).
Then we define L(~)
for any divisor £
~
=
{f L-elliptic: div(f) ~ -£
i
Div(E). Due to the preceding convention 0 E:L(£) ,
and it is easily seen that
L(~)
is a complex vector space (the nota-
tion L comes from the appellation
linear
system, sometimes used
for this space, or for the set of divisors of the form div(f) for f
E: L (£)
).
(2.12) Corollary (Riemann-Roch). If d is a divisor on E of degree then
dim[L(~)
I,
= deg(£) .
Let 9 be any fixed theta function with div(9) =
Proof
~
~
(we may
suppose 9 reduced, cf. (2.9) ) and let (h,a) be the type of 9. Define then
f
~
Because
~/9
f
on the space of theta functions of fixed type (h,a).
and 9 have same type, their quotient f,
=
~/9 is an
elliptic function, and div(f,) = div(f) - div(9)
div(4)) - d
We see that
= se(t1"
s(31
f ¢
of the open intersecting discs U« .
is a l-cochain s such that ~s
when U"'~
on
0, i.e.
1 on
u"Pt = u.nu,..nu, ·
(when this triple intersection
uo(~y
is non-empty). This is a compatibility condition on the transition functions
s.~
defined on the non-empty
.
u~~
A l-cocycle is said to
be a l-coboundary, or more simply to be trivial, when there exists a map
G":
Cl( I--+-
cr(aI)€t1>CCU",) with
5
=~r
C"',13)
:
1---+
rC~)/cr(",) on U
olp
To each l-cocycle s one can associate a holomorphic line bundle Fs
~
E on E as follows. We glue together holomorphically the
a:
trivial line bundles on the Uo( , Pee : U-e x
--+-
U-
{\ Uo( Th~
_
Uo(
compatibility conditions on the
s~~
[
n U~ (cocycle condition) ensure
the coherence of these patchings. The line bundle Fs will precisely be holomorphically trivial when s is trivial (existence of a global holomorphic, nowhere vanishing cross-section), so we have an isomorphism
Hl(E,~) ~ s
~
group of holomorphic line bundles /E (Fs~E)
We have already indicated the fact that this group is also isomorphic to the group of classes of divisors Div(E)/P(E) . The co boundary homomorphism
30
- 1.31 -
is given on the line bundles precisely by ~
: F(h,a) ~
(1/2i)ah
d·B
(where d is the degree of any theta function of type (h,a), or equivalentl~ the
degree of any meromorphic section of
F(h,a)~E
).
This justifies in some sense the terminology adopted for "coboundary ah" of
the homomorphism h : L
[ . (More precisely,
~
we should have defined the coboundary of h by C) h (4)1
' lcJz)
=
( 1I
Cc1 h ( w2 ) )
2i) (h (WI ) 4)2 -
If we identify 7l·B with 7l , the coboundary
0 gives what is known
c : Hl(E,~~) ~
under the name of Chern character
class c(F) of a holomorphic line bundle
, but... ! )
F~E
7l : the Chern
is computed (in theory
at least) by taking the degree of the divisor of any meromorphic section of
F~E
. With these interpretations, the exact sequence
(2.13) gives the (isomorphic) exact sequence (2.13)'
o
---+
L -+- [
--+-
deg group of 1ine bundles IE --... Z '--
~
--+-
0
./
or HI (E ,~), or Di v (E) IP (E) The subgroup of H1(E,eJJ consisting of line bundles of Chern class zero (or the subgroup of classes of divisors of degree zero) is thus isomorphic to E = [/L . This is the previously defined (1.24) isomorphism. These line bundles are also called flat line bundles on E and are precisely those which are topologically trivial. So we restate the definition : Pic (E)
"connected component" of Hl(E,~>c) group of holomorphic line bundles over E which are topologically trivial (trivializable!) .
It is important and interesting to note that Hl(E,~~ is also the group of invertible sheaves over E (~odules or O-modules, locally free of rank one), by associating to the bundle
F~E
the sheaf
of germs of its holomorphic cross-sections. The point is that this 31
- 1.32 -
sheaf can be defined without speaking of the line bundle (a real advantage in characteristic, 0
!) as follows. If s EHl(E,~~) is
a coherent system of transition functions, the sheaf by
~s(U)
s is defined
~
= r(U,fs ) = group (or ring) of holomorphic functions on U.
The restrictions are twisted as follows:
f
---+
s~oc
·restr(f) to Uoc(3
Equivalent formulations are, when starting with a divisor d = Ldk(a ), k to put { f:U-..([
d
fez) = (z-a k ) kh(i) in neighbd.
of a k if in U, wi th h hOlomorphiC} , and when starting with a theta function 9 of type (h,a), to put ta(U) = {f:U+L~[ holomorphic, satisfying
f(z+~)=a(~)enh(~)(z+l~)f(z) for Z€U+L} • The sheaf restrictions in these two definitions are the usual restrictions of functions. (2.14) Complement 1. We show why it is sufficient to consider multiplicators of the form Let indeed
F
~
B(W))
exp(A(~)z +
when treating theta functions.
E be any holomorphic line bundle over the elliptic
curve E = ([/L. We use again the basic existence theorem of Riemann surface theory which gives the existence of a meromorphic section s of this bundle. We choose a lifting of this section to the trivial line bundle over the universal covering has the property that (s' Is) (z
+ IU)
-
E = ([ of E. This section 5
(s' Is) (z) is holomorphic
everywhere (entire function) in z for a fixed element w of L. But ~'/s is a section of the canonical (cotangent) line bundle over E.
As the tangent, as well as the cotangent line bundles over E, are trivial, this proves that (s'/s)(z
+
32
~)
-
(s'/s)(z) is constant in z,
- 1.33 -
hence (5' Is)' is an elliptic function. This is what we wanted to prove. (2.15) Complement 2.
It is possible to give more explicitely than
has been done, the possible types of theta functions. Let 9 be a theta function of type (h,a). We have seen that a) h : L
~ ~
is an additive homomorphism,
= 2id·B where d
b) ~h
degdiv(9) ,
and a simple computation shows that a : L ~ ~~satisfies c) a (t.o + w') = (-1) dB(W ,t.t)') a (w) a ( W r) for t..), lAJ' E: L. In particular, fal
: L -+-
TIt.:
is a homomorphism, and this is what
was needed to decompose B = Bo·B red as after (Z.7). As the proof of (2.8) shows, we can write h(z) = (d/S)z + «·z with a complex constant ~.
Moreover, if
win ~L (for
f:
L
new)
denotes the biggest positive integer n such that
tV £L), we see that (_l)dafl(W) a(tJ)
~ [~
~ t - + Re(~tU)
is a homomorphism
the absolute value of which must be of the form for a suitable complex constant
~.
Summing up, the three
above properties a),b), and c) imply that
z + o(·z
h(z)
(diS)
a (c.))
(-1) d-n (c.)) eRe (~c.)) (w)
X
with complex constants
(U = [1 = unit
oC,
(5, and a unitary character ~: L~ V
circle in [). Conversely, all such functions satisfy
the three conditions a),b),c) and hence form the type of a theta function B (with divisor of degree d). (Z .16) In Jacobi t s theory, the variable q
plays as important
.a
= e 1Ci"t where -c =WZllU l
role as ~ itself. If we put t = q2, the
normalized exponential! gives an isomorphism ~
: [I
Z
"-".
[~,
whence also
~ : [I
1l + Z
--r """"'.
[)C I t'll
,
where t~ denotes the multiplicative subgroup of [~ generated by t 33
- 1.34 -
t
Obviously» 1m(1:) > 0
71
=
~t n
: nE
1
~
is equivalent to Itll, but not
It(
= 1).
We shall see that this point of
view is more suited to the study of p-adic elliptic curves.
34
- 1.35 3. Variation of the elliptic curve and modular forms
To compare different elliptic curves, we consider two lattices Land L' in [. Then we put E
= [/L , E' = [/L' and compare them by
means of holomorphic maps.
f:
(3.1) Proposition. Let
E ---+- E' be any holomorphic map. If
Cf
is not constant, it is induced by an affine linear transformation Z
...-.. 0( Z
C
: E
Il
+ (3
--+-
z ..-.. z +
and is surj ective. In particular, if we define
E' Q.r Z ....-.
f3
(cons tant map) and T~
~ (translation by
0/ is a homomorphism E ~
p in E'), then E' · (Ii f is not
ep
=
t
= Tp
+ C(3
fez
+ w) -
fez)
0
t
and
constant, it is surjective~
Proof: We choose a lifting, to the universal coverings and note that
.!?L
E' ---... E'
E = E'= [ ,
must be an element of L' for every
W€L. By continuity in z of this expression, we see that this diffe-
rence
is independent of z and by derivation,
L-elliptic function. By (l.l.a) we see that consequently ~(z) Obviously
0(
=
«z +
~
f'
must be an entire
~'must
be constant, and
with two complex constants
must send L into L'
O(·L eL'. When
0
0
(ad - bc) 'c~
+
by hypothesis
dl- z lm(1:)
=
det(o0 sufficiently small so as to enclose no pole of f'/f other than i), we get
r
JDE
(f' /f)dz _
-} (21tiord (f))
for
i
£~O.
The arcs BC and PG can be treated similarly and give 2 + (f' I f) (z) dz --+ --627t'i· ord (f) when lBC JpG ~
(r
r )
£,.... 0.
On CD, we suppose that the semi-circle chosen to avoid the pole of f'/f is the transform of that on EF (one of them having radius !), under the mapping z
~
-liz (this mapping transforms EF in DC).
We compute the integral over EF by making the change of variable ~ = -liz, so that z ~~
(f'/f)(z)dz
applies EF --+- DC. Thus (f'/f)(-I/{)d(-I/~) =
(f'/f)(-l/l)t-2d~ = (g'/g)(')d~ with
g(~) =
f(-1/0
= f((~ -~)~) = Z;2kf(~).
because f is supposed
to be a modular form of weight k. Hence (g' I g) (() d~
= [2k/~
+
(f' I f)
(l~d~
Now we gather the two integrals over CD and EF of f'/f :
(r
JCD
1
+
EF
)Cf' /f)Cz)dz
=
(r J
CD
+
r )Cf' /f)dz JrDC 2kd;~ +
i DC
Only the integral of 2k dl over DC remains to be computed, or at 1; least its limit when £~O. For the limit, we can 'rectify" the small semi-circle (of radius £) to put CD on Izi
=
I (this does not change
the value of the integral a bit !) and integrate from i to ~ . This is only the variation of the argument between i and' hence
(r
JeD
+
r ) (f' /f)dz
JEF
-
2k(2n:i/12)
21ri'~
The residue theorem gives on the other hand, the global result ;
(f' /f)dz
=
21[i
L
ordp(f) 42
- 1.43 -
with a summation over the poles P of f'lf enclosed by the contour, i.e. all the poles of f'lf in D, distinct from i and
~.
The compari-
son between the explicit computation and the global result gives the announced formula q.e.d. In the course of the proof, we have used the following result of function theory. Let g (g = f'lf in our case) be an analytic function (in a neighborhood of the origin) having a simple pole at the origin, and let CI ' Cz be two continuously differentiable arcs starting at the origin, having respective tangents d l ' d Z with angle ~. Then the integral of g(z)dz on circles of radii t between CI ' Cz and on circles of radii t between d l ' d Z (both in the positive direction if 0( is positive) have same limit
We are mainly interested in holomorphic modular
forms, so
we define Mk to be the complex vector space of holomorphic modular forms f of weight k with ordoo(f) ":1 0 (thus all vp(f) for p~1} are ~O). We are going to show that all M are finite dimensional, and defining k dk dim[(M k ), we shall show that we have the explicit formula L. d Tk (1 - TZ) -1 (1 - T3) -1 E:. 7l[T]] k k This formal series is called the Poincare series of the graded module M =
ek Mk
·
(3.8) Consequences of (3.7). Let fc:::Mo . Since the constants are in Mo ' f - f(i) will be in Mo . By construction, this function has a zero in i, so that the formula given by the proposition cannot hold. This proves f - f(i)
=
0, hence Mo
= ~
is of dimension 1.
Now if f, fE:M k have all same orders ordp(f) = ordp(r) (PE))), their quotient flf will be in Mo ' hence constant. This shows that a function of M is determined up to a multiplicative constant by its k ~
orders at the P6i}. Obviously Mk = W!when the integer k is negative. 43
- 1.44 -
For k
= 1, there is no possibility to satisfy (3.7) with integral
orders, so that M = {Ol, d = l l lity in (3.7):
ord~(f)
O.
For k = 2, there is only one possibi-
= 1, all other orders being o. This proves
= 1, and all functions in M2 vanish at 4. We know that g2 or G4 is Z a holomorphic modular form of weight two, so that M2 = [g2 = [G 4 and
d
g2 vanishes at { with order one (simple zero)! Similarly for k
1 all other
we have only one possibility to satisfy (3.7) : ord. (f) 1. orders being zero. Hence d
= 1, M3 = [g3 = [G 6 and g3 (or
3
= i.
one simple zero located at z on~
Ag~in
for k
=
3,
G ) has only 6
4, there is only
possibility to satisfy (3.7): ord{(f) = 2, all other orders being 2
so that d 4 1, M4 = [g2 = [G S and in particular, GS ' propor2 tional to G ,has a double zero at ~ (and no other zero). In full: 4 ~ero,
L.'(mz
=
+ n) - S
c(
r:: (m z I
4 + n) - ) 2
( 1m ( z)
> 0) ,
a dream of youth, realized up to a constant (which can be determined explicitely). Finally there is only one possibility to satisfy (3.7) when k
= 5: ordi(f) = ordt(f) = 1 so that d S = 1 and MS = [g2 g 3 = [G 10 .
For larger k's, there are several possibilities. Since vanishes in the upper half-plane, necessarily ord00 ~) 3
tions
~€M6
=
never
1. Other func-
2
in M are g2 and g3. We can quote £ E:M with any prescribed 6 6 simple zero P ; i, ~ in D • It is sufficient to construct Z 332 3 2 g3(P)·gZ - g2(P)·g3 ag 2 + bg 3 3
2
The space M6 is thus of dimension d 6 = 2, spanned by g2 and g3 . But more generall~ we have (3.9) Lemma. Let £:M f
k
~
1-+
[ be the linear map defined by =
~(f)
lim f(iy) y+eo
and 1:1'- : Mk ---... M + be defined by multiplication by the function 6. k 6 Then we have an exact sequence of complex vect~spaces
o
A-
£
--+- Mk
~
M +6 ~ [
and in particular d k + 6
= dk
+
k
1, for k
-. 0 , ~O
•
- 1.45 -
Proof: Because
0, it follows that multiplication by
~;
= Ker(£)CM k +6
tive map. Put M~+6 the composite f
=
£(f)
o
. Since ordoocA)
=
~
is an injec-
1, we see that
{6·) is zero. Conversely, if fE:M k +6 is such that
0, i.e. f vanishes at infinity, it can be divided by
~
(this
function will have the same orders at all P E:D and order one less at 00). Finally, GZ (k+6) does not vanish at infinity (we could also take g;g~ of weight Zoe
3[1 , with
+
C(
and ~ chosen so that ZO( + 3(3 = k+6,
possible since Z and 3 are relatively prime). The exactness gives d k - d k + 6 + 1 = O. From this lemma, and the explicit description of the M and d k
for
0
k
(k < 6, we can check that (1 -
r 2 ) (1
-
r 3)
LdkT
k
k~O
=1
We can also say the essentially equivalent result
1
[k/6 (integral part of k/6) if k51 mod 6 dk
= dima:(M k ) =
l(k/6] +
1 if k ~ 1 mod 6
Here, since these formulas are true for 0 ~ k < 6, they will be true by induction for any k because d k +6 = d k + 1 · We can state the general result. (3.10) Theorem 1. Let M
= E9 Mk be the
~raded algebra, sum of the
spaces M of holomorphic modular forms of weight k (bounded when
k
z
=
iy and y ~ ~). Let on the other hand
a:(x,y] be the algebra over
0:, of polynomials in two indeterminates X,Y with the degree d defined
by the two conditions d(X)
ep : (t[X, YJ is an isomorphism of
= Z and
dey)
= 3. Then
--+ M defined by X ~raded
modular form, bounded when z
t-+
&Z and Y ~ g3
algebras. In other words, every holomorphic
= iy is purely imaginary and
y
~~
is a polynomial in gz and g3 (or a polynomial in G4 and G6 ). Here the degree d has been defined ad hoc, but if a:
[x' ,y ~
is
a polynomiahalgebra in two indeterminates, and if we look at the 4S
- 1.46 -
polynomials subalgebra generated by X
=
X' 2 and y
=
will inheri t the degree d from the natural degree of case, d(Xnym) = 2n
+
y' 3, this subalgebra
a: [X' ,Y ,].
In any
3m by definition. Another way of saying the same
thing would be to consider the graded algebras [[X], d' being defined as the double of the usual degree, and
a: [y] ,
d" being defined as the
triple of the usual degree, and then [[X,Y] = [[X]0 [[Y]with the degree d = d'~ d". The Poincare series of ([ [X] ,d') is obviously 1 + T2 + r 4 + r 6 + = 1/(1 - r 2) because this algebra has one generator Xk in degree d' = 2k and no non-zero element of odd degree. Similarly the Poincare series of (a: [Y] ,d tt ) is (1 - T3 ) -1. I t is then well known (and easy to prove) that the Poicare series of the tensor product of two graded algebras is the product of their Poincare series, so that the Poincare series of ([ [X, Y] ,d) is peT)
= (1 -
T 2 )1(l - T 3 f1.
This proves the theorem completely, but we give the classical proof as well. Proof of Theorem 1. The elements of M for 0 ~ k < 6 have been checked k to be polynomials in gz and g3 Since ~ = g~ - 27g~ is also a polynomial in g2 ' and g3 ' induction applies by the lemma, and shows that any f €M k (any integer k) is a polynomial in g2 and g3 ' so ~ is surjective. We have to emphasize the fact that g2 and g3 satisfy no polynomial relation P(g2,g3) = 0 with 0 t: PE:[[X,Y]. Decomposing P into homogeneous components (for the degree d) shows that we may assume P d-homogeneous to start with. If we had \ ij Lc. ·g2 g 3 = 0 2i+3j=a 1J we could solve e.g. for g2 and get \ , i j L- c .. g2 g 3 j t: 0 1J
(after suitable division by g~g~)
which is impossible, because all monomials in the right hand side contain a positive power of &3 vanishing at z = i, and the left member 46
- 1.47 vanishes only at z = ~ (wi th order i o ). This concludes the proof. Now we call modular function a (meromorphic) modular form of weight
o.
Our example is the modular invariant J
ord.(A) = 1, we have
ord~(J)
3
= g2/~
. Because
= -1, but J is holomorphic in H.
(3.11) Theorem 2. The modular invariant J defines a holomorphic bijection of the fundamental domain D defined in (3.3)(and after) onto the complex line J : D ~~ [ , which is conformal except at z = i (ramification index 2) and at z This function is real on the sides of with the normalizing values
J(~)
=
~ D
(ramification index 3).
and on the imaginary axis,
0, J(i)
= 1, lim J(iy) = ~ . y+oo
Finall~
every modular function is a rational function of J. A
Proof: J has a simple pole at P = 00 E: D, so that the same will be true of any of the functions J - A
any complex value). In the formula
(~
of (3.7), only one compensation is possible, namely vp(J - A) for one (and only one) point P€D. According to
A the
1
following
possibilities occur : ordp(J
- A)
1 with p ~ i, {
A= J(i)
ordi(J -
~)
2 if
-
~)
3 if A= J (~)
ord~(J
(because &3 (i)
0)
0 (because &2({)
0)
1
This proves that J takes once and only once each complex value in the fundamental region
D
(not counting multiplicities). The second part
of the theorem follows from
- = G (z) - = Lr ' (mz- + n) - 2=kG- -(z) G2k (-z) 2k 2k hence J(-z) = J(z). In particular J(z) will be real for -z for z purely imaginary. When Re(z)
=
1 , we have
-z
= z, i.e.
= z - 1 , so that
the periodicity of J shows that it must also be real on the lines Re(z)
= tl. For w = (10 -10) and for z on. the unit circle, we have
-z = w(z) , so that J(z)
=
J(-Z)
J(w(z))
47
=
J(z) must also be real.
- 1.48 -
Since J preserves the orientations, it must apply the part Re(z)
~O
of the fundamental region D onto the closed upper half-plane. From that (and the normalizing conditions), it can be reconstituted globally by means of the symmetry principle of Schwarz. Finally, let f be any (meromorphic) modular function. The conditions on f show that there must exist a meromorphic factorization F f HV{iot) ~ a: u( 00) J ~ ,"~F a: u( 00)
with [u(~)= Wl(a:)
= S2 Riemann sphere. Since any meromorphic function
on the Riemann sphere is a rational function, it follows that F is rational and that f
=
F(J) is a rational function of J.
q.e.d.
The second part of the theorem shows that J gives a conformal representation of D/\ (Re(z) (0) deleted of i , ' still with
J(~)
=
0 , J(i)
=
onto (Im(z) ~ 0) - {O,l}
I (but no conformity at these points).
Then using the symmetry principle of Schwarz on Re(z) and then
J
-1 , +} , •••
Izi = I
0
will give the periodic function J in D + Z.
As the symmetry principle also applies along arcs of be applied to
= -} , ][),
it can
, and inductively, it will eventually lead to
a definition of J (by analytic extension) on H. Although this definition of J may be considered as more elementary than the one we have given (because it does not refer to elliptic curves in any way), it seems far from easy to derive from it the arithmetical properties of
J which we are going to consider later. Also we note that the action of SL 2 (R) on H has a natural extension to JPI (JR) = lRu(ioo) (considered as boundary of H) by fractional linear transformations. The transforms under SL ( 7l) are the rational points ~ em on the real 2 axis, so that a must vanish at all these points (tend to 0 when we
of ioo
approach these points vertically from above)
48
and
the real axis is
t
- 1.49 -
the natural boundary for the analytic extension of
~
(and of J).
(3.lZ) Corollary 1 (Little Picard Theorem). Let f : [
Proof: We may assume a = tion points of J
° and
[ be an
f ((I:)n {a, b} = ¢ , then
entire function. If a f b ~[ are avoided by f f must be constant.
~
b = 1. Let AcH be the set of ramifica-
-1
A
J{O,l}. Then J defines a topological analytical
covering J : H - A --+ [ - {O,l)
(local analytic isomorphism). By the
monodromy principle (G: is simply connected) f : [ be lifted to the covering I F : [
~
~
[ - {O,l} can
H - A. But Liouville's theorem
(applied to exp(iF)) shows that F must be constant (by continuity), and so must f be. More important for us is the following (3.13) Corollary Z. Every non-singular plane (projective) cubic curve yZt = x 3 + Axt Z + Bt 3 , is isomorphic (as analytical group) to an elliptic curve E = [/L , wi th a lattice L C[. Proof. The group law on this cubic is given by the condition that three points have sum zero if and only if they are on a line. If we make the transformation of variable y
~
y =
y/Z , we get an equation
for x,y in the Weierstrass form with gz = -4A and g3 = -4B. For this new equation, the non-singularity condition is gi - Z7g~ ~ O. By the theorem, there exists a z £H such that J(z) define a first lattice L
= gi/~i - z7g~)and we
Lz in ~ . Because J is homogeneous of degree 0 in the lattice, any homothetic lattice L would give the =
same value for J. We choose thus a complex constant ~~Osuch that moreover gZ(AL) = A-4 gZ(L) = gz = -4A. This is possible if A 1 0
A4 = -gZ(L)/(4A)
, and determines ~ up to a power of i
=1=1.
In this
case (A 1 0 implies J ; 0) 2
g3 ().L) = so that
g3(~L)
3
g2(~L) (J(~L)
-
1)/(27JC~L))
=
2
g3 '
= ±g3 · By homogeneity of g3 of degree six in the 49
- 1.50 -
lattice, changing A to iA if necessary, will give the equality. For this lattice L', (1.11) gives an isomorphism E
=
plane cubic curve X : y2 t
=
[/L'
~
X with the
4x 3 - g2xt2 - g3t3. In the case A
= g2
0,
we note that the non-singularity condition implies g3 1 0, so that one (at least) of the lattices
AL~
~E[~)
will do. It is indeed
sufficient to take for A any sixth root of g3(L~)/g3 . (3.14) Corollary 3. Two elliptic curves E
=
[/L and E' = [/L' are
isomorphic (as analytic spaces, or equivalently as analytic groups), if and only if J(E) • J(L) = J(L')
= J(E').
Proof: Indeed, two elliptic curves are isomorphic if and only if their lattices are homothetic. Now there are unique region D with L'VL't and
L'~
L"
~,~'
in the fundamental
. The result follows from the injec-
tivity of J on D. The corollary 2 is the basis of the algebraic study of elliptic curves. For example (3.15) Application. Let X be the plane cubic of equation (g~-27g~rO) 2
Y t = 4x
3
- g2xt
2
g3t
continuous). We look at
3
and take any frE:Aut([) (not necessarily (f
as mapping [2 ----. [2 defined by
(x,y) ~ (x~,yr). Then the image X~ of X X fl [2 is the affine o 0 r part of a plane cubic X , and the two curves X and X~ are isomorphic 3 3 exactly when G" leaves J = g2/(g2 - 27g~) fixed. tr
It is obvious that X
has the equation y2 t
4x 3 - gzxt tr
2
3 - g3CS"'t (non-
singular because X is assumed non-singular). Hence the invariant r of X is Jr where J is the invariant of X. This proves the assertion. r Note that if X is isomorphic to [/L and X to [/L r ' we have in general no simple way of determining Lr in function of L (we mean, no algebraic way of determining Lr in function of L: the connection is given by the transcendental function J).
50
- 1.51 -
This application suggests that the field be chosen to be equal to of E (because
~(J)
~(g2,g3)=>~(J)
might
for a special "model" of the equation
~(J)
is the fixed field of the group of automorphisms
~cAut(~) satisfying Jr = J). It is easy to show that this is true.
g'2 = g'3 = 27J/(J - 1) 323 4x - gxt - &t
E:~(J).
The curve
is isomorphic to E (it has the invariant J = J(E) ). If &3 = 0, any non-zero &2 will do, for instance g2 1, hence an equation y2 t = 4x 3 - xt 2 = 4x(x - }t)(x + }t) for E with coefficients in
=
~
(J = 1 in the case &3 = 0).
~(J)
(3.16) Remark 1. It is easy to check elementarily that G6 vanishes at i and that G vanishes at , . By definition 4 G (i) = (mi + n)-6 = l:'i 6 (-m + ni)-6 6
L.
(m~O
= (-1)3 G6 (i). Hence 2G (i) = O. Similarly since ~3 6 polynomial for 4), we see G (0 = 4
L.
(m/n)'0
(m~
n) -4
L' (2(m I:
Hence (1 - ,2)G4(~)
+
J
+
= 1 and ~2
o
+
~
+
L'z:8 (m{3
+
n(2)-4
n4 2)-4 = L'(2(m
1
+
(minimal
n(-1-4))-4
,2(m-n - n,)-4 = (2G4(~)'
= 0 and G4(~) = O. This method does not give
the orders of these zeros however, nor does it show that these "trivial" zeros are the only ones. (3.17) Remark 2. Let q
=
e 7tit: be Jacobi's variable (cf. proof of (2.10]
and t = q2 = ~('t'). Then ~ ~ t = ~(-r) maps the region -l~Re(~)O but b is not an integer, only the terms m = 0 give a contribution,
feb : L z ) _b- 2
L
+
L'(b
+
n)-2 - "E.'n- 2
=
(b - n) -2 - 2{(2) = x2 sin -2'J(b - 7(2/ 3
56
- 1.57 -
With the values
~(4)
= 7r4/90 and
'(6) =
6
-n:
/945, we find
= ( 21f) 6/216 •E3
&Z
(2lt) 4/12· E2
!:l
Z (21f) 12/1728 • (E~ _ E ) 3
&3
J
= E3Z/(E 3Z _ E3Z)
But the first coefficients of the t-expansion (Fourier expansion) of EZ and E3 may be computed easily by starting with the equalities ltcot<Z
=
= z-l
ni(t + l)/(t - 1)
+
r='~z - n)-l - n- 1].
and differentiating with respect to z three (resp. five) times both sides. We see then that E2 ( z)
=1
+ 24 0t +
C1 ( t 2 )
• E3 (z)
=
1 - 504 t +
O( t 2 )
(a much stronger result, giving all Fourier coefficients explicitely will be derived in detail later, in (4.1) ). This shows that E3Z - E3Z (3·Z40 + Z·504) t + O(t Z)
This finishes the proof of the proposition. We turn to a closer sudy of the function A of Legendre, and its dependence on the elliptic curve. By definition, if E is an elliptic curve, it has an equation of the form E : y2
and
~
= 4x
3
- &2 X
-
&3
3
4
IT (x i.1
- h( 1w.
a-
1.
))
= (e 3 - eZ)/(e l - e Z) with e i =
possible equation for E of the form : yZ
x(x - l)(x - A)
Ccf. Cl.13) ) with
A'
0 ,1 .
To bring a dependence in z eH, we define explicitely el(z)
P(1
ezCz)
PC1z: Lz )
e 3 (z)
P(lCz+l): Lz )
:
Lz )
and "C z)
Ce 3 Cz) - eZCz))/CelCz) - eZCz)) 57
- 1.58 -
If we take a direct basis all
tc)z
az + b
=
we shall obviously have
=0
, b :: c
a :: d :: 1
~
, tUz of Lz of the form , tell
cz + d
mod Lz , as soon as We define the subgroup f(Z) c r = 5L (1l) Z
4l =
mod Z
=
and ~WZ =
~z
0) mod Z. In other words, (1 (a c ~) - 0 1 kernel of the natural homomorphism of reduction mod Z
by the condition
5 L Z(Z)
---+- SL Z(II ZZ) ,
hence in particular is a normal subgroup of
r
r(Z). is the
with quotient isomorphic
to the non-abelian group of order 6 (necessarily isomorphic to When
~
E'3).
is in f(Z), we can thus write
Hence eZ(z) = J(o< ,z)eZ(O«z)) , and similarly for e l and e • Coming back to 3 property
=
A(O«Z))
We say that
A
for any
A(Z)
0(
A ,
this gives the
E:r(Z) .
is a modular function for the group
f(Z). To have a
full description of the behavior of this function under the fractional linear transformations coming from the full look at a system of representatives of observed that w
o
(1
=
-1 1 0) and n = (0
Z and nw of order 3 in
r
r,
it is sufficient to
mod f(Z). We have already
1 1) are such that w is of order
f/(±l); they will generate such a system of
representatives. Consequently, it is sufficient to connect A(-l/z) and ~
A(z+l)
under
r.
with
A(Z) to have the description of the behavior of
The first matrix corresponds to the change of basis Wz
= -1 ,
or 1~ = z I Z , 1lUZ This gives ~
-1
=z ,
~l
~
~
, so that e 1 and e Z are interchanged. 1 - A(Z)
(-liz) 58
•
- 1.59 -
Under the second transformation, "2 = z
+
1
~
hence e 2 and e 3 are interchanged ~(z + 1) = (e Z - e 3 )/(e l - e 3 )(z) To sum up
(3.25) Proposition. The function
~(Z)/(A(Z)
- 1)
,
•
of Legendre is a modular function
~
r
f(Z) (normal subgroup of
for the group
(z + 1)
of index 6 formed of all
matrices congruent elementwise to the unit matrix mod 2). Under a fractional linear transformation ~
the s ix fun c t ions
t - " , 1/(1
1/'). ,
t
of
r,
A is transformed in one of
- A), 1 - 1/').. = (~- 1)/ A, A/ (A - 1) ,
;\(-l/z) = 1 - A(Z) , A{Z + 1) =
and explicitely
A/(~ -1)
(z) .
To give an explicit relation with J, it is necessary to study the limit of
A
the matrix (~
when Im(z) -+
i)
in some detail. Since
00
contains
acting by a translation of 2 in H, the function
is periodic of period 2. As before we put q = e~iz expansion of"
r(2)
around q
sarily valid for Im(z)
"7
= 0
and the Laurent
gives the Fourier expansion of
0 because
~
(neces-
~
A has no pole in the upper half-
plane H). Using (3.24), we see that ~ 2 1t£ ( 1 - 1/ 3 ) + 'It' / 3
( e 1 - e 2) (z) ~
7(2
for 1m (z)
--+-
00
but we need more. (e 3 - e 2 ) (z) --+ 0 for Im(z) -+- 00 Coming back to the defini tion of e 2 and e by means of the ~ function,
Also
3
and grouping together the corresponding terms in the expansion of
p,
(this cancels the convergence factors and gives an absolutely convergent series), we find
~ (e 3 - eZ)(z) = L-m,n
(~+ ~z
[
~ ~
((m+~)z
m,n
'7(2
[
L
m
+ mz - n)
+ a - n)
-2
-2 -
-
(az + mz - n)
((m+a)z - n)
[cos -2 (m+ D'7(z - sin-2 (m+ D1CZ ]
59
-2]
-2] =
- 1.60 -
where we have used the classical expansion formula
L
1(Z /sinZ'lI'z ..
(z - n)-Z
"€Z Using the duplication formulas for the trigonometric functions, we
l/cosZ~ - 1/sin2~ = -4cosZ~ /sinZZ«
find that the difference
•
We have thus obtained -47tZ
L. cos (Zm+1)7tz/sin Z(2m+1)1t"z m
-81tZL. cos(2m+l)1rz/sinZ(2m+l)"Jrz m~O
Furthermore
I (q +
cosnz
161(Z
q
-1
1= fil
This gives the desired formula (e
3
- e ) (z) .. 16xZq + O(qZ) , Z
and so
A(z) q-1
(3.26)
=
A{z) e -xiz -.. 16
From this and the relation A(-l/z) )(iy)
~
1 when
gives A(l + iy)
y~O ~ ~
=1
(y real), and when
y~O
~(z
=
24
for Im(z)
-+-
00 •
- A(Z), we infer that + 1)
=
~(Z)/(A(Z)
- 1)
(ibid.).
We come to the explicit relation between
A and
J. Any symmetric
combination of the six functions appearing in (3.25) will be a modular function for
r
(the growth condition for Im{z) -+00 is satisfied by
what we have just seen). We construct the symmetric product of the translates by 1 of these functions : f = (1 + ")(1 + l/~)(l + (l-A))(l + 1/(1-~))(2 - 1/~)(2A-l)/(~-l).
By (3.11), f is a rational function of J. Since f has no pole in H, it must be a polynomial in J, and more precisely since f has a simple 60
- 1.61 -
pole at i
is a linear function in J : f = A
+
B·J • We determine these constants.
The limits q-2 f
e- 2niz f(z) ~ -4.16
q-2 J
e-2~izJ(z) ~ 1/1728
show that B
=
g3(i)
=
=-
_33
=
=
0, e 2 (i)
= 0, and quite generally e 1
-e 3 (i) gives
-1, and finally if e (i) 3
A(i) 0, A(i)
and since J(i) = 1 this proves
= 27 - 27J or
f
= 2- 6 3- 3 ,
27. To determine A, we look at the special z
0 implies e l e 2e 3 (i)
If el(i) A(i)
=
= _2 6
=
+
e2
+
=
e 3 = O.
2. If e 2 (i) = 0, similarly
= 1.
In all cases f(A(i)
A
27. We have obtained
J
1 - f/27.
=
A small computation gives now
=.!. (1
J
(3.27)
-
A2 (1
27
A
e
l
+
e
2
+
e
3
=
0
,
+ ),2)3
_ ~)2.
The relation between J and
th~
e.' s is easily found now. From 1
J
e.2 + e 2 + 2e e {1,2,3j, j k for li J j , k = k J e2 + e2 -2(e 1 e + e 2e + e 3 e l ). Then (3.27) 3 3 2 2
we get e~ 1
hence by summation e 12
+
gives immediately (3.28)
Since g2
-4(e l e 2
+
e e3 2
+
e 3 e 1 ), this expression shows that
A u. = l6(e -e ) 2 (e -e ) 2 (e -e ) 2 l 2 2 3 3 l
IT (e i -e j )
= -16.
J,..
lrJ
(it could,of course, have been derived more simply from it).
61
i.
•
0
- 1.62 -
We draw two possible fundamental regions for f(2) in H.
D(2)
D(2)'
(3.29)
" / /
,,---- " .....
'~.1
\ \1
o
-1
\
The behavior of a near
i~
r(2)-invariant (say meromorphic) function in H, is made by introducing as usual the variable q = e ?tiz . In
the neighbourhood of 0 (in D(2) or D(2)'), we introduce the variable 7Ci z q' e = e -Jrij z' (z' = -l/z near 0 means z near iot ). When z" is in a neighbourhood of 1 (in D(2) or D(2)'), z' = z"-l is near 0, so we take the variable q" = e 1tiz = e- 1Ciz ' = e- 1ti /(z"-l). (3.30) Definition. A modular function for
f(2) is a meromorphic
function f in H satisfying a)
f(~(z))
= fez) for all
~ ~
f(2) (when one member is defined),
b) f has poles at most at the points ioO,O,l with reference to the Laurent expansion in the variables The function
A has
these properties. Indeed,
(uniformly in Re(z)),
'A(z)
~
q,q~q"
A(Z)
~
respectively. 0 when
Im(z).~
1 when z + 0 in D(2), so only the
point 1 deserves verification. But there, we can use ~(-l/(z-l))
= 1/(1 -
A(Z))
We could also look at (3.27) because J has simple poles (in the 62
- 1.63 -
mentioned points) with respect to the variables t = q2 , t' = q,2 and ttl = qtl 2 The orders of a f(2)-modular function are thus · d at a 1 1 · we 11 d e f lne pOlnts
0
f ~ D(2)
D(2)U{ioc),0,1} (if the function
does not vanish identically) , by means of the corresponding Laurent expansions. As in (3.7), integrating f'lf on the boundary
~D(2),
making small semi-circles to avoid the zeros and poles of f and the points
° and 1,
we find
p~
(3.31)
ordp(f)
=
0 •
(This formula is simpler to establish than the corresponding one in
~.7)
because we have no ramification points like we had there in
7;, i , and we take here only the case k=O.) Because
and nowhere that
° in H,
~
is holomorphic
and because A has a simple zero at iaO, we see
A and all the ,,- c (c £
have a simple pole at 1. These ............... functions must have one and only one zero in D(2) to compensate in (3.31). This proves that "
~
° contrary to
v(~/_~) = v((x-~)
-
(x-~')) ~
the fact that v is assumed trivial
on K. Now write the equation of the affine part of V in the form
=
Fo(X,Y) and factor Fo(~ ,Y)
L. v(y
c
Fo(~'Y)
11 (Y
- '1i)
- (X -
=
~)A(X,Y)
°,
- ~i) in linear terms. We have
= vex - {)
+ v(A(x,y))
~ vex - 0 > 0
Because x and y have been chosen with yositive valuation, every polynomial in x and y will have positive valuation, and the above relation can only hold if for one i at least v(y -
~i)
>
0. But for
the same reason as above, this strict inequality can hold at most for one
~i
• By construction, the point P
=
( ~, 'Ji ,1)
Pv
is on
the affine curve Va of equation Fo ' and defines a point (still denoted by P) on V. After a translation, we may suppose that this point is at the origin Pv
=
(0,0,1) of the affine coordinate system chosen.
If fE:.R p , then f admits a representation AlB
with B(O,O,l) ;
Because now vex) >0 and v(y) >0, this implies v(Bo(x,y)) = v(f) = v(Ao(x,y)/Bo(x,y)) = v(Ao(x,y)) Ao(O,O) =
° or
>
° is
O.
° and
equivalent to
to f£M p as was to be shown.
We show a kind of converse to this lemma. (1.17) Theorem 3. Let P be a regular point of V. Then there is a
unique valuation v p valuation, f
E:R p
ring and Mp =
ord p E: X which is centered at P. For that
is equivalent to ordp(f)
(~p)
~ 0.
Thus
Rp
is a valuation
is principal with generator any element such that
n M~
ordpC't'p)
1. Also
determined
br ordp(7t~R;)
r)O
lo},
so that the valuation in guestion is
ordp(M~ - Mr 1 ) = r. (K = k, cL end of proaL)
If S = S(V) denotes the finite set (K is perfect,(1.13),(1.15)) of singular points of V, we see that ord : V - S p
~
ord p is inverse of
7t
(rc-v and
on the sets where they are defined). 90
V-1t
~
X defined by
are the identi ty mapping
- 11.17 -
Proof. Let again X,Y,Z be a permutation of the X.1 such that Z(P) , 0 and (dF/dY)(P) ,
° . Normalize Poe:P by Z(Po )
= 1 ,
and identify Po
with a point (~,~) of the affine plane K 2 • After a translation (replacing x -
~
by x' and y -
by y') we may assume that Po = (0,0)
ry
is at the origin of the affine coordinate system defined by X,Y • Thus the affine part Vo of V is defined by the polynomial Fo with no constant term. Supposing that Fo is not proportional to Y (trivial case), we may assume Fo(X,Y) with A(O)
=
YA(Y) - B(X,Y)X
1 (this is no restriction) and
B(X,Y) = Bo(X)
° (because
YBl(X)
+
+ •••
Fo is not divisible by V). We denote also by x and y the classes of X/Z and Y/Z resp. in K(V).
with Bo(X) ,
First step. Obviously Mp = (x,y) is generated (over Rp ) by x and y. We show (under the preceding hypotheses) that Mp is principal and generated by
x , or what amounts to the same, that y is a
rr p
multiple of x in Rp
By (*) we have yA(y)
•
y = xB(x,y)/A(y) e:xR p because A(O) ,
B(x,y)x, hence
=
° implies
l/A(y) E:R p •
Second step. We show more precisely that there is an integer r such that y€xrR;
= M~
-
If on the contrary B(O,O)
Mr
1
= 0,
• This is so with r
=
~
1
1 if B(O,O) "f O.
we proceed as follows: multiplying
by A(y), A(y)B(x,y) = A(y)Bo(x) and writing Bo(X) =
+
Bl(x)A(y)y
+
•••
XB~(X),
A(y)B(x,y) =
= x(B~
x(B~(x)A(y)
+
B1 (x)B(x,y)
+
y( ••• ))
B1Bo + y( ••• )) = x(~o + y( •.• )) with a new polynomial Ero(X) = B~(X) + Bo(X)Bl(X) containing exactly one power of X less than B (X). If Xr - l is the highest power of X +
o
dividing Bo ' we shall get after r-l steps 91
- 11.18 -
A(y)
r-l
B(x,y)
x
=
r-l
~
(Bo(x)
+
y( ••• ))
with fo(O) ~ 0 • This shows that y
= xrC(x,y)/A(y)r
°,
with C(O,O) !
r
hence the assertion (C(x,y) / A(y) E:R p ). )C
Third step. We show now that
n
~ = {O} mlO (this is Krull's theorem valid quite generally for local noetherian rings, but we indicate a direct proof in our particular case). Equivalently we show Rp -
{OJ
=
U
m)O
JCRR"
p p
=
U (~p _ ~+1) p
m~O
If fe:R p - {OJ we choose a representation C/D of f with D(O,O) ! 0. If C(O,O) ! 0 , fER; and we are done. If on the contrary C(O,O)
0 , it is rather obvious that a method similar to that m )( used in the second step will lead to f £ x Rp • =
The conclusion of the proof is now easy. Any normalized valuation (trivial on K) centered at P is such that vex)
=
1 and trivial on
R~ • This shows that there is at most one such valuation. Conversely
we can define for f = xDlg(x,y) with mE.7L ordp(f) = m. As every element f of
K(V)~
and g(x,y)€R; is of this form this gives
the existence of the required valuation, and concludes the proof. We note however, that we have only used the fact that k = K is algebraically closed to be able to suppose that the point under consideration was at the origin. In other words, to have x -
~Ek(V)
and Y -"]Ek(V).
Thus, the proof shows that the result is true if we replace K by the subfield k'
k(~,~) =
k(P) generated by the coordinates of Po
over any field of definition of V (i.e. containing the coefficients of the polynomial F0 (X, Y) defining V). This field k(P) is a fini te algebraic extension of k (independent of the choice of the line at
92
- 11.19 -
infinity, as long as we take the representant Po with one coordinate equal to 1). (1.18) Corollary 1.
Ii
V = VK is an absolutely irreducible non-singular plane curve, n: X ~ V is a bijection. (One could show in general that finite.
7t
is surjective and ~(S{V)) is
In fact X is in one-one correspondence with the normalized
- or any non-singular model - of the curve V.) One should be careful about the fact that even if element v, v{f) >
° (for f E:K{V))
-1 ~(P)
has only one
does not imply f G:M p • Take for example as in (1.14) the curve y2 =x 3 , p at the origin, and f = y/x.
Then there is a unique valuation v centered at P, and for that valuation v{f) Recall now that a subring A of a field F is called valuation ring if for any
°;: x
E:
F, either x or x-I belongs to A. The rings
associated to valuations (by v(f»O) as before, are obviously valuation rings in this sense. But conversely (1.19) Corollary 2. Rp is a valuation ring of K{V) if and only if the point P is regular (and then Rp is the ring of ord p ). Proof. Remains to show that if Rp is a valuation ring, then P is regular. So we suppose that ylx By definition we can write y/x
E:
=
Rp (otherwise interchange y and x) • A(x,y)/B(x,y) with formal polyno-
mials A(X,Y), B(X,Y)E:K[X,Y] and say
B(O~O)
= 1 (we suppose P at
the origin). This gives YB(X,Y) - XA(X,Y) £1 o (F o ). A degree consideration shows that up to a constant factor Fo = YB(X,Y) - XA(X,Y) so that (dFo/3Y)(0,0) = 1 ;:
° and P
(0,0) is regular.
We note explicitely that all f£K(V) are defined at all regular points PEV (possibly infinite at some of them). If f is defined and finite at the regular point P, f(P) is the unique 93
- 11.20 -
element aoof K such that ordp(f f
= ~mod
aJ >0. Equivalently we may write
Mp . From this it follows that f can be expanded in a
formal Taylor series 2
T(fj = a o + alxp + a 2np + ••• -1 where a l is determined by a l = (f - ao)~p mod Mp , and so on inductively. This series converges to f in Rp (or its completed~p) for the (1.20)
topology defined by taking the powers of the maximal ideal Mp as neighbourhoods of 0 (hence all ideals of Rp are neighbourhoods of 0 because Rp is principal). We can also look at this expansion as formal series T(f)
E: K[[1rp]]
• More generally, every f
E:
K(V) can be
expanded in a Laurent series at P, which will be an element of the field of fractions K((np )) of the former ring K[T~] gives the usual
Then ord p
notion of order of a Laurent series.
Suppose now that V is non-singular. Let Div(V) be the free abelian group generated over the set of points of V. Then we can define the divisor of f e: K(V)
by the formal expression L.ordp(f}(P) •
By (1.11) applied to f and l/f , only finitely ordp(f) will be different from 0 which assures div(f)
=r:
ordp(f). (P) £ Div(V) . PE-V (If V were not assumed to be non-singular, we could define the (1.21)
divisor of f on X using the fact that ~(S(V)) is finite, by div(f) =
L
veX
v(f)· (v)
94
E:
Div(X)
.)
- 11.21 -
Another consequence of Th.3 is the possibility of defining purely algebraically the order of contact of a line and the curve V at a regular point PE:V. Let o<X
+ ~Y +
l =
0 be the equation of
an affine line. We can define the order of contact of the curve and this line at the regular point P £V to be the positive integer ordp(C: (Z/3Z) . Proof. Had we proved Bezout's theorem, the first part would follow immediately, because the Hessian of a cubi.c is of degree 3 hence has 3x3
=
9 intersection points with the cubic. We proceed differen-
tly (and prove a more precise result: the nine flexes are always distinct). We use Lefschetz' principle and thus suppose that the cubic is defined over k c: ([, and we assume that the algebraic closure
k of k in K is itself embedded in ([. Because the group law is given by rational functions (we suppose that P has coordinates in k, or that the cubic is given in Weierstrass' form) with coefficients in k, the solutions of 3Q = 0 have coordinates in kc([. We are reduced
104
- 11.31 -
to the case K = [, where the situation is easy by the transcendental theory of the first chapter : points of order dividing three on the elliptic curve [/L
are the points (1/3)L/L
~
L/3L , which
form a subgroup of order 9 isomorphic to ~/~)Z as was to be proved. (Z.ll) Remark. The nine flexes on such a cubic curve have a special configuration. Indeed, if PI ' Pz are two distinct flexes, we have 3P l
= 0 , 3P Z = 0 hence 3(-Pl -P Z) = 0 , so that -PI -P Z is a third
flex colinear with PI and Pz
•
To sum up, any line joining two
distinct flexes contains a third (distinct) flex. However, the nine flexes are not on a line, because a line cannot cut the cubic in more than three points. In a suitable coordinate system, these nine points will be contained in an affine portion KZ• We cannot make a picture in R Z of this configuration, because any finite set of points of R Z such that a line joining two distinct of them contains a third point of the set is necessarily on a line (proof?).
(Z.lZ) Example. Let yZ = 4x 3 - gzx - g3 be the (affine) equation of a non-singular plane cubic with gz ' g3
~ ~
. Then the set of
points (x,y) £ ~Z on this curve, form an abelian group with the point at infinity as neutral element (for the group law described before (Z.8) ). It can be shown that this abelian group is finitely generated (Mordell-Weil theorem), hence isomorphic to the direct sum of a finite group and a free group
Zr of rank r ~O. It is still
an open problem to determine this rank r in terms of the cubic.
105
- 11.32 -
Let k be an algebraically closed field, L a function field of one variable over k, i.e. a 0eparable) finitely generated extension of k of transcendental degree one. Then it is well known that there exists a transcendental element xE:L such that L is a finite separable algebraic extension of k. Such an extension has a generator y, so that we can write L = k(x,y). Let F be the minimal polynomial of y over k(x) and multiply it by a suitable polynomial in x so as to get an irreducible polynomial F€k[X,y]. Then, L if V
key)
=
V(F) is the (affine) irreducible variety of zeros of F.
=
We denote by X the set of normalized valuations of L which are trivial on k (hoping that this X will not be confused with the former indeterminate X !). We have a mapping (1.16) 1t
X
:
~
xCv)
V
This mapping is bijective outside the set S(V) of singular points -1
of V. We assume that rr (S(V)) is finite (this could be proved, but we know at least that it is true if V is non-singular). Thus every fE'.L
has a well defined divisor div(f)
=
L.
v(f)· (v)
E:
Div(X)
VE:.X
For every divisor
~ E:
L(~)
=
Di v (X) we define the k-subspace L (~) of L by {fE.L : div(f) ?; -~ }
(remember the convention
o Ei L (~)
div(O)~ -~
for any divisor
~,
so that
).
(2.13) Definition. We say that L is of genus g over k preceding conditions are satisfied and dimkL(~)
Observe that
= deg(~)
h
~
fh
+
1 - g
when the
when
for all
~
E:Div(X) of
deg(~)
defines a k-linear isomorphism of
> 2g
- 2.
L(~)
onto L(i - div(f)). In particular, these spaces have the same dimension and this proves that the degree of the divisors of the form div(f)
106
- 11.33 -
are always zero in a field of genus g.
o.
(2.14) Example 1. The rational field k(x) is of genus
Indeed, in
this case, the set X is in one to one correspondence with the projective line over k, and divisors d over X are thus expressions of the form -d = d00 (~)
+
~ d a (a) with rational integers d a vanishing for
ae:
all but a fini te number of a we observe that f
J---+-
f
E:
k. To compute the dimension of
TT (x - a) d a
L(~),
is a k-linear isomorphism of
a£k L(i) into the vector space of polynomials k[x] , having as image the subspace of polynomials F such that -deg(F)
= ord~(F) ~
or equivalently, such that
-d_ -
~
ae:k
deg(F)~d
polynomials is of dimension d
+
=
da
= -deg(~)
deg(~).
I
This subspace of
1 , so that the conditions of the
definition (2.13) are satisfied with g =
o.
(2.15) Example 2. The function field key) over a singular plane cubic curve V is of genus O. It is sufficient to look at the two special curves defined in (2.4). If the singular cubic has an . . db y ratlona . I functIons, . equatIon y 2 = x 3, . I t can b e parametrIze 3 e.g. y = t and x = t 2 • If the singular cubic has an equation
y2
x 2 (x - 1) , it can also be parametrized by rational functions,
e.g. x
=
1
+
t2 , y
=
t(l
+
t 2 ) • In both cases, key)
k(x,y) ck(t)
is a rational field (LUroth's theorem). It has genus 0 by the example 1 above. (2.16) Example 3. We have proved in (1.2.12) that if L is a lattice in [, then the field of L-elliptic functions is a field of genus one over the complex field.
107
- 11.34 -
0,1 in
The above examples exhaust the possibilities for g the following more precise sense.
(2.17) Theorem 3. Let k be an algebraically closed field, L a function field of one variable over k. If L is of genus 0 over k, k(x) (L is a rational field).
then L is purely transcendental L
If L is of genus one over k, then L is the field of rational functions over a non-singular irreducible plane cubic with a flex. In this case (g = 1), if the characteristic p of k is different from 2, the cubic can be taken in Legendre's form y2 = x ex - 1) (x - A) , and if the characteristic p of k is different from 2 and 3, the cubic can be taken in Weierstrass' form y2 = x 3
+
ax
b . In these
+
forms, the non-singularity amounts to AI: 0,1 and respectively 3 Z 4a + 27h I 0 . Proof. Observe once for all that X is not empty: L contains a rational subfie1d and all valuations from this rational subfield (trivial on k) can be extended to L (this shows more precisely that X is infinite because k is infinite). First part. Assume g = 0, and take any v£X, putting
= (v) for
~
the corresponding divisor on X of degree one. By hypothesis L(Q) is of dimension 1, hence equal to k, and contained in dimension 2. Select any
- L(Q)cL, so that necessarily
xk . By construction vex) = -1 because x ~ L(Q), and consequently v(x n ) = -n for all integers n. By induction on n, one
L(~)
=k
xe:L(~)
of
L(~)
$
sees that L(n~) = k • xk e ... e xnk . I claim that the rational subfield k(x) of L is the whole field L. Take any f in L - k and write its divisor div(f) where
(f)~ ~
div(f)
~
=
(£)0 -
(f)~
-f
-f
(symbolically f (0) - f (-) ),
0 is the polar part of the divisor of f. In particular
- (f)oo and so f E::L( (£)00). This shows that the 2 (n
108
+
1)
- 11.35 -
functions
1 , x , x2
,
...
, x
n
f , xf, x 2 f, are in
L(n~ +
n
+
+
d oO
By hypothesis this last space is of dimension
(f)~).
1 where doo = deg (f)eo . As soon as n ~ d oo
inequali ty n
dIDO
+
1
+
+
w t ) ';p Inf(ordp(w) ,ordp(w t ) )
(with equality if these orders are distinct) , b) ordp(fw)
=
ordp(f)
+
ordp(W) for f t:K, WE:Diff(K) .
From the proof of (3.10) we can also see that (3.13)
c) ordp(df)
=
ordp(f) - 1 if ordp(f)
The points P e: V where a differential
W
r
0
(f
E::
K).
has order 0 are called zeros of
W.
(3.14) Proposition. Let K be the function field of k-rational functions over a non-singular plane curve V defined over k, and be a differential form of K. Then div(lU)
L
=
PE:V
ordp(w) (P)
E:
tA)
I: 0
Div(V) .
In other words, a non-zero differential of K has only finitely many zeros and poles. Proof. By choice, x is transcendant over k, so that V is not a vertical line in the (X,Y) coordinate system and dx rential of a uniformizing variable
d (x - x (P)) is the diffe-
at all P with non-vertical
tangent (this lasr set is finite by (1.10)). Thus if
fdx, we
W=
shall have ordp(w) = ordp(f) at nearly all points P and this is zero except if P appears in the divisor of f. (3.15) Definition. A differential form W€Diffk(K) is said to be abelian, or of first kind when it has no pole : ordp(w)
~
0 for
all points P ~ V (which is supposed non-singular). The abelian differentials make up a vector space differential'df can be abelian only when f
€
~t
k and df
over k. An exact =
0 (3.13).
An example of non-zero abelian differential is given by the following (3.16) Proposition. The dimension (over k) of the space
~1
of
abelian differentials of K is 0 if K is of genus 0 over k (K rational over k) and 1 if K is of genus lover k. Proof. Take first the case K = k(x) is rational over k. Then V is the 129
11.56 -
projective line
ku{~J
(identified with the X-axis). Because x - x(p)
is a uniformizing variable at all finite points P and dx = d(x - x(P)), we see that the condition ordp(fdx)
~o
for all finite points P
implies that fl::.k[x] is a polynomial in x. At the point at infinity, we have 'TtoO
ord~(x)
= -1 so that we can take the uniformizing variable
= 1/x. The n x = 1/rr 00
has order ordco(c.J)
~
,
dx =
-2 d Tr
-It06
oO
and
£J
-2 if frO. This shows
= f dx = - f ( l/TL) / 7t002d Tr
oO
=
~1
o.
We turn to the
case of genus one and suppose K = k(x,y) is the function field over the cubic of equation y2 = x 3 + ax + b (for some a, b
Eo
k). I claim
that w = dx/y = (l/y)dx has order 0 at all points P of the cubic V. This is obvious if P is not among the E. (i = 0, ... ,3 , points of 1
order 1 or 2) because then x -x(P) is a uniformizing variable at P and
0
r dP (dx / y ) =
0
r d P ( d (x - x ( P) ) J -
0
r dP ( y) =
0
r dP (x - x (P)) - 1 - 0 = 0
(because yep) 1: 0 at these points). At the points E. = (e. ,0) (i=1,2,3) 1
we ha v e
0
rd. (y) 1
1 and
0
rd. (dx) =
0 rd. (d (x - e. )) 1 1 1
1
= 2 - 1 = 1 whi c h
gives ord.1 (dx/y) = O. At the flex at infinity E0 (neutral element), we choose the uniformizing parameter ~00 = y/x 2 (2.21) (we could also choose x/y). Small computations give
dn~
3 (1/x 2 )dy -2(y/x 3 )dx = (2ydy)/2x 2y - 2(x 3+ax+b)/x .dx/y (-1/2 - 3a/(2x 2) - 2b/x 3 ).dx/y .
We can write this in the form wi th f
(3.17) This proves in particular
E
ord~(dx/y)
X
R00
and f (00) =
o.
=
-l .
Had we only needed this
order, we could have proceeded more expediently as follows (using (2.21) and (3.13)) ordco(dx/y)
ord~(dx)
-
ord~(y)
=
ord~(x)
- 1 + 3 = 0
but we shall use the more precise form given in (3.17). The conclusion of the proof is now straightforward : if ~'= f(dx/y) is an abelian differential form, we must have div(f) = 0, so that f 130
E:
k.
- 11.57 We are now able to give a purely algebraic definition of the residue of a differential form at a regular point P. (3 18) Proposition. Let K be the function field over the plane curve
V, and let P be a regular point on V. Then there exists a unique linear form
res p :
Diffk(K)
~
k
having the following properties a) resp(€A»)
=
0 if w£Diffk(K) wi th ord p (4))
o
b) resp(df) df
n
=
~p
for all exact differentials (f
be a uniformizing variable at P :
and wri te the Laurent series in f
= ~ a i 7ti
1(a) - P"(a)p(J)(a) - P' (a)pCf>(a) 24
+
(3 )(
p" ( a ) p
a) )
+
0
The assiduous reader will check it for g3 ! Then we say that the polynomial pg is in normal form when it is given by pg(x)
=
4x 3
+
ax
+
b
With the notations introduced in (3.42), this gives a 4 = 0 , a 3 = 1 , a = 0 , a l = a/4 , a o = b Z and if we compute the invariants defined in (3.43) we get -b This shows that pg is in normal form precisely when pgex) = 4x
3
- g2 ep )x - g3 ep )
The important point is that if P has no multiple roots, it has a transform pg (for some unimodular complex matrix g) in normal form and g2 ep )3 - 27g 3 ep )2 phy of the form x
~
f
0 • To see this, we first make a homogra-
II (x·- a) which has the effect of reversing
the coefficients of P(x - a) (x -a)4 P(1/(x -a))
P(a)x 4
142
+
On o If we denote by ord
anpn p
with
O~an0 and take a uni tary polynomial P E: K[X] of
the same degree as P and wi th coefficients
a.1
E::
K close to the
r·
corresponding coefficients a i of P : lai ail < Because K is supposed to be algebraically closed, all its roots ~.1 lie in K and we can write P (X) =
IT (X
Then on one hand P(~) =
P(~) =
-
~i )
n(~ - ~
(P - P)
(~i E: K)
•
i) and on the other
(~) = L (ai
- a ) i
~i
so that we get the following estimate for the absolute values
This implies that for one i at least
I~ - ~il < as soon as
o
(o,l/n < e
is chosen according to
argument shows that ~
(;i e:K)
,
d < f.n/M1. The preceding
belongs to the closure of K : ~
E
i< .
Consequently J( is algebraically closed. As we shall use several ultrametric properties of
Jlp ,
we
recall some of them briefly here. An ultrametric space in general, is a metric space equipped with a distance d satisfying the stronger inequality d(x,z) < Max(d(x,y),d(y,z))
for all x,y,z .
Let us introduce the following convenient notations and terminology. The closed ball B'(a) of radius r and center a is defined by r d(x,a) ~r. The open ball Br(a) of radius r and center a is defined by d(x,a)1
'ail
-1
>1
if
Thus for x E: U one has
m
B Tf
j=l
wi th two constants
11 -
A ~l, B ~ 1 •
x/b-I J
We now look at the balls B; (a i )
(disj oint if distinct) centered at the roots of f in U (1 and choose d> t
close to
(,
~
i , n)
so that the balls BS (a i ) are distinct
(or disjoint) at the same time as the corresponding smaller B~(ai). We let x vary in the region (annulus) D1 = B cr ( a 1)
- B; ( a 1 ) c U (c
149
K 0 r .ft p )
,
- 11.76 -
e S' ~ (, for
defined equivalently by
those i and we get
n
IT Ix
If(x)l = A
i=l
- ail
all< f
>
>
A~n
gn
for x E:D I •
But If(x) - g(x)/< £n for the same x, hence by the ultrametric property, necessarily
If
(x) I = Jg (x)1 for x
€ D
I
• More precisely, still
for x£D I ' if we denote by n l the number of roots of f in ITlx - a. I
1"=1
1
IT
=
If(x) I
This shows that
n'pe -
Ix - a. I 1, =/x -all
a"~B'(a ). 1~ ( I
,
~
= Allx - al(l
a. v
b j e:.
I
=(a i -all for x
e: DI
' and similarly if
I
B~
,
1,
ml denotes the number of roots of g in B£(a l ), we get ml g(x) = Bllx - all for x €D I (note that Ix - bjl = Ix - all
I
B~(al)
if
(a l ) ). Since we know that f and g have same modulus on DI '
we get Ix - all
n - m l
l
independent of x in DI ·
Taking x,x'E:D I with distinct distances in ]f,cf[ from a l (this is possible because the set of absolute values is dense in ]~,E[ proves n l = ml · This shows more generally that f and g have same number of roots in all closed balls
B~(ai).
For any other ball
B~
n
contained in U, the inequality If(x)l>c will still be valid in B£ hence )g (x)1
=
If(x)' in B~
and nei ther f nor g have roots in B£
This proves the statement completely. We can now study power series on a formal series f(X)
= L. anx n~O
If
n
, n Lanx converges for some x
np .
Let
f
E:
.12p [[X]] be
(an E: IIp) n
Eo .1L
X
p
, this means anx
n
---+-
0 , or
J Ixl n ~ 0 and consequently la )Iyl n --.. 0 for every y l:: 12 in n n " p the closed ball Iyf ~ lxl and the series~will converge normally on
la
that closed ball (absolutely at each point and uniformly on the ball). 150
- 11.77 -
(This attractive situation is well-known to fail with ordinary complex power series!) Under these circumstances, f defines a mapping B', x I (0)
11p · If all coefficients a n lie in a finite
~
sub-extension K of
~p
, and if y is also in K with
'Y/~lx(,
the
value fey) will also be in K because this field is complete (and all partial sums are in K). The whole theory is based on the fact that the modulus If(x)f can be given more or less explicitely, at least as far as x is not on certain exceptional spheres. We have to remember that, in a finite (or infinite) sum, a term alone of maximal size (absolute value) carries all responsability for the modulus of
I
the whole sum ! For instance, if a o to, necessarily If (x) = lao I in a neighbourhood (an open disc) of x o. More generally, if n is the order of f
~
0 at the origin, i.e. the smallest index i
with a i ; 0, we shall have )f(x)I = lanllxl n in a neighbourhood of m x = o. If however f(X) r a Xn , the size of a term a x (m >n) with n m n am.r 0 will necessarily overtake the size of anx for lx' large enough. This leads, to the notion of cri tical radius. (4.4) Definition. Suppose that the power series defined by f converges in an open ball B = BR(O) with some positive R>O. Then a R radius 0 N. This implies that all an wi th index n > n i vanish and proves the proposi tion.
152
- 11.79 -
We have to consider a slightly more general situation, namely
=
that of Laurent series. Let thus f(X)
00
•
2:c.X 1 be a formal series
-00
1.
with coefficients c. = c.(f) in a fixed finite extension K of 1.
1.
~
P
(the reader will observe that K could be replaced by the universal domain ..Q.p if wedid not want to prove rationality conditions) . We suppose that f(x) is convergent in a non-empty open annulus and we denote by
... < r_ 1
0) into the mul tiplica ti ve
~(T)
E ([)(,
\q I < 1). (Strictly speaking
to keep perfect coherence with the first
chapter. However, this would lead to using throughout even powers of q, and the reader will easily convince himself that this new normalization is more natural.) But now, there are plenty of discrete subgroups (of rank one) in each
O~~'l:Z
mod K
-~(gZ/g3)
·
This is a well-defined element of K~/K~Z, independent of the particular Weierstrass equation chosen with coefficients in K which (together with the absolute invariant j) characterizes completely the K-isomorphism class of the elliptic curve. The reason for the choice of the factor
-~
will be apparent below. To compute the relative invariant
of Tate's curve Eq , we need a lemma. (5.28) Lemma. Let K be a p-adic field, x 1 + 4x is a square of K : 1 + 4x
E
E:
K with \xl < 1. Then
>eZ
K
Proof. We start with the formal series expansion (1+4X)1 = 1 + !(4X) + !(-I)(4X)Z/Z! + with coefficient of Xk (up to the sign) given by 1.3-s ... (Zk-3)Zk/k!
=
z.
1·3·s ... (Zk-3)·Z·4·6 ... (Zk-Z)/(k!(k-l}!)
~ (2(k-l))
(Zk-Z)! = Z k!(k-l)! But (2;)
eC:_-l1)).
k
k-l
Z (k-l)) = (Zk-l)-k (2 k-l
is an integer
and k is prime to Zk-l (e.g. because Zk - (Zk-l) = 1) so that k must divide
2
This proves that the coefficients of (1
all integers: (1 + 4X)1
=
power series expression Because both 1
+
l:-a Xn (a n~O
n
L.a n x n
n
+
4X) 1 are
€~) which gives a convergent for the square root of 1 + 4x .
Z40 s 3 and 1 - 504 s 5 are of the form 1 + 4x wi th
IxlO € Tp(G) means that pX n + l for all integers n ~ o. This is a Z -module because t neG) is a
o
Let, for a moment, G be any additive p-divisible group We define the Tate module (1.6)
T (G) p
By definition, x
= l~
t neG) p
n
=
module over Z/p~ = ~ p Ip~ p a canonical isomorphism
p and lim ~ Ip~
Hom(p-~~,G)
+-p
p
p = ~ p . Because there is
=t
(G) pn another possible definition for Tate's module would have been -n (1.7) l~ Horn (p ll/ll, G) Tp(G) Horn ( 1im --..,.. p -Ilz IZ , G)
Horn (~ I'll , G)
P P
We could still replace G by its subgroup G = t (G) of elements p pOD having a power of p as order (p-primary component of G) in this last formula. Now we consider the projective sequence of homomorphisms of multiplication by p ... ~ G ~ G ~ G p p p We put VP (G) = Vp(Gp ) = lim(G ~ G ) +-p p and call it the extended Tate module of G (or G , relative to the (1.8)
p
prime p). By definition Tp(G) C Vp(G) is the submodule consisting of sequences x
= (x n ) with Xo = 0, and the projection x
178
~
Xo onto the
- 111.7 -
o~ component gives an exact sequence
o
Gp ~ 0 . Because each x ~Vp(G) is such that pmx has O~ component pmxo ~
Tp(G)
~
Vp(G)
~
0
for m sufficiently large, we see that
U
V (G) = p -m T (G) p m~O p
and because Tp(G) has no torsion (multiplication by p is the shift operator), we have (1.9)
Vp(G) = Tp(G)
~ ~p
II
p
In particular, this extended module is a vector space over
~
p
.
Let us take in particular for G a torus OR/~)n ~ ([l)n. Then (1.10)
Tp(G) ~ HOm(~p/7lp,R/71)n ';;;' HOm(~p/zp,a:1)n =
(top.dual of~ II )n ~ ll n p p p (and this is also isomorphic to Hom((~ II )n,a: 1 ) ~ top.dual of t 1lt
P
P
~
(G)).
This is a free Zp -module of rank n and consequently Vp (G) is a vector space of dimension n over ~ p in this case. These spaces can be looked at as p-adic analogues of the tangent space (elements of order p are "closer to the origin" than elements of order p2, in the
= ~,~ p =
algebraic sense). For p Voo(G)
=
Hom(R,G)
=
~~
= R and we could put
Lie(G)
To come back to our case, we let G
,
E
=
E be the group of
k
k-rational points on our elliptic curve. There are two canonical representations attached to the space Vp(E) (2-dimensional over
~p).
The first one is a representation of the ring of endomorphisms of E (1.11)
End(E)
~
End(Vp(E))
and the second one is a representation of the Galois group of the algebraic (separable) closure of k over k (1.12)
Gal (k/k) --.. Aut(V (E)) . P
These two representations are the main reason for introducing the
179
- 111.8 -
p-adic modules Tp(E) and Vp(E).
In particular, the Galois module
Vp(E) is isomorphic to the vector dual (over
~p)
of the etale cohomo-
logy group Hl(E) defined by Artin-Grothendieck. It could thus be p
called first p-adic homology group of E. In a particular case (transcendental invariant j), the image of Gal(k/k) in Aut(Vp(E)) will be determined explicitely in the next section. From a somewhat different point of view, let us add a few general considerations on rational division points over an elliptic curve E defined over a number field k, i.e. those with coordinates in k. More precisely, we can show that the k-rational torsion subgroup t(E k ) of E is finite (this would follow from the Mordell-Weil theorem asserting that Ek is finitely generated, but we prove this corollary directly by a local method). Let (10) be a prime ideal in the
1
ring of integers
~k
of k, which we assume prime to (2) for the sake
of simplicity. We denote by K
=
k?
its
r
-adic completed field, by
R the ring of integers of K and by P the maximal ideal of the local ring R. The announced result will follow from the following local uniformization theorem for
~
-adic elliptic curves.
(1.13) Proposition. Let E be an elliptic curve defined over the p-adic field K, say by a Weierstrass equation y2 = 4x 3 + ax + b with integral coefficients a,b
£
R. Then there is an open subgroup
U of EK,isomorphic both algebraically and topologically to the (additive) group of integers R of K. Proof. The neighbourhood in question on the elliptic curve E will be defined by x large, or l/x small, and because infinity is a ramification point of index two for x we shall take t = l/x l as local uniformizing variable. To be able to do that, we have to check that l/x is indeed a square in our p-adic field K. But
180
- 111.9 -
and this is a square in K as soon as x is big, say x -1 ~ the formal series for (1
+
~,
because
X)l is convergent in ~ (the denominators
of its coefficients have only powers of 2 in their denominators, hence are in R : we use the fact that
1
is prime to (2) here, otherwise
the convergence radius of that series would be smaller). Thus let us put x = t- 2 , whence y 2 = 4t- 6 + at- 2 + b = t- 6 (4 + at 4 + bt 6 ) and l/y = :!:t 3 (4
+
at 4
+
bt 6 )-1 = ±(t 3 /2
+
higher order terms)
is given by a power series with coefficients in R. More precisely, the coefficients of this series are in
~[l,a,b]~ R
(polynomials in
a and b with rational coefficients having only powers of two in their denominators). Let us now define by a formal term by term integration z(t) = d; (t 3/Z + ••• )(-Zt- 3 )dt = -t + •••
J
J
th coefficient of that series, nan Now if we call an the n-
E
7l[l,a,b] ,
because the integration has introduced the denominator n. This series has same convergence radius as that giving l/y because n ord x-n = nord x - ord n ~ 00 and so Ixn/nl =lxl n /n --. 0 if lxl< will find t = -z + bzz Z + b 3 z 3 + b i recursively as polynomials in the
if ord x > 0 1 . If we solve now for t, we and the equations giving the
b j (j < i) and a k (k ~ i) (compare with the expansions of the
show that n!b n e ~[l,a,b] logarithm and the exponential functions). Thus the series giving t = t(z) will converge in the same disc as the exponential series t(z)
=
I:
n>l
£~ zn n.
Pn{a,b) ~ Z[l,a,b] .
But the exponential series has a non-zero convergence radius as follows from the well-known formula ord (n!) = n - Sen) p
p - 1
if Sen) = [no1 for n = [n.pi 1
181
- 111.10 -
I cIa im now t hat z
t-+
(x ( t ( z) ) , y ( t ( z) ))
= (( t
( z) - 2, - 2t ( z) - 3 , ...)
is a group isomorphism in the domain of convergence of these series. But this assertion amounts to a lot of identities between the coefficients Pn(a,b) of t= t(z). To check these identities, we choose an embedding a
~
A,b
B of
~
then we just observe that
~[!,a,b]
into the complex field [ •
they resul t from the classical (complex)
theory of elliptic curves, where they are true formally (i.e. when A and B are indeterminates) because we could choose A and B transcendental, independent. This proves the polynomial identities and the isomorphism in the domain of convergence of the series t = t(z), where all p-adic series have a meaning. (1.14) Corollary 1. Let j
€
K be fixed. Then there is a constant M.
such that the order Card t(E K) defined over K of invariant j.
J
Mj for every elliptic curve E
~
Proof. The projective space p2(K) is compact because it can be covered by the three compact charts, images of the compact sets
Jlj
=
{ex i ) i = 0 , 1 ,2
: xi E: R , x j = I} ·
This implies that the closed subset EK is also compact. On the other hand EK has an open subgroup U isomorphic to R,hence without torsion. Consequently is finite Now the K-isomorphism classes of elliptic curves of invariant j are parametrized by the finite sets K)'C/(K~)2
if j
~
0,1728
(relative invariant
K)&/(K~)4 if j
1728
K~/(K>c)6 if j
0
(l[. 5 . 27) )
,
This gives the uniformity of the bounds for the orders of t(E ) for K fixed invariant j.
182
- 111.11 -
(1.15) Corollary 2. Let E be an elliptic curve defined over the number field k. Then the order of the rational torsion subgroup t(E k ) is finite. Proof. Observe that with the above notations t(E ) k
C
t(E K )
(1.16) Remark. It has been conjectured for some time that the finite number of rational torsion points Card t(E k ) on elliptic curves defined over k is bounded by a constant Mk depending only on the number field k and not the elliptic curve E over k (with
Mk ~ ~ for increasing k c k = ~). Manin first proved a weak form of that conjecture, showing that for any prime number p, the th component of that order is bounded (uniformly in E defined over pk). Recently, the conjecture in its strong form has been proved by Demjanenko.
183
- 111.12 -
2. An t-adic representation of a Galois group
Let E be an elliptic curve defined over a field k of characteristic ~(j)
o.
If j = j(E) is the absolute invariant of E, necessarily
c k. We suppose that k c [ is embedded in the complex field.
For any integer N
~
1, we denote by tN(E)
t of E having an order dividing N : N·t module of rank 2. There is a canonical
=
NE the subgroup of points
O. This is a free
~-bilinear
~/~
-
form over this
module, with values in the group ~(N-~~)of N~ roots of 1 in [ which can be defined as follows. For t E:NE, the divisor N((t) -(0)) over E is of degree 0 and satisfies Abel's condition, hence is principal. Take a rational function over E, f t £[(E) with this divisor (f t is determined up to a multiplicative constant by this condition) div(f t ) = N((t) - (0)). Select t'
£
E with N·t' = t (noting that two
such p.oints t' and t" must differ by a point in NE: t" = t' +u with u
E:
NE). The divisor
~t =
L..
[(t' +u) - CU)] €. Div(E)
u~E
depends only on t and not on the choice of t' with N·t'
t, has
degree 0, and satisfies Abel's condition: (t'+u-u) = t' = N2t' = N·t = 0 . UENE UENE This proves that ~t is also principal, and we can find a rational
L
L..
function Ft £ [(E) on E with divisor ~t (and this condition determines Ft up to a multiplicative constant). Then
diV(F~)
=
N~t
=
L
E N
N( (t' +u) - (u))
is the divisor of the rational function fteN : v there exists a constant c € Ft ) with get
F~ (v
~
ft(N.v) so that
(depending on t, the choices of f t and Ft(v) = cft(N.v), and replacing v by v + s (for s e: NE), we [~
N
+
s)
= F~ (v)
• Hence there exis ts a well-defined Nt_h root
- 111.13 eN(t,s) with Ft(V + s)
eN(t,s)Ft(v) (for all v
=
E:
E). This mapping
eN has the following properties. (2.1) Proposition. The mapping eN
and satisfies
~-bilinear
a) eN is antisymmetrical : eN(s,t)
= eN(t,s) -1
b) eN is non-degenerate : eN(t,s) = 1 for all s
=
t
£
NE implies
0 E:E
c) for any automorphism
F
of [ trivial on k (or on any field
of definition of E) eN(t,s)~ = eN(tr,sr) Proof. For brevity we write e = eN in this proof, hoping that no confusion will arise! By definition, it is clear that e(t,s
= e(t,s)e(t,st). (t)
=
+
s ')
Let us prove that also e(t +t' ,s) = e(t,s)e(t' ,s).
For that, put ttt = t + t' and take a rational function F div(F)
+
E:
a: (E) wi th
(t') - (t") - (0). Then
N diV(F ) = N(t) + N(t') - N(t") - N(O)
=
div(ftft,/f t ")
with certain choices of functions ft' f t , and f t " corresponding to the points t, tt, ttl. Thus
(F~F~,/F~II)(V) = and hence
(cftc'ft,/C"ft")(NoV)
(FtFt,/Fttt)(v)
=
C'F(N.v)
is invariant under the substi tution v e(t,s)e(t' ,sJ/e(t",s) = I hence the proof of a). For n ft,n(v)
=
=
C·F N (N·v)
~
v + s (s
E::
~-bilinearity.
NE). This proves We turn to the
1, ... ,N define the translate ft,n of f t by
ft(v -nt), and compute the divisor of the product
N N div(TI f t n) = N [(t n=l' n=l N Hence f C is constant and n=l t,n
L.
+
nt) - (nt)]
TIft,n
=0 ·
IT
IT ft,n (N ov) = IT f t (Nv - nt) IT f t (N(v - nt')) definition to TT CnF~(V -nt') and must be constant.
ITft,n (v) is equal by
185
- 111.14 N
IT
Extracting the N1J1 root, we see that
n=l
and replacing v by v + t' we get
N
IT
n=l
Ft (v - nt ') must be constant
N
Ft(v-nt') = N-1
IT
n n=l
Ft(v+t' -nt')
Ft (v - nt' )
n=O
and after simplification by the common factors, Ft(v) = Ft(v -Nt') Ft(v - t) is invariant by translation of t
: e(t,t) = I
(for any
t E:: NE). From there, replacing t by t + s and using the bi1ineari ty of the symbol e = eN ' we derive 1
=
e(t + s,t + s) = e(t,s)e(s,t)
For the proof of b), note that if e(t,s) Ft (v + s)
=
. for all s
I
~NE,
we have
= Ft (v) for all these s and so Ft (v) = ep(Nv) implying
F~ = (~.N)N = ftoN so that div(~)
=
div(~oN) = N-ldiV(fiN) and
div(ft)/N
=
(t) - (0) .
Abel's condition gives t = 0 so that e
cr
Finally, for c), we observe that
=
eN is non-degenerate.
acts on E and on the set X of
normalized valuations of [(E) (trivial on [) according to r
f"
(ord p €oX or p 6.E).
(ordpJ (f ) = ordp(f)
In particular, if x and yare coordinate functions on E regular at P (and defined over k) r
r
~
~
(ord p ) (x - x (P )) = (ordpJ (x - x (P) )
and similarly (ord p { (y -y(pll"))), 0, centered at P
r
r
: (ord p )
=
r
that if f ~ [(E), div(f )
ordpr , ordpr (f )
=
=
e(tr ,sr)Ftcr(v r
=
ordp(f) .
t
, F
t
=
r
= Ftr(vG""+sr) =
F~((V + s)O"'")
Ft (v)eT" = e ( t , s)cr Ft cr (vcr)
e(t~,s~)
~ on
are chosen as before, we see that
r
)
is
This shows
(div f f (with the natural action of
(f ) , F tr = (F ) so that t t
= e ( t , s)O""
This proves e(t,s)r
ord p (x - x (P)) >- 0 ,
which proves that (ordpt cr
divisors). If t,s £NE and f we can choose f tr
=
as asserted in c).
186
- III.lS To be able to go to the inverse limit in the symbols eN (with N
= tn) we have to give a connection between two of them.
(2.2) Proposition. Let M,N be two (strictly) positive integers, t
tN(E)
£
C
tMN(E) , s E: tMN(E). Then one has eMN(t,s) = eN(t,Ms)
( Ms E:tNCE) ).
Proof. We simplify the notations for the proof, denoting by e = eN ' e'
= e MN
(putting primes' to all notions relative to MN). We have
used the notations div f t = N(t) From there we deduce
cf oN
and
N(O)
t
(F oM)MN = (F No M)M t
t
Hence we may choose f' F
,
(F t'MN )
MN(t) - MN(O) as it should be),
c " f t MN . Now by definition of e ,
=
F~(v + s) giving
div f t
, e' = eM
FtoM
t
so that
,
(implying
t
Ft(Mv+Ms)
e' (t,S)F~(V)
=
e'(t,s)Ft(Mv). But by definition of e(t,s)
=
Ft (w + Ms)
=
e (t ,Ms) Ft (w) ,
hence the result. Taking now t
~
tMN(E), s
~tMN(E),
using the bilinearity, we
get immediately eMN(t,s)
M
=
eMN(Mt,s)
=
eN(Mt,Ms).
Now we take for M and N powers of the prime t . Remembering that an element of tn
E::
t
~n
T~(E)
can be identified with a sequence
(E) and .t·t
e(t,s) (for t
=
(t n ) and s
tive limit
=
n
t
n-l
(tn)n~O
with
,we define a bilinear symbol
(t,s) = (e~n(tn,sn))n~o =
(sn) ) and we consider the result in the projec-
~ -tnt-'- ~ ~ 7l/tn 71. = 7l-t, • Thus 0 and consider the complex torus [/L T • Its division points of order dividing N are the images of the points
a~/N +
bIN with integers
a and b. Introducing the line vector i = (a/N,b/N) e: (N-7;ll) 2 we can denote them by i(i) (product of a line vector by a column vector, resulting in a scalar). Or, if we prefer to introduce the integral line vector i' = Ni = (a,b) sion points in the form by
f i (t:)
=
g2:
€
(7lINll) 2, we can also write these divi-
., "tIN
1
(liN). Then Weber's functions are defined
3 (1:)
P(i(i) : Lt:)
if 0 ~ i E: (N-7;?Z) Z
(we discard an integral multiplicative constant which would play no
189
- 111.18 -
role here). We also put f o = j (the modular function). In more algebraic terms, these Weber functions are normalized first projection of division points : Cf : G:/L"( - - E c p2 (G:) defined by y2 = 4x
3
- g2 (t:)x - g3 ('r)
~~x
pI (([)
is a commutative diagram, and if we put ~ = gZg3(-c)x for the norrna-
a
lized first projection of E, then
(Z.7) where t l = ~(~/N)
, tz
=
f(l/N)
£
E ·
We observe that the rational function x on E being defined over k =
(field of definition of E) : x ek(E), the nor-
~(gZ(t),g3(~))
malized first projection
~
is also rational over k :
~£k(E).
Had we taken any other Weierstrassian model E' of ([ILL
= ~4g2(L)
coefficients g;
~:
E
~
,
with other
g~ = ~6g3(L)' the isomorphism
E' , (xo'Yo) ..-..+ (tL2xo'p:'yo)
,
shows that the normalized first projection ~' of E would take the same values as
(Z.8)
~
~ ~'(t')
on corresponding points :
=
~(t) if
o/(t)
t'
(or
t
= 0 (as in the proof of (1.3.4) we have more precisely
1m (t')
1m ('e) Ic 1:
dl- 2det (0
=
~Z'o(lI with
o{y=
(pel,e Z = ~~«p with «p = (~ Lp are p + 1 lattices of index p in ~2. and
b) Let
r
=
~). In particular, there
SL Z (~), and define the matrices
DC y
(0
~ y ~
p) as in a).
Then we have the disjoint coset decomposition
(11 :
disjoint union) .
These are just two ways of expressing the same result as we shall see. Let
r
operate at right in 71 2 (we consider elements of
Z2 as line
vectors, and we multiply them at right by square matrices, getting again line vectors). The mapping
0c(n)e(n"C+ v ) = p
~
c(n)l;lIn qn/ p
lO~
y
< p-1)
,
Lc(n)qpn n~O
with integers c(n)
£~
(the coefficients of the q-expansion of j are
integers (1.4.4) ) . This shows that the coefficients a k
in the ring 7l [q . But if f integer r, 1 , r
~
EO:
=
ak(n) are
Gal (ll (~) Ill), then ~f = t r for some
p-l and applying the automorphism
wise to the q-expansions permutes the jv (1
0)
,
hence, dividing by p, also identically Fp(j(T/p),j(~))
=0
This shows that the two polynomials Fp(j,j*), Fp(j* ,j) have j*
£ [(j){j*]
jo as common root, and by irreducibility of the polynomial
Fp(j,j*)
= (j*)P+I + ~p{j*)p + ••.
there must exist a polynomial P(j,j*) £ll[j,j*] such that
Fp(j*,j)
= P(j,j*)Fp(j,j*)
Iterating this procedure of inversion will give
Fp(j,j *) = PU* ,j)Fp(j * ,j) = PU* ,j)P(j,j *)FpU,j *) , and hence
P(j*,j)P(j,j*) = lin
ll[j,j*] : P(j*,j) = P(j,j*)
205
!l.
- 111.34 -
If P(j,j*) = -1, we have Fp (0J * , j ) = -F P (j , j *) and giving the value .* j , we would get identically F (j,j) = 0, so that Fp(jJj*) would J p
have the root J. * = j and be divisible by j * the
irreducibility of Fp(j,j*) in
modular polynomial F
p
j , a contradiction to
7l[jl[j*] • This proves that the
is symmetrical in j and j*. -
These polynomials can be very difficult to determine exp1icitely as the case p
= 2 already shows (Bateman t.3,p.25) j 3 + j * 3 - ( j j *) 2 + 3 4 5 3 4027 j j * + 2 4 3 • 31 j j * (j + j *)
F 2 (j , j *)
_ 24 3 4 5 3 (j2
+
j*2)
28 3 7 56 (j
+
j*) _ 212 3 9 5 9
+
It can be shown however, that F (j,j*) :: (j - j*p)(jP - j*) P
mod p
(3.15) Theorem 3. When L is a singular lattice, jell is an algebraic integer. Proof. We observe that L is isogenic to the ring of integers in the quadratic field
End(L~) =
K, so by transitivity of the notion of
integrality, it is sufficient to prove that
itself is an
j(~K)
integer. There exists an element a € K having a prime norm N(a) = p. For example, take a prime p which is not inert in K (i.e. which does not generate a prime ideal of ideals
f'
c- K),
P'&K =
PCf
with principal
q , necessarily of norm p. Then N(r) I
element of prime norm p. This shows that
~K
Inf N(a) gives an
=
ae.p
is isomorphic to a
sub1attice (namely a 1)-K) of index P in C"K : p
N(a) = Card (-e}K/ a ~K)
=
[-6-K: a -()..K]
By definition of the modular polynomial, we have
Fp(j(~K),j(~K))
o.
But the polynomial Fp(j,j) ~~[j] has leading term -1 as the q-expansions of the j -jy show: (j - i,,) tr)
q
-1
-q-P
higher order terms
+ +
(for
0, y
~
p-l) ,
higher order terms
(these expansions are to be regarded as power series in q1/p), and so
206
- 111.35 -
p _q-2p
+
terms
higher order terms
=
_j2p
+
•••
This proves that-Fp(j,j) is a unitary polynomial of degree 2p in j giving an integral equation of dependance of
j(~K)
over
~.
Now we turn to the p-adic proof of Th.3, not using Th.2 (but using Th.l which is comparatively very simple). For that, we shall prove (3.16) Theorem 4. Let L be a finite algebraic extension of the p-adic field (Q p ,q E: J:C such that lql < 1 -and Eq = Eq (L) be the corresponding elliptic curve of Tate. Then End(E ) = ~ .
--
--
q
Proof. (We are using L instead of K for the p-adic field, keeping
K for End(E q ) 0 (Q .) First step
For every algebraic extension L'
of L we have a parametrization
'f :
--+- Eq(L ' ) C
L'" X
p2(L ' )
~ (P(x),DP(x),l)
if x ¢ q~
giving isomorphisms
€
~rI]
c
~p (p prime ~ 2), and these numbers are in p~ when
I(p-l) so that we have
t
00
(-l)!H (A) =L.(t)2 Ak =L(-I)2>.k = F(},}:l:,A) €F [fA]] p k=O k k=O k P But this hypergeometric function satisfies the differential equation (hypergeometric differential equation with a
" + (1 -
A(1 - A ) F
2~)
F
,
- 1F
b = I, c = 1) 0,
Where we consider this equation as differential equation for formal series. This proves that the roots of this function are simple ( F(A) = F'(A) = 0 implies F"(A)
o because
we have already checked
that the roots are neither 1 nor 0, and differentiating, we would get F
" , (A) = 0, ... , hence finally F = 0 as formal series). It only
remains to prove the formula for the number of supersingular j's . I
recall that (1.4.3) j = 28 X- 2 (1 - A) -2(1 - A +tX 2 )3
(and this formula defines j in characteristic p = 3), and thus the affine
X-line is a covering of the affine j-line (Luroth's theorem)
of degree 6, with ramified points over j
o of
ramification indices
3 corresponding to the roots A.1 (i = 1,2) of 1 - .A + .A 2 = 0, and over j = 12 3 with ramification indices 2 at the points A= -1,},2 . Here we have implicitely assumed p ; 3 (and p ~ 2) because if p = 3, 12 3 • 0 mod 3 shows that all ramification points collapse over j = 0 3 and give an index of ramification 6 at A = -lover j = 0 = 12 E: F 3 • Before giving the general discussion, let us treat the case of charac-
234
- IV.Il -
teristic p
=
3. Then H is of degree one, has only one root and 3
there is only one supersingular invariant j. But all values of
A
corresponding to this value of j must also be roots, so that the
A= -1 of ramification index 6 above
only possibility is
Indeed, H3 (A) = - 1 - A case p
>3
such that
H
=
= 0 = 12 3 .
j
A = -1 ! Consider now the'
0 also gives
along similar lines. Suppose that there is a prime p p
=
0
does not have roots corresponding to j =
(I do not claim that such p exist,
0
or 12 3
but it will be a consequence of
the discussion that they do exist!). Then, for each value of supersingular j, there will be six values of A above j, and we shall have the formula
~
= l(p-l) = 6h, or p
l2h
+
1 and this shows that
P would necessarily be congruent to 1 mod 12 in this case. Suppose now that p is such that j = 0 is supersingular but j = 12 3 is not supersingu1ar in characteristic p. Then the h-l supersingular values of j distinct from 0 must come from six roots
Aof
there will be only two corresponding values for
t
1 (p-l)
6(h-l)
=
~
Hp and above j
=0
:
+ 2
In this case, we would thus have p
l2(h-l)
+
5
Similarly, if p is such that j
-
5 mod 12. 0 and 12
3
are supersingular in
characteristic p, we get
l = 1(p-l)
6(h-2)
hence p
=
l2(h-2)
+
11
+
= -1
2
+
3
mod 12 .
Finally suppose that j = 12 3 is supersingular but j
o
is not, in
characteristic p, then
leads to
t
l(p-l)
p
l2(h-l)
6(h-l)
= +
7
+
= -5
3 mod 12 .
Because these different conditions for p are mutually exclusive, they
235
- IV.12 -
must characterize the different cases and all occur (if we use Dirichlet's theorem about primes in arithmetic progressions, we see that all these cases occur infinitely many times - with the same density). Moreover, we have found the following formulas (where
E
p
denotes the finite set of supersingular invariants j in characteristic p) h
Card(r ) p
1 1 2 (p-1) if P _ 1 (12) case
I:
~
0,12
1 1 2 (p+7) if P _
0 €I:
, 12
5
IJ
l2(P+13) if p :-1
II
11 (p+5) if P 2
JJ
-5
;0
p
p
0,12 0
~I:p
3
3
3
~
I:
p
,
E:I: p
3 , 12 E:I:
p
This concludes the proof of the theorem, giving more precisely the cases when 0 or 12 3 are supersingular (in characteristic 2, there is only one supersingular curve, namely y2
+
Y
= x3,
and with a sui table
definition for its invariant - given by Deuring - it gives
E
Z=
{a} )
Although we have already used the term "Hasse's invariant" of the curve
E~
as being the value Hp(A) of Deuring's polynomial, let
us give a formal definition now for this terminology. First, we observe that by definition, S'(dx) y
=
H (A)l/p dx p
y
where S' denotes
the modified Cartier operator, hence also
S(dx) = H (A)(dx)p
(1.17)
y
p
y
In general, if E is an elliptic curve given with a first kind differential
W;
0, we define the Hasse invariant
by
S(w) = H(E,W) wP
(1.18)
If we replace
H(E,~)
W
by any other first kind differential, say
fAJ'
= aw
with ae k~, then obviously H(E,aw) = al-PH(E,~), so that we can say that the Hasse invariant is defined up to multiplication by elements of the form a l - p independently of the choice of the basic first
236
- IV.l3 -
kind differential. When the elliptic curve is given by an equation yZ
f(x) with a cubic polynomial f having all its roots distinct,
the choice of first kind differential
W=
dx is sometimes implicitely y
made, and we have dx p-1 (1.19) H(E,)') = H(E) = coefficient of x in f(x)}(p-l) as in the proof of Th.l (l.lZ). Let us take now E in Weierstrass form y2
4x 3 - &2 X
-
&3 and
let us compute its Hasse invariant (with respect to d;) in function of &2 and &3. We have to find the coefficient of x 2t before) in (4x
3
1,
- gzx - g3)
(t = !(p-l) as
· Using twice the binomial expansion
formula in a suitable way we get t -t-m 3 t ~ ~ t! 4 t-m-n m()n J.l-2rn-31\ (4x - gzx -g3) = l - L- m!n!(t-m-n)! (-gZ) -&3 x m=O n=O and because 3n = t - Zm implies n ~ ~ -m, the required coefficient is
L
1 m+n 4 t-m-n t! mn (- ) m! n! (.t-m-n)! gZ&3 Zm+3n=.e This is an isobaric polynomial of degree in gz and g3 if these (l.ZO) H(E)
e
elements are given the respective weights Z and 3. On the other hand, the Eisenstein series
E~
can also be expressed as isobaric
polynomial of the same form in g2 and g3 ' but coefficients in characteristic 0 (1.3.10). The result is that (1.21) L~t
(Deligne) .
us just sketch a proof of this formula. We compare the q-expan-
sions of both sides. By Ek(z)
=1
+
with
(1
.4.1)
¥k ~ 02k_l(n)qn
(q
e 2xiz
~(z)
)
n~l
(B I
= 1/6, B2 =
1/30, B 3
= 1/42, ... ).
But the denominator of Bk is the product of the primes m such that m-l divides 2k (von Staudt), so that~ is always divisible by p : The q-expansion of (E
t mod p) is identically 1. We show that the
237
- IV.14 -
same is true for the q-expansion of H(g2,g3). To find this q-expansion, we evaluate the value of the Hasse invariant of Tate's curve K>CI q1l
=
with the local field (of characteristic p) Fp((q)) valuation of this field is defined by ord(q)
=
K (the additive
1, hence Iql
!p
1
m degq:» u mdeg(1))
deg(~)l un/n
deg(~) In
f
means that [lFq(P) : lF q]
In,
,and is equivalent to P ~ YOF ). If we recall qn qn is precisely the number of distinct conjugates of P,
we obtain the expression (2.4) with
log Zy(u) N n
=
=
L.
n)l
Card Y (IF) qn
N unln n
L.
deg(~)/n
deg (r)
Two equivalent formulas are (2.5)
0)
(2.6) Because N
n
Card Y(lF n) q
~ Card p2 (IF n) q
q 2n + qn + 1 ,
we see that the series (2.4-6) converge absolutely for lui
2, thereby proving the convergence of (2.1) for large Re(s) and the legitimacy of the formal computations in that domain. Also, if V is a straight line, we see from (2.6) that 241
- IV.18 1
(l-u)(l-qu) because Card pI (IF) q
qn
n
1
+
Thus the zeta function of a line has a meromorphic extension to the complex s-plane as rational function in q-s. This fact is general, and one can prove that the zeta function of a curve of genus g has the form P
(2.7)
z
(u)
= (l-U~(l
Zy(u)
-qu)
with a polynomial Pzg(u) E:. Z [uJ of degree Zg in u, satisfying P2g(O) = 1,
P
2g
(1) = N , 1
P
2g
(u)
qgu2gP2g(1/ qu)
This implies the functional equation 'y(l - s)
=
q (1 - 2s) (1 - g) 'yes)
A. Weil has also proved that the zeros of these zeta functions all lie on the critical line Re(s) = ! (for g = 1, this was due to H. Hasse). The main interest of the explicit knowledge of the zeta function of a curve as rational function in u resides in the fact that it is equivalent to the knowledge of all number of points Nn of V (over the extension of degree n of k
= F
q
). Actually, using the
functional equation, we see that the determination of the g coefficients of u, •.• , u g in the polynomial P
suffices to determine
2g
completely the zeta function (and hence all Nn for n
~
1). For
an elliptic curve, the rationality of Zv is very easy to show. (Z.8) Theorem. Let E be an elliptic curve defined over k
=
Fq
(i.e. an absolutely non-singular projective plane cubic over F q with one rational point chosen as neutral element). The zeta function 'E(s) extends as meromorphic function of s in the whole complex plane and satisfies
4E (1
- s)
=
(E(s). This function is rational in u 1 - (1 +q - N1 )
U +
(1 -u)(l -qu)
242
qu
Z
=
q-S
- IV.19 -
Proof. The function field keEl of k-rational functions over E is of genus one (IT.2.26) (when p f 2, which we will suppose for this proof; because we assume that there is a rational point P
~E(Fq)'
the proof of the reference given is still valid although k is not algebraically closed). Let DiV~(E) denote the set of k-rational divisors on E of degree n ~ 0, and Div~(E)/P(E) the set of classes of divisors of degree n mod principal divisors (this has a meaning because the principal divisors have degree zero). From Abel-Jacobi's condition, we see that this set is parametrized by the rational points of E (the sum of the affixes of the points in a rational divisor over k is a point of E which must be equal to all its EQf ) ), and in particular has a finite conjugates, hence be in E k q number N of elements equal to the number of rational points NI on E (and in particular independent of the degree n in Div n ). On the
other hand, it is easy to compute the number of positive divisors in a given class (of degree n ~ 0). For ~ £DiV~(E) and f ~ +
div(f)
~
0
~
~ ~
-div(f)
f
E:
€
k(E),
L (~)
and thus the number of positive divisors in the class of d is the number of principal divisors div(f) with f
~ L(~),
f f
o.
Since a
function is determined up to a multiplicative constant by its divisor, this number is qdim L (~)
_ I
q - I
Because the field keEl is of genus one, dim deg(~)
> o.
This shows that ~
d=deg
d
q
L-
~~1
- ds .9-...:...l N q - I
243
L(~)
deg(~)
if
- IV.20 -
(I-S/(1 -q 1-5) -q -5/(1 -q -5)) . = -N- q q - 1
q-S we get (adding the term 1 corresponding to
Hence putting u d
0) 1 + q
~ 1 (qui (1
- qu) - ul (1 - u))
1 - (9 + 1 - N)u + (1 -u)(l -qu)
qu 2
which is the desired expression. One checks immediately the functional equation on this "explicit" formula. On this explicit formula we see
< l/q Re(s) > 1
that the zeta function ZE converges for luI
Iq-sJ < l/q
, qRe(s) =
IqSj > q ,
or for
(we had only checked the convergence for luI 2
after (2.6) ) • There is a formula for the number of rational points for an elliptic curve over F p given in Legendre form y2 = x (x - l) (x -
A)
(A E: F ) • p
For each finite x £F p ' there will be no point, one point, two points on E (F p ) wi th first coordina te x if x (x - I) (x - A) is resp. #
not a square, zero, a square of F p . Using the quadratic residue symbol, this number is (x(x-l) (X-A)) + l , and if we add the point at p
infinity, we find (in this case) ( 2 • 9)
N = 1 +
L.. { (x (x -1 ) (x - ).))
xEF
p
(We use the convention (~)
+
1}
P
= 0 when a = 0 for the quadratic residue
symbol.) (2.10) Remark. If we write the numerator of the zeta function of
the elliptic curve E in the form Z (2.I ) I - cu + qu Hasse has shown first that the inverse roots
~. 1
have absolute value
ql . This proves that the zeros of this zeta function are allan the
244
- IV.21 -
critical line Re(s)
= }.
The corresponding general property for the
zeta functions of curves (any genus) over finite fields has been established by A. Weil (see the references concerning this chapter). Finally there is result connecting the zeta functions of two isogenous curves:
if E and E' are two elliptic curves defined over
F q , and if there exists a non-zero homomorphism E
--+
E' (defined
over some extension of Fq ), then the zeta functions of E and of E' over Fq are the same.
245
- IV.22 -
3. Reduction mod p of rational elliptic curves
Let k be a number field (finite algebraic extension of ~
~)
and
a (non-zero) prime ideal of the ring of integers of k. By elliptic
curve over k, we always mean a non-singular (over
k) projective
plane cubic defined by a polynomial with coefficients in k,with one rational point over k chosen as neutral element for the group law (usually, this neutral rational element will be on the line at infinity with a suitable coodinate system in the projective plane). (3.1) Definition. The elliptic curve E over k is said to have good reduction mod
~
if there exists a suitable coordinate system in
the projective plane p2 in which E is given by an equation with
coefficients in the ring of 1-integers ,
of k, this equation mod
~
still defining an elliptic curve (non-singularity condition) over
the residual finite field :IF q
=
-
