Arithmetic on Elliptic Curves with Complex Multiplication

99 Pages·1980·1.48 MB·English

by  Benedict H. Gross (auth.)

Lecture Notes ni
Mathematics
Edited by .A Dold dna .B Eckmann
776
Benedict .H Gross
Arithmetic no Elliptic Curves
with Complex Multiplication
With na Appendix by .B Mazur
galreV-regnirpS
Berlin Heidelberg New York 1980
Author
Benedict H. Gross
Mathematics Department
Princeton University
Princeton, NJ 08544
USA
AMS Subject Classifications (1980): 10B10, 10D25, 12A25, 14K22
ISBN 3-540-09743-0 Springer-Verlag Berlin Heidelberg NewYork
ISBN 0-38?-09743-0 Springer-Verlag NewYork Heidelberg Berlin
Publication in Cataloging Congress of Library ataD
,ssorG H Benedict, -0591
multiplication. with complex elliptic on curves Arithmetic
erutceL( in notes ;scitamehtam 7?6)
Bibliography: .p
Includes .xedni
.1 Elliptic. Curves, .2 Multiplication, .xelpmoC .I Title. .1I in notes Lecture Series:
scitamehtam 776. (Berlin);
QA3.L28 .on 776 [QA567] 510s ]5'3.615[ 80-334
NBSI 0-387-09743-0
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To my father --
on his 75 th birthday
Table of Contents
0. Introduction . . . . . . . . . . . . . . . . i
I. Acknowledgements . . . . . . . . . . . . . . . 2
2. Notation and Conventions . . . . . . . . . . . . 3
Chapter i: The theory of complex multiplication
3. Elliptic curves . . . . . . . . . . . . . . . 4
4. Elliptic curves over ~ and ~ . . . . . . . . . . 8
5. The analytic theory of complex multiplication . . . . . 12
6. Elliptic curves over p-adie fields . o • , 14
7. g-adic Galois representations • . • • 17
8. The arithmetic theory of complex multiplication . . . . . 20
Chapter 2: A classification
9. Curves over H . . . . . . . . . . . . . . . . 23
I0. Descended curves . . . . . . . . . . . . . . . 29
ll. ~-curves . . . . . . . . . . . . . . . . . . 32
Chapter 3: Local arithmetic
12. A classification over F . . . . . . . . . . . . 34
13. A rational p-isogeny . . . . . . . . . . . . . . 38
14. Local invariants and global torsion . . . . . . . . . 42
Chapter 4: Global a~ithmetic
15. Restriction of Scalars . . . . . . . . . . . . . 45
16. The ~-rank . . . . . . . . . . . . . . . . . 49
17. The first descent . . . . . . . . . . . . . . . 53
18. A factorization of the L-series . . . . . . . . . . 57
19. The sign in the functional equation . . . . . . . . . 60
20. ~-curves and modular forms . . . . . . . . . . . . 64
Chapter 5: The ~-curve A(p)
21. Periods . . . . . . . . . . . . . . . . . 67
22. The rank of A(p) . . . . . . . . . . . . . . . 72
23. Global models . . . . . . . . . . . . . . . . 80
24. Computational examples . . . . . . . . . . . . . 82
Appendix (by B. Mazur)
25. The cohomology of the Fermat group scheme • , , 87
26. Bibliography . . . . . . . . . . . . . . . . 92
27. Index . . . . . . . . . . . . . . . . . . 94
0. Introduction.
Let K be an imaginary quadratic field, with ring of integers 0 and class-
number h . Let A be an elliptic curve over ~ with complex multiplication by 0,
and let j (A) be the modular invariant of A . Then j (A) is an algebraic integer
of degree h over Q : its conjugates generate the Hilbert class~field H of K .
This fundamental result has its practical drawbacks. For example, the curve A
cannot be defined over ~ when the class-number of K is greater than one. We can
often circumvent this problem by passing to the category of elliptic curves up to
isogeny. Specifically, when the discriminant of K is odd, one has a large supply
of curves defined over their field of moduli F = ~(j (A)) which are iso~enous to
all of their Galois conjugates over H . Arithmetically these curves behave as if
they were defined over ~ . I call them ~-curves,and these notes are devoted to
their study.
In Chapter I we recall some of the general theory of elliptic curves with com-
plex multiplication. The treatment will be brief: many excellent references on
this subject have already appeared in print. In Chapter 2 we classify elliptic
curves A over H with complex multiplication by 0 . We show which curves de-
scend to F = ~(j(A)) and which descended curves are actually ~-curves.
In Chapter 3, we study the arithmetic of descended curves at all completions of
the field F . For simplicity, we restrict to the case where K has prime discrimi-
nant -p . In Chapter 4 we investigate the global arithmetic of ~-curves. Chapter
5 is devoted to a detailed study of the Q-curve A(p) , with multiplication by the
integers 0 of ~(-~p) and good reduction at all places of F not dividing p .
We end with a discussion of some questions which remain open.
.i Acknowledgements.
It is a pleasure to acknowledge the mathematical assistance I received from
Joe Buhler, Pierre Deligne, Ken Kramer, Barry Mazur, Gilles Robert, David Rohrlich~
Jean-Pierre Serre, and Don Zagier. I also wish to thank Lauri M. Hein for the ex-
cellent job of typing. Much more than thanks are due my family and friends --
Debby Gans, lan Morrison, and Jane Reynolds -- who kept me distracted during the
write-up. Finally, I want to thank my teacher, John Tare, for all the inspiration
and support he gave me in the course of this work.
Princeton, New Jersey
July, 1979
2. Notation and ConventioDs.
Groups G will always act on the left. If M is a G-module and ~ e G , this
o
action will be written either as m~--+ ~(m) or m~ ~ m . We let G M denote the
sub-module of G-invariants. If M and N are G-modules, so is Hom(M,N) : the
action of ~ on a homomorphism f:M---+ N is given by
~(f)(m) = ~(f(~-im)) .
Rings will also act on the left. If M is an R-module and r c R we let
M = {m e M : rm = O} denote the sub-module of "r-torsion."
r
If A and B are elliptic curves (or, more generally, abelian varieties) de-
fined over the field F , we let HomF(A,B) be the group of algebraic homomorphisms
¢:A B ~ which are defined over F . If S is any F-algebra, we let A(S) de-
note the abelian group of all S-rational points of A . If ~ is any automorphism
of F we write ~A and ~B for the conjugate varieties, and o@ for the conju-
gate homomorphism from °A to ~B .
If F is a field, ~ denotes an algebraic closure of it. We shall always use
the isomorphism of local class field theory which takes a uniformizing parameter to
an arithmetic Frobenius in the Galois group.
Chapter l: The thgQry of complex multiplication
3. Elliptic clarves (References: Deligne [6], Tate [29]).
3.1. An elliptic curve A over the field F is a complete, irreducible, non-
singular curve of genus I over F , furnished with a F-rational point A 0 . By
the theorem of Riemann-Roch, there exist functions x and y of degree 2 and 3
on A which are regular outside of A 0 . These functions, when suitably normalized,
satisfy an equation
(3.1.1) y2 + alxY + a3Y 3 = + x a2x2 + a4x + a6
where the coefficients a. lie in F . We call such an equation a generalized
I
Weierstrass model for A . Then x and y generate the function field F(A) ,
and the above model is unique up to a change of coordinates of the form:
x' = u2x + r
(3.1.2)
y' = u3y + su2x + t ,
where u is in F* and r,s,t are in F . Associated to the model (3.1.1) we
have the non-zero differential of the first kind
(ml.3) dx dy
2y+alx+a 3
3x 2a2x+a4-alY
This gives a basis for the F-vector space H°(A,~ I) • Under a change of coordinates
(3.1.2) we find
-I
(3.1.4) ~' = u - ~ .
3.2. Given a generalized Weierstrass model for A over F , define the ele-
ment s
(3.2.l)
2 b I a = 2 + 4a 2 e4 2 = - b 2 2464
4 b = ala 3 + 2a 4 6 c = -b 3 + 2 36b2b4 - 216b 6
b6 = a~ + 4a 6 A =-b~b 8 - 8b~ - 27b~ + 9b2b4b 6
2 2
8 = b b2a 6 - ala3a 4 + a2a 3 4 - a
Then we have the relation
(3.2.2)
1728A : c~ - c~ .
The condition that A ~ 0 is equivalent to the assertion that the curve de-
fined by (3.I.i) is non-singular. This being the case, we may define the "modular
invariant":
(3.2.3) j : (o~/~) : ~728 + (e~/~>.
Under a change of coordinates (3.1.2) we find
4
(3.2.h) %' : u 4 e
6
c~' = u
o 6 c
A' = uI2A
j, = j
Hence the quantities c4, 6 c and A depend only on the pair (A,m) , and the in-
variant J = J(A) depends only on the curve A . If s is any automorphism of F :
(~.2.5) j(~A) = ~($(A)) .
3.3 If A and B are two curves over F with j(A) = j(B) , then A and B
become isomorphic over ~ . More generally, assume that F is a perfect field and
6
let G = GaI(~/F) . Then there is a bijection between the pointed sets
(3.3.1) {Isomorphism with j =sessalC)B( J (A)°f B/F} ( ~ HI(G,Aut~(A))
which takes A to the trivial class. This bijection is constructed as follows.
If j(B) = j(A) we may choose an isomorphism ¢:A ~ B over ~ , The
assignment
(3.3.2)
@:G ~ Auto(A)
01 + 4 -1 o ~¢
is a continuous l-cocycle on G whose cohomology class depends only on the F-isomor-
phism class of B . Conversely, given such a cocycle @ one can construct a "twist"
B @ = A over F with
(3.3,3) j(B) = J(A)
B(F) = {P ~ A(7) : o(P) = ~(~) o p}
The isomorphism class of B over F depends only on the cohomology class of ~ .
3.4. Since the curve A is isomorphic to its Jacobian over F , it inherits
the structure of an abelian variety. The distinguished point A 0 corresponds to
the identity in the algebraic group. Any non-zero homomorphism ¢:A---+ B is
called an isogeny. The (separable) degree of ¢ is the (separable) degree of the
corresponding field extension: [F(A) : ¢ o F(B)] .
The set HomF(A,B) inherits the structure of an abelian group from the addition
law on B . The group End(A) forms a ring with multiplication given by the com-
position of homomorphisms. For any m ~ ~ we let A m be the endomorphism "multi-
2
plication by m " in the algebraic group of A . This isogeny has degree m and is
separable if and only if m is prime to char(F) . If char(F) = p , the isogeny
PA may have separable degree either p or 0 . In the first case A is said to be
ordinary, in the second case A is said to be supersingular.