We will use, when possible, similar notation to the notation Kolyvagin uses in his papers (e.g., [Kol91]). If is an abelian group let . Kolyvagin writes for the -torsion subgroup, but we will instead write for this group.
Let be an elliptic curve over with no constraint on the rank of . Fix a modular parametrization , where is the conductor of . Let be a quadratic imaginary field with discriminant that satisfies the Heegner hypothesis for , so each prime dividing splits in , and assume for simplicity that .
Let be the ring of integer of . Since satisfies the
Heegner hypothesis, there is an ideal in such that
is cyclic of order . For any positive integer
, let be the ray class field of
associated to the conductor (see Definition 3.13).
Recall that is an abelian extension of
that is unramified outside , whose existence is
guaranteed by class field theory.
Let
be the order in
of conductor , and let
.
Let
Let
, and
let be the set of primes in
that do not divide the discriminant of and
are such that the image of the representation
sage: E = EllipticCurve('11a') sage: E.non_surjective() [(5, '5-torsion')] sage: E = EllipticCurve('389a') sage: E.non_surjective() []
Fix a prime . We next introduce some
very useful notation. Let denote the set of all
primes
such that , remains prime in ,
and for which
Fix an element
, with , and
consider the -power
If is any Galois extension, we have
(see Section 2.1.2 for most of this)
an exact sequence
Thus (3.4.1) with becomes
Thus to construct
, it suffices
to construct a class
that is invariant under the action of .
We will do this by constructing an element of
and using the inclusion
Recall that
. Unfortunately,
there is no reason that the class
Let be the Hilbert class field of .
Write
, and for each let
where is the ray class
field associated to . Class field theory implies
that the natural map
Finally, let be a set of coset representatives
for
in
,
and let
Let
Before proving that we can use to define a cohomology class in , we state two crucial facts about the structure of the Heegner points .
We have now constructed an element of that is fixed by . Via (3.4.3) this defines an element . But then using (3.4.2) we obtain our sought after class .
We will also be interested in the image of in .
Next we consider a consequence of Proposition 3.27 when is not a torsion point. Note that nontorsion implies that for all but finitely many . Moreover, the Gross-Zagier theorem implies that is nontorsion if and only if .
After Kolyvagin proved his theorem, independently Murty-Murty, Bump-Friedberg-Hoffstein, Waldspurger, each proved that infinitely many such quadratic imaginary always exists so long as has analytic rank or . Also, Taylor and Wiles proved that every over is modular. Thus we have the following theorem:
The author has computed the upper bound of the theorem for all elliptic curves with conductor up to and .
William 2007-05-25