Given an elliptic curve of analytic rank at most one, Kolyvagin proved that the Tate-Shafarevich group
is finite using his theory of Euler systems. In the higher rank cases, Kolyvagin gave a conjectural description of the structure of the Selmer group in terms of the Euler system. This conjecture has recently been proved by Wei Zhang for a large class of elliptic curves. We shall explain Kolyvagin's conjecture, discuss some of its remarkable consequences and overview the strategy of the proof. To put things in context, we shall begin with a brief survey on the current status of the BSD conjecture.
This is a note prepared for an Alcove Seminar talk at Harvard, Spring 2014. Our main references are [1] and [2].
Let be an elliptic curve over a number field
. Recall that the BSD conjecture relates the arithmetic invariants of
with the an analytic object, the
-function
(see a lowbrow introduction).
There has been vast progress on the BSD conjecture in recent years. We give a very brief (highly incomplete) summary.
Analytic continuation. Due to the recent work of someone in the audience, we now know has analytic continuation for any totally real field
of degree
.
Rank. To make clear the rank part of the conjecture, we recall the -descent exact sequence
The
-Selmer group is a direct sum of several (denoted by
) copies of
and a finite abelian
-group. Conjecturally,
is finite, and hence conjecturally
for any
. Consider the following implications for
,
The implication 1) is trivial. The implication 2) is due to Gross-Zagier and Kolyvagin (80s) for all
. The implication 3) is due to Skinner-Urban (2000s) for
good ordinary with surjective mod
representation
. Now consider the analogous statement for rank 1 case,
The implication 1) is trivial. The implication 2) is again due to Gross-Zagier and Kolyvagin. The implication 3) is due to Wei Zhang (2013) for
good ordinary with surjective
under some mild ramification hypothesis.
BSD formula. In the case, the
-part of the BSD formula is known for
(under similar hypothesis) due to the work of Kato (2000s) and Skinner-Urban on the Iwasawa main conjecture for elliptic curves. In the
case, the
-part of the BSD formula is proved by Wei Zhang for
under similar hypothesis.
We now brief review the Euler system of Heegner points to motivate the statement of Kolyvagin's conjecture. In the next section, we shall state Wei Zhang's result on Kolyvagin's conjecture and deduce some of its consequences mentioned above.
Let be an elliptic curve over
of conductor
. Let
be an imaginary quadratic extension with discriminant
coprime to
(we will assume
for simplicity). Let
be the quadratic character associated to
. We assume the Heegner condition: every prime factor of
splits in
. Then
and the BSD conjecture predicts that
is odd (in particular,
). How do you construct a point?
The key thing is that under the Heegner condition, we have an abundant supply of algebraic points on over the ring class fields of
. Recall that
classifies cyclic
-isogenies between two elliptic curves
. For
, let
be the order of
of conductor
. Under the Heegner condition, there exists an ideal
with norm
. Then we have a pair of elliptic curves with CM by
,
with kernel
. This defines a point (a Heegner point)
, which is defined over the ring class field
of
by the theory of complex multiplication. Here
corresponds to the open compact subgroup
of
under class field theory (so
is the Hilbert class field of
). Using the modular parametrization
(mapping
to 0) we obtain algebraic points
. Taking the trace using the group law on
, we construct a Heegner point
.
The big theorem of Gross-Zagier we went through last semester is the following.
It follows from the Gross-Zagier formula that is infinite order if and only if
. When
is of infinite order, Kolyvagin used his theory of Euler system to prove the finiteness of
. A simple version of his theorem looks like
The rough idea of Kolyvagin's proof is that he constructed a system of cohomology classes in derived from the Heegner points
satisfying nice norm and congruence relations. The local information about these explicit cohomology classes were good enough to bound the Selmer group
(via the local and global Tate duality and the Chebotarev density). More precisely, let
be the sign of the functional equation, then it was actually shown that
and
(the latter is contributed by
).
We now briefly recall the construction of Kolyvagin's Euler system. The key thing is the following assumption:
The reason for this key condition is the following. Let Since
is inert in
, we have
.
Then is indeed invariant under
for
. To check this, we need to show that
lies in
. Namely
By assumption
, so it remains to show that
. This is true since by assumption
: the trace of the Heegner point
on
is exactly the Hecke operator
acting on
, which projects to
. The above relation is also the origin of the name Euler system:
appears in the Euler factor of
at
.
More generally,
Notice is nothing but the cohomology class of
. In higher rank cases, the class
is indeed trivial by Gross-Zagier. Kolyvagin's Euler system constructs more classes
in
. One can ask whether some of these classes could be nontrivial. An obvious but crucial observation is that this is the same as asking whether some
is not infinitely divisible by
.
Kolyvagin proved that in fact So increasing the number of prime factors of
may help bring down the
-divisibility! We define
The hope is that even in the case that is torsion, the cohomology classes
eventually becomes nontrivial.
Assuming this key conjecture, Kolyvagin's work allows us to understand the refined structure of .
Starting from a nontrivial element by assumption , Kolyvagin constructed the whole
-Selmer group (even in the higher rank case!). Moreover, its
-eigenspace can be completely determined.
Now we are ready to state Wei Zhang's theorem precisely and then deduce some remarkable consequences.
Wei Zhang's proof beautifully blends various ideas and ingredients:
[1]Selmer groups and the divisibility of Heegner points, 2013, www.math.columbia.edu/~wzhang/math/online/Kconj.pdf.
[2]Kolyvagin's work on modular elliptic curves, $L$-functions and arithmetic (Durham, 1989), London Math. Soc. Lecture Note Ser., 153 Cambridge Univ. Press, Cambridge, 1991, 235--256.