What about curves with
? Suppose
that is an elliptic curve over with
.
In the short paper [Kol91],
Kolyvagin states an amazing structure
theorems for Selmer groups assuming the following unproved conjecture,
which is the appropriate generalization of
the condition that has infinite order.
So far nobody has been able to show that Conjecture 3.32
is satisfied by every elliptic curve over , though several
people are currently working hard on this problem (including
Vatsal and Cornut). Proposition 3.28
above implies that Conjecture 3.32 is true for
elliptic curves with
.
Kolyvagin also goes on in [Kol91]
to give a conjectural construction
of a subgroup
for which
.
Let be an arbitrary prime, i.e., so we do not necessarily
assume .
One can construct cohomology class
,
so long as
, where
, and
is the compositum of all class field
for
. For any ,
, and , let
be the subgroup generated by the images of the
classes
where runs through
.
Recall that
and
Let be the maximal nonnegative
integer such that
.
Let
if
,
and
otherwise.
For any , let
and let be the minimal such that
is finite.
Proposition 3.34
We have if and only if has infinite order.
Let , where
If
is a
-module and
.
then
Assuming his conjectures, Kolyvagin deduces that
for every prime number there exists
integers and such that for
any integer we have
Here the exponent of means the or eigenspace
for the conjugation action.
Conjecture 3.35 (Kolyvagin)
Let be any elliptic curve over
and any prime.
There exists and a subgroup
such that
Let
.
Then for all sufficiently large and all ,
one has that
Assuming the above conjecture for all primes , the group is
uniquely determined by the congruence condition in the second part of
the conjecture. Also, Kolyvagin proves that if the above conjecture
is true, then the rank of
equals the rank of ,
and that
is finite. (Here is
or its quadratic twist.)
When has infinite order, the conjecture is true
with and
. (I think here has
.)
William
2007-05-25