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A *major* very central conjecture in modern number theory is that
if
is an elliptic curve over
then

is finite. This is a theorem when
, and is
*not known in a single case* when
.
Proving finiteness of
for any curve of rank
would be a
massively important result that would have huge ramifications.
Much work toward the Birch and Swinnerton-Dyer conjecture (of
Greenberg, Skinner, Urban, Nekovar, etc.) assumes finiteness
of
. Note that if
or
,
then it is a *theorem* that
is finite; there
isn't even a single curve with
for which
finiteness of
is known.
As far as I can tell nobody has even the slightest clue how
to prove this. However, we can at least try to do some computations.

**Remark 8.8.6** (From Christian Wuthrich.)
Seems doable. I quickly run shark up to p
= 53, it does not take too long. As far as I remember I never
actually checked if shark is optimal when computing the p-adic
L-function.

**Remark 8.8.7**
In theory one can verify that

for
any

using a

-descent. In practice this does not seem
practical except for

. For

use mwrank
or

`simon_two_descent` in

*SAGE*. For

use

`three_selmer_rank` in

*SAGE* (this command just
calls MAGMA and runs code of Michael Stoll).

**Remark 8.8.8**
The work of Cristian Wuthrich and Stein mentioned in
Section

8.8.1 could be used to verify finiteness for many

. And Perrin-Riou does exactly this in the supersingular case in
[

PR03]. (In fact, she does much more, in that
she computes

in the whole

tower. Shark contains now
her computations with a few modifications. - from Christian
Wuthrich.)

** Next:** Fun with Visualizing Modular
** Up:** The Shafarevich-Tate Group
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William Stein
2006-10-20