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## Finiteness of

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.

Problem 8.8.5   Let be the elliptic curve defined by of conductor . This curve has rank .
1. Verify that is finite for 5 primes .
2. Verify that is finite for all primes .

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 Previous: Verifying the Full Conjecture   Contents
William Stein 2006-10-20