Finiteness of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}$

A major very central conjecture in modern number theory is that if $E$ is an elliptic curve over $\mathbb{Q}$ then

$${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontsh... ...}\selectfont Sh}}}(E) = \ker(\H ^1(\mathbb{Q},E)\to\prod \H ^1(\mathbb{Q}_v,E))$$

is finite. This is a theorem when ${\mathrm{ord}}_{s=1} L(E,s)\leq 1$ , and is not known in a single case when ${\mathrm{ord}}_{s=1} L(E,s)\geq 2$ . Proving finiteness of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}$ for any curve of rank $>1$ 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 ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}$ . Note that if $L(E,1)\neq 0$ or $L'(E,1)\neq 0$ , then it is a theorem that ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E/\mathbb{Q})$ is finite; there isn't even a single curve with $L(E,1)=L'(E,1)=0$ for which finiteness of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E)$ 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 $E$ be the elliptic curve defined by $y^2 + y = x^3 + x^2 - 2x$ of conductor $389$ . This curve has rank $2$ .

1. Verify that ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E)[p]$ is finite for 5 primes $p$ .
2. Verify that ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E)[p]$ is finite for all primes $p<100$ .

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 ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E)[p] = 0$ for any $p$ using a $p$ -descent. In practice this does not seem practical except for $p=2,3$ . For $p=2$ use mwrank or simon_two_descent in SAGE. For $p=3$ 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 $p$ . And Perrin-Riou does exactly this in the supersingular case in [PR03]. (In fact, she does much more, in that she computes ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(p)$ in the whole $\mathbb{Z}_p$ tower. Shark contains now her computations with a few modifications. - from Christian Wuthrich.)