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## Verifying the Full Conjecture for Elliptic Curves

Stein has done substation work with many students on explicit verification of the full Birch and Swinnerton-Dyer conjecture (the exact formula) for elliptic curves over . This verification amounts to compute the exact order of the Shafarevich-Tate group for elliptic curves.

Problem 8.8.1   Aron Lum and the Stein (mostly!) wrote a paper on explicit verification of BSD for CM elliptic curves up to some conductor. Help finish this paper and get it ready for publication. We are OK with sharing publication credit.

Problem 8.8.2   Cristian Wuthrich and Stein (mostly Wuthrich) have written a bunch of code related to using Peter Schneider's work on -adic analogues of the BSD conjecture to compute at certain primes where the methods of Kolyvagin and Kato fail.

Remark 8.8.3 (From Christian Wuthrich.)   Note that the paper mentioned above, as far as I have written it is, to my taste, more or less done. I should add some data of numerical results which you can of course ask the students to produce. But there is no need or interest for a long list. I have not written yet the introduction nor the part I named technical details (but I am not sure if I actually want to do that).

Of course, I am very happy that part (or the whole of) shark will be included in SAGE.

Remark 8.8.4 (From Christian Wuthrich.)   Schneider's (and simultanoeously Perrin-Riou's work) is strictly speaking not on the p-adic BSD. The most important result to use is Kato's which links the algebraic to the analytic side. Look in the article we write together for a tigher bound in the case is not zero. Your katobound in sage is not sharp.

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William Stein 2006-10-20