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Computing Shafarevich-Tate Groups of Elliptic Curves


Sha(E)=Ker�H1(Q;E)!MH1(Qv;E)�:

Joint work with Christian Wuthrich, students (Grigor Grigorov, Andrei Jorza, Stefan Patrikis, and Corina Tarnita), with input from John Cremona, Ralph Greenberg, Robert Pollack, Michael Stoll, and Stephen Donnelly.


1. The Birch and Swinnerton-Dyer Conjecture

2. The Explicit Upper Bounds of Kolyvagin and Kato

3. Iwasawa theory and p-adic Analogues of the BSD Conjecture

4. BSD is true for 141a

5. Summary of Results and The Goal



r!L(r)(E;1)=#E(Q)2tor#Sha(E)��E�RegE�YpjNcp;















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