Possibilities for Shafarevich-Tate Groups of Modular ... - William Stein

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What if the abelian variety A is not an elliptic curve? Assume is finite. Overly optimistic literature: • Page 306 of [Tate, 1963]: If A is a Jacobian then • Page 149 of [Swinnerton-Dyer, 1967]: Tate proved that
Michael Stoll’s Computation During a grey winter day in 1996, Michael Stoll sat puzzling over a computation in his study on a majestic embassy-peppered hill near Bonn overlooking the Rhine. He had implemented an algorithm for computing 2-torsion in Shafarevich-Tate groups of Jacobians of hyperelliptic curves. He stared at a curve X for which his computations were in direct contradiction to the previous slide! What was wrong????
Page 1 and 2: Possibilities for Shafarevich-Tate
Page 3 and 4: Abelian Varieties Abelian Variety:
Page 5 and 6: The Modular Jacobian J 1 (N) • J
Page 7 and 8: Extra structure • A is an abelian
Page 9 and 10: Weil Restriction of Scalars
Page 11 and 12: BSD Conjecture
Page 13 and 14: The Shafarevich-Tate Group
Page 15 and 16: The Dual Abelian Variety The dual o
Page 17: Cassels-Tate Pairing
Page 21 and 22: Poonen-Stoll Theorem Theorem (Annal
Page 23 and 24: In 2002 I finally found Sha of orde
Page 25 and 26: Constructing Nonsquare Sha While at
Page 27 and 28: What is the Abelian Variety A? Let
Page 29 and 30: Proof (2): Mazur’s Etale Cohomolo
Page 31 and 32: Proof (4): Apply Kato’s Finitenes
Page 33: Acknowledgements: Thank you for com

Possibilities for Shafarevich-Tate Groups of Modular ... - William Stein

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