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

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Proof (3): Long Exact Sequence The long exact sequence of étale cohomology begins Take the p-power torsion in this exact sequence then use Mazur’s theorem. Next analyze the cokernel of δ…
Proof (4): Apply Kato’s Finiteness Theorems (To see this requires chasing some diagrams.) Both of these steps use Kato’s finiteness theorem in an essential way. Putting everything together yields the claimed exact sequence
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 and 18: Cassels-Tate Pairing
Page 19 and 20: Michael Stoll’s Computation Durin
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: Proof (2): Mazur’s Etale Cohomolo
Page 33: Acknowledgements: Thank you for com

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

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