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Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero


Authors: Amod Agashe and William Stein; with an Appendix by J. Cremona; B. Mazur
Journal: Math. Comp. 74 (2005), 455-484
MSC (2000): Primary 11G40; Secondary 11F11, 11G10, 14K15, 14H25, 14H40
Published electronically: May 18, 2004
MathSciNet review: 2085902
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Abstract: This paper provides evidence for the Birch and Swinnerton-Dyer conjecture for analytic rank $0$ abelian varieties $A_f$ that are optimal quotients of $J_0(N)$ attached to newforms. We prove theorems about the ratio $L(A_f,1)/\Omega_{A_f}$, develop tools for computing with $A_f$, and gather data about certain arithmetic invariants of the nearly $20,000$ abelian varieties $A_f$ of level $\leq 2333$. Over half of these $A_f$ have analytic rank $0$, and for these we compute upper and lower bounds on the conjectural order of  $\mbox{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}(A_f)$. We find that there are at least $168$ such $A_f$ for which the Birch and Swinnerton-Dyer conjecture implies that $\mbox{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}(A_f)$is divisible by an odd prime, and we prove for $37$ of these that the odd part of the conjectural order of $\mbox{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}(A_f)$ really divides $\char93 \mbox{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}(A_f)$ by constructing nontrivial elements of $\mbox{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}(A_f)$ using visibility theory. We also give other evidence for the conjecture. The appendix, by Cremona and Mazur, fills in some gaps in the theoretical discussion in their paper on visibility of Shafarevich-Tate groups of elliptic curves.


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Additional Information

Amod Agashe
Affiliation: Department of Mathematics, University of Texas, Austin, Texas 78712
Email: agashe@math.utexas.edu

William Stein
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Email: was@math.harvard.edu

J. Cremona
Affiliation: Division of Pure Mathematics, School of Mathematical Sciences, University of Nottingham, England
Email: john.cremona@nottingham.ac.uk

B. Mazur
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts
Email: mazur@math.harvard.edu

DOI: https://doi.org/10.1090/S0025-5718-04-01644-8
Keywords: Birch and Swinnerton-Dyer conjecture, modular abelian variety, visibility, Shafarevich-Tate groups
Received by editor(s): May 17, 2002
Received by editor(s) in revised form: June 9, 2003
Published electronically: May 18, 2004
Article copyright: © Copyright 2004 American Mathematical Society