Abstract:

This paper provides evidence for the Birch and Swinnerton-Dyer conjecture for 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 the arithmetic invariants of the nearly $20000$ abelian varieties $A_f$ of level $\leq 2333$. Over half of these $A_f$ have 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 $168$ such that ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A_f)$ should be divisible by an odd prime, and we prove for $33$ of these $168$ 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 nonzero elements of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A_f)$ using visibility theory.

Visible Evidence for the Birch and Swinnerton-Dyer Conjecture for Rank 0 Modular Abelian Varieties

Department of Mathematics, Harvard University, Cambridge, Massachussetts 02138

Date: January 22, 2002

Department of Mathematics, Harvard University, Cambridge, Massachussetts 02138

was@math.harvard.edu

Key words and phrases: Birch and Swinnerton-Dyer conjecture, modular abelian variety, visibility, Shafarevich-Tate groups

1991 Mathematics Subject Classification: Primary 11G40; Secondary 11F11, 11G10, 14K15, 14H25, 14H40.