Introduction

In this paper we give 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. For such abelian varieties, the conjecture asserts that $A_f(\mathbf{Q})$ is finite, and gives a formula for $\char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A_f)$.

It is a theorem [KL89,KL92] that $A_f(\mathbf{Q})$ is finite, but very little work has been done on the explicit formula for $\char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A_f)$, especially in the case when $\dim A_f > 1$. In [KL92, §1.6] Kolyvagin and Logachev say that if one were able with a compute the height of a certain Heegner point, their methods could be used to give a bound on $\char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A_f)$. This we have not done, and to the best of our knowledge nobody has done this for any abelian variety $A_f$ with $\dim A_f > 1$.

Upon learning of [CM], we had the idea to prove in some cases that $\char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A_f)$ is at least as big as predicted by the Birch and Swinnerton-Dyer conjecture by turning Mazur's idea of visibility on its head. Instead of assuming that ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A_f)$ is as predicted by the conjecture and trying to understand whether or not it is visible in $J_0(N)$, we instead proved the main theorem of [AS02b], which allowed us to sometimes construct the odd part of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A_f)$ without assuming any conjectures. After developing algorithms that allow us to compute the conjectural order of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A_f)$ in most cases, we analyzed the $19608$ abelian varieties $A_f$ of level $\leq 2333$, and constructed the tables of Section 5. This resulted in the first systematic experimental evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of dimension $>2$ (see [FpS+01] for dimension $2$).

This paper is organized as follows. In Section 2 we review background about modular abelian varieties and state the Birch and Swinnerton-Dyer conjecture. Section 3 explains the basic facts about quotients $A_f$ of $J_0(N)$ that one needs to know in order to compute with them. In Section 4 we discuss a generalization of the Manin constant, derive a formula for the ratio $L(A_f,1)/\Omega_{A_f}$, and bound the denominator of this ratio. Section 5 reports on our construction of a table of $168$ rank 0 abelian varieties $A_f$ of level $\leq 2333$ such that the Birch and Swinnerton-Dyer conjecture predicts that ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A_f)$ is divisible by an odd prime, and discusses what we computed to show that for $33$ of the $A_f$ there are at least as many elements of the odd part of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A_f)$ as predicted.

Acknowledgment. It is a pleasure to thank Birch, Coleman, Gross, Lenstra, Lorenzini, Merel, Poonen, Ribet, and Tate for many helpful comments and discussions. Special thanks go to Barry Mazur for guiding our ideas on visibility and purchasing the second author a powerful computer, and to Allan Steel and David Kohel at MAGMA for their crucial computational support.