Introduction

Let $A$ be an abelian variety over the rational numbers  $\mathbf{Q}$. Birch and Swinnerton-Dyer found a conjectural formula for the order of the Shafarevich-Tate group of $A$. The Tamagawa numbers $c_p$ of $A$ are among the quantities that appear in this formula. We now recall the definition of the Tamagawa numbers of an abelian variety (the definition of Néron model and component groups is given in Section 2).

Definition 1.1 (Tamagawa number)   Let $p$ be a prime, let $\mathcal{A}$ be the Néron model of $A$ over the $p$-adic integers  $\mathbf{Z}_p$, and let $\Phi_{A,p}$ be the component group of  $\mathcal{A}$ at $p$. Then the Tamagawa number $c_p$ of $A$ at $p$ is the order of the subgroup $\Phi_{A,p}(\mathbf{F}_p)$ of $\mathbf{F}_p$-rational points in $\Phi_{A,p}(\overline{\mathbf{F}}_p)$.

Remark 1.2   The Tamagawa number is defined in a different way in some other papers, but the definitions are equivalent.

When $A$ has dimension one, $A$ is called an elliptic curve, and $A$ can be defined by a Weierstrass equation $y^2=x^3+ax+b$. Using that elliptic curves (and their related integral models) can be described by simple equations, Tate found an efficient algorithm to compute all of the Tamagawa numbers of $A$ (see [18]). In the case when $A$ is the Jacobian of a genus $2$ curve, [7] discusses a method for computing the Tamagawa numbers of $A$. In this paper, we consider the situation in which $A$ has purely toric reduction at $p$, with no constraint on the dimension of $A$. For such $A$ we give an explicit description of the order of the group of connected components of the closed fiber of the Néron model of $A$. In the case when $A=A_f$ is a quotient of $J_0(N)$ attached to a newform $f\in{}S_2(\Gamma_0(N))$ and  $p\mid\mid{}N$, our method is completely explicit, and yields an algorithm to compute the Tamagawa number $c_p$ of $A$ (up to a bounded power of $2$).

This paper is structured as follows. In Sections 2-6 we state and prove an explicit formula involving component groups of fairly general abelian varieties. Then in Section 7 we turn to quotients of modular Jacobians $J_0(N)$. We give some tables and discussed the arithmetic of quotients of $J_0(N)$ when $N$ is prime. In Section 8 we prove a couple of facts about toric reduction that are used in the proof of Theorem 6.1.

Acknowledgement: This paper was inspired by lectures of R. Coleman and K. Ribet, and a letter from Ribet to Mestre (see [17]), which contains some of the results of the present paper in the case when $A$ has dimension $1$. The second author would like to thank A. Abbes, A. Agashe, D. Kohel, and D. Lorenzini, for helpful conversations. Both authors were partially supported by the NSF and the Clay Mathematics Institute during work on this paper.