Explicit Approaches to Modular Abelian Varieties

We use the algorithms of this section to enumerate the  $A_f$, compute information about the invariants of $A_f$ that appear in Conjecture 2.2, and to verify the hypothesis of Theorem 3.13 in order to construct nontrivial subgroups of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A_f)$. The second author has implemented the algorithms discussed in this paper, and made many of them part of the MAGMA computer algebra system [BCP97].

In Section 3.1, we discuss modular symbols, which are the basic tool we use in many of the computations, and in Section 3.2 we discuss how we systematically enumerate modular abelian varieties. There is an analogue for $A_f$ of the usual elliptic-curve modular degree, which we discuss in Section 3.3, and which we use to rule out the existence of visible elements of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A_f)$ of a certain order. In Section 3.4 we describe how to compute the intersection of two abelian varieties, which will be needed to verify the hypothesis of Theorem 3.13. In Sections 3.5 and 3.6, we describe standard methods for bounding the torsion subgroup of an abelian variety above and below. Section 3.7 reviews an algorithm for computing the odd part of the Tamagawa number $c_p$ when $p\mid\mid N$, and discusses the Lenstra-Oort bound in the case when $p^2\mid N$.

Unless otherwise stated, $f$ is a newform, $I_f$ its annihilator, and $A_f$ the corresponding quotient of $J_0(N)$.