Arithmetic of Totally Split Modular Jacobians and Enumeration of Isogeny Classes of Prime Level Simple Modular Abelian Varieties

Abstract

In this thesis, we aim to give algorithms for computing two key invariants of the modular Jacobians $J_0(N)$. We first give methods for computing the rational torsion order of rank-0 Jacobians $J_0(N)$ that are isogenous to a product of elliptic curves. We call these the rank-0 totally split Jacobians. The rational torsion is an important invariant of the Generalized BSD conjecture so being able to compute the rational torsion order will provide evidence towards this conjecture. We will provably enumerate the set of totally split $J_0(N)$, give an algorithm for computing the rational torsion subgroup, and later give techniques for computing the rational torsion order for rank-0 totally split Jacobians $J_0(N)$. Next we will give an algorithm for computing the rational odd-isogeny class of a simple abelian subvariety $A$ of $J_0(N)$ for prime $N$, under mild conditions. This is done by showing every $G_\QQ$-submodule of $A(\QQbar)_\odd$ is a Hecke module and then attacking the non-Eisenstein and Eisenstein isogenies separately. }