Modular Abelian Variety of Odd Modular Degree

TL;DR

This paper investigates modular Abelian varieties with odd congruence numbers, characterizing their conductors and classifying elliptic curves with odd modular degree, providing evidence for related conjectures.

Contribution

It characterizes conductors of modular Abelian varieties with odd congruence numbers and classifies elliptic curves with odd modular degree, supporting Stein and Watkins's conjecture.

Findings

Conductor of such Abelian varieties must be of specific types, like p^α or pq.

Classified many elliptic curves with odd modular degree.

Provided evidence supporting Stein and Watkins's conjecture.

Abstract

We will study modular Abelian varieties with odd congruence numbers, by studying the cuspidal subgroup of $J_0(N)$. We show the conductor of such Abelian varieties must be of a special type, for example if $N$ is odd then $N=p^\alpha$ or $N=pq$ for some prime $p$ and $q$. We then focus our attention to modular elliptic curves, and using result of Agashe, Ribet, and Stein, we try to classify all elliptic curves of odd modular degree. Our studies prove many cases of the Stein and Watkins's conjecture on elliptic curves with odd modular degree.