Modular abelian varieties of odd modular degree

Abstract

We study modular abelian varieties with odd congruence number by examining the cuspidal subgroup of $J_0\left(N\right)$. We show that the conductor of such abelian varieties must be of a special type. For example, if $N$ is the conductor of an absolutely simple modular abelian variety with odd congruence number, then $N$ has at most two prime divisors, and if $N$ is odd, then $N=p^{\alpha }$ or $N=pq$ for some primes $p$ and $q$. In the second half of the paper, we focus on modular elliptic curves with odd modular degree. Our results, combined with the work of Agashe, Ribet, and Stein for elliptic curves to have odd modular degree. In the process we prove Watkins’ conjecture for elliptic curves with odd modular degree and a nontrivial rational torsion point.