Modular Abelian Varieties of Odd Modular Degree

TL;DR

This paper investigates modular Abelian varieties with odd congruence numbers, characterizing their conductors and establishing conditions for elliptic curves with odd modular degree, including a proof of Watkins's conjecture in specific cases.

Contribution

It provides new constraints on the conductors of modular Abelian varieties with odd congruence numbers and proves Watkins's conjecture for elliptic curves with odd modular degree and rational torsion.

Findings

Conductor of simple modular Abelian varieties with odd congruence number has at most two prime divisors.

If the conductor is odd, it is either a prime power or a product of two primes.

Proves Watkins's conjecture for elliptic curves with odd modular degree and nontrivial rational torsion.

Abstract

In this paper, we will study modular Abelian varieties with odd congruence numbers by examining the cuspidal subgroup of $J_0(N)$. We will 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 an 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 prime $p$ and $q$. In the second half of this paper, we will focus on modular elliptic curves with odd modular degree. Our results, combined with the work of Agashe, Ribet, and Stein, finds necessary condition for elliptic curves to have odd modular degree. In the process we prove Watkins's conjecture for elliptic curves with odd modular degree and a nontrivial rational torsion point.