Modular abelian varieties of odd modular degree

Yazdani, Soroosh

doi:10.2140/ant.2011.5.37

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Modular abelian varieties of odd modular degree

Soroosh Yazdani

Vol. 5 (2011), No. 1, 37–62

DOI: 10.2140/ant.2011.5.37

Abstract

We study modular abelian varieties with odd congruence number by examining the cuspidal subgroup of $J_{0} (N)$ . 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 $N = p^{α}$ or $N = p q$ 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.

Keywords

modular form, modular curve, elliptic curve, congruence number

Mathematical Subject Classification 2000

Primary: 11F33

Secondary: 11G05

Milestones

Received: 23 November 2009

Revised: 17 September 2010

Accepted: 5 December 2010

Published: 22 August 2011

Authors

Soroosh Yazdani
Department of Mathematics and Statistics McMaster University Hamilton, ON L8S 4L8 Canada

Vol. 5, No. 1, 2011