Vol. 5, No. 1, 2011

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Modular abelian varieties of odd modular degree

Soroosh Yazdani

Vol. 5 (2011), No. 1, 37–62

We study modular abelian varieties with odd congruence number by examining the cuspidal subgroup of J0(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 = pα 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.

modular form, modular curve, elliptic curve, congruence number
Mathematical Subject Classification 2000
Primary: 11F33
Secondary: 11G05
Received: 23 November 2009
Revised: 17 September 2010
Accepted: 5 December 2010
Published: 22 August 2011
Soroosh Yazdani
Department of Mathematics and Statistics
McMaster University
Hamilton, ON  L8S 4L8