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2011 Modular abelian varieties of odd modular degree
Soroosh Yazdani
Algebra Number Theory 5(1): 37-62 (2011). DOI: 10.2140/ant.2011.5.37


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.


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Soroosh Yazdani. "Modular abelian varieties of odd modular degree." Algebra Number Theory 5 (1) 37 - 62, 2011.


Received: 23 November 2009; Revised: 17 September 2010; Accepted: 5 December 2010; Published: 2011
First available in Project Euclid: 20 December 2017

zbMATH: 1262.11069
MathSciNet: MR2833784
Digital Object Identifier: 10.2140/ant.2011.5.37

Primary: 11F33
Secondary: 11G05

Keywords: congruence number , Elliptic curve , modular curve , modular form

Rights: Copyright © 2011 Mathematical Sciences Publishers


Vol.5 • No. 1 • 2011
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