Algebra & Number Theory

Modular abelian varieties of odd modular degree

Soroosh Yazdani

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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.

Article information

Algebra Number Theory, Volume 5, Number 1 (2011), 37-62.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F33: Congruences for modular and $p$-adic modular forms [See also 14G20, 22E50]
Secondary: 11G05: Elliptic curves over global fields [See also 14H52]

modular form modular curve elliptic curve congruence number


Yazdani, Soroosh. Modular abelian varieties of odd modular degree. Algebra Number Theory 5 (2011), no. 1, 37--62. doi:10.2140/ant.2011.5.37.

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