Algebra & Number Theory
- Algebra Number Theory
- Volume 5, Number 1 (2011), 37-62.
Modular abelian varieties of odd modular degree
We study modular abelian varieties with odd congruence number by examining the cuspidal subgroup of . We show that the conductor of such abelian varieties must be of a special type. For example, if is the conductor of an absolutely simple modular abelian variety with odd congruence number, then has at most two prime divisors, and if is odd, then or for some primes and . 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.
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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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. https://projecteuclid.org/euclid.ant/1513729629