Modular Abelian Variety of Odd Modular Degree

Details Modular Abelian Variety of Odd Modular Degree Modular Abelian Variety of Odd Modular Degree

2007-07-03

arxiv.org/abs/0707.0437v1

Mathematics

Number Theory

44 Pages, Ph.D. Thesis

Scientific paper

We will study modular Abelian varieties with odd congruence numbers, by studying the cuspidal subgroup of $J_0(N)$. We show the conductor of such Abelian varieties must be of a special type, for example if $N$ is odd then $N=p^\alpha$ or $N=pq$ for some prime $p$ and $q$. We then focus our attention to modular elliptic curves, and using result of Agashe, Ribet, and Stein, we try to classify all elliptic curves of odd modular degree. Our studies prove many cases of the Stein and Watkins's conjecture on elliptic curves with odd modular degree.

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