Modular Abelian Variety of Odd Modular Degree

2007/07/03 by Shahram Yazdani, S. Yazdani, Yazdani, S.
Engineering · Mathematics · #11F11 #11F33 #11G18 #11G40 #Advanced Numerical Analysis Techniques #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11F11 #msc:11F33 #msc:11G18 #msc:11G40

paper · pdf · doi:10.48550/arxiv.0707.0437

44 Pages, Ph.D. Thesis

arxiv created 2007/07/03 · openalex publication_date 2007/07/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We will study modular Abelian varieties with odd congruence numbers, by studying the cuspidal subgroup of J₀(N). We show the conductor of such Abelian varieties must be of a special type, for example if N is odd then N=p^α 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.

Related

• Modular Abelian Varieties of Odd Modular Degree
• Elliptic Curves of Odd Modular Degree
• Elliptic curves of conductor 2^m p, quadratic twists, and Watkins' conjecture
• The modularity of an abelian variety
• On a special case of Watkins' conjecture