Powers of 2 in modular degrees of modular abelian varieties

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• 作者：Neil Dummigan^a ; ^{n.p.dummigan@shef.ac.uk} ; Srilakshmi Krishnamoorthy^b ; ^{lakshmi@imsc.res.in}
• 关键词：Modular abelian variety ; Modular degree ; Modular form ; 2-Adic Galois deformations
• 刊名：Journal of Number Theory
• 出版年：2013
• 出版时间：February, 2013
• 年：2013
• 卷：133
• 期：2
• 页码：501-522
• 全文大小：358 K

文摘

An analogue, for modular abelian varieties A, of a conjecture of Watkins on elliptic curves over , would say that divides the modular degree, where R is the rank of the Mordell-Weil group . We exhibit some numerical evidence for this. We examine various sources of factors of 2 in the modular degree, and the extent to which they are independent. Assuming that a certain 2-adic Hecke ring is a local complete intersection, and is isomorphic to a Galois deformation ring (a 2-adic ¡°dn.com/content/image/1-s2.0-S0022314X12002703-si4.gif""/>¡± theorem), we show how the analogue of Watkins?s conjecture follows, under certain conditions on A, extending and correcting earlier work on the elliptic curve case.