Powers of 2 in modular degrees of modular abelian varieties

Abstract

An analogue, for modular abelian varieties A , of a conjecture of Watkins on elliptic curves over Q , would say that 2 R divides the modular degree, where R is the rank of the Mordell–Weil group A ( Q ) . 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 “ R ≃ T ” theorem), we show how the analogue of Watkinsʼs conjecture follows, under certain conditions on A , extending and correcting earlier work on the elliptic curve case.