On Atkin–Swinnerton-Dyer congruence relations

doi:10.1016/j.jnt.2004.08.003

Journal of Number Theory

Volume 113, Issue 1, July 2005, Pages 117-148

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Abstract

In this paper, we exhibit a noncongruence subgroup $Γ$ whose space of weight 3 cusp forms $S_{3} (Γ)$ admits a basis satisfying the Atkin–Swinnerton-Dyer congruence relations with two weight 3 newforms for certain congruence subgroups. This gives a modularity interpretation of the motive attached to $S_{3} (Γ)$ by Scholl and also verifies the Atkin–Swinnerton-Dyer congruence conjecture for this space.

Keywords

Modular forms for noncongruence subgroups

Atkin–Swinnerton-Dyer congruence relations

l-adic representations

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¹: Supported in part by an NSF Grant DMS 99-70651 and an NSA Grant MDA904-03-1-0069.

²: Supported in part by an NSF Grant No. DMS 97-29992 and a Liftoff grant from the Clay Mathematical Institute.