On l-adic representations for a space of noncongruence cuspforms

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This paper is concerned with a compatible family of 4-dimensional l-adic representations ρl of GQ:= Gal(Q/Q) attached to the space of weight-3 cuspforms S3(Γ) on a noncongruence subgroup Γ ∩ SL2(Z). For this representation we prove that: 1. It is automorphic: the L-function L(s, ρ lv) agrees with the L-function for an automorphic form for GL4(AQ), where ρ lv is the dual of ρl. 2. For each prime p ≥ 5 there is a basis hp = {h +p, h -p } of S3(Γ) whose expansion coefficients satisfy 3-term Atkin and Swinnerton-Dyer (ASD) relations, relative to the q-expansion coefficients of a newform f of level 432. The structure of this basis depends on the class of p modulo 12. The key point is that the representation ρl admits a quaternion multiplication structure in the sense of Atkin, Li, Liu, and Long. © 2011 American Mathematical Society.

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Proceedings of the American Mathematical Society

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