On -adic Representations for a Space of Noncongruence Cuspforms

This paper is concerned with a compatible family of 4-dimensional -adic representations ρ 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∨ ) agrees with the L-function for an automorphic form for GL4(AQ), where ρ ∨ is the dual of ρ . 2. For each prime p ≥ 5 there is a basis hp = {hp , hp } 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 ρ admits a quaternion multiplication structure in the sense of Atkin, Li, Liu, and Long.