#### Title

On l-adic representations for a space of noncongruence cuspforms

#### Document Type

Article

#### Publication Date

5-1-2012

#### Abstract

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.

#### Publication Source (Journal or Book title)

Proceedings of the American Mathematical Society

#### First Page

1569

#### Last Page

1584

#### Recommended Citation

Hoffman, J., Long, L., & Verrill, H.
(2012). On l-adic representations for a space of noncongruence cuspforms.* Proceedings of the American Mathematical Society**, 140* (5), 1569-1584.
https://doi.org/10.1090/S0002-9939-2011-11045-1