On l-adic representations for a space of noncongruence cuspforms

TL;DR

This paper proves that certain 4-dimensional l-adic Galois representations attached to noncongruence cuspforms are automorphic and exhibit specific ASD relations, revealing deep connections between noncongruence forms and automorphic theory.

Contribution

It establishes the automorphy of l-adic representations for noncongruence cuspforms and identifies ASD relations in their basis expansions, with a quaternion multiplication structure.

Findings

Proves automorphy of the l-adic representations.

Identifies ASD relations in basis coefficients.

Shows quaternion multiplication structure in the representations.

Abstract

This paper is concerned with a compatible family of 4-dimensional \ell-adic representations \rho_{\ell} of G_\Q:=\Gal(\bar \Q/\Q) attached to the space of weight 3 cuspforms S_3 (\Gamma) on a noncongruence subgroup \Gamma \subset \SL. For this representation we prove that: 1.)It is automorphic: the L-function L(s, \rho_{\ell}^{\vee}) agrees with the L-function for an automorphic form for \text{GL}_4(\mathbb A_{\Q}), where \rho_{\ell}^{\vee} is the dual of \rho_{\ell}. 2.) For each prime p \ge 5 there is a basis h_p = \{h_p ^+, h_p ^- \} of S_3 (\Gamma) 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 $\rho_{\ell}$ admits a quaternion multiplication structure in the sense of a recent work of Atkin, Li, Liu and Long.