On $\ell$-adic representations for a space of noncongruence cuspforms

This paper is concerned with a compatible family of 4-dimensional $\ell$-adic representations $\rho _{\ell }$ of $G_{\mathbb{Q}}:=\mathrm {Gal}(\overline {\mathbb{Q}}/\mathbb{Q})$ attached to the space of weight-3 cuspforms $S_3(\Gamma )$ on a noncongruence subgroup $\Gamma \subset \mathrm {SL}_2(\mathbb{Z})$. 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 $\text {GL}_4(\mathbb{A}_{{\mathbb{Q}}})$, 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 Atkin, Li, Liu, and Long.