Potentially $\text{GL}_2$-type Galois representations associated to noncongruence modular forms

TL;DR

This paper explores Galois representations linked to noncongruence modular forms, demonstrating their potential automorphy and applying recent automorphy lifting theorems to specific examples, including infinite families of weight 3 cusp forms.

Contribution

It establishes potential automorphy results for Galois representations associated with noncongruence modular forms, extending the scope of automorphy lifting techniques.

Findings

Potential automorphy results for Galois representations of noncongruence forms.

Automorphy lifting theorems applied to infinite families of weight 3 cusp forms.

Connections between Scholl representations and automorphic representations.

Abstract

In this paper, we consider Galois representations of the absolute Galois group $\text{Gal}(\overline {\mathbb Q}/\mathbb Q)$ attached to modular forms for noncongruence subgroups of $\text{SL}_2(\mathbb Z)$. When the underlying modular curves have a model over $\mathbb Q$, these representations are constructed by Scholl and are referred to as Scholl representations, which form a large class of motivic Galois representations. In particular, by a result of Belyi, Scholl representations include the Galois actions on the Jacobian varieties of algebraic curves defined over $\mathbb Q$. As Scholl representations are motivic, they are expected to correspond to automorphic representations according to the Langlands philosophy. Using recent developments in the automorphy lifting theorem, we obtain various automphy and potential automorphy results for potentially $\text{GL}_2$-type Galois representations associated to noncongruence modular forms. Our results are applied to various kinds of examples. Especially, we obtain potential automorphy results for Galois representations attached to an infinite family of spaces of weight 3 noncongruence cusp forms of arbitrarily large dimensions.