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arXiv · 2501.17276

The moduli space of representations of the modular group into $G_2$

Angelica Babei·Andrew Fiori·Cameron Franc
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Abstract

In this paper we construct a large four-dimensional family of representations of the modular group into $G_2$. Precisely, this family is an etale cover of degree $96$ of an open subset of the moduli space of such representations. This moduli space has two main components, of dimensions one and four. The one-dimensional component consists of well-studied rigid representations, in the sense of Katz. We focus on the four-dimensional component which consists of representations that are not rigid. We also provide algebraic conditions to ensure that the specializations surject onto $G_2(\mathbf{F}_p)$ for primes $p\geq 5$. These representations give new examples of $\phi$-congruence subgroups of the modular group as introduced in previous work.

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Angelica Babei, Andrew Fiori, Cameron Franc. 2025-01-28. The moduli space of representations of the modular group into $G_2$. https://arxiv.org/abs/2501.17276

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