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

TL;DR

The paper constructs a large, non-rigid four-dimensional component of the moduli space of representations of the modular group into $G_2$ and provides an explicit étale cover of a patch of this space. It develops a detailed invariant-theoretic framework for a double coset description, yielding a four-parameter open patch with a finite cover of degree $48$ by a versal family of representations and giving precise invariants $(t,u)$ that classify specialization equivalence. A thorough analysis identifies conditions ensuring surjectivity onto $G_2(\mathbf{F}_p)$ for primes $p\ge 5$, by ruling out embeddings into maximal subgroups and through explicit reducibility obstructions; these yield new examples of $\varphi$-congruence subgroups. The work integrates explicit octonion models, subgroup structure of $G_2$, and computational invariant theory to produce concrete arithmetic-geometric data on non-rigid $G_2$-local systems related to the modular group.

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 $φ$-congruence subgroups of the modular group as introduced in previous work.

Theorems & Definitions (36)

• Theorem 1.1
• Theorem 1.2
• Remark 2.1
• Remark 2.2
• Remark 2.3
• Remark 3.1
• Lemma 4.1
• proof
• Remark 4.2
• Lemma 4.3