The moduli space of representations of the modular group into G₂

2025/01/28 by Angelica Babei, Babei, Angelica, Andrew Fiori +3
Mathematics · #Advanced Algebra and Geometry #Mathematical Analysis and Transform Methods #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2501.17276

Abstract

In this paper we construct a large four-dimensional family of representations of the modular group into G₂. 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 G₂(F_p) for primes p≥ 5. These representations give new examples of ϕ-congruence subgroups of the modular group as introduced in previous work.

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