The consideration of perverse sheaves on genus zero moduli spaces of curves lead to some arithmetic results of two Galois natures: the inverse Galois realization of symplectic and orthogonal groups (Dettweiler-Reiter & Völklein), and some Tannakian computations of periods relations (Deligne-Terasoma). The goal of this talk is to present similar results in genus one.

Having briefly introduced the inverse Galois and perverse frameworks, I will present how some explicit monodromy computations and a 2-level principle lead to the identification of G2 as Tannaka group of a certain category of perverse sheaves, and how these methods allow going beyond the Fried-Völklein rigidity barrier in Regular Inverse Galois theory.

This is a joint work with M. Dettweiler, S. Reiter and W. Sawin.