On Galois Action on the Inertia Stack of Moduli Spaces of Curves

Abstract

We establish that the geometric action of the absolute Galois group $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ on the étale fundamental group of moduli spaces of curves induces a Galois action on its stack inertia subgroups, and that this action is given by cyclotomy conjugacy. This result extends the special case of inertia without étale factorisation previously established by the authors. It is here obtained in the general case by comparing deformations of Galois actions.

Since the cyclic stack inertia corresponds to the first level of the stack stratification of the space, this result, by analogy with the arithmetic of the Deligne–Mumford stratification, opens the way to a systematic Galois study of the stack inertia through the corresponding stratification of the moduli stack.