Séverin Philip: The l-monodromy fixed fields of curves

Abstract:

From the étale homotopy exact sequence of a curve one gets an outer Galois representation. The study of these representations has led to remarkable developments such as the Grothendieck–Teichmüller group. After introducing these, I will present a pro-\(\ell\) variant and the associated \(\ell\)-monodromy fixed fields. These come with conjectures from the Japanese school such as Anderson–Ihara's “san=ten?” question, the Rasmussen–Tamagawa conjecture and the main topic of the talk Oda's problem. Oda's problem is concerned with \(\ell\)-monodromy fixed fields defined from the moduli spaces of curves and asks if those depend on the genus and the number of marked points. For the second half of the talk I will give a variant of Oda's problem for special loci, some substacks of the moduli spaces of curves of those curves that admit a faithful cyclic action, and give the main steps of the proof.