Michel Emsalem

HARBATER–MUMFORD COMPONENTS AND TOWERS OF MODULI SPACES

A method of choice for realizing finite groups as regular Galois groups over Q(T ) is to find Q-rational points on Hurwitz moduli spaces of covers. In another direction, the use of the so-called patching techniques has led to the realization of all finite groups over Qp(T ). Our main result shows that, under some conditions, these p-adic realizations lie on some special irreducible components of Hurwitz spaces (the so-called Harbater-Mumford components), thus connecting the two main branches of the area. As an application, we construct, for every projective system (Gn)n≥0 of finite groups, a tower of corresponding Hurwitz spaces (HGn )n≥0, geometrically irreducible and defined over Q, which admits projective systems of Qur p -rational points for all primes p not dividing the orders |Gn| (n≥0). 2000 MSC. Primary 12F12 14H30 14H10 ; Secondary 14D15 14G22 32Gxx

Twisting by a Torsor

Twisting by a G-torsor an object endowed with an action of a group G is a classical tool. For instance one finds in the paragraph 5.3 of the book [17] the description of the “operation de torsion” in a particular context. The aim of this note is to give a formalization of this twisting operation as general as possible in the algebraic geometric framework and to present a few applications. We will focus in particular to the application to the problem of specialization of covers addressed by P. Debes et al. in a series of papers.

Lifting Galois sections along torsors

The cuspidalization conjecture, which is a consequence of Grothendiecks section conjecture, asserts that for any smooth hyperbolic curve X over a finitely generated field k of characteristic 0 and any non empty Zariski open , every section of lifts to a section of . We consider in this article the problem of lifting Galois sections to the intermediate quotient introduced by Mochizuki [10]. We show that when and is an union of torsion sub-packets every Galois section actually lifts to . One of the main tools in the proof is the construction of torus torsors and over X and the geometric interpretation .