Béranger Seguin

Hurwitz spaces are moduli spaces that classify ramified covers of the projective line on which a fixed group G acts. Their geometric and arithmetic properties are related to number theoretical questions, particularly the inverse Galois problem. In this thesis, we study the connected components of these spaces. Firstly, we prove results concerning the asymptotical behaviour of the count of connected components of Hurwitz spaces as the number of branch points of the covers they classify grows. Secondly, we establish stability results for fields of definitions of connected components of Hurwitz spaces under the gluing operation. These results relate topological and arithmetical properties of covers. Three expository chapters, devoid of original statements, present the various objects. In an appendix, we summarize the thesis for the general public..

For a fixed finite group $G$, we study the fields of definition of geometrically irreducible components of Hurwitz moduli schemes of marked branched G-covers of the projective line. The main focus is on determining whether components obtained by "gluing" two other components, both defined over a number field $K$, are also defined over $K$. The article presents a list of situations in which a positive answer is obtained. As an application, when $G$ is a semi-direct product of symmetric groups or the Mathieu group M₂₃, components defined over ℚ of small dimension ($6$ and $4$, respectively) are shown to exist.

We study a variant of the ring of components of Hurwitz moduli spaces for covers, introduced by Ellenberg, Venkatesh and Westerland. For G-covers of the projective line, the ring of components is a commutative algebra of finite type. We study it using tools from algebraic geometry. We relate the geometry of its spectrum to group-theoretical properties and combinatorial aspects of Galois covers. When G is a symmetric group, we fully describe the geometric points of the spectrum.