Abstract. 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 Q p (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 (G n ) n≥0 of finite groups, a tower of corresponding Hurwitz spaces (H G n ) n≥0 , geometrically irreducible and defined over Q, which admits projective systems of Q ur p -rational points for all primes p not dividing the orders |G n | (n≥0).
Let $S$ be a Dedekind scheme, $X$ a connected $S$-scheme locally of finite
type and $x\in X(S)$ a section. The aim of the present paper is to establish
the existence of the fundamental group scheme of $X$, when $X$ has reduced
fibers or when $X$ is normal. We also prove the existence of a group scheme,
that we will call the quasi-finite fundamental group scheme of $X$ at $x$,
which classifies all the quasi-finite torsors over $X$, pointed over $x$. We
define Galois torsors, which play in this context a role similar to the one of
Galois covers in the theory of \'etale fundamental group.
Comment: in French. Final version (finally!)
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