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Let the field $K$ be complete w.r.t. a non-archimedean valuation. Let $X / K$ be a Mumford curve, i.e. the irreducible components of the stable reduction of $X$ have genus 0. The abelian etale coverings of $X$ are constructed using the analytic uniformization $Ω \to X$ and the theta-functions on $X$ . For a local field $K$ one rediscovers $G$ . Frey’s description of the maximal abelian unramified extension of the field of rational functions of $X$ .

Étale Galois covers of affine smooth cuves. The geometric case of a conjecture of Shafarevich. On Abhyankar's conjecture.

Florian Pop (1995)

Inventiones mathematicae

Explicit Selmer groups for cyclic covers of ℙ¹

Michael Stoll, Ronald van Luijk (2013)

Acta Arithmetica

For any abelian variety J over a global field k and an isogeny ϕ: J → J, the Selmer group $S e l^{ϕ} (J, k)$ is a subgroup of the Galois cohomology group $H ¹ (G a l (k^{s} / k), J [ϕ])$ , defined in terms of local data. When J is the Jacobian of a cyclic cover of ℙ¹ of prime degree p, the Selmer group has a quotient by a subgroup of order at most p that is isomorphic to the ‘fake Selmer group’, whose definition is more amenable to explicit computations. In this paper we define in the same setting the ‘explicit Selmer group’, which is isomorphic...

Exposants $p$ -adiques et théorèmes d’indice pour les équations différentielles $p$ -adiques

François Loeser (1996/1997)

Séminaire Bourbaki

Factorisations explicites de g(y) - h(z)

Pierrette Cassou-Noguès, Jean-Marc Couveignes (1999)

Acta Arithmetica

Families of elliptic curves with genus 2 covers of degree 2.

Claus Diem (2006)

Collectanea Mathematica

We study genus 2 covers of relative elliptic curves over an arbitrary base in which 2 is invertible. Particular emphasis lies on the case that the covering degree is 2. We show that the data in the "basic construction" of genus 2 covers of relative elliptic curves determine the cover in a unique way (up to isomorphism).A classical theorem says that a genus 2 cover of an elliptic curve of degree 2 over a field of characteristic ≠ 2 is birational to a product of two elliptic curves over the projective...

Familles de Hurwitz et cohomologie non abélienne

Pierre Dèbes, Jean-Claude Douai, Michel Emsalem (2000)

Annales de l'institut Fourier

Nous nous intéressons à la question de l’existence de familles de Hurwitz au-dessus d’un espace de modules de revêtements de la droite. On sait que de telles familles existent dans le cas où les revêtements n’ont pas d’automorphismes. Dans le cas général, il y a une obstruction cohomologique, de nature non-abélienne. Nous donnons une double description de cette obstruction : la première en termes de gerbe, l’outil le mieux adapté à des situations cohomologiques non-abéliennes et la deuxièmes en...

Familles de revêtements de la droite projective

Michel Emsalem (1995)

Bulletin de la Société Mathématique de France

Field of moduli versus field of definition for cyclic covers of the projective line

Aristides Kontogeorgis (2009)

Journal de Théorie des Nombres de Bordeaux

We give a criterion, based on the automorphism group, for certain cyclic covers of the projective line to be defined over their field of moduli. An example of a cyclic cover of the complex projective line with field of moduli $ℝ$ that can not be defined over $ℝ$ is also given.

Fields of moduli of three-point $G$ -covers with cyclic $p$ -Sylow, II

Andrew Obus (2013)

Journal de Théorie des Nombres de Bordeaux

We continue the examination of the stable reduction and fields of moduli of $G$ -Galois covers of the projective line over a complete discrete valuation field of mixed characteristic $(0, p)$ , where $G$ has a cyclic $p$ -Sylow subgroup $P$ of order $p^{n}$ . Suppose further that the normalizer of $P$ acts on $P$ via an involution. Under mild assumptions, if $f : Y \to ℙ^{1}$ is a three-point $G$ -Galois cover defined over $\bar{ℚ}$ , then the $n$ th higher ramification groups above $p$ for the upper numbering of the (Galois closure of the) extension $K / ℚ$ vanish,...

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