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Über dieses Buch

This Lecture Notes volume is the fruit of two research-level summer schools jointly organized by the GTEM node at Lille University and the team of Galatasaray University (Istanbul): "Geometry and Arithmetic of Moduli Spaces of Coverings (2008)" and "Geometry and Arithmetic around Galois Theory (2009)". The volume focuses on geometric methods in Galois theory. The choice of the editors is to provide a complete and comprehensive account of modern points of view on Galois theory and related moduli problems, using stacks, gerbes and groupoids. It contains lecture notes on étale fundamental group and fundamental group scheme, and moduli stacks of curves and covers. Research articles complete the collection.​



Algebraic Stacks with a View Toward Moduli Stacks of Covers

Stacks arise naturally in moduli problems. This fact was brilliantly foreseen by Mumford in his wonderful paper about Picard groups of moduli problems [47]a nd further amplified by Deligne and Mumford in their seminal work about the moduli space of stable curves [15]. Even if the theory of stacks is somewhat technical due to the predominance of a functorial language, it is important to be able to use stacks without a complete knowledge of all intricacies of the theory. In these notes our aim is to explain the fundamental ideas about stacks in rather concrete terms. As we will try to demonstrate in these notes, the use of stacks is a powerful tool when dealing with curves, or covers, or more generally when we are trying to classify objects with non-trivial automorphisms, abelian varieties, vector bundles etc. Many people think that stacks should be considered as basic objects of algebraic geometry, like schemes, and [62]is an example of a convincing and heavy set of notes toward this goal. We hope to show how to use them in various concrete examples, especially the moduli stack of stable pointed curves of fixed genus g ≥ 2, with a view toward the moduli stack of covers between curves of fixed genera, the so-called Hurwitz stacks. Hurwitz stacks appear basically as correspondences between moduli stacks of pointed curves.
José Bertin

Models of Curves

The main aim of these lectures is to present the stable reduction theorem with the point of view of Deligne and Mumford. We introduce the basic material needed to manipulate models of curves, including intersection theory on regular arithmetic surfaces, blow-ups and blow-downs, and the structure of the jacobian of a singular curve. The proof of stable reduction in characteristic 0 is given, while the proof in the general case is explained and important parts are proved. We give applications to the moduli of curves and covers of curves.
Matthieu Romagny

Galois Categories

These notes describe the formalism of Galois categories and fundamental groups, as introduced by A. Grothendieck in [SGA1, Chap. V]. This formalism stems from Galois theory for topological covers and can be regarded as the natural categorical generalization of it. But, far beyond providing a uniform setting for the preexisting Galois theories as those of topological covers and field extensions, this formalism gave rise to the construction and theory of the étale fundamental group of schemes −one of the major achievements of modern algebraic geometry.
Anna Cadoret

Fundamental Groupoid Scheme

This article is an overview of the original construction by Nori of the fundamental group scheme as the Galois group of some Tannaka category EF(X) (the category of essentially finite vector bundles) with a special stress on the correspondence between fiber functors and torsors. Basic definitions and duality theorem in Tannaka categories are stated. A paragraph is devoted to the characteristic 0 case and to a reformulation of Grothendieck΄s section conjecture in terms of fiber functors onEF(X)..
Michel Emsalem

Extension of Galois Groups by Solvable Groups, and Application to Fundamental Groups of Curves

The issue of extending a given Galois group is conveniently expressed in terms of embedding problems. If the kernel is an abelian group, a natural method, due to Serre, reduces the problem to the computation of an étale cohomology group, that can in turn be carried out thanks to Grothendieck-Ogg-Shafarevich formula. After introducing these tools, we give two applications to fundamental groups of curves.
Niels Borne

On the “Galois Closure” for Finite Morphisms

We give necessary and sufficient conditions for a finite flat morphism of schemes of characteristic Hurwitz stacks.p > 0 to be dominated by a torsor under a finite group scheme. We show that schemes satisfying this property constitute the category of covers for the fundamental group scheme.
Marco A. Garuti

Hasse Principle and Cohomology of Groups

In a recent article, Colliot−Théléne, Gille and Parimala have considered fieldsK of cohomological dimension 2, of geometric type, analogous to totally imaginary numbers fields. One standard example is the field C((x,y)). Using previous results of Borovoi and the author, they compute the cohomology of K in degree one and two with coefficients in a semi−simpleK−group. The aim of our paper is to extend their results to fields K of cohomological dimension 2 that are not of geometric type but satisfy the Hasse principle; by Efrat, extensions of PAC fields of relative transcendence degree 1 are examples of such fields. For such fields K, we show that it is possible to calculate the non abelian cohomology in degree two with coefficients in a semi−simple K−group (the cohomology in degree one is calculated by Serre′s conjecture about the fields of cohomological dimension 2). We also show, in the case that K is of transcendence degree 1 over a PAC field, that if the group is semi−simple and a direct factor of a K−rational variety, then its Shafarevitch group is trivial, thus getting an analog of a result of Sansuc for number fields. For the field C((x,y)), the analogous result was established by Borovoi−Kunyavskii.
Jean-Claude Douai

Periods of Mixed Tate Motives, Examples, l-adic Side

One hopes that the ℚ-algebra of periods of mixed Tate motives over SpecZ is generated by values of iterated integrals on ℙ1(ℂ) \ {0, 1,8} of sequences of one-forms dz⁄z and dz⁄z-1 from ⃗ 01 to⃗ 10. These numbers are also called multiple zeta values. In this note, assuming motivic formalism, we give a proof, that the ℚ-algebra of periods of mixed Tate motives over SpecZ is generated by linear combinations with rational coefficients of iterated integrals on ℙ1(ℂ) \ {0, 1,-1,8} of sequences of one-forms dz⁄z , dz⁄z -1 and dz⁄z +1 from ⃗ 01 to⃗ 10, which are unramified everywhere. The main subject of the paper is however the l-adic Galois analogue of the above result. We shall also discuss some other examples in thel-adic Galois setting.
Zdzisłlaw Wojtkowiak

On Totally Ramified Extensions of Discrete Valued Fields

We give a simple characterization of the totally wild ramified valuations in a Galois extension of fields of characteristic p. This criterion involves the valuations of Artin‐Schreier cosets of the 𝔽×pr̵translation of a single element. We apply the criterion to construct some interesting examples.
Lior Bary-Soroker, Elad Paran

An Octahedral Galois-Reflection Tower of Picard Modular Congruence Subgroups

Between tradition (Hilbert́s 12th Problem) and actual challenges (coding theory) we attack infinite two-dimensional Galois theory. From a number theoretic point of view we work over ℚ(x). Geometrically, one has to do with towers of Shimura surfaces and Shimura curves on them. We construct and investigate a tower of rational Picard modular surfaces with Galois groups isomorphic to the (double) octahedron group and of their (orbitally) uniformizing arithmetic groups acting on the complex 2-dimensional unit ball 𝔹.
Rolf-Peter Holzapfel, Maria Petkova
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