# École - Courbes et groupes en familles

Cours :

- Tim Dokchitser (Univ. Bristol): regular models of curves, computational and theoretical aspects.
- Stefan Wewers (Univ. Ulm): stable models of curves, computational and theoretical aspects.
- Michel Brion (Univ. Grenoble): Structure of algebraic groups over arbitrary fields.
- Fabio Tonini, Lei Zhang (Freie Univ., Berlin): Fundamental group schemes in characteristic p.

Conférencier(ère)s :

- Alessandra Bertapelle (Univ. Padova)
- Peter Bruin (Univ. Leiden)
- Joao Pedro dos Santos (Institut de Mathématiques de Jussieu)
- Anne Frühbis Krüger (Univ. Hannover)
- Margarida Melo (Univ. Roma Tre)
- Céline Maistret (Univ. Bristol)
- Giulio Orecchia (Univ. Rennes 1)
- Antonella Perucca (Univ. Luxembourg)
- Srimathy Srinivasan (Institute for Advanced Study, Princeton)

Michel Brion:

Every smooth connected algebraic group over a perfect field is an extension of an abelian variety by a smooth connected linear algebraic group, which in turn is an extension of a reductive group by a smooth connected unipotent group. These very useful results do not extend to an imperfect field. But there are substitutes, based on the notions of pseudo-reductive groups and pseudo-abelian varieties. The lectures will give an overview of these structure results, illustrated by many examples.

Stefan Wewers:

- lecture: Introduction to arithmetic surfaces and models of curves. Regular and semistable models with some examples. Applications.
- lecture: computing with models looking at valuations, which one can explicitly describe via inductive valuations à la Mac Lane. Then one refines this using Berkovich spaces.
- lecture: details the algorithm to compute the semistable reduction of p-cyclic covers of PP^1, computing some interesting examples, demonstration of the software MCLF.

Tim Dokchitser:

- lecture: how to (often) construct a regular model from the Newton polygon of the defining equation and examples.
- lecture: Review of l-adic Galois representations attached to curves over local fields.
- lecture: Relation between geometry and arithmetic: How to to compute Galois representations attached to curves from semistable and regular models.

Fabio Tonini and Lei Zhang:

In the course we will discuss some fundamental group schemes defined for algebraic varieties and compare them with the classical étale fundamental group introduced by Grothendieck. The key tool to study those group schemes is Tannaka duality, which transfers the study of a group scheme into the study of its representations. Some of the fundamental groups we will talk about are the Nori fundamental group, its etale and local variants, the algebraic fundamental group and the crystalline fundamental group.

Margarida Melo : « Universal Compactified Jacobians »

Abstract : Given a reduced (but possibly reducible) curve with locally planar singularities, one can consider several possible compactifications of its Jacobian, depending on a stability condition. The geometry of these objects is very rich and they satisfy a number of properties which are well known for smooth curves, as e.g. autoduality statements. Some of these compactifications glue over the moduli space of stable curves, and they can be used to describe geometrical loci of the moduli space of stable curves itself. In the talk, I will describe a number of different universal compactifications of the Jacobian, describing some of their properties and how they relate with different constructions. I will then indicate a number of possible applications of the subject, possibly mentioning how could some of them be attacked from the point of view of tropical geometry.

Giulio Orecchia : « A monodromy criterion for existence of Neron models »

Abstract : Néron models are central objects to the study of degenerations of abelian varieties over Dedekind schemes. However, over bases of higher dimension, they do not always exist. In this talk I will introduce a criterion, called toric additivity, for an abelian family with semistable reduction to admit a Neron model. It can be expressed in terms of monodromy action on the l-adic Tate module, and, in the case of jacobians of curves, in terms of finiteness of the tropical jacobian.

João Pedro P. dos Santos : « Finite torsors on schemes defined over a DVR »

Abstract : Determining unramifed coverings over various base spaces is a classical activity, which can take place in many contexts: topological, analytic, algebraic or arithmetic. In this talk I shall report on a theory proposed by myself and P. H. Hai accounting for the case of finite principal bundles (analogues of unramified coverings) over projective schemes defined over a discrete valuation ring A. I shall begin by briefly explaining Nori's theory of the fundamental group scheme through three approaches (semi-stability, “filtering’’ and “trivializing’’) and then introduce the analogous questions for schemes over A. After talking about the filtering version proposed by Gasbarri, I will explain how the “trivializing’’ alternative allows to identify certain Tannakian categories of coherent modules on a projective A-scheme. Following this, I comment on how to put all the aforementioned categories together to form a single Tannakian category and a single flat group scheme ∏^{tr}. In the rest of the talk I will put forward the salient properties of ∏^{tr} as strict pro-finiteness, fibre-by-fibre characterisation and its relation to the other approaches.

Peter Bruin : « Dual pairs of algebras and finite commutative group schemes »

Abstract : We consider finite locally free commutative group schemes over a ring. I will explain how Cartier duality can be used to give a concise description of such a group scheme that is useful for explicit calculations with, for example, torsion subschemes of elliptic curves and modular Galois representations.

Srimathy Srinivasan : « Motivic Decomposition of Projective Pseudo-homogeneous varieties »

Abstract : I will describe a broader class of projective homogeneous varieties that occur over perfect fields of positive characteristic. I will also discuss some of their properties and their decomposition in the category of Chow motives using a generic decomposition theorem.

Anne Frühbis-Krüger : « Hironaka-style resolution of 2-dimensional schemes over ℤ »

Abstract : Lipman's approach to desingularization of 2-dimensional schemes uses both normalization and blowing up. It is hence incompatible with the embedded situation. In 2009 Cossart, Jannsen and Saito published a different approach for this task which is closer to Hironaka's famous proof in characteristic zero. In particular, it does not rely on normalization and its heart is a smart choice of centers for blowing up. This implies that it is suitable for embedded desingularization. In this talk, I will present a new variant of this approach, in which all steps have been made sufficiently explicit to allow implementation (both in a sequential and a massively parallel environment). This is joint work with Bernd Schober and Lukas Ristau.

Céline Maistret : « Arithmetic of hyperelliptic curves over local fields »

Abstract : Let C:y² = f(x) be a hyperelliptic curve over a local field K of odd residue characteristic. We show how several arithmetic invariants of the curve and its Jacobian, including its potential stable reduction, Galois representation and (in the semistable case) Tamagawa numbers, can be simply extracted from combinatorial data coming from the roots of f(x). This is a joint work with Tim Dokchitser, Vladimir Dokchitser and Adam Morgan.

Alessandra Bertapelle : « Serre-Tate theorem for 1-motives »

Abstract : In the first part of the talk I will present the classical Serre-Tate theorem on deformation of abelian schemes. In the second part I will discuss its extension to 1-motives; this is joint work with N. Mazzari.

Antonella Perucca : « Reductions of Abelian varieties »

Abstract : If A is an abelian variety (for example, an elliptic curve) defined over a number field K, then we may consider its reductions modulo the primes of K. All but finitely many primes p of K are of good reduction, and we have a family {A mod p} of abelian varietes defined over finite fields. In this talk we illustrate various problems concerning these reductions (among others, the support problem and the problem of detecting linear dependence).