# School - Curves and groups in families

Lecturers :

- 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 en caracteristic p.

Speakers:

- 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.