Program

The school is one week long in June 2018.

There are four mini-courses, each consists of 5 x 90 minute lectures. Several tutorial sessions are planned if necessary.

Lecture 1: An introduction to algebraic theory of difference equations.

Lecturer: Lucia Di Vizio

Abstract: This course will provide an introduction to algebraic theory of difference equations, with a particular attention to the recent development of Galois theory. It will be presented as a tool, meaning that the aim of the course is to provide the students with enough knowledge to be able to apply the theory.

Lecture 2: An introduction to Drinfeld modular forms

Lecturer: Federico Pellarin

Abstract: A theory of moduli objects for global function fields was introduced by Drinfeld in the years 1970s. Let C_∞ be the completion of an algebraic closure of the local field K_∞ := F_q((1/T)). In his Ph. D. Thesis, David Goss introduced a theory of modular forms with values in C_∞. These functions are defined over a rigid analytic space called the Drinfeld upper-half plane. They present similarities with automorphic functions over the Bruhat-Tits tree of K_∞, but are fundamentally different, as they don’t take their values in C_∞. In fact, the analytic, algebraic and arithmetic theories of these functions are very rich and have motivated many important researches. Today, this topic is rapidly expanding, being at the center of an active scientific community, and the course is an invitation to it. We will analyze the structure of these functions and compare the results with the corresponding results on classical modular forms and, if the time allows, we will also discuss a generalization of Drinfeld modular forms where one has, more generally, a C_∞-Banach algebra instead that just C_∞, with applications to L and zeta values.

Lecture 3: A p-adic study of zeta functions of varieties over finite fields

Lecturer: Jérôme Poineau

Abstract: In this course, we will introduce zeta functions of varieties over finite fields and explain their main properties according to the Weil conjectures (proved by Alexander Grothendieck and Pierre Deligne). The first proof of the rationality of those functions has been obtained by Bernard Dwork using p-adic methods. We will provide a detailed account of it, including all the necessary background in p-adic analysis.

Lecture 4: From Galois theory to Frobenius modules and shtukas

Lecturer: Lenny Taelman

Abstract: A Frobenius module is a module M over a ring P of characteristic p, equipped with a semi-linear endomorphism F (satisfying F(rm)=r^pF(m)). Such objects (and variations on them) play an important role in geometry and arithmetic in characteristic p. In this lecture series we will survey the basic properties of such modules, and show how they arise in various contexts. This includes: Galois theory in characteristic p, point counting modulo p on varieties over F_p, the theory of Drinfeld modules, and Goss' characteristic p zeta values.