Modul: MAT075 Zurich Graduate Colloquium

## What is.. geometric class field theory?

Vortrag von Dr. Alberto Merici

**Datum:** 03.11.20 **Zeit:** 16.30 - 17.30 **Raum:**

Number theory is "the study of equations with coefficients in Q", and in more high-level terms this can be rephrased as "the study of the absolute Galois group of Q", i.e. the Galois group G of the algebraic closure of Q. Class field theory is then "the study of the abelianization of G". The main theorems of class field theory, due to Artin, Tate and others, give a very satisfactory description of this group in terms of reciprocity maps. The aim of Geometric class field theory is then to apply this theory to geometric objects: the field Q is substituted with the field of rational functions of an algebraic curve over a finite field, using the well-established philosophy that says that "Q is the field of rational functions of an algebraic curve over an hypothetical field with one element". If time permits, I will give a glimpse of some generalizations of this theory to the higher dimensional case and its link to motives.