Arithmetic and Geometry over Local Fields VIASM 2018

This volume introduces some recent developments in Arithmetic Geometry over local fields. Its seven chapters are centered around two common themes: the study of Drinfeld modules and non-Archimedean analytic geometry. The notes grew out of lectures held during the research program "Arithmetic and geometry of local and global fields" which took place at the Vietnam Institute of Advanced Study in Mathematics (VIASM) from June to August 2018.

The authors, leading experts in the field, have put great effort into making the text as self-contained as possible, introducing the basic tools of the subject. The numerous concrete examples and suggested research problems will enable graduate students and young researchers to quickly reach the frontiers of this fascinating branch of mathematics.

- Some Elements on Berthelot’s Arithmetic D-Modules. - Difference Galois Theory for the “Applied” Mathematician. - Igusa’s Conjecture on Exponential Sums Modulo pm and the Local-Global Principle. - From the Carlitz Exponential to Drinfeld Modular Forms. - Berkovich Curves and Schottky Uniformization I: The Berkovich Affine Line. - Berkovich Curves and Schottky Uniformization II: Analytic Uniformization of Mumford Curves. - On the Stark Units of Drinfeld Modules.