Two lectures on the arithmetic of K3 surfaces

TL;DR

This paper reviews various aspects of the arithmetic properties of K3 surfaces, including rational points, Picard number, Tate conjecture, zeta functions, and modularity, providing a comprehensive overview of current research topics.

Contribution

It offers a detailed synthesis of recent developments and open problems related to the arithmetic of K3 surfaces, connecting multiple key areas in the field.

Findings

Discussion of rational points and their distribution on K3 surfaces

Analysis of Picard number variations and Tate conjecture implications

Exploration of zeta functions and evidence for modularity of K3 surfaces

Abstract

In these lecture notes we review different aspects of the arithmetic of K3 surfaces. Topics include rational points, Picard number and Tate conjecture, zeta functions and modularity.