K3 surfaces of high Picard number and arithmetic applications.

摘要

The subject of my thesis is finding moduli spaces of K3 surfaces that fiber as families of elliptic curves of high rank, or higher genus curves with many points. In particular, we find a family of genus 2 curves over an elliptic curve with 226 sections, as well as some interesting elliptic K3 surfaces with constant j-invariant j = 0. Since K3 surfaces play a role in many other parts of algebraic geometry, explicit knowledge of these moduli spaces will be of interest in several areas of algebraic geometry and number theory.