Categoricity and covering spaces

TL;DR

This thesis explores the model theory of algebraic curve covers, linking the behavior of the modular j-function with properties of Galois representations in Tate-modules of abelian varieties.

Contribution

It establishes an equivalence between model-theoretic properties of covers and the openness of Galois representations, connecting algebraic geometry and model theory.

Findings

Equivalence between model-theoretic behavior and Galois representation openness

Insights into the structure of covers of algebraic curves

Connections between the modular j-function and Galois actions

Abstract

This thesis develops some of the basic model theory of covers of algebraic curves. In particular, an equivalence between the good model-theoretic behaviour of the modular j-function, and the openness of certain Galois representations in the Tate-modules of abelian varieties is described.