Hurwitz spaces and Inverse Galois Theory

TL;DR

This survey reviews fifty years of research on Hurwitz spaces and their role in inverse Galois theory, highlighting recent advances in arithmetic and geometric aspects.

Contribution

It provides a comprehensive overview of developments from classical to modern methods, including recent breakthroughs in arithmetic geometry of Hurwitz spaces.

Findings

Construction of components over ${f Q}$

Application of Ellenberg-Venkatesh-Westerland approach

Insights into rational points over finite fields

Abstract

Hurwitz spaces which parametrize branched covers of the line play a prominent role in inverse Galois theory. This paper surveys fifty years of works in this direction with emphasis on recent advances. Based on the Riemann-Hurwitz theory of covers, the geometric and arithmetic setup is first reviewed, followed by the semi-modern developments of the 1990--2010 period: large fields, compactification, descent theory, modular towers. The second half of the paper highlights more recent achievements that have reshaped the arithmetic of Hurwitz spaces, notably via the systematic study of the ring of components. These include the construction of components defined over ${\mathbb Q}$, and the Ellenberg-Venkatesh-Westerland approach to rational points over finite fields, applied to the Cohen-Lenstra heuristics and the Malle conjecture over function fields ${\mathbb F}_q(T)$.