Hurwitz spaces and Inverse Galois Theory
Pierre Dèbes
TL;DR
This survey traces fifty years of Hurwitz-space theory and its role in inverse Galois theory, foregrounding the Riemann–Hurwitz construction, descent from $\mathbb{C}$ to $\mathbb{Q}$, and patching as complementary realizations of Galois extensions. It highlights semi-modern developments—integral models, compactifications, large-field phenomena, and modular towers—before turning to recent advances on components defined over $\mathbb{Q}$ (via the ring of components) and rational points over finite fields (via the Ellenberg–Venkatesh–Westerland program). The latter yields arithmetic applications to Cohen–Lenstra heuristics and the Malle conjecture over function fields, while Seguin and Cau advance the understanding of $\mathbb{Q}$-defined components through braid-action, lifting invariants, and patching. Together, these threads push forward the regular inverse Galois problem over $\mathbb{Q}$ and illuminate the distribution of covers over global and finite fields, dramatically reshaping the arithmetic of Hurwitz spaces.
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 ${\mathbb F}_q(T)$.