Malle's conjecture over function fields - A. Landesman

Nov 6, 2025  ·   In Kyoto Japan  ·   Event ``AHGT Seminar''

Abstract

The inverse Galois problem asks whether every finite group \(G\) can be realized as the Galois group of a field extension of the rational numbers. Malle's conjecture is a refined version of the inverse Galois problem which predicts the asymptotic number of such extensions. In joint work with Ishan Levy, we prove a version of Malle's conjecture, computing the asymptotic growth of the number of Galois \(G\) extensions of \(\mathbb F_q(t)\), for \(q\) sufficiently large and relatively prime to \(|G|\). We use tools from algebraic geometry to relate this conjecture to a question in topology about the cohomology of certain Hurwitz spaces. We then complete the proof by solving the topological question using techniques from homotopy theory.

1. A. Landesman, I. Levy, Homological stability for Hurwitz spaces and applications, 2025 [ArXiv]
2. A. Landesman, I. Levy, The Cohen--Lenstra moments over function fields via the stable homology of non-splitting Hurwitz spaces, 2025 [ArXiv]
3. A. Landesman, I. Levy, The stable homology of Hurwitz modules and applications, 2025 [ArXiv]