The leading constant in Malle's conjecture over function fields

2025/12/14 by Santens, Tim
#11R58 #14D23 #14F22 (secondary) #14G05 (primary) #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2512.12838

Abstract

We prove the analogue of Malle's conjecture for the global function field \F_q(t) with q sufficiently large, including a precise formula for the leading constant. The main ingredients are the recent breakthrough of Landesman--Levy on the stable homology of Hurwitz spaces, a novel interpretation of the Frobenius fixed components of Hurwitz spaces in terms of the Brauer group of the stack BG and an interpretation of the number of \F_q points of configuration spaces as a certain Tamagawa volume.

Citations

• The stable homology of Hurwitz modules and applications
• The imaginary case of the nonabelian Cohen--Lenstra heuristics
• Homological stability for Hurwitz spaces and applications
• Malle's conjecture and Brauer groups of stacks
• The Cohen--Lenstra moments over function fields via the stable homology of non-splitting Hurwitz spaces
• Homological stability for generalized Hurwitz spaces and Selmer groups in quadratic twist families over function fields
• Malle's conjecture with multiple invariants
• Fox-Neuwirth-Fuks cells, quantum shuffle algebras, and Malle's\n conjecture for function fields
• Geometry-of-numbers methods over global fields I: Prehomogeneous vector spaces
• Homological stability for Hurwitz spaces and the Cohen-Lenstra\n conjecture over function fields, II
• Composantes connexes et irréductibles en familles
• Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields
• Igusa integrals and volume asymptotics in analytic and adelic geometry
• Connected Components of Hurwitz Schemes and Malle's Conjecture
• Uniformization of Deligne-Mumford curves

Related

• Counterexamples for Türkelli's Modification on Malle's Conjecture
• Connected Components of Hurwitz Schemes and Malle's Conjecture
• Malle's conjecture and Brauer groups of stacks
• Fox-Neuwirth-Fuks cells, quantum shuffle algebras, and Malle's\n conjecture for function fields
• Homology of Hurwitz spaces and the Cohen--Lenstra heuristic for function\n fields (after Ellenberg, Venkatesh, and Westerland)