Hyperelliptic curves, the scanning map, and moments of families of quadratic L-functions

2023/02/15 by Jonas Bergström, Adrian Diaconu, Bergström, Jonas +5 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Analytic Number Theory Research #FOS: Mathematics #Geometric Topology (math.GT) #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2302.07664

openalex publication_date 2023/02/15 · openalex created_date 2023/02/18 · openalex updated_date 2026/07/28

Abstract

We compute the stable homology of the braid group with coefficients in any Schur functor applied to the integral reduced Burau representation. This may be considered as a hyperelliptic analogue of the Mumford conjecture (Madsen--Weiss theorem) with twisted coefficients. We relate the result to the function field case of conjectures of Conrey-Farmer-Keating-Rubinstein-Snaith on moments of families of quadratic L-functions. Combined with a recent homological stability theorem of Miller-Patzt-Petersen-Randal-Williams, our homological calculations confirm the Conrey-Farmer-Keating-Rubinstein-Snaith predictions for all large enough prime powers q.

Cited by

• Refinements of Katz-Sarnak theory for the number of points on curves over finite fields

Related

• Uniform twisted homological stability
• Families of superelliptic curves, complex braid groups and generalized Dehn twists
• Point pushing, homology, and the hyperelliptic involution
• Burau maps and twisted Alexander polynomials
• Homology of the braid group with coefficients in the reduced Burau representation