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

Jonas Bergström, Adrian Diaconu, Dan Petersen, Craig Westerland

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 $q$.

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

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 -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 .
Paper Structure (55 sections, 93 theorems, 340 equations)

This paper contains 55 sections, 93 theorems, 340 equations.

Table of Contents

  1. Introduction
  2. Summary of results
  3. Asymptotics for moments of quadratic L-functions
  4. Braid groups and moduli spaces
  5. Outline of arguments
  6. Acknowledgements
  7. Degree conventions
  8. Symmetric function formalism
  9. Polynomial and analytic functors
  10. The ring of symmetric functions
  11. Plethystic exponential and logarithm
  12. Graded plethystic algebra
  13. A primer on lambda-rings
  14. Schur--Weyl duality, Weyl's construction, and symplectic Schur functions
  15. Recollections on moduli of surfaces
  16. ...and 40 more sections

Key Result

Theorem 1.1.6

There is an equality

Theorems & Definitions (251)

  • Conjecture 1.1.2: Conrey--Farmer--Keating--Rubinstein--Snaith
  • Theorem 1.1.6
  • Remark 1.1.8
  • Theorem 1.1.10
  • Definition 1.1.13
  • Theorem 1.1.14
  • Theorem 1.1.16: Miller--Patzt--Petersen--Randal-Williams
  • Conjecture 1.2.6
  • Conjecture 1.2.9
  • Remark 1.3.3
  • ...and 241 more