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Big monodromy for higher Prym representations

Aaron Landesman, Daniel Litt, Will Sawin

TL;DR

The paper proves that for finite covers with group $H$, the monodromy of the higher Prym representations on $H^1(\, arget{Σ}{g'}}$, induced by the mapping class group, is as large as possible when the base genus $g$ is large relative to the largest irreducible $H$-representation. Central to the approach is a novel, functorial reconstruction of unitary local systems from the derivative of the period map (a generic Torelli theorem with coefficients), together with global-generation techniques to control deformations. The authors show that the connected monodromy is the commutator subgroup of the centralizer of $H$ in the ambient symplectic group, yielding consequences for Mumford–Tate groups and endomorphism algebras of Jacobians, and they extend these results to Kodaira–Parshin fibrations. They also discuss a large-$n$ regime and connections to arithmetic statistics, and they pose several open questions and potential extensions, including analogues for free groups. Overall, the work provides a comprehensive Hodge-theoretic framework for establishing big monodromy in a broad family of geometric settings with substantial arithmetic and geometric consequences.

Abstract

Let $Σ_{g'}\to Σ_g$ be a cover of an orientable surface of genus g by an orientable surface of genus g', branched at n points, with Galois group H. Such a cover induces a virtual action of the mapping class group $\text{Mod}_{g,n+1}$ of a genus g surface with n+1 marked points on $H^1(Σ_{g'}, \mathbb{C})$. When g is large in terms of the group H, we calculate precisely the connected monodromy group of this action. The methods are Hodge-theoretic and rely on a "generic Torelli theorem with coefficients."

Big monodromy for higher Prym representations

TL;DR

The paper proves that for finite covers with group , the monodromy of the higher Prym representations on , induced by the mapping class group, is as large as possible when the base genus is large relative to the largest irreducible -representation. Central to the approach is a novel, functorial reconstruction of unitary local systems from the derivative of the period map (a generic Torelli theorem with coefficients), together with global-generation techniques to control deformations. The authors show that the connected monodromy is the commutator subgroup of the centralizer of in the ambient symplectic group, yielding consequences for Mumford–Tate groups and endomorphism algebras of Jacobians, and they extend these results to Kodaira–Parshin fibrations. They also discuss a large- regime and connections to arithmetic statistics, and they pose several open questions and potential extensions, including analogues for free groups. Overall, the work provides a comprehensive Hodge-theoretic framework for establishing big monodromy in a broad family of geometric settings with substantial arithmetic and geometric consequences.

Abstract

Let be a cover of an orientable surface of genus g by an orientable surface of genus g', branched at n points, with Galois group H. Such a cover induces a virtual action of the mapping class group of a genus g surface with n+1 marked points on . When g is large in terms of the group H, we calculate precisely the connected monodromy group of this action. The methods are Hodge-theoretic and rely on a "generic Torelli theorem with coefficients."
Paper Structure (37 sections, 40 theorems, 117 equations)

This paper contains 37 sections, 40 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Statement of results
  3. The large $n$ regime
  4. Previous work
  5. Innovations of the proof
  6. Setup for the proof
  7. A novel technique: functorial reconstruction
  8. Proving \ref{['theorem:main-thm-2']} by repeatedly applying functorial reconstruction
  9. Proving \ref{['theorem:main-thm-1']}
  10. A new technical ingredient: global generation
  11. Acknowledgments
  12. Notation and preliminaries on moduli
  13. Notation
  14. Preliminaries on moduli
  15. Review of parabolic bundles and period maps
  16. ...and 22 more sections

Key Result

Theorem 1.3

Let $H$ be a finite group and let $\Sigma_{g',n'}\to \Sigma_{g,n}$ be an $H$-cover associated to a surjection $\varphi: \pi_1(\Sigma_{g,n}, x)\twoheadrightarrow H$, where $x$ is a base point of $\Sigma_{g,n}$. Let $\overline{r}$ be the maximal dimension of an irreducible representation of $H$. Suppo Let $\operatorname{Mod}_\varphi$ be the stabilizer of $\varphi$ inside $\operatorname{Mod}_{g,n+1}$

Theorems & Definitions (103)

  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Definition 1.8
  • Theorem 1.9
  • Corollary 1.10
  • Corollary 1.11
  • Conjecture 1.13
  • ...and 93 more