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A geometric approach to functional equations for general multiple Dirichlet series over function fields

Matthew Hase-Liu

TL;DR

The paper resolves a key gap in Sawin’s framework by proving analytic convergence and meromorphic continuation for general multiple Dirichlet series over $\mathbb{F}_{q}(T)$, whose coefficients arise as Frobenius traces on perverse-sheaf stalks. It develops a geometric approach that combines a precise description of intermediate extensions, explicit Fourier-transform-type relations for $a$-coefficients, and cohomological bounds via Kontsevich moduli spaces to control coefficients. The authors derive functional equations in a multi-variable setting, distinguished by a parity analysis and a fudge-factor twist that mirrors Dirichlet-character conjugation, and then establish convergence and region-of-convergence results using Grothendieck–Lefschetz and decomposition theorems. Together, these results provide a solid analytic underpinning for the constructed general multiple Dirichlet series and illuminate the interplay between arithmetic, geometry, and automorphic-type symmetries in the function-field setting.

Abstract

Sawin recently gave an axiomatic characterization of multiple Dirichlet series over the function field $\mathbb{F}_{q}(T)$ and proved their existence by exhibiting the coefficients as trace functions of specific perverse sheaves. However, he did not prove that these series actually converge anywhere, instead treating them as formal power series. In this paper, we prove that these series do converge in a certain region, and moreover that the functions obtained by analytically continuing them satisfy functional equations. For convergence, it suffices to obtain bounds on the coefficients, for which we use the decomposition theorem for perverse sheaves, in combination with the Kontsevich moduli space of stable maps to construct a suitable compactification. For the functional equations, the key identity is a multi-variable generalization of the relationship between a Dirichlet character and its Fourier transform; in the multiple Dirichlet series setting, this uses a density trick for simple perverse sheaves and an explicit formula for intermediate extensions from the complement of a normal crossings divisor.

A geometric approach to functional equations for general multiple Dirichlet series over function fields

TL;DR

The paper resolves a key gap in Sawin’s framework by proving analytic convergence and meromorphic continuation for general multiple Dirichlet series over , whose coefficients arise as Frobenius traces on perverse-sheaf stalks. It develops a geometric approach that combines a precise description of intermediate extensions, explicit Fourier-transform-type relations for -coefficients, and cohomological bounds via Kontsevich moduli spaces to control coefficients. The authors derive functional equations in a multi-variable setting, distinguished by a parity analysis and a fudge-factor twist that mirrors Dirichlet-character conjugation, and then establish convergence and region-of-convergence results using Grothendieck–Lefschetz and decomposition theorems. Together, these results provide a solid analytic underpinning for the constructed general multiple Dirichlet series and illuminate the interplay between arithmetic, geometry, and automorphic-type symmetries in the function-field setting.

Abstract

Sawin recently gave an axiomatic characterization of multiple Dirichlet series over the function field and proved their existence by exhibiting the coefficients as trace functions of specific perverse sheaves. However, he did not prove that these series actually converge anywhere, instead treating them as formal power series. In this paper, we prove that these series do converge in a certain region, and moreover that the functions obtained by analytically continuing them satisfy functional equations. For convergence, it suffices to obtain bounds on the coefficients, for which we use the decomposition theorem for perverse sheaves, in combination with the Kontsevich moduli space of stable maps to construct a suitable compactification. For the functional equations, the key identity is a multi-variable generalization of the relationship between a Dirichlet character and its Fourier transform; in the multiple Dirichlet series setting, this uses a density trick for simple perverse sheaves and an explicit formula for intermediate extensions from the complement of a normal crossings divisor.
Paper Structure (14 sections, 32 theorems, 137 equations)

This paper contains 14 sections, 32 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and Sawin's construction
  4. An explicit formula for a particular intermediate extension
  5. Elementary combinatorial lemmas on trees
  6. Bounds for the cohomology of lisse sheaves on the moduli space of stable maps
  7. Relationship between $a$-coefficients and Fourier transforms
  8. Derivation of functional equations
  9. $\prod_{i=s+1}^{m}\left(\frac{-}{f_{i}}\right)_{\chi}^{M_{1,i}}$ is trivial on $\mathbb{F}_{q}^{\times}$
  10. $\prod_{i=s+1}^{m}\left(\frac{-}{f_{i}}\right)_{\chi}^{M_{1,i}}$ is not trivial on $\mathbb{F}_{q}^{\times}$
  11. Putting everything together
  12. A short verification
  13. Bounds on $a$-coefficients and their sums
  14. Meromorphic continuation of multiple Dirichlet series

Key Result

Theorem 1

Let $n,m,$ and $s$ be positive integers, and $M$ a symmetric $m\times m$ matrix with coefficients in $\mathbb{Z}/n\mathbb{Z}$. Then, $L\left(u_{1},\ldots,u_{m};M\right)$ is an analytic function with a non-empty region of convergence.

Theorems & Definitions (67)

  • Theorem 1
  • Definition 2
  • Definition 3
  • Remark 4
  • Remark 5
  • Theorem 6
  • Remark 7
  • Remark 8
  • Remark 9
  • Theorem 11
  • ...and 57 more