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

TL;DR

This paper proves the convergence and functional equations of multiple Dirichlet series over function fields, building on Sawin's axiomatic framework and using advanced geometric and sheaf-theoretic tools.

Contribution

It establishes convergence and functional equations for these series, extending Sawin's axiomatic approach with geometric and analytic proofs.

Findings

Series converge in a specific region

Functional equations are satisfied after analytic continuation

Bounds on coefficients are obtained via perverse sheaves

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.