General multiple Dirichlet series from perverse sheaves

TL;DR

This paper characterizes multiple Dirichlet series over function fields using axioms and constructs them via trace functions of perverse sheaves, unifying many known series under a common framework.

Contribution

It provides an axiomatic framework for multiple Dirichlet series over function fields and constructs them explicitly using perverse sheaves, extending previous work.

Findings

Axiomatic characterization of multiple Dirichlet series over $\\mathbb{F}_q(T)$

Construction of series via trace functions of perverse sheaves

Unification of previously known series within this framework

Abstract

We give an axiomatic characterization of multiple Dirichlet series over the function field $\mathbb F_q(T)$, generalizing a set of axioms given by Diaconu and Pasol. The key axiom, relating the coefficients at prime powers to sums of the coefficients, formalizes an observation of Chinta. The existence of multiple Dirichlet series satisfying these axioms is proved by exhibiting the coefficients as trace functions of explicit perverse sheaves and using properties of perverse sheaves. The multiple Dirichlet series defined this way include, as special cases, many that have appeared previously in the literature.