General multiple Dirichlet series from perverse sheaves

2022/09/26 by Will Sawin, Sawin, Will
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2209.13072

openalex publication_date 2022/09/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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