General multiple Dirichlet series from perverse sheaves (Q6556225)

This paper gives a uniform construction of a class of multiple Dirichlet series in \(r\) complex variables over the rational function field \(\mathbb{F}_q(T)\), parametrized by the finite field \(\mathbb{F}_q\), a character \(\chi\) of \(\mathbb{F}_q\), and a symmetric \(r\times r\) integer matrix \(M\). The matrix \(M\) specifies how the coefficients at powers of an individual prime (i.e., irreducible monic polynomial) are mixed to create the general coefficients, by a generalization of the `twisted multiplicativity' found in prior constructions (and involving \(\chi\) as well). The author introduces a set of axioms for the coefficients, generalizing a set of axioms given by \textit{A. Diaconu} and \textit{V. Paşol} [``Moduli of hyperelliptic curves and multiple Dirichlet series'', Preprint, \url{arXiv:1808.09667}]. The author's key axiom, relating the coefficients at prime powers to sums of coefficients, a local-to-global property, formalizes an observation due to \textit{G. Chinta} [Acta Arith. 132, No. 4, 377--391 (2008; Zbl 1165.11048)]. The existence and uniqueness of multiple Dirichlet series satisfying these axioms is proved by constructing the coefficients as trace functions of explicit perverse sheaves. The multiple Dirichlet series defined this way include, as special cases, many that have appeared previously in the literature. The series constructed here are not continued beyond a region of absolute convergence, but investigating this is a natural next step.\N\NTo put this work in context, \textit{D. Bump} et al. [Bull. Am. Math. Soc., New Ser. 33, No. 2, 157--175 (1996; Zbl 0865.11043)] proposed the construction of a broad class of Dirichlet series in two complex variables attached to automorphic \(L\)-functions, explained how Bochner's theorem could (under certain conditions) be used to continue the series beyond the region of absolute convergence, and noted a connection to moments of \(L\)-functions. The theory was extended to more than two variables by \textit{A. Diaconu} et al. [Compos. Math. 139, No. 3, 297--360 (2003; Zbl 1053.11071)], who related it to moment conjectures obtained by random matrix theory. \textit{B. Brubaker} et al. [Proc. Symp. Pure Math. 75, 91--114 (2006; Zbl 1112.11025)] introduced multiple Dirichlet series attached to a reduced root system, where the twisted multiplicativity property depended on the root system. In all cases the coefficients in the multisums were defined for squarefree coprime indices but the adjustments for general indices were delicate, and in many cases, conjectural. The author explains that the use of the trace function on perverse sheaves gives a natural way to extend a function from `generic' values such as values on squarefree coprime integers to all values. For multiple Dirichlet series, he suggests that ``every extension that satisfies nice analytic properties likely comes from a suitable perverse sheaf.'' This may open the way to a new method for fully defining and then studying the multiple Dirichlet series attached to higher rank automorphic forms.\N\NThe characterization of various multiple Dirichlet series over the rational function field by means of axioms is carried out in several works including [Chinta, loc. cit.; \textit{G. Chinta} et al., Proc. Symp. Pure Math. 75, 3--41 (2006; Zbl 1124.11023), see Section 5; \textit{G. Chinta} and \textit{J. B. Mohler}, Acta Arith. 144, No. 1, 53--68 (2010; Zbl 1217.11108); \textit{A. Bucur} and \textit{A. Diaconu}, Mosc. Math. J. 10, No. 3, 485--517 (2010; Zbl 1220.14020); \textit{I. Whitehead}, Compos. Math. 152, No. 12, 2503--2523 (2016; Zbl 1407.11103)].