2018/08/29 by Adrian Diaconu, Diaconu, Adrian, Vicenţiu Pașol +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1808.09667
openalex publication_date 2018/08/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we provide an explicit construction of a distinctive multiple Dirichlet series associated to products of quadratic Dirichlet L-series, which we believe should be tightly connected to a generalized metaplectic Whittaker function on the double cover of a Kac-Moody group. To do so, we first impose a set of axioms, independent of any group of functional equations, which the aforementioned object should satisfy. As a consequence, we deduce that the coefficients of the p-parts of the multiple Dirichlet series satisfy certain recurrence relations. These relations lead to a family of identities, which turns out to be encoded in the combinatorial structure of certain moduli spaces of admissible double covers. Finally, via this crucial connection, we apply Deligne's theory of weights to express inductively the coefficients of the p-parts in terms of the eigenvalues of Frobenius acting on the ℓ-adic étale cohomology of local systems on the moduli \mathscrHg[2] of hyperelliptic curves of genus g with level 2 structure.