Moduli of Hyperelliptic Curves and Multiple Dirichlet Series

TL;DR

This paper constructs a special multiple Dirichlet series linked to hyperelliptic curves and quadratic L-series, revealing deep connections with moduli spaces, cohomology, and automorphic forms.

Contribution

It introduces an explicit construction of a multiple Dirichlet series satisfying new axioms, connecting its coefficients to moduli spaces and cohomological data.

Findings

Coefficients satisfy specific recurrence relations.

Identities are encoded in moduli space combinatorics.

Coefficients expressed via Frobenius eigenvalues and étale cohomology.

Abstract

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 $\ell$-adic \'etale cohomology of local systems on the moduli $\mathscr{H}_{g}[2]$ of hyperelliptic curves of genus $g$ with level 2 structure.