Moduli of Hyperelliptic Curves and Multiple Dirichlet Series

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.