A class of functional identities associated to curves over finite fields

Abstract

Goss zeta values can be found, in some cases, as evaluations of a new type of rigid analytic function on projective curves $X$ over a finite field $\mathbb{F}_q$, called "Pellarin $L$-series". In the case of genus $0$ and $1$, Pellarin and Green--Papanikolas further determined functional identities for Pellarin $L$-series, in partial analogy with the functional equation of Dirichlet $L$-series. The aim of this paper is to prove that a generalization of these functional identities holds in arbitrary genus. Our proof exploits the topological nature of divisors on the curve $X$, as well as the introduction of an "adjoint shtuka function". This allows us to reinterpret Pellarin $L$-series as dual versions of the special functions studied by Angl\`es, Ngo Dac, and Tavares Ribeiro.