A class of functional identities associated to curves over finite fields

TL;DR

This paper generalizes functional identities for Pellarin L-series associated with curves over finite fields to arbitrary genus, using divisor topology and an adjoint shtuka function.

Contribution

It extends known identities from genus 0 and 1 to all genera, introducing new techniques involving divisor topology and an adjoint shtuka function.

Findings

Functional identities hold for Pellarin L-series in arbitrary genus.

The proof uses divisor topology and an adjoint shtuka function.

Pellarin L-series are interpreted as duals of certain special functions.

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.