Functional equations for Selberg zeta functions with Tate motives

TL;DR

This paper investigates the functional equations of Selberg zeta functions associated with Tate motives on compact Riemann surfaces, establishing that these equations hold precisely when the motives exhibit absolute automorphy.

Contribution

It provides a characterization of when functional equations for Selberg zeta functions with Tate motives are valid, linking them to the motives' automorphy properties.

Findings

Functional equations hold iff the Tate motives are absolutely automorphic.

Characterization of automorphy conditions for Tate motives in the context of Selberg zeta functions.

Advances understanding of the symmetry properties of zeta functions attached to motives.

Abstract

For a compact Riemann surface $M$ of genus $g\ge 2$, we study the functional equations of the Selberg zeta functions attached with the Tate motives $f$. We prove that certain functional equations hold if and only if $f$ has the absolute automorphy.