Zeta Functions on Arithmetic Surfaces

TL;DR

This paper develops a two-dimensional adelic integral representation of zeta functions for arithmetic surfaces using lifted harmonic analysis, providing new insights into mean-periodicity and automorphicity conjectures.

Contribution

It introduces a novel adelic integral approach to zeta functions on arithmetic surfaces, linking harmonic analysis with automorphicity conjectures.

Findings

Derived a two-dimensional adelic integral representation of zeta functions.

Connected mean-periodicity with automorphicity conjectures.

Provided an adelic interpretation of the mean-periodicity correspondence.

Abstract

We use a form of lifted harmonic analysis to develop a two-dimensional adelic integral representation of the zeta functions of simple arithmetic surfaces. Manipulations of this integral then lead to an adelic interpretation of the so-called mean-periodicity correspondence, which is comparable to the better known automorphicity conjectures for the generic fibre.