Automorphicity and Mean-Periodicity

TL;DR

This paper investigates the deep connection between automorphicity and mean-periodicity of L-functions associated with algebraic curves over number fields, proposing a novel approach to understanding their analytic properties and automorphic nature.

Contribution

It establishes a dual relationship between automorphicity and mean-periodicity, and introduces a technique emulating the Rankin-Selberg method to analyze L-functions.

Findings

Orthogonality of matrix coefficients to certain vector spaces

Mean-periodic functions analogous to Eisenstein series

Spectral interpretation of zeros of automorphic L-functions

Abstract

If C is a smooth projective curve over a number field k, then, under fair hypotheses, its L-function admits meromorphic continuation and satisfies the anticipated functional equation if and only if a related function is X-mean-periodic for some appropriate functional space X. Building on the work of Masatoshi Suzuki for modular elliptic curves, we will explore the dual relationship of this result to the widely believed conjecture that such L-functions should be automorphic. More precisely, we will directly show the orthogonality of the matrix coefficients of GL_{2g}-automorphic representations to certain vector spaces which are constructed from the Mellin transforms of specified products of arithmetic zeta functions corresponding to a model of the curve and the base number field. To compare automorphicity and mean-periodicity, we use a technique emulating the Rankin-Selberg method, in which the function mean-periodic function plays the role of an Eisenstein series, exploiting the spectral interpretation of the zeros of automorphic $L$-functions.