Automorphicity and Mean-Periodicity

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.