Automorphicity and mean-periodicity

摘要

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 (￡)-mean-periodic for some appropriate functional space (￡). 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 the vector spaces T(h(S, {k_i}, s)), which are constructed from the Mellin transforms f(S, {k_i}, s) of certain products of arithmetic zeta functions ζ(S, 2s) ∏_i ζ(k_i, s), where S → Spec(O_k) is any proper regular model of C and {k_i} is a finite set of finite extensions of k. To compare automorphicity and mean-periodicity, we use a technique emulating the Rankin-Selberg method, in which the function h(S, {k_i｝, s)) plays the role of an Eisenstein series, exploiting the spectral interpretation of the zeros of automorphic L-functions.