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 $\mathfrak{X}$-mean-periodic for some appropriate functional space $\mathfrak{X}$. 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 the vector spaces $\mathcal{T}(h(\mathcal{S},\{k_i\},s))$, which are constructed from the Mellin transforms $f(\mathcal{S},\{k_i\},s)$ of certain products of arithmetic zeta functions $\zeta(\mathcal{S},2s)\prod_{i}\zeta(k_i,s)$, where $\mathcal{S}\rightarrow {\rm Spec}(\mathcal{O}_k)$ 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 $h(\mathcal{S},\{k_i\},s))$ plays the role of an Eisenstein series, exploiting the spectral interpretation of the zeros of automorphic $L$-functions.