Mean-periodicity and automorphicity

If CC is a smooth projective curve over a number field kk, then, under fair hypotheses, its LL-function admits meromorphic continuation and satisfies the anticipated functional equation if and only if a related function is XX-mean-periodic for some appropriate functional space XX. 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 LL-functions should be automorphic. More precisely, we will directly show the orthogonality of the matrix coefficients of GL2gGL2g-automorphic representations to the vector spaces T(h(S,{ki},s))T(h(S,{ki},s)), which are constructed from the Mellin transforms f(S,{ki},s)f(S,{ki},s) of certain products of arithmetic zeta functions ζ(S,2s)∏iζ(ki,s)ζ(S,2s)∏iζ(ki,s), where S→Spec(Ok)S→Spec(Ok) is any proper regular model of CC and {ki}{ki} is a finite set of finite extensions of kk. To compare automorphicity and mean-periodicity, we use a technique emulating the Rankin–Selberg method, in which the function h(S,{ki},s))h(S,{ki},s)) plays the role of an Eisenstein series, exploiting the spectral interpretation of the zeros of automorphic LL-functions.