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 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.example, if S → Spec(O k ) is a proper, regular model of a smooth projective curve C over a number field k, then one haswhere ζ(S, s) = x∈S 0 1 1−|k(x)| −s , the product being taken over the closed points of the arithmetic surface. Ignoring temporarily the term n(S, s), the expression on the right hand side is an "Euler characteristic of L-functions". Each term in the quotient is a Hasse-Weil L-function, in which the Euler factor is the reciprocal characteristic polynomial of the action of the Frobenius operator on inertia invariants ofétale cohomology, i.e.Full details for the definition of Hasse-Weil L-functions are given in [16]. The function n(S, s) depends on the choice of model S of C. In fact, it is a finite product of functions rational in variables of the form p −s , where p ranges over the residual characteristics of bad reduction. An exercise inétale cohomology shows that (the appropriate completion of) n(S, s) satisfies the correct functional equation with respect to s → 2 − s. We thus see that L(C, s) admits meromorphic continuation if and only if ζ(S, s) does, and their functional equations are equivalent. In this paper we will focus on the zeta functions of arithmetic surfaces. Within modern harmonic analysis, long established as an effective tool in the study of number theory, one finds the notion of mean-periodicity. First introduced in 1935 [3], the general theory of mean-periodic functions is exposed in [9]. Recent research places mean-periodicity at the heart of some of the basic open questions concerning the analytic properties of zeta functions of arithmetic schemes 1 [5], [6], [4]. In fact, under fair hypotheses, the meromorphic continuation and functional equation is equivalent to the mean-periodicity of a related object. In light of converse theorems, and their limitations, it is natural to ask what the relationship is between mean-periodicity and the expected ...

Zeta integrals on arithmetic surfaces

“…Remark 7.4. The mean-periodicity correspondence may be compared to automorphicity of the Hasse-Weil L-functions appearing in the motivic decomposition of the zeta function, for example [27], [22].…”