Open Access
January, 2017 Automorphicity and mean-periodicity
J. Math. Soc. Japan 69(1): 25-51 (January, 2017). DOI: 10.2969/jmsj/06910025


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.


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Thomas OLIVER. "Automorphicity and mean-periodicity." J. Math. Soc. Japan 69 (1) 25 - 51, January, 2017.


Published: January, 2017
First available in Project Euclid: 18 January 2017

zbMATH: 1370.11106
MathSciNet: MR3597546
Digital Object Identifier: 10.2969/jmsj/06910025

Primary: 11M41 , 11M99 , 11R39
Secondary: 11G40 , 11K70 , 14G10

Keywords: $L$-functions , arithmetic schemes , Automorphic representations , mean-periodicity , zeta functions

Rights: Copyright © 2017 Mathematical Society of Japan

Vol.69 • No. 1 • January, 2017
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