Mean-periodicity and zeta functions





Submitted on 10 April, 2008.

Accepted for publication on 23 September 2011, by Annales de l’Institut Fourier.




The general admitted expectation is that the right objects parametrizing L-functions are automorphic representations.


In this joint work with Ivan Fesenko and Masatoshi Suzuki, it is suggested that the right objects parametrizing Hasse zeta functions of arithmetic schemes are mean-periodic functions over the real line, which have at most polynomial growth.


Such Hasse zeta functions are conjecturally ratios of L-functions. As a consequence, the traditional way to prove the expected analytic properties of such Hasse zeta functions is to prove automorphic properties of each of the conjectural L-factors, which is not entirely satisfactory. 


It is shown in this work that establishing the expected analytic properties of these zeta functions boils down to proving the mean-periodicity of some explicit functions on the real line.


The case of regular models of zeta functions of elliptic curves is carefully analysed in this paper.


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