Mean-periodicity
and zeta functions

#### State

Submitted
on 10 April, 2008.

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

### Abstract

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.

mpprint.pdf mpview.pdf

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