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.
The previous texts are
not the published one. For they should not be quoted.
Reprints of published versions may be asked by e-mails.