*
A holomorphic version of the Tate-Iwasawa method and its applications*

Abstract

The Tate-Iwasawa method deals with the problem of meromorphic continuation
and functional equation for zeta- and L-functions of one-dimensional
arithmetic schemes. In the talk we introduce a new version of the method
following the lines of my paper ˙˙Notes on the Poisson formula˙˙ in SPb
Math. J. 23:5(2011) (arXiv:math/1011.3392) and a remark made by M.
Kapranov in his preprint on S-duality (arXiv:math/0001.005). The
construction is applied to a new proof of the functional equation for
the L-functions. The proof avoids completely analytic arguments and
discloses purely algebraic construction which is responsible for the
analytic continuation and the functional equation. We consider
unramified L-functions and the case of a curve over a finite field but
there are reasons to think that the method can be extended to the number
field case as well.

Publication: Math. Sbornik, 205:10(2014), 107-124