A.N. Parshin

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

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