Alexey Parshin (SMI, Russian Academy of Sciences, Moscow)
Tuesday, December 4, 2018 - 13:15
Humboldt-Universitรคt zu Berlin, Institut fรผr Mathematik
Rudower Chaussee 25, 12489 Berlin-Adlershof, Raum 006, Haus 3, Erdgeschoss
Forschungsseminar Arithmetische Geometrie
Prof. Jรผrg Kramer / Prof. Thomas Krรคmer
Let X be an algebraic projective smooth variety defined over a finite field Fq. Then one can consider the formal power series ๐œ๐‘‹(๐‘ )=โˆ‘๐‘›โ‰ฅ1#๐‘‹(๐”ฝ๐‘ž๐‘›)๐‘žโˆ’๐‘›๐‘ /๐‘›. If dimX = n then the series will convergent for R (s) > n. The main problems are to extend this function of s to the whole s-plane, to locate the singularities and to find the analytic behavior around them. For any n, these problems can be solved by the Grothendieck cohomological method. In the case when n = 1, there is an adelic method which works also for zeta-functions of the fields of algebraic numbers. We will explain a new version of this method which is purely algebraic and functorial comparing with the classical version going back to Riemann and where one uses a manipulation with formulas.
submitted by Marion Thomma (thomma@math.hu-berlin.de, 2093-5815)