Hebrew University Amitsur Algebra Seminar ---------------------------------- Time: Thursday, Nov 22 , 12:00-13:15 Place: math 209 Speaker: Michael Schein (Bar-Ilan) Title: Zeta functions of Heisenberg groups over number rings Abstract: This is a report on work in progress with Mark Berman (Bar-Ilan) and Christopher Voll (Bielefeld). Let G be a finitely generated group, and let a_n be the number of subgroups of G of index n, which is always finite. The zeta function Z(s) = \sum a_n n^{-s} counts the finite index subgroups of G and has been an object of active study for the past 25 years. The zeta function splits into an Euler product of local factors, and in some cases these factors possess a striking symmetry (a functional equation). It is an interesting and deep problem to explain this symmetry in terms of the algebraic properties of G. We consider the special case of a Heisenberg group over a number ring. Let K be a number field with ring of integers O. The Heisenberg group H(O) consists of upper triangular matrices with entries in O and ones on the diagonal. We have studied the local zeta factors of the group H(O). In some cases we have explicit formulae for these factors. In the remaining cases, we study an algorithm for computing them that provides an interesting connection to the combinatorics of Dyck words. The local zeta factor at any prime p appears to satisfy a functional equation that depends on the ramification of p in K. You are cordially invited! <http://math.huji.ac.il/amitsur.html> --------------------------------------------------------- Technion Math Net-2 (TECHMATH2) Editor: Michael Cwikel <techm@math.technion.ac.il> Announcement from: Gili Schul <gili.schul@mail.huji.ac.il>