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
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!
Technion Math Net-2 (TECHMATH2)
Editor: Michael Cwikel   <techm@math.technion.ac.il> 
Announcement from: Gili Schul   <gili.schul@mail.huji.ac.il>