Speaker: Michael Schein
(Bar-Ilan University):
Title:
#Subgroup growth in Heisenberg groups over number rings

Abstract:
Let G be a finitely generated group, and let a_n be the number of normal 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 consider the local zeta factors of the group H(O).  If p is either unramified or non-split in K, we have explicit formulae for these factors, expressed as a sum of terms parametrized by Dyck words of length 2[K:Q].  The local zeta factor at any prime p appears to satisfy a functional equation that depends on the ramification of p in K.  This is joint work with Christopher Voll.