Pure Maths Colloquium: Functional equations for zeta functions of groups and rings
29 January 2007 16:00 in E101
A finitely generated group has only finitely many subgroups of each
finite index. The group's zeta function is the Dirichlet series
encoding these numbers. If the group is nilpotent its zeta function
admits an Euler product decomposition into local factors, indexed by
the primes. These local factors have remarkable arithmetic properties.
One of their features is a beautiful palindromic symmetry: I shall
report on recent work, establishing certain functional equations for
the local factors of zeta functions of nilpotent groups. I will show
how this can be achieved using techniques from the theory of Igusa's
local zeta function associated to polynomial mappings, generalizing
work of Denef and Meuser's and others.
I will also discuss variants of these zeta functions. Among them are
zeta functions counting subgroups up to conjugacy, or counting just
normal subgroups in nilpotent groups, representation zeta functions of
nilpotent groups and zeta functions of torsion-free rings.
Contact email@example.com for more information