Abstract. In this talk, I will present joint work with
Uri Onn, Mark Berman, and Pirita Paajanen.
Let G be a linear algebraic group defined over the integers.
Let O be a compact discrete valuation ring with a finite residue field of
cardinality q and characteristic p. The group
G(O) has a filtration by congruence subgroups
G_m(O) (which is by definition the kernel of reduction
map modulo P^m where P is the maximal ideal of O).
Let c_m=c_m(G(O)) denote the number of conjugacy
classes in the finite quotient group G(O)/G_m(O) (which is called the mth
congruence quotient of G(O)). The conjugacy class zeta function of
G(O) is defined to be the Dirichlet series
Z_{G(O)}(s)=\sum_{m=0,1,...} c_m q^_{-ms}, where s is a complex number with
Re(s)>0. This zeta function was defined by du Sautoy when G is a p-adic
analytic group and O=Z_p, the ring of p-adic integers, and he proved that in
this case it is a rational function in p^{-s}. We consider the question
of dependence of this zeta function on p and more generally on the ring O.
We prove that for certain algebraic groups, for all
compact discrete valuation rings with finite residue field of cardinality q and
sufficiently large residue characteristic p, the conjugacy class zeta function
is a rational function in q^{-s} which depends only on q and not on the
structure of the ring. Note that this applies also to positive characteristic
local fields.
A key in the proof is a transfer principle. Let \psi(x)
and f(x) be resp.
definable sets and functions in Denef-Pas language.
For a local field K, consider the local integral
Z(K,s)=\int_\psi(K)
|f(x)|^s dx, where | | is norm on K and dx normalized
absolute value
giving the integers O of K volume 1. Then there is some
constant
c=c(f,\psi) such that for all local fields K of
residue characteristic larger than c and residue field of cardinality q, the
integral Z(K,s) gives the same rational function in q^{-s} and takes the same
value as a complex function of s.
This transfer principle is more general than the
specialization to local fields of the special case when there is no additive
characters of the motivic transfer principle of Cluckers and Loeser since their
result is the case when the integral is zero.
The conjugacy class zeta function is related to the
representation zeta function which counts number of irreducible complex
representations in each degree (provided there are finitely many or finitely many
natural classes) as was shown in the work of Lubotzky and Larsen, and gives
information on analytic properties of latter zeta function.
