Forthcoming events in this series
11:00
"Henselianity as an elementary property".
Abstract
Following Prestel and Ziegler, we will explore what it means for a field
to be t-henselian, i.e. elementarily equivalent (in the language of
rings) to some non-trivially henselian valued field. We will discuss
well-known as well as some new properties of t-henselian fields.
11:00
Model Theory of Adeles, Adele Classes, and Residual Hyperfields of Valued Fields
Abstract
This is joint work with Angus Macintyre. We study model-theoretic properties of
the ring of adeles, the hyperring of adele classes (studied by Connes-Consani),
and residual hyperfields of valued fields (in the sense of Krasner).
11:00
"Valued fields, integration: future and past directions."
Abstract
I'll sketch some context for future and past research around valued fields
and motivic integration, from a model theoretic viewpoint, leaving out technical details.
The talk will be partly conjectural.
11:00
``Relative CM-triviality and interpretable groups in the bad field''
Abstract
I shall present a geometric property valid in many Hrushovski
amalgamation constructions, relative CM-triviality, and derive
consequences on definable groups: modulo their centre they are already
products of groups interpretable in the initial theories used for the
construction. For the bad field constructed in this way, I shall
moreover classify all interpretable groups up to isogeny.
11:00
"Motivic Integration and counting conjugacy classes in algebraic groups over number fields"
Abstract
This is joint work with Uri Onn. We use motivic integration to get the growth rate of the sequence consisting of the number of conjugacy classes in quotients of G(O) by congruence subgroups, where $G$ is suitable algebraic group over the rationals and $O$ the ring of integers of a number field.
The proof uses tools from the work of Nir Avni on representation growth of arithmetic groups and results of Cluckers and Loeser on motivic rationality and motivic specialization.
11:00
"Model theoretic properties of S-acts and S-poset".
Abstract
An S-act over a monoid S is a representation of a monoid by tranformations of a set, analogous to the notion of a G-act over a group G being a representation of G by bijections of a set. An S-poset is the corresponding notion for an ordered monoid S.
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"Conjugacy classes in quotients of algebraic groups, model theory, and a transfer principle".
Abstract
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.