11:00

### Does decidability go up in finite field extensions?

## Abstract

We will follow a suggestion by Udi to construct a decidable field which has an undecidable finite extension.

Forthcoming events in this series

Thu, 15 Jun 2017

11:00

11:00

C5

Kesavan Thanagopal

(Oxford)

We will follow a suggestion by Udi to construct a decidable field which has an undecidable finite extension.

Thu, 08 Jun 2017

11:00

11:00

L6

Haden Spence

I will discuss my ongoing project towards a version of the Modular Andre-Oort Conjecture incorporating the derivatives of the j function. The work originates with Jonathan Pila, who formulated the first "Modular Andre-Oort with Derivatives" conjecture. The problem can be approached via o-minimality; I will discuss two categories of result. The first is a weakened version of Jonathan's conjecture. Under an algebraic independence conjecture (of my own, though it follows from standard conjectures), the result is equivalent to the statement that Jonathan's conjecture holds.

The second result is conditional on the same algebraic independence conjecture - it specifies more precisely how the special points in varieties can occur in this context.

If time permits, I will discuss my most recent work towards making the two results uniform in algebraic families.

Thu, 25 May 2017

11:00

11:00

C5

Jamshid Derakhshan

(Oxford)

I will talk about a result on meromorphic continuation of Euler products over primes p of definable p-adic or motivic integrals, and applications to zeta functions of groups. If time permitting, I'll state an analogue for counting rational points of bounded height in some adelic homogeneous spac

Thu, 18 May 2017

11:00

11:00

C5

Philip Dittmann

(Oxford)

We discuss a recent preprint by Aschenbrenner, Khélif, Naziazeno and

Scanlon, giving a positive solution to the ring-analogue of Pop's

problem on elementary equivalence vs isomorphism.

Thu, 27 Apr 2017

11:00

11:00

C5

Adam Topaz

(Oxford)

This talk will discuss the so-called ``generic cohomology’’ of function fields over algebraically closed fields, from the point of view of motives and/or Zariski geometry. In particular, I will describe some interesting connections between cup products, algebraic dependence, and (geometric) valuation theory. As an application, I will mention a new result which reconstructs higher-dimensional function fields from their generic cohomology, endowed with some additional motivic data.

Everyone welcome!

Thu, 02 Mar 2017

11:00

11:00

C5

Philip Dittmann

(Oxford)

Generalising previous definability results in global fields using

quaternion algebras, I will present a technique for first-order

definitions in finite extensions of Q(t). Applications include partial

answers to Pop's question on Isomorphism versus Elementary Equivalence,

and some results on Anscombe's and Fehm's notion of embedded residue.

Thu, 16 Feb 2017

11:00

11:00

C5

Sebastian Eterovic

Given a Shimura variety, I will show how to define a corresponding two-sorted structure. Based on work of Chris Daw and Adam Harris, we will study what is needed for the class of this structures to be categorical. Of course, an introduction to Shimura varieties will be given.

Thu, 16 Feb 2017

11:00

11:00

C5

Sebastian Eterovic

Given a Shimura variety, I will show how to define a corresponding two-sorted structure. Based on work of Chris Daw and Adam Harris, we will study what is needed for the class of this structures to be categorical. Of course, an introduction to Shimura varieties will be given.

Thu, 17 Nov 2016

11:00

11:00

C5

Chris Daw

(Oxford)

In this talk, we will explain how the counting theorems of Pila and Wilkie lead to a conditional proof of the aforementioned conjecture. In particular, we will explain how to generalise the work of Habegger and Pila on a product of modular curves.

Habegger and Pila were able to prove that the Zilber-Pink conjecture holds in such a product if the so-called weak complex Ax and large Galois orbits conjectures are true. In fact, around the same time, Pila and Tsimerman proved a stronger statement than the weak complex Ax conjecture, namely, the Ax-Schanuel conjecture for the $j$-function. We will formulate Ax-Schanuel and large Galois orbits conjectures for general Shimura varieties and attempt to imitate the Habegger-Pila strategy. However, we will encounter an additional difficulty in bounding the height of a pre-special subvariety.

This is joint work with Jinbo Ren.

Thu, 03 Nov 2016

11:00

11:00

C5

Thu, 10 Mar 2016

11:00

11:00

C5

Thu, 03 Mar 2016

11:00

11:00

C5

H.Schmidt

(Oxford)

We will discuss families of Pell's equation in polynomials

with one complex parameter. In particular the relation between

the generic equation and its specializations. Our emphasis will

be on families with a triple zero. Then additive extensions enter

the picture.

Thu, 21 Nov 2013

11:00

11:00

C5

Thu, 24 Oct 2013

11:00

11:00

C5

Thu, 17 Oct 2013

11:00

11:00

C5

Levon Haykazyan

(Oxford)

Thu, 16 May 2013

11:00

11:00

SR2

Thu, 07 Feb 2013

11:00

11:00

SR1

Thu, 02 Jun 2011

11:00

11:00

L3

Franziska Jahnke

(Oxford)

The class of fields with a given absolute Galois group is in general not an elementary class. Looking instead at abstract elementary classes we can show that this class, as well as the class of pairs (F,K), where F is a function field in one variable over a perfect base field K with a fixed absolute Galois group, is abstract elementary. The aim is to show categoricity for the latter class. In this talk, we will be discussing some consequences of basic properties of these two classes.

Thu, 04 Nov 2010

11:00

11:00

SR2

Jamshid Derakhshan

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.

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