London Number Theory Seminar

Abstract: Recently L. Taelman has constructed a finite F_q[T]-module attached to a Drinfel'd module which is an analogue of the ideal class group of a number field. Taelman has proved an analytic class number formula for these modules and an analogue of Ribet's Theorem. In this talk, we will consider an analogue of the Kummer-Vandiver problem for these modules and we will present examples which give a negative answer to this problem. This talk is based on a joint work with L. Taelman.

Speaker: Kevin Buzzard

Title: Reduction of crystalline representations

Abstract: 2-dimensional crystalline Galois representations of the absolute Galois group of Q_p can be completely determined by linear algebra data. One can also classify 2-dimensional mod p Galois representations easily. This leads us to the following question: given a piece of linear algebra data, what is the reduction of the corresponding p-adic Galois representation? This question turns out to be a little subtle. I'll survey the state of the art and talk about the most recent method of attack -- the p-adic and mod p Langlands Program.

Speaker: Jamshid Derakhshan

Title: Model theory of the adeles and connections to number theory
Abstract: This is joint work with Angus Macintyre. Model theory studies definable subsets of a structure in a specific language. For many important structures, definable sets turn out to have a rich geometry in a natural language. Once the family of definable sets has a 'a direct image theorem', the structure of definable sets becomes transparent. This usually implies decidability, but there are also applications to geometry and arithmetic; and structures enjoying such properties can be thought of as 'tame' in some sense.

In model-theoretic terms, such a direct image theorem is called 'quantifier-elimination'. In the language of rings, this has been achieved by A. Tarski for the fields of real and complex numbers, and A. Macintyre for the field of p-adic numbers. An example where there is no direct image theorem, is the ring of rational integers. This is an undecidable territory where the definable sets become increasingly complicated as follows from the work of Godel. Even the 'existential' theory of the integers, which is the theory of Diophantine equations, is undecidable as is known from the negative solution of Hilbert's tenth problem.

We prove a quantifier-elimination for the ring of adeles of a number field in the language of rings. To get this, we use quantifier-elimination theorems for the local fields. It follows that the definable subsets of the adeles are Borel, and their measures are related to values of zeta functions in many cases. Time permitting, I shall end with some speculations on a general picture on this connection and applications to number theory.

Speaker: Toby Gee

Title: p-adic Hodge-theoretic properties of etale cohomology with mod p coefficients, and the cohomology of Shimura varieties

Abstract: I will discuss some new results about the etale cohomology of varieties over a number field or a p-adic field with coefficients in a field of characteristic p, and (if time permits) give some applications to the cohomology of unitary Shimura varieties. (Joint with Matthew Emerton.)

Speaker: Andreas Langer

Title: An integral structure on rigid cohomology

Abstract: For a quasiprojective smooth variety over a perfect field k of char p we introduce an overconvergent de Rham-Witt complex by imposing a growth condition on the de Rham-Witt complex of Deligne-Illusie using Gauus norms and prove that its hypercohomology defines an integral strcuture on rigid cohomology, i.e. its image in rigid cohomology is a canonical lattice. As a corollary we obtain that the integral Monsky-Washnitzer cohomology (considered before inverting p) of a smooth k-algebra is of finite type modulo torsion. This is joint work with Thomas Zink.

Speaker: Francois Loeser

Title: Motivic height zeta functions and the Poisson formula

Abstract: Recently, Chambert-Loir and Tschinkel obtained asymptotic formulas for the number of integral points of bounded height on partial equivariant compactifications of vector groups. In this talk we shall present a motivic version of these results. We show the rationality of the corresponding motivic height zeta functions and determine its leading pole and the residue. Our approach relies on "motivic harmonic analysis". In particular a motivic Poisson formula due to Hrushovski and Kazhdan plays a key role. This is joint work with Antoine Chambert-Loir.

Speaker: Ambrus Pal

Title: Around de Jong’s conjecture

Abstract: I will talk about one item of my current work in progress, which gives a new proof of de Jong’s conjecture in the rank two case, and a closely related analogue of Serre’s conjecture for function fields.

Speaker: Alena Pirutka

Title: On some aspects of unramified cohomology

Abstract: During this talk, I would like to explain different interactions of unramified cohomology groups with other problems, such as the study of Chow groups or some local-global principles.

Speaker: Peter Scholze

Title: On the cohomology of compact unitary group Shimura varieties at ramified split places

Abstract: Generalizing our previous methods, we give a description of the cohomology of Shimura varieties for which the reductive group G is locally at p a product of general linear groups, allowing arbitrary signature at infinity and arbitrary ramification at p. As applications, we give a complete description of the semisimple local Hasse-Weil zeta function in terms of automorphic L-functions in nonendoscopic cases, and reconstruct the l-adic Galois representations attached to RACSD cuspidal automorphic representations, using endoscopic cases. This is joint work with Sug Woo Shin.

Speaker: Ivan Tomasic

Title: Twisted Galois stratification

Abstract: The aim is to present the development of difference algebraic geometry and its applications to counting solutions of difference polynomial equations over fields with powers of Frobenius. We prove a twisted version of Chebotarev's theorem for a Galois covering of difference schemes, and use it to deduce an important direct image theorem: the image of a “twisted Galois formula” by a morphism of difference schemes is again a twisted Galois formula.

Speaker: Teruyoshi Yoshida

Title: The Hecke action on the weight spectral sequences

Abstract: We review the question of making algebraic correspondence act on the weight spectral sequence for l-adic cohomology of semistable schemes. To do this we need some intersection theory and cycle classes on regular schemes over the ring of integers. This approach works for the unitary Shimura varieties considered by Harris-Taylor/Shin (as the Hecke correspondences are finite and flat).