UCSD Number Theory Seminar (Math 209)

Thursday 2-3pm, AP&M 7421

Spring Quarter 2013

April 11

Alina Bucur (UCSD)
Counting points on curves over finite fields


April 18

Hadi Hedayatzadeh (Caltech)
Drinfeld displays and tensor constructions of π-divisible modules in equal characteristic

Using results of Drinfeld and Taguchi, we establish an equivalence of categories between the category of "Drinfeld displays" (objects to be introduced) and the category of π-divisible modules. We define tensor products of π-divisible modules and using the above equivalence, we prove that the tensor products of π- divisible modules over locally Noetherian base schemes exist and commute with base change. If time permits, we will show how this will provide tensor products of Lubin-Tate groups and formal Drinfeld modules.

April 25


May 2

Veronika Ertl (University of Utah)
Overconvergent Chern classes

For a proper smooth variety over a perfect field of characteristic p, crystalline cohomology is a good integral model for rigid cohomology and crystalline Chern classes are integral classes which are rationally compatible with the rigid ones. The overconvergent de Rham-Witt complex introduced by Davis, Langer and Zink provides an integral p-adic cohomology theory for smooth varieties designed to be compatible with rigid cohomology in the quasi-projective case. The goal of this talk is to describe the construction of integral Chern classes for smooth varieties rationally compatible with rigid Chern classes using the overconvergent complex.


Rudy Perkins (Ohio State University)
On certain operators resembling polylogarithms in function field arithmetic

Building on the work which will appear in my thesis and certain formalism introduced in F. Pellarin's paper Values of certain L-series in positive characteristic, I will introduce a family arising from the Carlitz module in the ring of twisted power series over the ring K of rational functions in one indeterminate with coefficients in a finite field. I will describe how this family gives rise to both the explicit calculation of the rational functions appearing (inexplicitly) in Pellarin's main result in the paper above and also to certain K-relations between Thakur's multizeta values originally introduced in his book Function Field Arithmetic. This is a joint work with F. Pellarin.

May 9

Cristian Popescu (UCSD)
Explicit l-adic models of Tate sequences and applications

I will discuss an explicit Iwasawa theoretic construction of Tate (exact) sequences and give some of its applications to various conjectures on special values of Artin L-functions. This is based on joint work with Greither and Banaszak.

May 16

David Helm (The University of Texas at Austin)
A derived local Langlands correspondence for GLn

We describe joint work (with David Ben-Zvi and David Nadler) that constructs an equivalence between the derived category of smooth representations of GLn(Qp) and a certain category of coherent sheaves on the moduli stack of Langlands parameters for GLn. The proof of this equivalence is essentially a reinterpretation of K-theoretic results of Kazhdan and Lusztig via derived algebraic geometry. We will also discuss (conjectural) extensions of this work to other quasi-split groups, and to the modular representation theory of GLn.


Liang Xiao (UC Irvine)
Goren-Oort stratifications of Hilbert modular varieties mod p, and the Tate conjecture

I will report on an ongoing joint project with David Helm and Yichao Tian. Let p be a prime unramified in a totally real field F. The Goren-Oort strata are defined by the vanishing locus of the partial Hasse invariants; they furnish an analog of the stratification of modular curves mod p by the ordinary locus and the supersingular locus. We give an explicit global description of the Goren-Oort stratification of the special fiber of the Hilbert modular variety for F. An interesting application of this result is that, when p is inert of even degree, certain generalizations of the strata considered by Goren-Oort contribute non-trivially as Tate cycles to the cohomology of the special fiber of the Hilbert modular varieties. Under some mild conditions, they generate all the Tate cycles.

May 23

Fucheng Tan (Michigan State University)
The Breuil-Mezard conjecture for split residual representations

I will explain how to prove the Breuil-Mezard conjecture for split (non-scalar) residual representations by local methods. Combined with the cases previously proved by Kisin and Paskunas, this completes the proof of the conjecture for GL2(Qp). As an application, we can prove the Fontaine-Mazur conjecture in the cases that the global residual representation restricts to the decomposition group at p as an extension of the trivial character by the mod p cyclotomic character. These are the cases complementary to Kisin's. This is a joint work with Yongquan Hu.

May 30

Shahed Sharif (CSU San Marcos)
Geometric Shafarevich-Tate groups of certain elliptic threefolds

Let Z be a variety and A an elliptic curve over the function field of Z. I. Dolgachev and M. Gross define the geometric Shafarevich-Tate group of A over Z to classify the set of isomorphism classes of principal homogeneous spaces for A which are locally trivial in the étale topology. In joint work with Chad Schoen, we describe how to compute the Shafarevich-Tate group when A is the generic fiber of a class of elliptic threefolds and Z is the base. We also obtain results on the Brauer groups of such threefolds.

June 6