UCSD Number Theory Seminar (Math 209)

Thursday 2-3pm, AP&M 7421


Spring Quarter 2014

For previous quarters' schedule, click here.


April 3

Stephan Garcia (Pomona College)
Supercharacters and their super powers

The theory of supercharacters, which generalizes classical character theory, was recently developed in an axiomatic fashion by P. Diaconis and I.M. Isaacs, based upon earlier work of C. Andre. When this machinery is applied to abelian groups, a wide variety of applications emerge. In particular, we develop a generalization of the discrete Fourier transform along with several combinatorial tools. This perspective illuminates several classes of exponential sums (e.g., Gauss, Kloosterman, and Ramanujan sums) that are of interest in number theory. We also consider certain exponential sums that produce visually striking patterns of great complexity and subtlety. (Partially supported by NSF Grants DMS-1265973, DMS-1001614, and the Fletcher Jones Foundation.)

April 10

Claus Sorensen (UCSD)
Ihara's lemma and local Langlands in families

In their attempt to mimic the proof of Fermat's Last Theorem for GL(n), Clozel, Harris, and Taylor, were led to a conjectural analogue of Ihara's lemma -- which is still open for n>2. In this talk we will revisit their conjecture from a more modern point of view, and reformulate it in terms of local Langlands in families, as currently being developed by Emerton and Helm. At the end, we hope to sketch how this can be used to obtain a factorization of completed cohomology for U(2). [The last part is joint work with P. Chojecki.]

April 17

Liang Xiao (UC Irvine) 2-3 in 7421
Galois representations and torsion in the cohomology of Hilbert modular varieties

Let F be a totally real number field, p a prime number, and M the (splitting model of) Hilbert modular variety for F (of some fixed level) defined over a finite field of characteristic p. I will explain how exploiting the geometry of M, and in particular the existence of the partial Hasse invariants, one can attach Galois representations to Hecke eigensystems occurring in the coherent cohomology of M. This is a joint work with Matthew Emerton and Davide Reduzzi.

Chan-Ho Kim (UC Irvine) 3-4 in 7421?
On congruences of anticyclotomic p-adic L-functions of Hida families

We construct anticyclotomic p-adic L-functions of Hida families in a more controlled way using a multiplicity one result arising from arithmetic of Shimura curves. Using this construction, we can calculate the difference of Iwasawa invariants of p-adic L-functions of congruent modular forms in different weights. As an application, we can see the equivalence of the main conjectures of two congruent forms under certain conditions. This is joint work in progress with Francesc Castella and Matteo Longo.

April 24

Kiran Kedlaya (UCSD)
Witt vectors in mixed characteristic and p-adic Hodge theory

The functor of p-typical Witt vectors is most well known for lifting perfect fields of characteristic p into complete discrete valuation rings. However, it is a well-defined functor on arbitrary rings; we will indicate how applying this functor to local rings of mixed characteristic gives some new perspectives on p-adic Hodge theory. We will also touch briefly upon some mysterious links to algebraic K-theory coming from the work of Hesselholt. Based on joint papers with Chris Davis (Copenhagen).

May 1

Cristian Popescu (UCSD)
Hecke characters and the Quillen K-theory of number fields

First, I will describe how our results (joint with Greither) on the Brumer-Stark conjecture lead to a new construction of Hecke characters for CM number fields, generalizing A. Weil's Jacobi sum Hecke characters. Second, I will show how the values of these characters can be used to construct special elements in the even K-groups of CM and totally real number fields. Several applications ensue: a general construction of Euler Systems in the even K-theory of CM and totally real number fields; a K-theoretic reformulation (and potential proof strategy) of a classical and wide open conjecture of Iwasawa on class groups of cyclotomic fields; potential new insights into Hilbert's 12th problem for CM number fields etc. Time permitting, I will touch upon some of these applications as well. This is based on joint work with G. Banaszak (Poland).

May 8

Cristian Popescu (UCSD)
Hecke characters and the Quillen K-theory of number fields - Part 2

I will briefly review the material covered in last week's lecture (May 1) and will continue with a more detailed description of the K-theoretic constructions and their arithmetic applications mentioned in last week's abstract. Joint work with G. Banaszak (Poland.)

May 15

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May 22

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May 29

Ryan Rodriguez (UCSD)
Acyclicity of preperfectoid algebras

It is desirable for the spectrum of a uniform Banach algebra to have a structure sheaf. This happens when the algebra is acyclic. We will discuss what it means for a uniform Banach algebra to be acyclic. I will explain how to show preperfectoid algebras are acyclic.

June 5

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June 12

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