University of Chicago Number Theory Seminar

Fall 2015: Tuesday 1:30-2:50pm, E202


This is the homepage of the Number Theory Seminar at the University of Chicago. To get on or off the mailing list, you can either go to lists.uchicago.edu or contact Brandon Levin.

NOTE: the room for this quarter is E202.

Click here to see the location of Eckhart Hall, and here for directions to the University of Chicago.

Click here to see the schedule of previous quarters: Winter 2013 / Spring 2013 / Fall 2013 / Spring 2014 / Fall 2014 / Winter 2015 / Spring 2015

Fall 2015 Schedule

Date

Speaker

Title

Sept 29

Cheng-Chiang Tsai
(MIT)

Computing p-adic orbital integrals
We sketch an inductive algorithm to compute p-adic orbital integral of a general orbit on nice test functions. Our emphasis will be on geometric structures, that is, varieties over the residue field on which we have to count rational points. We also give some formal results such as uniform bounds for orbital integrals and a local constancy result. If time permits, we can formulate an endoscopic conjecture regarding Borel-Moore homology of the above varieties. (Show Abstract)

Oct 6

Ana Caraiani
(Princeton)

On vanishing of torsion in the cohomology of Shimura varieties
I will discuss joint work in progress with Peter Scholze showing that torsion in the cohomology of certain compact unitary Shimura varieties occurs in the middle degree, under a genericity assumption on the corresponding Galois representation. (Show Abstract)

Oct 13

Christian Johansson
(IAS)

Overconvergent modular forms using perfectoid modular curves
We give a new definition of overconvergent modular forms roughly analogous to the complex analytic definition of modular forms, and discuss an application to the overconvergent Eichler-Shimura isomorphism of Andreatta-Iovita-Stevens. Joint with Przemyslaw Chojecki and David Hansen. (Show Abstract)

Oct 20

Jessica Fintzen
(Harvard)

Stable vectors in the Moy-Prasad filtration
Reeder and Yu gave recently a new construction of certain supercuspidal representations of p-adic reductive groups (called epipelagic representations). Their construction relies on the existence of stable vectors in the first Moy-Prasad filtration quotient under the action of a reductive quotient. We will explain these ingredients and present a theorem about the existence of such stable vectors for all primes p. This builds on a result of Reeder and Yu about the existence of stable vectors for large primes. Some of the above work forms part of a joint research project with Beth Romano. (Show Abstract)

Oct 27

Erick Knight
(Harvard)

A p-adic Jacquet-Langlands correspondence
I will construct a p-adic Jacquet-Langlands correspondence, which is a correspondence between Banach space representations of GL_2(Q_p) and Banach space representations of the unit group of the quaternion algebra D over Q_p. The correspondence satisfies local-global compatibility with the completed cohomology of Shimura curves, as well as a compatibility with the classical Langlands correspondence, in the sense that the representations of the unit group of D can often be shown to have the expected locally algebraic vectors. (Show Abstract)

Nov 3

Samit Dasgupta
(UC Santa Cruz)

On the higher rank Gross-Stark conjecture
In 1980, Gross stated a conjecture relating the leading term of the p-adic L-function of a ray class character of a totally real field at s=0 to a p-adic regulator of p-units in the field cut out by the character. In previous joint work with Darmon and Pollack, we proved this conjecture in the rank one case under certain assumptions; these assumptions were later removed by Ventullo. In this talk, we describe work in progress with Ventullo and Kakde on the higher rank case. In particular, we present a proof in the rank two setting under a certain assumption. As a corollary of our result, we obtain an unconditional proof of the conjecture when the ground field is real quadratic. (Show Abstract)

Nov 10

Florian Herzig
(UToronto)

On de Rham lifts of local Galois representations
It is an open problem to show that a given n-dimensional mod p local Galois representation \rho has a (regular) de Rham lift. We discuss several results concerning the existence of de Rham lifts of \rho of prescribed weights and types, assuming that \rho admits a nice lift to start with (for example, a Fontaine-Laffaille lift). Our arguments combine local and global methods. This is joint work with T. Gee, T. Liu, and D. Savitt. (Show Abstract)

Nov 17

Akshay Venkatesh
(Stanford)

Height of modular forms
I will define the notion of "height" of a (cohomological) automorphic form, for any group over a number field, and give some example of where this notion comes up naturally. I will then speculate (rather wildly) that it should be tightly related to the height of the associated motive (as recently defined by Kato). I have only a tiny bit of evidence for this speculation but if true it is rather surprising, because the two notions of height are quite different. (Show Abstract)

Nov 24

No Seminar

No Seminar
(Show Abstract)

Dec 1

Bjorn Poonen
(MIT)

Most odd degree hyperelliptic curves have only one rational point
We prove that the probability that a curve of the form y^2 = f(x) over Q with deg f = 2g+1 has no rational point other than the point at infinity tends to 1 as g tends to infinity. This is joint work with Michael Stoll. (Show Abstract)

One may also want to check out:

  • Geometric Langlands Seminar Monday and Thursday 4:30pm;
  • Algebraic Geometry Seminar biweekly on Wednesday 4:30pm-6pm;
  • Northwestern University Number Theory Seminar Monday 4pm;
  • UIC Number Theory Seminar Tuesday 1pm.

    This page is maintained by Brandon Levin; it was shamelessly copied from Davide Reduzzi page, which in turn was shamelessly copied from Liang Xiao's page, which was shamelessly copied from Kiran Kedlaya's page, which in turn was shamelessly copied from Jason Starr's page, which in turn was shamelessly copied from Ravi Vakil's page, which in turn was shamelessly copied from Pasha Belorousski's page at the University of Michigan. For more sites with a similar pedigree, see Michael Thaddeus's list or Jim Bryan's list.