## University of Chicago Number Theory Seminar

### Spring 2016: Tuesday 1:30-2:50pm, E206

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 E206.

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: Fall 2013 / Spring 2014 / Fall 2014 / Winter 2015 / Spring 2015 / Fall 2015 / Winter 2016 /

## Spring 2016 Schedule

Date

Speaker

Title

March 29

Henri Darmon
(McGill)

Generalised Kato classes and arithmetic applications
I will discuss some of the arithmetic applications of the eponymous cohomology classes of the title, notably, their relations with the so-called Stark-Heegner points" that are conjecturally defined over ring class fields of real quadratic fields. ( Hide Abstract)

April 5

Matthew Emerton
(UChicago)

Moduli stacks of local p-adic Galois representations
I will describe work in progress with Toby Gee, in which we construct moduli stacks parameterizing p-adic representations of the absolute Galois groups of p-adic local fields. (Hide Abstract)

April 14 (Thur)
Note special day!

Toby Gee
(Imperial)

Potentially automorphic components of local deformation rings
We show that there is a well-behaved notion of a "potentially automorphic component" of a potentially semi stable universal deformation ring of an n-dimensional mod p representation of the absolute Galois group of a finite extension of Q_p, whenever n is odd (or n=2) and p>2(n+1). This is joint work with Frank Calegari and Matthew Emerton. (Hide Abstract)

April 19

Kartik Prasanna
(UMichigan)

Hodge classes on products of quaternionic Shimura varieties
I will discuss the relation between Langlands functoriality and the theory of algebraic cycles in one of the simplest instances of functoriality, namely the Jacquet-Langlands correspondence for Hilbert modular forms. In this case, functoriality gives rise to a family of Tate classes on products of quaternionic Shimura varieties. The Tate conjecture predicts that these classes come from an algebraic cycle, which in turn should give rise to a Hodge class that is compatible with the Tate classes. While we cannot yet prove the Tate conjecture in this context, I will outline an unconditional proof of the existence of such a Hodge class. This is joint work (in progress) with A. Ichino. (Hide Abstract)

April 26

Preston Wake
(UCLA)

Level structures beyond the Drinfeld case
Drinfeld level structures are a key concept in the arithmetic study of the moduli of elliptic curves. They also play an important role in the moduli of 1 dimensional p-divisible groups, and related Shimura varieties studied by Harris and Taylor. I'll explain why Drinfeld level structures (and the related "full set of sections" defined by Katz and Mazur) are not adequate for studying more general Shimura varieties. I'll discuss two examples of a satisfying theory of level structure outside the Drinfeld case: i) full level structures on the group \mu_p x \mu_p; ii) Gamma_1(p^r)-type level structures on an arbitrary p-divisible group (joint work with R. Kottwitz). (Hide Abstract)

May 3

Jared Weinstein
(BU)

The absolute Galois group of Q_p as a geometric fundamental group
We construct an object defined over an algebraically closed field, whose fundamental group equals the absolute Galois group of Q_p. Formally, this object is a quotient of a perfectoid space, and is closely related to the "fundamental curve of p-adic Hodge theory" of Fargues and Fontaine. Time permitting, we will explain the connect between this object and the moduli spaces of Rapoport-Zink, which are linked to local Langlands. (Hide Abstract)

May 10

Hansheng Diao
(Princeton)

Log adic spaces and overconvergent modular forms
The main objects of the talk are adic spaces with logarithmic structures. We study their Kummer etale and Kummer pro-etale topologies. In particular, we show that log adic spaces are locally "perfectoid". As an application, we establish an oveconvergent Eichler-Shimura morphism connecting modular symbols and overconvergent modular forms. (Hide Abstract)

May 17

Ellen Eischen
(Oregon)

p-adic families of Eisenstein series and applications
I will discuss a construction of a p-adic family of Eisenstein series. I will also describe how it feeds into a program to construct p-adic L-functions associated to automorphic forms (in particular, on unitary groups). The latter part is joint with Michael Harris, Jian-Shu Li, and Chris Skinner. (Hide Abstract)

May 24

Gil Moss
(Oklahoma St.)

A local converse theorem and the local Langlands correspondence in families
In 2012 it was conjectured by Emerton and Helm that the local Langlands correspondence for GL(n) of a p-adic field should interpolate in \ell-adic families, where \ell is a prime different from p. Recently, Helm showed that the conjecture follows from the existence of an appropriate map from the integral Bernstein center to a Galois deformation ring. Such a map would connect moduli spaces of representations on either side of the correspondence. In this talk we will present recent work (joint with David Helm) showing the existence of such a map and describing its image. (Hide Abstract)

May 31

Brandon Levin
(UChicago)

Weight elimination in Serre-type conjectures
I will discuss recent results towards the weight part of Serre's conjecture for GL_n as formulated by Herzig. The conjecture predicts the set of weights where an odd n-dimensional mod p Galois representation should appear in cohomology (modular weights) in terms of the restriction to decomposition group at p. We show that the set of modular weights is always contained in the predicted set in generic situations. This is joint work with Daniel Le and Bao V. Le Hung. (Hide 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.