? University of Chicago Number Theory Seminar

University of Chicago Number Theory Seminar

Spring 2015: Tuesday 1:30-2:50pm, E203

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 E203.

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 2012 / Winter 2013 / Spring 2013 / Fall 2013 / Spring 2014 Fall 2014 / Winter 2015

Spring 2015 Schedule




March 31

Xin Wan

Iwasawa main conjecture for supersingular elliptic curves
We prove the +- main conjecture formulated by Kobayashi, for supersingular elliptic curves with a_p=0. (Show Abstract)

April 7

John Bergdall

Arithmetic properties of Fredholm series
The slopes of modular forms are encoded in the slopes of the Newton polygon of the Fredholm determinant of Up acting on spaces of overconvergent p-adic modular forms. In this talk I will discuss an old technique, due to Koike, of computing this characteristic power series, and give new results regarding the mod p reduction of this series. This is joint work with Rob Pollack. (Show Abstract)

April 14

Francesc Castella

p-adic heights of Heegner points and Beilinson-Flach elements
About 10 years ago, Ben Howard proved a Lambda-adic Gross-Zagier formula relating the p-adic heights of Heegner points over ring class fields of p-power conductor to the derivative of a two-variable p-adic L-function. In this talk, we will explain a strategy for extending Howard’s theorem to higher weights. Rather than on calculations inspired by the original work of Gross and Zagier, our approach is via Iwasawa theory, based on the connection between Heegner points and Beilinson-Flach elements, and their variation in p-adic families. (Show Abstract)

April 21

Martin Luu

Numerical local Langlands duality and Weil’s Rosetta Stone
The analogy between number fields, curves over finite fields, and Riemann surfaces has a long and fruitful history, in particular with respect to the various Langlands dualities. More recently, through the work of Kapustin and Witten, quantum physics has been added as a fourth pillar to the story. In this talk I will describe some local aspects of these analogies. In particular, I will explain how the numerical local Langlands duality and the T-duality of 2D quantum gravity can be derived from the same symmetry principle of local Langlands parameters. (Show Abstract)

April 28

Junecue Suh
(UC-Santa Cruz)

New vanishing theorems for mixed Hodge modules and applications
We'll review various vanishing theorems (of Kodaira, Nakano, Kawamata, Viehweg, Esnault, Illusie, ...) and then present new vanishing theorems, with coefficients in mixed Hodge modules. If time permits, we'll mention applications to the cohomology of Shimura varieties. (Show Abstract)

April 30 (Thurs)
Note special day!
(1:30-3PM in E203)

Nike Vatsal

Congruences for modular forms of half-integer weight
Suppose F and G are holomorphic cuspidal newsforms of even weight and trivial characters of levels M and N respectively, such that F and G are congruent modulo a prime P in the algebraic closure of Q. We can then pose the question of whether or not the modular forms associated to F and G by the Shimura-Waldspurger correspondence are also congruent modulo P. In considering this question, one quickly realizes that the in the most naive form the answer to this question is negative, but the reason for the failure turns out to be quite subtle. One is faced with the obvious fact that there’s no evident way to single out a specific form on the metaplectic group that corresponds to F or G, but a more subtle issue is that the usual Shimura-Waldspurger correspondence does not even yield a canonical bijection on the level of automorphic representations. In attempting to formulate a statement that might conceviably be true, one has to consider in some detail the structure of the Waldspurger packets on the metaplectic group, and the existence of a congruence on the metaplectic side is related to a hypothetical multiplicity one theorem for metaplectic modular forms in positive characteristic. Informal speculations along these lines were first made some years ago by K. Prasanna, and we will attempt to make some of his speculations more precise and state an actual conjecture. (Show Abstract)

May 5

Bianca Viray

Obstructions to the Hasse principle on degree 4 del Pezzo surfaces
In 1970, Manin showed that the Brauer group can obstruct the existence of rational points. Colliot-Thélène and Sansuc have conjectured that this obstruction completely explains the failure of rational points on del Pezzo surfaces. We show that on degree 4 del Pezzo surfaces, this Brauer-Manin obstruction manifests itself through linear projections. As a consequence of the proof, we obtain a simple and efficient for computing the Brauer classes of a degree 4 del Pezzo surface. This is joint work with Anthony Várilly-Alvarado. (Show Abstract)

May 12

Rebecca Bellovin

Local epsilon-isomorphisms in families
Given a representation of Gal_{Q_p} with coefficients in a p-adically complete local ring R, Fukaya and Kato have conjectured the existence of a canonical trivialization of the determinant of a certain cohomology complex. When R=Z_p and the representation is a lattice in a de Rham representation, this trivialization should be related to the \varepsilon-factor of the corresponding Weil--Deligne representation. Such a trivialization has been constructed for certain crystalline Galois representations, by the work of a number of authors. I will explain how to extend these trivializations to certain families of crystalline Galois representations. This is joint work with Otmar Venjakob. (Show Abstract)

May 19

Liang Xiao

Eigencurve over the boundary of the weight space
Eigencurve was introduced by Coleman and Mazur to parametrize modular forms varying padically. It is a rigid analytic curve such that each point corresponds to an overconvegent eigenform. In this talk, we discuss a conjecture on the geometry of the eigencurve: over the boundary annuli of the weight space, the eigencurve breaks up into infinite disjoint union of connected components and the weight map is finite and flat on each component. This was first verified by Buzzard and Kilford by an explicit computation in the case of p = 2 and tame level 1. We will explain a generalization to the definite quaternion case with no restriction on p (except p > 2) or the tame level. This is a joint work with Ruochuan Liu and Daqing Wan, based on an idea of Robert Coleman. (Show Abstract)

May 26

No Seminar

No Seminar
(Show Abstract)

June 2

Ariane Mézard

Genetic of local p-adic Galois representations
In this talk, we define a combinatorial data, said the gene, associated to a modulo $p$ Galois representation $\bar{\rho}$ and a Galois type. We prove that this gene encodes explicit information on geometric deformations of $\bar{\rho}$. First, the gene provides an explicit easy instant description of Kisin variety parametrizing Breuil-Kisin modules associated to potentially Barsotti-Tate deformations of modulo $p$ Galois representations of dimension 2. Then we explain how we may deduce the associated deformation ring in non generic cases. This is joint work with Xavier Caruso and Agnes David. (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.