? University of Chicago Number Theory Seminar

University of Chicago Number Theory Seminar

Winter 2015: Tuesday 1:30-2:50pm, E308


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 Davide Reduzzi.

NOTE: the room for this quarter is E308.

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 Schedule

Date

Speaker

Title

January 6

Chao Li
(Harvard)

Level raising mod 2 and arbitrary 2-Selmer ranks
We prove a level raising mod p=2 theorem for elliptic curves over Q, generalizing theorems of Ribet and Diamond-Taylor. As an application, we show that the 2-Selmer rank can be arbitrary in level raising families. We will begin by explaining our motivation from W. Zhang's approach to the p-part of the BSD conjecture. Explicit examples will be given to illustrate different phenomena compared to odd p. This is joint work with Bao V. Le Hung. ( Hide Abstract)

January 13

David Zywina
(Cornell)

Elliptic surfaces and the Inverse Galois Problem
By studying the Galois action on etale cohomology groups arising from elliptic surfaces, we will prove several new cases of the Inverse Galois Problem. In particular, we will explain why each of the simple groups PSp_4(F_p) occur as the Galois group of some Galois extension of Q. The key ingredients will be a big monodromy result along with some known cases of the Birch and Swinnerton-Dyer conjecture. (Hide Abstract)

January 20

Takashi Suzuki
(UChicago)

Grothendieck's pairing on Neron component group
The component group of the special fiber of the Neron model of an abelian variety is called the Neron component group. In SGA7, Grothendieck constructed a canonical pairing between the Neron component groups of an abelian variety and its dual over a local field with perfect residue field. He conjectured that it is perfect. In this talk, we prove his conjecture. A key tool is the category of fields viewed as a Grothendieck site. The cohomology of an abelian variety can be regarded as a sheaf on this new site. This allows us to treat Neron models completely functorial in the derived category of sheaves. From the known case of semistable abelian varieties, we deduce the perfectness in full generality. Time permitting, we also discuss a global function field version of this duality, generalizing the Cassels-Tate pairing to the case that the base field is perfect. (Hide Abstract)

January 27

Matthew Emerton
(UChicago)

Moduli spaces of local Galois representations
I will describe work in progress, joint with Toby Gee. Our goal is to construct, for a finite extension K of Q_p and a natural number n, to construct a formal Artin stack over the formal spectrum of Z_p which parameterizes families of n-dimensional representation of the absolute Galois group G_K. The underlying reduced closed subscheme of our formal scheme will be an equidimensional Artin stack over F_p whose F_p-bar points naturally correspond to representations G_K ---> GL_n(F_p-bar). My aim in the talk is to motivate our construction, to sketch some of the ideas underlying the construction, and also to explain what points remain to be worked out in order to complete the construction. If time permits, I will also explain some applications of the construction that we have in mind. (Hide Abstract)

February 3

Simion Filip
(UChicago)

Hodge theory and arithmetic in Teichmuller dynamics
The dynamics of a billiard ball in a polygon is a classical dynamical system for which many questions remain open. These questions are intimately related to a natural action of the group SL(2,R) on the tangent bundle to the moduli space of Riemann surfaces. It can be viewed as a "complexified" geodesic flow. By recent results of Eskin and Mirzakhani, this action of SL(2,R) enjoys rigidity properties akin to Ratner's theorems - in particular, orbit closures are submanifolds. In this talk, I will explain that these orbit closures are in fact algebraic varieties with interesting arithmetic properties. For instance, they parametrize algebraic curves with real multiplication and torsion conditions on (factors of) their Jacobian. These results depend on extending results about variations of Hodge structures to this special setting (in particular, giving a different approach to some results of Schmid). (Hide Abstract)

February 10

Brandon Levin
(UChicago)

Moduli of finite flat groups schemes with descent
In Kisin’s work on modularity lifting, he resolves flat deformation rings by moduli spaces of finite flat group schemes. The geometry of this resolution can be related to local models of Shimura varieties. I will discuss a generalization of this story to potentially flat Galois deformation rings. Time permitting I will also discuss how this relates to the moduli spaces of local Galois representations constructed by Matt Emerton and Toby Gee. This is joint work with Ana Caraiani. (Hide Abstract)

February 17

Keerthi Madapusi Pera
(UChicago)

Heights of special divisors on orthogonal Shimura varieties
The Gross-Zagier formula relates two complex numbers obtained in seemingly very disparate ways: The Neron-Tate height pairing between Heegner points on elliptic curves, and the central derivative of a certain automorphic L-function of Rankin type. I will explain a variant of this in higher dimensions. On the geometric side, the intersection theory will now take place on Shimura varieties associated with orthogonal groups. On the analytic side, we will find Rankin-Selberg L-functions involving modular forms of half-integral weight. This is joint work with Fabrizio Andreatta, Eyal Goren and Ben Howard. (Hide Abstract)

February 24

Kai-Wen Lan
(UMN)

Compactifications of PEL-type Shimura varieties in ramified characteristics
I will report on what we know about the compactifications of PEL-type Shimura varieties in ramified characteristics, allowing arbitrary levels, when they are simply defined by taking normalizations in certain auxiliary good reduction integral models. While we cannot expect them to be smooth, I will explain that they still enjoy many nice properties comparable to the good reduction cases. If time permits, I will also talk about a construction for the splitting models introduced by Pappas and Rapoport. (Hide Abstract)

March 3

Mohammad Hadi Hedayatzadeh
( Purdue University )

Exterior powers of Lubin-Tate groups
After defining exterior powers of p-divisible groups, we prove that the exterior powers of p-divisible groups of dimension at most one over any base scheme exist and their construction commute with arbitrary base change. If time permits, we will prove the similar results for Lubin-Tate groups in mixed and equal characteristic cases. (Hide Abstract)

March 10

Vaidehee Thatte
(UChicago)

Ramification Theory for Arbitrary Valuation Rings: Equal Characteristic Case
In the classical ramification theory we consider complete discrete valued field extensions with perfect residue fields and separable residue extensions. It is interesting to see if we can have ramification theory for arbitrary valuations, that is compatible with the classical theory. A lot of work has been done for complete discrete valued fields with arbitrary residue fields. In this talk, we will discuss Artin-Schreier extensions of arbitrary valued fields (with arbitrary residue fields, possibly with non-discrete value groups). Special difficulties arise in the defect case (i.e., a ramified extension with no extension of the residue field and no extension of the value group). Complete discrete valued fields are defectless, but the defect occurs often when we study arbitrary valuations.We will also introduce a generalization and further refinement of Kato's refined Swan conductor. (Hide Abstract)

March 17

Michael Rapoport
(Univ. of Bonn)

On an Arithmetic Transfer conjecture for formal moduli spaces
In the relative trace formula approach to the arithmetic Gan-Gross-Prasad conjecture, there is a Arithmetic Transfer analogue of the Arithmetic Fundamental Lemma conjecture of Wei Zhang. In joint work with B. Smithling and W. Zhang we prove this conjecture in the case of a unitary group in three variables. (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.