University of Chicago Number Theory Seminar

Winter 2018: Tuesdays 2:00-3:20pm, Eckhart 308

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Click here to see the schedule of previous quarters: Spring 2013 / Autumn 2013 / Spring 2014 / Autumn 2014 / Winter 2015 / Spring 2015 / Autumn 2015 / Winter 2016 / Spring 2016 / Autumn 2016/ Winter 2017/ Spring 2017/ Autumn 2017

Winter 2018 Schedule




January 9th

Shrenik Shah

Class number formulae for some Shimura varieties of low dimension
The class number formula connects the residue of the Dedekind zeta function at s=1 to the regulator, which measures the covolume of the lattice generated by logarithms of units. Beilinson defined a generalized regulator morphism and conjectural class number formula in the “motivic” setting. His formula provides arithmetic meaning to the orders of “trivial” zeroes of L-functions at integer points as well as the value of the first nonzero derivative at these points. We study this conjecture for the middle degree cohomology of the Shimura varieties associated to unitary groups of signature (2,1) and (2,2) over Q. We construct explicit “Beilinson-Flach elements” in the motivic cohomology of these varieties and compute their regulator. This is joint work with Aaron Pollack. ( Hide Abstract)

January 16th

No seminar

January 23rd

Sean Howe

Classicality, Mayer-Vietoris, and deRham local-global compatibilty
Coleman's classicality theorem says that an overconvergent modular form of weight k is classical if it has slope <k-1. We explain how to recover this theorem using the admissible representation theory of GL_2 and a Mayer-Vietoris sequence for the de Rham cohomology of the tower of modular curves which separates the contributions of the ordinary and supersingular loci. Time permitting, we will also explain connections with work of Breuil-Emerton and Saito and make a bold conjecture about the de Rham cohomology of the Lubin-Tate tower. ( Hide Abstract)

January 30th

Liang Xiao

Cycles on the special fiber of some Shimura varieties and Tate conjecture
We describe the irreducible components of the basic locus of Shimura varieties of Hodge type at a place with good reduction, when the basic locus is of middle dimension. Under certain genericity condition, we show that they generate the Tate classes of the special fiber of the Shimura varieties. This is a joint work with Xinwen Zhu. ( Hide Abstract)

February 6th

Lue Pan

Fontaine–Mazur conjecture in the residually reducible case.
We prove the modularity of some two-dimensional residually reducible p-adic Galois representations over Q under certain hypothesis on the residual representation at p. To do this, we generalize Emerton’s local-global compatibility and devise a patching argument for completed homology in this setting. ( Hide Abstract)

February 13th

Lucia Mocz

A new Northcott property for Faltings heights
The Faltings height is a useful invariant for addressing questions in arithmetic geometry. In his celebrated proof of the Mordell and Shafarevich conjectures, Faltings shows the Faltings height satisfies a certain Northcott property, which allows him to deduce his finiteness statements. In this work we prove a new Northcott property for the Faltings height. Namely we show, assuming the Colmez Conjecture and the Artin Conjecture, that there are finitely many CM abelian varieties of a fixed dimension which have bounded Faltings height. The technique developed uses new tools from integral p-adic Hodge theory to study the variation of Faltings height within an isogeny class of CM abelian varieties. In special cases, we are able to use these techniques to moreover develop new Colmez-type formulas for the Faltings height. ( Hide Abstract)

February 20th

Karl Schaefer

Unramified p-Extensions of Kummer Fields via Galois Representations
In their paper "On the Ramification of Hecke Algebras at Eisenstein Primes", Calegari and Emerton prove a relationship between Merel's number and the rank of the p-part of the class group of Q(N^1/p) for primes N = 1 mod p. We answer a question of Calegari-Emerton on this rank and give an effective method for computing this rank in the case p = 5. Our results are obtained by relating the existence of Galois representations satisfying certain local conditions to generalizations of Merel's number. This is joint work with Eric Stubley. ( Hide Abstract)

February 27th

Brandon Levin

Models for Galois deformation rings
An important input into modularity lifting theorems is an understanding of the geometry of Galois deformation rings, especially local deformation rings with p-adic Hodge theory conditions at l = p. Outside of a few cases (ordinary, Fontaine-Laffaille,...), the detailed structure of these deformation rings is very mysterious. I will introduce models which have the same singularities in generic situations as those of a class of potentially crystalline deformation rings. I will then discuss applications to the Breuil-Mézard conjecture and the weight part of Serre's conjecture. This is joint work in progress with Daniel Le, Bao V. Le Hung, and Stefano Morra. ( Hide Abstract)

March 6th


One may also want to check out:

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

    This page is maintained by Jack Shotton; it was shamelessly copied from Brandon Levin's page, which in turn was shamelessly copied from Davide Reduzzi page, which 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.