University of Chicago Number Theory Seminar

Winter 2016: Tuesday 1:30-2:50pm, R358

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NOTE: the room for this quarter is R358.

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

Winter 2016 Schedule




January 5

Jack Shotton

Local deformation rings when l is not equal to p
Given a mod p representation of the absolute Galois group of Q_l , consider the universal framed deformation ring R parametrising its lifts. When l and p are distinct I will explain a relation between the mod p geometry of R and the mod p representation theory of GL_n(Z_l), that is parallel to the Breuil-Mézard conjecture in the l = p case. I will give examples and say something about the proof, which uses automorphy lifting techniques. ( Hide Abstract)

January 12

Jeff Achter
(Colorado State)

On descending cohomology geometrically
Mazur has drawn attention to the question of determining when the cohomology of a smooth, projective variety over a number field can be modeled by an abelian variety. I will discuss recent work with Casalaina-Martin and Vial which constructs such a "phantom" abelian variety for varieties with maximal geometric coniveau. In the special case of cohomology in degree three, we show that the image of the (complex) Abel-Jacobi map admits a distinguished model over the base field, and that an algebraic correspondence realizes this descended intermediate Jacobian as a phantom. (Hide Abstract)

January 19

Melanie Wood

The Cohen-Lenstra Heuristics and and Random Groups
We will introduce the Cohen-Lenstra Heuristics that conjecturally give the distribution of class groups of imaginary quadratic fields, and discuss features of this probability distribution on finite abelian groups motivating this conjecture. In particular, we will explain a new theorem that this distribution is ``universal'' (in the sense that the Central Limit Theorem shows that the normal distribution is universal). Further we will explain non-abelian analogs of these conjectures and the related random non-abelian groups, based on conjectures of Boston, Bush, and Hajir and joint work with Boston, including evidence for these non-abelian analogs in the function field case. (Hide Abstract)

January 26

Yunqing Tang

Algebraic solutions of differential equations over the projective line minus three points
The Grothendieck–Katz p-curvature conjecture predicts that an arithmetic differential equation whose reduction modulo p has vanishing p-curvatures for almost all p, has finite monodromy. It is known that it suffices to prove the conjecture for differential equations on the projective line minus three points. In this talk, I will first focus on this case and introduce a p-adic convergence condition, which would hold if the p-curvature is defined and vanishes. Using the algebraicity criteria established by Andr\'e, Bost, and Chambert-Loir, I will prove a variant of this conjecture for the projective line minus three points, which asserts that if the equation satisfies the above convergence condition for all p, then its monodromy is trivial. I will also prove a similar variant of the p-curvature conjecture for a certain elliptic curve with j-invariant 1728 minus its identity point. (Hide Abstract)

February 2

Bao V. Le Hung

Some computations with potentially crystalline deformation rings
We explain how to explicitly compute some potentially crystalline deformation ring for three dimensional local Galois representation, and explain some applications. This is joint work in progress with D.Le, B.Levin and S.Morra. (Hide Abstract)

February 9

Keerthi Madapusi Pera

On the average height of abelian varieties with CM
In the 90s, generalizing the classical Chowla-Selberg formula, P. Colmez formulated a conjectural formula for the Faltings heights of abelian varieties with multiplication by the ring of integers in a CM field, which expresses them in terms of logarithmic derivatives at 1 of certain Artin L-functions. Using ideas of Gross, he also proved his conjecture for abelian CM extensions. In this talk, I will explain a proof of Colmez's conjecture in the average for an arbitrary CM field. This is joint work with F. Andreatta, E. Goren and B. Howard. (Hide Abstract)

February 16

Yifeng Liu

Bad reduction of Hilbert modular varieties and application to Bloch-Kato conjecture
In this talk, we will study the reduction of Shimura varieties attached to certain quaternion algebras at some ramified prime. We explain how the global structure of the bad reduction is related to the level raising phenomenon for modular forms. As an application, we will use this to bound the Selmer groups of certain motives of high rank, providing new cases of the Bloch-Kato conjecture. (Hide Abstract)

February 23

Stefan Patrikis
(Univ. of Utah)

Deformations of Galois representations and exceptional monodromy
I will explain how to realize the exceptional algebraic groups as algebraic monodromy groups of geometric (in the sense of Fontaine-Mazur) Galois representations, in particular obtaining the first such examples in types F4 and E6. The argument relies on lifting well-chosen mod p representations to characteristic zero, using a generalization (to essentially any reductive group) of a technique developed by Ravi Ramakrishna for type A1. (Hide Abstract)

March 1

Simon Marshall

The asymptotic behaviour of periods of automorphic forms
If f is an automorphic form on a group G, the integral of f over the adelic points of a subgroup of G is known as a period of f. I will describe how period integrals can give us useful information about the cohomology or harmonic analysis of arithmetic manifolds, and present results on the asymptotics of certain periods that can be interpreted as upper and lower bounds for the sup norms of Maass forms. I will discuss the links between these results and topics such as theta lifting, special L-values, trace formulae, and the work of Sakellaridis and Venkatesh on the spectra of spherical varieties. Part of this is joint work with Farrell Brumley. (Hide Abstract)

March 8

Matthias Strauch

Arithmetic differential operators on the p-adic upper half plane and p-adic representations of GL(2)
The p-adic upper half plane comes equipped with a remarkable tower of GL(2)-equivariant etale covering spaces, as was shown by Drinfeld. It has been an open question for some time whether the spaces of global sections of the structure sheaf on such coverings provide admissible locally analytic representations. Using global methods and the p-adic Langlands correspondence for GL(2,Qp), this is now known to be the case by the work of Dospinescu and Le Bras. For the first layer of this tower Teitelbaum exhibited a nice formal model which we use to provide a local proof for the admissibility of the representation (when the base field is any finite extension of Qp). The other key ingredients are suitably defined sheaves of arithmetic differential operators and D-affinity results for formal models of the rigid analytic projective line, generalizing those of Christine Huyghe. This is joint work with Deepam Patel and Tobias Schmidt. (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.