University of Chicago Number Theory Seminar

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

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Spring 2018 Schedule




March 27th

Ellen Eischen

p-adic L-functions for unitary groups
I will discuss a construction of p-adic L-functions, especially in the setting of unitary groups. I will highlight how this construction relates to those of Serre, Katz, and Hida, and I will emphasize the role of properties of certain automorphic forms (analogous to the role played by modular forms in their work). This includes joint work with Michael Harris, Jian-Shu Li, and Christopher Skinner. ( Hide Abstract)

April 3rd

Ari Shnidman
(Boston College)

A higher order Gross-Kohnen-Zagier formula
I'll present a formula relating the intersection of two different Heegner-Drinfeld cycles to the r-th derivative of an automorphic L-function. This is a "higher-order" generalization of the Gross-Kohnen-Zagier formula, in the function field setting, and is inspired by recent work of Yun and W. Zhang. Our formula gives strong evidence that all Heegner-Drinfeld cycles are colinear in cohomology. This is joint work with Ben Howard. ( Hide Abstract)

April 10th

Jack Shotton

Ihara's lemma for Shimura curves via patching
Ihara's lemma is a statement about the structure of the mod l cohomology of modular curves that was a key ingredient in Ribet's results on level raising. It was generalised to Shimura curves over Q by Diamond and Taylor, by a totally different method. Under an assumption on the image of the residual Galois representation, we give another proof of Diamond and Taylor's result that extends to some new cases and that we hope will generalise to Shimura curves over totally real fields. This proof uses the Taylor--Wiles--Kisin `patching' method and the geometry of local deformation rings to reduce to the case of definite quaternion algebras. This is joint work in progress with Jeffrey Manning. ( Hide Abstract)

April 17th

Anthony Várilly-Alvarado

Vojta's conjecture and uniform boundedness of full-level structures on abelian varieties over number fields
In 1977, Mazur proved that the torsion subgroup of an elliptic curve over Q is, up to isomorphism, one of only 15 groups. Before Merel gave a qualitative generalization of this result to arbitrary number fields, it was known that variants of the abc conjecture would imply uniform boundedness of torsion on elliptic curves over number fields of bounded degree. In this talk, I will explain how, using Vojta's conjecture as a higher-dimensional generalization of the abc conjecture, one can deduce similar uniform boundedness statements for full-level structures on abelian varieties of fixed dimension over number fields. This is joint work with Dan Abramovich and Keerthi Madapusi-Pera. ( Hide Abstract)

April 24th

Ilya Khayutin

Joint equidistribution of CM points
A celebrated theorem of Duke states that Picard/Galois orbits of CM points on a complex modular curve equidistribute in the limit when the absolute value of the discriminant goes to infinity. The equidistribution of Picard and Galois orbits of special points in products of modular curves was conjectured by Michel and Venkatesh and as part of the equidistribution strengthening of the André-Oort conjecture. I will discuss the proof of a recent theorem making progress towards this conjecture. Currently, this problem does not seem to be amenable to methods of automorphic forms even assuming GRH. Nevertheless, assuming a splitting condition at two primes the joining rigidity theorem of Einsiedler and Lindenstrauss applies. As a result the obstacle to proving equidistribution is the potential concentration of mass on graphs of Hecke correspondences and translates thereof. I will present a method to discard this possibility using a geometric expansion of a relative trace, description of the relative orbital integrals in terms of integral ideals and a sieve argument. ( Hide Abstract)

May 1st

Kannan Soundararajan

Weak subconvexity for automorphic L-functions
I will discuss recent work with Jesse Thorner which establishes a weak subconvexity for central values of automorphic L-functions. I will explain what the subconvexity problem is, why we care (in some situations), and give an overview of the ideas behind our recent work. ( Hide Abstract)

May 8th

Kai-Wen Lan

Nearby cycles of automorphic étale sheaves
I will explain that, in most cases where integral models are available in the literature, the automorphic étale cohomology of a (possibly noncompact) Shimura variety in characteristic zero is canonically isomorphic to the cohomology of the associated (l-adic) nearby cycles in positive characteristics. If time permits, I will also talk about some applications or related results. (This is joint work with Stroh.) ( Hide Abstract)

May 15th

Robin Bartlett
(King's College London)

The reduction modulo p of crystalline representations of small weight
I will discuss an extension to GLn of a result of Gee--Liu--Savitt on the reduction modulo p of crystalline representations with Hodge--Tate weights in the interval [0,p]. The methods are purely local, using tools from p-adic Hodge theory. I will explain how these results are related to the weight part of Serre's conjecture beyond dimension 2. ( Hide Abstract)

May 22nd

Hiroki Kato

Wild ramification and restrictions to curves
I will discuss the characteristic cycle of an l-adic sheaf defined by Saito, which knows the Euler characteristic, Milnor formula for isolated singularities, and many other invariants attached to the l-adic sheaf. Saito and Yatagawa proved that it is determined by wild ramification. I will explain it is determined by the conductors of the restrictions to curves. ( Hide Abstract)

May 29th

Martin Weissman
(UC Santa Cruz)

L-groups and covering groups
To extend the Langlands program to covering groups, such as the metaplectic group, one needs to construct a dual group G^\vee (a connected reductive group) and an L-group (an extension of the absolute Galois group, or Weil group, by G^\vee). In this talk I will introduce covering groups in a framework suggested by Brylinski and Deligne, and describe the construction of the dual group and L-group for such covering groups. To finish the talk, I will describe the current state of the art -- the evidence that this L-group is the "correct" choice, and some open questions. ( Hide Abstract)

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.