Abstracts

Sept. 16
Title: Occult period mappings and their boundary divisors.
Speaker: Michael Rapoport (Bonn)
Abstract: We will discuss various examples of hidden period maps occuring in the literature with values in moduli spaces of abelian varieties of Picard type. The complements of the images of these maps can be described in terms of KM-divisors.

Sept. 23
Title: What the proof of the Fundamental Lemma gives us.
Speaker: James Arthur (Toronto)
Abstract: The recent proof of the Fundamental Lemms by Ngo Bau Chau is a breakthrough in the theory of automorphic forms. We shall try to motivate the origins of the problem in the trace formula and to say something of the history of its proof, and then to describe how its resolution opens the way for progress on several major fronts.

Sept. 30
Title: Exact averages of central values of triple product L-functions.
Speaker: Brooke Feigon (Toronto)
Abstract: In this talk, I will discuss exact formulas for averages of central L-values obtained using the relative trace formula and relations between periods and L-functions. I will focus on an example involving triple product L-functions.

Oct. 14
Title: Distribution algebras on a p-adic group and Lie algebra.
Speaker: Allen Moy (Hong Kong University of Science and Technology)
Abstract: An indispensable tool in the representation theory of reductive Lie groups is the universal enveloping algebra of the Lie algebra of a Lie group. For p-adic groups, the Hecke algebra is a partial analogue of the enveloping algebra. A much better analogue is the distribution algebra of left essentially compact distributions. Applied to a reductive p-adic group the center of the this distribution algebra is known as the Bernstein center. We consider the situation for a semisimple Lie algebra and describe a canonical family of G-invariant essentially compact distributions on the Lie algebra associated to the Killing form.

Oct. 21
Title: Quartic fields with large class numbers.
Speaker: Peter Cho (Toronto)
Assuming the Generalized Riemann Hypothesis and Artin conjecture for Artin L-functions, Duke showed the upper bound of class number of a totally real field of degree n whose normal closure has S_n, the symmetric group of n letters, as its Galois group. Again assuming the GRH and Artin conjecture, he constructed totally real number fields whose Galois closure is S_n with the class number having the same size as the upper bound up to constant. We prove, unconditionally, that there exists a totally real quartic field with arbitrary large discriminant whose normal closure has S_4 as its Galois group and its class number is the same size as Duke's upper bound up to constant. The main key ingredient is that the 3-dimensional representation of S_4 is the symmetric square of 2-dimensional representation of \widetilde{S_4}.

Oct. 28
Title: An arithmetic fundamental lemma for the unitary group in three variables.
Speaker: Wei Zhang (Harvard)
Abstract: In this talk I will present a relative trace formula approach to the Gross-Zagier formula and its high dimensional generalization (a derivative version of the global Gross-Prasad conjecture) for unitary Shimura variety. In particular, an arithmetic fundamental lemma (AFL) is proposed. Some results proved recently will be presented, including the AFL for the unitary group in three variables.

Nov. 4
Title: Sharp upper bounds on cubic character sums.
Speaker: Leo Goldmakher (Toronto)
Abstract: Because of their intimate connection to Dirichlet L-functions, character sums have long been a central subject in number theory. In this talk I will describe some classical results (dating back to 1918), sketch a new approach to the field devised by Granville and Soundararajan, and demonstrate some consequences of their ideas, including the result given in the title. The talk should be accessible to a fairly general mathematical audience, although familiarity with Dirichlet characters and L-functions will be helpful.

Nov. 18
Title: Abelian varieties isogenous to Jacobians.
Speaker: Ching-Li Chai (UPenn)
Abstract:   We explain a consequence of the Andre-Oort conjecture (recently shown to follow from GRH).  The statement is as follows. For a fixed integer g at least 4, there exists a constant C(g) such that no Jacobians of genus g has complex multiplication by a  CM field of degree 2g with discriminant at most C(g). (joint work with F. Oort)

Nov. 25
Title: Relatively supercuspidal representations of reductive p-adic groups.
Speaker: Fiona Murnaghan (Toronto)
Abstract: Let G be a reductive p-adic group and let H be the fixed points of an involution of G. The coset space G/H is a p-adic symmetric variety. The H-distinguished irreducible smooth representations of G are the ones that contribute to harmonic analysis on G/H. Among the H-distinguished representations, the relatively supercuspidal representations are the basic building blocks. We will describe some examples of relatively supercuspidal representations, and discuss work in progress concerning construction of relatively supercuspidal representations.