Sporadic Number Theory Seminars

In this talk we first review Poisson summation formula on GL(n) and its application to global functional equation in Godement-Jacquet integrals. We then discuss potential applications of such summation formula to derive global functional equations in other integral representations of L-functions.

In this talk, we will introduce a reciprocal problem of the Gan-Gross-Prasad conjectures, and explain an approach using the twisted descent method. In particular, we will give both local and global examples in the case of special orthogonal groups.

Automorphic forms are well-behaved functions on a group (such as GL(n,R)) which are invariant under the action of a discrete arithmetic subgroup (such as SL(n,Z)). They encode vital arithmetic information. Voronoi formulas are Poisson-style summation formulas for automorphic forms. They have been powerful tools in analytic number theory for a long time (with applications to the divisor problem, the circle problem, subconvexity of L-functions, etc). I will talk about Voronoi formulas and their recent development and applications. Some results are joint work with Stephen D Miller, Eren M Kiral.

This talk is based on my Thesis. By constructing (non-canonical) deformations for the associated p-divisible groups of the special fibre S of a Shimura variety, we manage to construct a morphism from S to some quotient sheaf of the loop group associated with S. We show that the geometric fibre of this morphism gives back the Ekedahl-Oort strata of S: this also gives a conceptual interpretation of Viehmann's new invariant "elements of truncation of level one". I will recall the Ekedahl-Oort stratification of Shimura varieties (of good reduction) and the classification of p-divisible groups in terms of filtered Breuil-Kisin modules (or Breuil-Kisin windows, in my term), and then present the strategy of constructing the morphism aforementioned.

We will first discuss a local trace formula for the Ginzburg-Rallis model. This trace formula allows us to prove a multiplicity formula for the Ginzburg-Rallis model, which implies the multiplicity one theorem on the Vogan L-packet. Then we will talk about some generalizations of this trace formula to other models.