Joint IAS/PU Number Theory
In this talk, I will present a formulation of the Gross-Zagier formula over Shimura curves using automorphic representations with algebraic coefficients. It is a joint work with Shou-wu Zhang and Wei Zhang.
A. Ghosh and P. Sarnak have recently initiated the study of so-called real zeros of holomorphic Hecke cusp forms, that is zeros on certain geodesic segments on which the cusp form (or a multiple of it) takes real values. In the talk I'll first introduce the problem and outline their argument that many such zeros exist if many short intervals contain numbers whose all prime factors belong to a certain subset of primes. Then I'll speak about new results on this sieving problem which lead to improved lower bounds for the number of real zeros.
JOINT IAS/PU NUMBER THEORY SEMINAR
We give a survey of recent results on conjectures of Heath-Brown and Serre on the asymptotic density of rational points of bounded height. The main tool in the proofs is a new global determinant method inspired by the local real and p-adic determinant methods of Bombieri-Pila and Heath-Brown.
This is a report on some joint work with Mark Reeder and Jiu-Kang Yu. I will review the theory of parahoric subgroups and consider the induced representation of a one-dimensional character of the pro-unipotent radical. A surprising fact is that this induced representation can (in certain situations) have finite length. I will describe the parahorics and characters for which this occurs, and what the Langlands parameters of the corresponding irreducible summands must be.