# Joint IAS/PU Number Theory

## The hyperbolic Ax-Lindemann conjecture

## Complex analytic vanishing cycles for formal schemes

## Genus of abstract modular curves with level \(\ell\) structure

## Independence of \(\ell\) and local terms

## \(G\)-valued flat deformations and local models

## The Landau-Siegel zero and spacing of zeros of L-functions

## Moduli of Representations and Pseudorepresentations

A continuous representation of a profinite group induces a continuous pseudorepresentation, where a pseudorepresentation is the data of the

characteristic polynomial coefficients. We discuss the geometry of the resulting map from the moduli formal groupoid of representations to the moduli formal scheme of pseudorepresentations.

## A Converse to a Theorem of Gross-Zaqier-Kolyvagin

The theorem of the title is that if the L-function L(E,s) of an elliptic curve E over the rationals vanishes to order r=0 or 1 at s=1 then the rank of the group of rational rational points of E equals r and the Tate-Shafarevich group of E is finite. This talk will describe an approach to the converse. The methods are mostly p-adic.

## An Analogue of the Ichino-Ikeda Conjecture for Whittaker Coefficients of the Metaplectic Group

A few years ago Ichino-Ikeda formulated a quantitative version of the Gross-Prasad conjecture, modeled after the classical work of Waldspurger. This is a powerful local-to-global principle which is very suitable for analytic and arithmetic applications. One can formulate a Whittaker analogue of the Ichino-Ikeda conjecture. We use the descent method of Ginzburg-Rallis-Soudry to reduce the Whittaker version to a purely local identity which we prove in the p-adic case under some mild hypotheses. Joint work with Zhengyu Mao