Joint IAS/PU Number Theory

A Converse to a Theorem of Gross-Zaqier-Kolyvagin

Christopher Skinner
Princeton University; Member, School of Mathematics
April 4, 2013

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

Erez Lapid
Hebrew University of Jerusalem and Weizmann Institute of Science
March 14, 2013

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

Abelian varieties with maximal Galois action on their torsion points

David Zywina
January 24, 2013

Abstract:
Associated to an abelian variety A/K is a Galois representation which describes the action of the absolute Galois group of K on the torsion points of A. In this talk, we shall describe how large the image of this representation can be (in terms of a number field K and the dimension of A). We achieve this by considering abelian varieties in families and then using a special variant of Hilbert's irreducibility theorem. Some results of Serre on the mod ell Galois image will also be reviewed. (This is joint work with David Zureick-Brown)

Local Global Principles for Galois Cohomology

Julia Hartmann
RWTH Aachen University; Member, School of Mathematics, IAS
December 13, 2012

We consider Galois cohomology groups over function fields F of curves that are defined over a complete discretely valued field.
Motivated by work of Kato and others for n=3, we show that local-global principles hold for
$H^n(F, Z/mZ(n-1))$ for all n>1.
In the case n=1, a local-global principle need not hold. Instead, we will see that the obstruction to a local-global principle for $H^1(F,G)$, a Tate-Shafarevich set, can be described explicitly for many (not necessarily abelian) linear algebraic groups G.

Open-Closed Gromov-Witten Invariants of Toric Calabi-Yau 3-Orbifolds

Chiu-Chu Melissa Liu
Columbia University
December 7, 2012

We study open-closed orbifold Gromov-Witten invariants of toric Calabi-Yau 3-orbifolds with respect to Lagrangian branes of Aganagic-Vafa type. We prove an open mirror theorem which expresses generating functions of orbifold disk invariants in terms of Abel-Jacobi maps of the mirror curves. This is a joint work with Bohan Fang and Hsian-Hua Tseng.