## Local-Global Compatibility in the p-Adic Langlands Program for GL(2) over Q

Matthew Emerton
Northwestern University
November 2, 2010

I will outline the proof of various cases of the local-global compatibility statement alluded to in the title, and also explain its applications to the Fontaine—Mazur conjecture, and to a conjecture of Kisin.

## A Semistable Model for the Tower of Modular Cures

Jared Weinstein
The usual Katz-Mazur model for the modular curve $X(p^n)$ has horribly singular reduction. For large n there isn't any model of $X(p^n)$ which has good reduction, but after extending the base one can at least find a semistable model, which means that the special fiber only has normal crossings as singularities. We will reveal a new picture of the special fiber of a semistable model of the entire tower of modular curves. We will also indicate why this problem is important from the point of view of the local Langlands correspondence for $GL(2)$ .