# Number Theory Days

## 6th April - 7th April 2006

The Number Theory Days are organised for the third time by Eva Bayer Fluckiger (EPFL) and Richard Pink (ETHZ) , and will be held this year in Zurich.

## Schedule

The talk on Thursday will take place in ... to be announced. On Friday the lectures will take place in room HG G43 (Hermann-Weyl Room).

### Thursday 6th April

 15h15 - 16h15 On rational curves on affine subsets of ${\bf P}_2$ and conjectures of Vojta 17h00 - 18h00 On the André-Oort conjecture 19h00 Dinner

### Friday 7th April

 10h00 - 11h00 Introduction to p-adic Langlands programm 11h15 - 12h15 Henri Darmon On the ranks of Mordell-Weil groups over towers of Kummer exensions 12h30 - 14h00 Lunch break 14h15 - 15h15 On some problems of Linnik

## Travel information

How to reach the ETH Zurich.

## Contacts

Christina Buchmann (Secretary):

## Abstracts

Speaker:
Umberto Zannier (Scuola Normale Superiore di Pisa)
Title:
On rational curves on affine subsets of {\bf P}_2 and conjectures of Vojta

Abstract:

### In recent joint work with P. Corvaja we investigate the regular morphism f: P_1\setminus S\to P_2\setminus D, where S is a finite subset and D is the sum of two lines and a conic. This case is special but significant, since it lies at the boundary of what is known concerning a conjecture of Vojta for integral points on affine subsets of P_2. Our results go in several directions. On the one hand, we give a bound for the degree of the image curve, proving Vojta's conjecture for function fields in this special case. On the other hand we sharpen the S-unit theorem in function fields, for sums of four S-units, going towards another conjecture of Vojta. In concrete terms, we can completely classify for instance the solutions of y^2=1+u+v where ,y,u,v are regular functions on a given curve, and u,v have zeros/poles in a prescribed finite set S. The methods rely heavily on a bound for the degree of \gcd(1-u,1-v), for S-units u,v in a function field; this sharpens a previous result of us in the arithmetic case.

Speaker:
Andrei Yafaev (University College London)
Title:
On the André-Oort conjecture

Abstract:

This is a joint work with Bruno Klingler.
The Andre-Oort conjecture predicts that irreducible components of the Zariski closure of a set of special points in a Shimura variety is a special subvariety.
We will present a proof of this conjecture under the assumption of the generalised Riemann Hypothesis.

Speaker:
Ariane Mézard (École normale supérieure)
Title:

Abstract:

We present an introduction to the p-adic Langlands program, which was intiated by C. Breuil. We expect that it will be a correspondence between certain p-adic Galois representations of Gal(\overline{{\bf Q}}_p/{\bf Q}_p) in dimension 2 and certain p-adic representations of Gl2({\bf Q}_p).

Speaker:
Henri Darmon (McGill Mathematics and Statistics)
Title:
On the ranks of Mordell-Weil groups over towers of Kummer exensions

Abstract:

Let E be an elliptic curve over the rationals. I will discuss some results inspired by a question of Coates concerning the growth of the rank of E over certain towers of Kummer extensions of the form {\bf Q}(\zeta_{p^n}, q^{1/p^n}). As a numerical illustration of this result in a special case, the rank of X_0(11)  over {\bf Q}(11^{1/3^n}) is shown to be equal to n. This is part of a joint work, in progress, with Ye Tian.

Speaker:
Philippe Michel (Institut de Mathématiques et de Modélisation de Montpellier)
Title:

On some problems of Linnik

Abstract:

The problems of Linnik alluded to in the title ask for the distribution properties of the set of representations of a large integer by various ternary quadratic forms. By now, these problems can be approached by various methods: via harmonic analysis, modular forms and L-functions or, as Linnik did originally, via ergodic theory. In this talk, we survey some recent developments and generalizations of Linnik’s problems which follow either of the above approaches or which combine them. These are joint works with Elon Lindenstrauss, Manfred Einsiedler and Akshay Venkatesh.