Number Theory Days

15h15 - 16h15

17h00 - 18h00

19h00

10h00 - 11h00

11h15 - 12h15

12h30 - 14h00

14h15 - 15h15

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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.

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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.

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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).

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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.

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On some problems of Linnik

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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.