We will try to have notes for all lectures in this program available on this webpage as soon as possible after the lecture has been given.
The lectures on September 23 and October 7 are in common with the Intercity Number Theory Seminar, after that the two seminars go on seperately.
The lectures will be accessible to algebraic geometers, (algebraic) number theorists and Lie group theorists who know about p-adic numbers.
Comments and suggestions on this sub-program are welcome, suggestions for good references especially. Those who are interested to give a lecture should contact Bas Edixhoven (edix@math.leidenuniv.nl).
Let K be a number field, and S a finite set of places of K. Let K_{S} be a maximal algebraic extension of K in which all places in S are completely split. Let X be a smooth irreducible quasi-projective scheme over K, such that for every v in S the set X(K_{v}) is non-empty. Then X(K_{S}) is Zariski dense in X.
Reference: see Moon and Taguchi in the references below. Or the very recent article by Moon and Ono.
Reference: Khare 2 in the references below.
In this lecture we will explain a generalization of this result, joint with Jason Starr, to the case when the base of the family is a surface.
In the other three lectures (October 14, November 11 and November 25) we will explain the idea behind the proof of the theorem, the analogy with Tsen's theorem, the connection with weak approximation and the connection with the period-index problem for Brauer groups.
References: Taylor 4 and T. Saito in the references below.
References: Taylor 4, Carayol, Bump and T. Saito in the references below.
Reference: Langlands in the references below.