Arithmetic geometry: rational points
TCC course, Autumn 2013
notes of the course
This course is about rational points on surfaces and higher-dimensional algebraic varieties over number fields, which is an active topic of research in the last thirty years. Let us call a collection of local points on a projective variety, one over each completion of the ground number field, an adelic point. The main idea (due to Manin) is that class field theory gives general conditions on adelic points which are satisfied by all rational points. These so called Brauer–Manin conditions come from elements of the Brauer group of the variety, as defined by Grothendieck. The method of descent, well known in the case of elliptic curves, can be applied to conic bundles (families of conics parameterised by the projective line). In certain cases one can prove that any adelic point satisfying the Brauer–Manin conditions can be approximated by a rational point. One of the first achievements of this theory and the main result of the course is a theorem of Colliot-Thélène–Sansuc–Swinnerton-Dyer on rational points on Châtelet surfaces, a particular type of conic bundles. This method also allows one to deduce results on rational points on general conic bundles over Q with split discriminant from recent spectacular work of Green, Tao and Ziegler in additive combinatorics. I will try to give proofs that are simplified compared to the existing literature, and I will try to make all constructions as explicit as possible.
Background: good knowledge of algebra and algebraic number theory, some familiarity with algebraic geometry.
Recommended books: part 1 of Serre “A course in arithmetic”, the beginning of Gille and Szamuely “Central simple algebras and Galois cohomology”, chapter 7 of my book “Torsors and rational points”.
Further reading: Serre “Local fields” and “Galois cohomology”, Cassels and Fröhlich “Algebraic number theory”.
For background in geometry, see Shafarevich “Basic algebraic geometry”. For a short introduction see lecture notes of my old course in Algebraic Geometry.