Rational Points on Varieties
The question of whether a given variety has a rational point is one mathematicians have studied for centuries and remains to be a key aspect of number theory. While modern results such as the Mordell-Weil theorem and Falting’s theorem provide great insight to the structure of rational points of curves, to determine if a variety has a rational point still remains a difficult problem. We will study methods to showing such a variety has no rational point and how these can be realised through cohomology of the variety. We will closely follow Bjorn Poonen’s ‘Rational Points on Varieties’. A first course in algebraic number theory and algebraic geometry (including scheme theory) will be expected but the necessary cohomology theories (group, etale and fppf) will be developed in the course.
The course will be weekly on Wednesdays 1pm-3pm in
- ARC 5125 in Glasgow
- 2.06 Appleton Tower in Edinburgh.
Organizer: Ruth Raistrick
The following is a provisional schedule. If stuck to we plan to cover further topics, in particular del Pezzo surfaces.
Schedule:
- 21/01 – Week 1 (Ruth Raistrick): Overview and algebraic background.
- 28/01 – Week 2 (Mia Lam): Geometric background I.
- 04/02 – Week 3: Geometric background II.
- 11/02 – Week 4: Faithfully flat descent.
- 18/02 – Week 5: Etale and fppf cohomology.
- 25/02 – Week 6: Brauer groups of varieties.
- 11/03 – Week 7: Functorial obstructions and Brauer-Manin obstruction.
- 17/03 – Week 8: Descent.