## UW-PIMS Mathematics Colloquium: Bianca Viray

- Date: 01/13/2014
- Time: 14:30

*Brown University*

University of Washington

The local to global principle for rational points

Let X be a connected smooth projective variety over Q. If X has a

Q point, then X must have local points, i.e. points over the reals and

over the p-adic completions Q_p. However, local solubility is often not

sufficient. Manin showed that quadratic reciprocity together with higher

reciprocity laws can obstruct the existence of a Q point (a global

point) even when there exist local points. We will give an overview of

this obstruction (in the case of quadratic reciprocity) and then show

that for certain surfaces, this reciprocity obstruction can be viewed in

a geometric manner. More precisely, we will show that for degree 4 del

Pezzo surfaces, Manin's obstruction to the existence of a rational point

is equivalent to the surface being fibered into genus 1 curves, each of

which fail to be locally solvable. This talk will be suitable for a

general audience.

Location: Thomson Hall 119