Cohomological methods in arithmetic geometry.

Arithmetic geometry studies solution sets of systems of polynomial equations over fields that are not algebraically closed. The most famous from arithmetic geometry is probably Fermat’s Last Theorem, which states that for every integer n > 2, and for every pair of rational numbers a en b such that a^n + b^n = 1, we have ab = 0. This result was proven by Andrew Wiles in 1995. The most fundamental question in arithmetic geometry is whether a given system S of polynomial equations over a field F admits a solution over F. More generally, one would like to understand the structure of the set of solutions, and relate it to the properties of the geometric object defined by S (an algebraic variety). My goal is to improve our understanding of what the cohomology of an algebraic variety - which contains a lot of geometric information - can say about its set of rational points, i.e. the set of solutions of S over the base field F. I will study this question over global fields, using the so-called Brauer-Manin obstruction to the existence of rational points, and over local fields, using techniques from logarithmic geometry and ramification theory.