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## Project

# Rational points on varieties over discretely valued fields.

The general theme of this thesis is the interaction between the behaviour of rational points on certain classes of algebraic varieties over global and local fields, and the étale cohomology of these varieties.

Part I studies cohomological obstructions to the validity of local-global principles (such as the Hasse principle and weak approximation)coming from the Brauer group of a variety. The BrauerManin obstructionhas become an essential tool to understand when such local-global principles can (or cannot) hold. Our contribution to this line of research istwofold. In a first piece of work, we investigate families of torsors under a constant torus defined over a number field using different methods (such as the descent method and the fibration method), thereby provingthat the Brauer-Manin obstruction to the Hasse principle and weak approximation is the only one for certain families. In a second piece of work, we construct new examples of varieties for which the étale Brauer-Manin obstruction (a refinement of the classical Brauer-Manin obstruction) does not suffice to explain the failure of the Hasse principle. We construct the first such examples with trivial Albanese variety; assumingthe abc conjecture, we even construct examples which are geometrically simply connected.

In Part II, we investigate the relationship between the rational volume of a smooth, projective variety defined over a strictly local field, and the trace of the tame monodromy operator on the étale cohomology of this variety. The motivation is work of NicaiseSebag on a trace formula for the motivic Serre invariant, inspired by the GrothendieckLefschetz trace formula for varieties over finite fields. We study this relationship using the framework of logarithmic geometry. We give explicit formulae for both the rational volume and the tamemonodromy zeta function in terms of the logarithmic stratification of alog smooth integral model (assuming that such a model exists). The maintools which we use in the proofs are Nakayamas description of the complex of nearby cycles in the log smooth case, and Katos formalism of logblow-ups.

Part I studies cohomological obstructions to the validity of local-global principles (such as the Hasse principle and weak approximation)coming from the Brauer group of a variety. The BrauerManin obstructionhas become an essential tool to understand when such local-global principles can (or cannot) hold. Our contribution to this line of research istwofold. In a first piece of work, we investigate families of torsors under a constant torus defined over a number field using different methods (such as the descent method and the fibration method), thereby provingthat the Brauer-Manin obstruction to the Hasse principle and weak approximation is the only one for certain families. In a second piece of work, we construct new examples of varieties for which the étale Brauer-Manin obstruction (a refinement of the classical Brauer-Manin obstruction) does not suffice to explain the failure of the Hasse principle. We construct the first such examples with trivial Albanese variety; assumingthe abc conjecture, we even construct examples which are geometrically simply connected.

In Part II, we investigate the relationship between the rational volume of a smooth, projective variety defined over a strictly local field, and the trace of the tame monodromy operator on the étale cohomology of this variety. The motivation is work of NicaiseSebag on a trace formula for the motivic Serre invariant, inspired by the GrothendieckLefschetz trace formula for varieties over finite fields. We study this relationship using the framework of logarithmic geometry. We give explicit formulae for both the rational volume and the tamemonodromy zeta function in terms of the logarithmic stratification of alog smooth integral model (assuming that such a model exists). The maintools which we use in the proofs are Nakayamas description of the complex of nearby cycles in the log smooth case, and Katos formalism of logblow-ups.

Date:1 Oct 2010
→
14 Oct 2014

Keywords:Arithmetic geometry, Rational points, Discretely valued fields, Trace formula

Disciplines:Algebra

Project type:PhD project