Rational points on curves and surfaces and the Brauer–Manin obstruction

Abstract

A central problem in arithmetic geometry is understanding the existence and structure of rational points on varieties. Over fields with interesting arithmetic, the study of rational points is tied deeply to both the number theoretic properties of the field as well as the geometric properties of the variety. In this thesis, we present two projects involving rational point problems. The first investigates superelliptic curves over Henselian fields and explains the structure of their degree sets, an object which captures a shadow of the arithmetic properties of the curve. The second proves the Hasse principle, a method of establishing the existence of rational points, over even degree extensions of the ground field for a large class of conic bundles over the projective line.