On obstructions to the Hasse principle

Summary

Introduction

A basic problem in arithmetic geometry is to decide if a variety defined over a number field k has a k-rational point. This is only possible if there is a k_v-point on the variety for each completion k_v of k. It remains to decide if there is a k-point on a variety with a k_v-point at each place v of k. The first positive results were obtained by Hasse for quadrics and varieties defined by means of certain norm forms. A class of varieties, therefore, is said to satisfy the Hasse principle if each variety in the class has a k-point as soon as it has k_v-points for all places v. The corresponding property for the smooth locus is called the smooth Hasse principle. It is also natural to ask if weak approximation holds. This means that the set of k-points is dense in the topological space of adelic points on the smooth locus.

There are counterexamples to the Hasse principle and weak approximation already for smooth cubic curves and cubic surfaces. These counterexamples can be explained by means of a general obstruction to the Hasse principle due to Manin based on the Brauer group of the variety and the reciprocity law in class field theory. Most but not all of the known counterexamples can be explained by this obstruction (Skorobogatov). It is likely that Manin's obstruction is the only obstruction to the (smooth) Hasse principle for rational varieties.