Simple Counterexamples to the Local-global Principle

After Hasse had found the first example of a Local-Global principle in the 1920s by showing that a quadratic form in n variables represented 0 in rational numbers if and only if it did so in every completion of the rationals, mathematicians investigated whether this principle held in other situations. Among the first counterexamples to the Hasse principle were curves of genus 1 constructed by Lind and Reichardt: these were curves without rational points but with points in every completion of Q. In this article we will show that the technique of rational parametrization of conics is powerful enough to derive Reichardt’s result.