The local-global principle for diophantine equations

Abstract

The p-adic numbers were first introduced by Kurt Hensel in 1897 in an attempt to carry the techniques of power series methods from complex analysis into number theory. In this thesis we discuss the relationship between solutions to equations over the field Q of rational numbers and solutions over the p-adic fields Qp. In order to do this, we begin by defining the p-adic absolute value and investigating some of its properties. The purpose of defining this absolute value is constructing the p-adic field Qp as the completion of Q with respect to it. Then, we explore Qp (its ring of integers, the analogy with power series …), before stating and proving Hensel's lemma. The first aspect of the local-global relationship will be Hasse's local-global principle for quadratic forms. We will show some results of Hensel's lemma and the local-global principle, and then move to the counterexamples to the Hasse principle. An equation is said to be a counterexample to the Hasse principle if it has solutions in every completion of Q, yet it fails to have a rational solution. In the thesis, we will state several counterexamples to the Hasse principle developed in the last century, classify them geometrically, and finally, we will prove a counterexample by Birch and Swinnerton-Dyer.