Congruent numbers, elliptic curves, and the passage from the local to the global

Abstract

The ancient unsolved problem of congruent numbers has been reduced to one of the major questions of contemporary arithmetic: the finiteness of the number of curves over $\bf Q$ which become isomorphic at every place to a given curve. We give an elementary introduction to congruent numbers and their conjectural characterisation, discuss local-to-global issues leading to the finiteness problem, and list a few results and conjectures in the arithmetic theory of elliptic curves.