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 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.
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