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

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