| 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 rationals 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.
Note : Related article by the speaker entitled "Congruent numbers,
elliptic curves, and the passage from the local to the global"
can be found at : http://arxiv.org/abs/0704.3783 |