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We show that a nontrivial abelian variety over a Hilbertian field in which the weak Mordell–Weil theorem holds admits infinitely many torsors with period any givenn  &gt;  1n&gt;1    that is not divisible by the characteristic. The corresponding statement with “period” replaced by “index” is plausible but open, and it seems much more challenging. We show that for every infinite, finitely generated field  K , there is an elliptic curveE/  KE_{/K}    which admits infinitely many torsors with index any givenn  &gt;  1n&gt;1   .

There are genus one curves of every index over every infinite, finitely generated field