  Every nontrivial abelian variety over a Hilbertian field in which the weak
Mordell-Weil theorem holds admits infinitely many torsors with period any $n >
1$ which is not divisible by the characteristic. The corresponding statement
with "period" replaced by "index" is plausible but much more challenging. We
show that for every infinite, finitely generated field $K$, there is an
elliptic curve $E_{/K}$ which admits infinitely many torsors with index any $n
> 1$.
