There are genus one curves over Q of every odd index
The index of a genus one curve X over a field K
is the smallest degree of an extension
L of K such that X(L) is nonempty.
Let K be a number field.
We prove that for every integer r not divisible
by 8, there is a genus one curve X over K of index r.
Our proof involves an analysis of Kolyvagin's Euler system of Heegner points
combined with explicit computations on the modular curve X0(17).
Here is the PDF version of the paper
that appeared in
Volume 547 (2002) of Crelle's Journal.