# There are genus one curves over Q of every odd index

**Abstract**

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 X_{0}(17).

Here is the PDF version of the paper
that appeared in
Volume 547 (2002) of Crelle's Journal.

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