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The index of an algebraic curve *C* over
**Q** is the
order of the cokernel of the degree map
;
rationality of the canonical divisor implies that the index
divides 2*g*-2, where *g* is the genus of *C*.
When *g*=1 this is no condition at all; Artin conjectured, and
Lang and Tate [14] proved, that for every integer *m*there is a genus one curve of index *m* over some number field.
Their construction yields genus one curves over
**Q** only for a few
values of *m*, and they ask whether one can find genus one curves
over
**Q** of every index. I have answered
this question for odd *m*.

**Theorem 8**
Let *K* be any number field. There are genus one curves over *K*
of every odd index.

The proof involves showing that enough cohomology classes in
Kolyvagin's Euler system of Heegner points do not vanish
combined with explicit Heegner point computations.
I hope to show that curves of every index occur, and
to determine the consequences of my
nonvanishing result for Selmer groups. This can be viewed
as a contribution to the problem of understanding
*H*^{1}(**Q**,*E*).

*William A. Stein*

*1999-12-01*