  We show that there exist genus one curves of every index over the rational
numbers, answering affirmatively a question of Lang and Tate. The proof is
"elementary" in the sense that it does not assume the finiteness of any
Shafarevich-Tate group. On the other hand, using Kolyvagin's construction of a
rational elliptic curve whose Mordell-Weil and Shafarevich-Tate groups are both
trivial, we show that there are infinitely many curves of every index over
every number field.
