There are genus one curves of every index over every number field

Details There are genus one curves of every index over every number field There are genus one curves of every index over every number field

2004-11-18

arxiv.org/abs/math/0411413v1

Mathematics

Number Theory

5 pages

Scientific paper

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.

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