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There are genus one curves of every index over every number field. (English) Zbl 1097.14024
Let $$K$$ be a number field and $$C/K$$ be a genus one curve defined over $$K$$. The index of $$C$$ is the least degree of a field extension $$L/K$$ such that $$C$$ has an $$L$$-rational point. The period of $$C$$ is the order of the cohomology class corresponding to $$C$$ in the Weil-ChĂ˘telet group $$H^1\bigl(K,\text{Jac}(C)\bigr)$$, Jac$$(C)$$ being the Jacobian of $$C$$. The period divides the index and these two integers have the same primes divisors. These two invariants quantify to what extent $$C$$ fails to have a $$K$$-rational point. It is a theorem of Shafarevich and Cassels that for any elliptic curve $$E$$ over $$K$$ and any integer $$n>1$$, there are infinitely many classes in the group $$H^1(K,E)$$ of period $$n$$.
About fifty years ago, Lang and Tate have asked the question of whether there are genus one curves of every index over $$\mathbb Q$$. They showed that if $$E$$ is an elliptic curve defined over $$K$$, with a $$K$$-rational point of order $$n$$, then $$H^1(K,E)$$ contains infinitely many classes of index $$n$$. A few years ago, Stein has proved that there are infinitely many genus one curves over $$K$$ of index equal to any number not divisible by $$8$$.
In this paper, the author proves the following result:
Let $$E$$ be an elliptic curve over $$K$$ such that the group $$E(K)$$ is trivial. Then, for every integer $$n\geq 1$$, there exists an element in $$H^1(K,E)$$ of index $$n$$. Since there are infinitely many elliptic curves over $$\mathbb Q$$ with trivial Mordell-Weil group, he deduces a complete answer to the question of Lang and Tate, by proving that for all $$n\geq 1$$ there are infinitely many genus one curves over $$\mathbb Q$$ of index $$n$$. Furthermore, using the fact that there exists an elliptic curve over $$\mathbb Q$$ whose Mordell-Weil and Shafarevich-Tate groups are both trivial, he proves that there are infinitely many genus one curves of every index over every number field.

##### MSC:
 14H52 Elliptic curves 11G05 Elliptic curves over global fields
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##### References:
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