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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-Châtelet 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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