Period, index and potential \(\text Ш\). (English) Zbl 1200.11037

Clark, Pete L.; Sharif, Shahed

Algebra Number Theory 4, No. 2, 151-174 (2010).

For a positive integer \(P\) let \(P^*\) be equal to \(P\) if \(P\) is odd, and \(2P\) if \(P\) is even. Let also \(K\) be a global field, and let \(E/K\) be an elliptic curve such that \(\# E(K)[P^*] = (P^*)^2.\)
The main result of the paper states that, for every positive divisor \(D\) of \(P\), there exist infinitely many classes \(\eta\in H^1(K,E)\) of period \(P\) and index \(P\cdot D\), and these classes can be chosen so as to be locally trivial except possibly at two places of \(K\).
It is well known that, for a curve \(C\) of genus one over an arbitrary field \(K\), the index \(I\) and the period \(P\) satisfy the relation \( P \,|\,I\,|\,P^2\). As a consequence of the main result, which generalizes a theorem in [P. L. Clark, J. Number Theory 114, 193–208 (2005; Zbl 1087.11036)], any pair \((P,I)\) satisfying the preceding relation appears as the period and index of a curve of genus one defined over some number field (depending on \(P\)). Moreover, the fact that the classes can be constructed with support of at most two places answers a question raised by Çiperiani.
As to the case of curves of genus one defined over a fixed global field, the author also prove the following: Let \(E/K\) be an elliptic curve. For any positive integer \(r\), there exists an extension \(L/K\) of degree \(P\) such that \(\text Ш (L/E)\) contains at least \(r\) elements of order \(P\).
The main tools are Lichtenbaum-Tate duality and the functorial properties of O’Neil’s period-index obstruction map under change of period.

Reviewer: Roberto Dvornicich (Pisa)

MSC:

11G05

Elliptic curves over global fields

Keywords:

period; index; Tate-Shafarevich group