Clark, Pete L.; Sharif, Shahed Period, index and potential \(\text Ш\). (English) Zbl 1200.11037 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) Cited in 2 ReviewsCited in 192 Documents MSC: 11G05 Elliptic curves over global fields Keywords:period; index; Tate-Shafarevich group Citations:Zbl 1087.11036 PDF BibTeX XML Cite \textit{P. L. Clark} and \textit{S. Sharif}, Algebra Number Theory 4, No. 2, 151--174 (2010; Zbl 1200.11037) Full Text: DOI arXiv Link OpenURL