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

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.

### MSC:

 11G05 Elliptic curves over global fields

### Keywords:

period; index; Tate-Shafarevich group

Zbl 1087.11036
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