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The period-index obstruction for elliptic curves. (English) Zbl 1033.11029
J. Number Theory 95, No. 2, 329-339 (2002); erratum ibid. 109, No. 2, 390 (2004).
Consider an elliptic curve \(E\) defined over a field \(K\) with absolute Galois group \(G_K\). The elements of \(H^1(G_K,E)\) can be interpreted as isomorphism classes of principal homogeneous spaces \(C\) of \(E\). The order \(n\) of such an element \(C\) in \(H^1(G_K,E)\) is called its period; the index of \(C\) is the smallest positive integer \(d\) such that there exists a \(K\)-rational line bundle of degree \(d\) on \(C\). It is known that \(n | d\) and that they have the same prime factors [S. Lang and J. Tate, Am. J. Math. 80, 659–684 (1958; Zbl 0097.36203)]; moreover, if \(K\) is a local field, then we have \(n = d\) by results of S. Lichtenbaum [Am. J. Math. 90, 1209–1223 (1968; Zbl 0187.18602)].
In this article, the author constructs a map \(\text{Ob}\) from \(H^1(G_K,E[n])\) to the Brauer group \(Br(K)\) whose properties are then used to study the relation between the period \(n\) and the order \(d\) of principal homogeneous spaces over fields \(K\) containing the \(n\)-torsion points \(E[n]\) of \(E\). Almost trivial consequences of the existence of \(\text{Ob}\) are the inequality \(d \leq n^2\), or J. W. S. Cassels’ result [J. Lond. Math. Soc. 38, 244–248 (1963; Zbl 0113.03701)] that, over global fields \(K\), we have \(n=d\) for elements of the Tate-Shafarevich group of \(E\). In addition, the author shows that the function \(\text{Ob}\) is quadratic on the \({\mathbb Z}\)-module \(H^1(G_K,E[n])\), and the works out relations with Hilbert symbols and the Tate pairing.

MSC:
11G05 Elliptic curves over global fields
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