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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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##### References:
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