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The period-index problem in WC-groups. I: Elliptic curves. (English) Zbl 1087.11036
The main theorem of the paper is the following. Let $$p$$ be a prime number, let $$E$$ be an elliptic curve defined over a number field $$K$$ such that $$E(K)$$ has full $$p$$-torsion, and let $$r$$ be a positive integer. Then there are infinitely many extensions $$L/K$$ of degree $$p$$ such that $$\dim_{{\mathbb F}_p} \text Ш (E/L)[p]\geq r$$.
The theorem is deduced from another result, which makes use of the period-index obstruction map introduced in C. O’Neil [“The period-index obstruction for elliptic curves”, J. Number Theory 95, No. 2, 329–339 (2002); erratum ibid. 109, No. 2, 390 (2004; Zbl 1033.11029)]. Under the same hypotheses of the main theorem, this result says that there exists an infinite subgroup $$G$$ of $$H^1(K,E)[p]$$ such that every nonzero element of $$G$$ has index $$p^2$$ (the index of $$g\in G$$ is defined as the greatest common divisor of all degrees of finite extensions $$L/K$$ such that $$g_{| _L}=0$$).

##### MSC:
 11G05 Elliptic curves over global fields 11R34 Galois cohomology
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##### References:
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