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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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