Let S be an infinite set of rational primes and, for some p ∈ S, let \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb{Q}_S^{(p)}}$$\end{document} be the compositum of all extensions unramified outside S of the form \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb{Q}(\mu_p,\sqrt[p]{d})}$$\end{document}, for \documentclass[12pt]{minimal}
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\begin{document}$${d \in \mathbb{Q}^\times}$$\end{document}. If \documentclass[12pt]{minimal}
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\begin{document}$${(\sigma) = (\sigma_{1},\ldots,\sigma_{n}) \in {\rm Gal} {(\overline{\mathbb{Q}}/\mathbb{Q})}^n}$$\end{document}, let \documentclass[12pt]{minimal}
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\begin{document}$${(\mathbb{Q}_S^{(p)})^{(\sigma)}}$$\end{document} be the intersection of the fixed fields by \documentclass[12pt]{minimal}
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\begin{document}$${{\langle\sigma_{i}\rangle}}$$\end{document}, for i = 1, . . , n. We provide a wide family of elliptic curves \documentclass[12pt]{minimal}
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\begin{document}$${E/\mathbb{Q}}$$\end{document} such that the rank of \documentclass[12pt]{minimal}
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\begin{document}$${E((\mathbb{Q}_S^{(p)})^{(\sigma)})}$$\end{document} is infinite for all n ≥ 0 and all \documentclass[12pt]{minimal}
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\begin{document}$${(\sigma) \in {\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q})^n}$$\end{document}, subject to the parity conjecture. Similarly, let \documentclass[12pt]{minimal}
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\begin{document}$${(A/\mathbb{Q},\phi)}$$\end{document} be a polarized abelian variety, let K be a quadratic number field fixed by \documentclass[12pt]{minimal}
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\begin{document}$${(\sigma) \in {\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q})^n}$$\end{document}, let S be an infinite set of primes of \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb{Q}}$$\end{document} and let \documentclass[12pt]{minimal}
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\begin{document}$$K^{p-{\rm dihe}}_S$$\end{document} be the maximal abelian p-elementary extension of K unramified outside primes of K lying over S and dihedral over \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb{Q}}$$\end{document}. We show that, under certain hypotheses, the \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb{Z}_p}$$\end{document} -corank of selp∞(A/F) is unbounded over finite extensions F/K contained in \documentclass[12pt]{minimal}
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\begin{document}$${(K^{p-{\rm dihe}}_S)^{(\sigma)}/K}$$\end{document}. As a consequence, we prove a strengthened version of a conjecture of M. Larsen in a large number of cases.
