# Estimating the growth in Mordell–Weil ranks and Shafarevich–Tate groups over Lie extensions

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## Abstract

Let \(E_{/\mathbb {Q}}\!\) be an elliptic curve, \(p>3\) a good ordinary prime for *E*, and \(K_\infty \) a *p*-adic Lie extension of a number field *k*. Under some standard hypotheses, we study the asymptotic growth in both the Mordell–Weil rank and Shafarevich–Tate group for *E* over a tower of extensions \(K_n/k\) inside \(K_\infty \); we obtain lower bounds on the former, and upper bounds on the latter’s size.

## Keywords

Elliptic curves Mordell–Weil ranks Noncommutative Iwasawa theory## Mathematics Subject Classification

Primary 11R23 11G05 22E20 22E05## Notes

### Acknowledgments

The majority of this work was carried out during the first named author’s visit to Université Laval in May–June 2015, and he would like to thank them for their generous hospitality, and in particular Hugo Chapdelaine. The authors would also like to thank Wei Lu for his comments on an earlier version of this article.

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