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arXiv · 2109.07457

Asymptotic growth of Mordell-Weil ranks of elliptic curves in noncommutative towers

Anwesh Ray
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Abstract

Let $E$ be an elliptic curve defined over a number field $F$ with good ordinary reduction at all primes above $p$, and let $F_\infty$ be a finitely ramified uniform pro-$p$ extension of $F$ containing the cyclotomic $\mathbb{Z}_p$-extension $F_{cyc}$. Set $F^{(n)}$ be the $n$-th layer of the tower, and $F^{(n)}_{cyc}$ the cyclotomic $\mathbb{Z}_p$-extension of $F^{(n)}$. We study the growth of the rank of $E(F^{(n)})$ by analyzing the growth of the $\lambda$-invariant of the Selmer group over $F^{(n)}_{cyc}$ as $n\rightarrow \infty$. This method has its origins in work of A.Cuoco, who studied $\mathbb{Z}_p^2$-extensions. Refined estimates for growth are proved that are close to conjectured estimates. The results are illustrated in special cases.

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BibTeXRIS

Anwesh Ray. 2021-09-15. Asymptotic growth of Mordell-Weil ranks of elliptic curves in noncommutative towers. https://doi.org/10.4153/s0008439522000108

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