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

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 $F_{cyc}^{(n)}$ 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 $F_{cyc}^{(n)}$ as $n\to \infty$. 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.

12 pages, final version, to appear in Canadian Math. Bull

References in corpus (2)

• Statistics for Iwasawa invariants of elliptic curves
• Some remarks on Kida's formula when $\mu \ne 0$

Cited by in corpus (2)

• Fine Selmer groups and ideal class groups
• Rational points on algebraic curves in infinite towers of number fields