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 ∞ be a finitely ramified uniform pro-𝑝 extension of 𝐹 containing the cyclotomic Z 𝑝 -extension 𝐹 cyc . Set 𝐹 (𝑛) be the 𝑛-th layer of the tower, and 𝐹 (𝑛) cyc the cyclotomic Z 𝑝 -extension of 𝐹 (𝑛) . We study the growth of the rank of 𝐸 (𝐹 (𝑛) ) by analyzing the growth of the 𝜆-invariant of the Selmer group over 𝐹 (𝑛) cyc as 𝑛 → ∞. This method has its origins in work of A. Cuoco, who studied Z 2 𝑝 -extensions. Refined estimates for growth are proved that are close to conjectured estimates. The results are illustrated in special cases. 2022/01/21 16:40 This is a ``preproof'' accepted article for Canadian Mathematical Bulletin This version may be subject to change during the production process.