Congruent elliptic curves over some $p$-adic Lie extensions

TL;DR

The paper investigates how Iwasawa invariants attached to dual $p^{\infty}$-Selmer groups and the fine Selmer group behave for residually isomorphic ($E_1[p]\simeq E_2[p]$) elliptic curves over $p$-adic Lie extensions, extending beyond the cyclotomic setting. It establishes criteria linking the vanishing of the $\mu$-invariant to the vanishing of $H^2(K_S/K_\infty,E[p])$ for general $\mathbb{Z}_p$-extensions and derives congruence-invariant results for $\mu$ and $\lambda$ across congruent curves, including explicit comparisons of coranks and ranks. The False-Tate extension case is treated in depth, with a distinction between regular and rank-1 growth regimes, providing explicit growth formulas and showing that $\mu=\lambda=0$ can persist across congruent curves under suitable hypotheses. Overall, the work broadens the understanding of how arithmetic invariants attached to Selmer-type groups behave under p-adic Lie extensions and congruences, with concrete applications to rank growth and regularity phenomena in noncyclotomic settings.

Abstract

Let $p$ be an odd prime number. In this article, we study the variation of Iwasawa invariants among $p$-congruent elliptic curves over certain $p$-adic Lie extensions. We investigate both the classical Selmer group as well as the fine Selmer group.

Theorems & Definitions (53)

• Definition 2.1
• Definition 2.2
• Theorem 3.1
• proof
• Remark 3.2
• Example 3.3
• Definition 3.4
• Theorem 3.5
• Corollary 3.6
• Lemma 3.7