Iwasawa Theory of Elliptic Curves in Quadratic Twist Families

TL;DR

This work investigates how Iwasawa invariants of elliptic curves behave in quadratic twist families using two complementary frameworks. The analytic approach leverages half-integral weight modular forms and Shimura lifts to relate twist behavior to $p$-adic $L$-functions, yielding constancy results for the analytic $\lambda$-invariant under precise conditions, which by the Iwasawa Main Conjecture translate into algebraic invariants. The algebraic approach uses BDP-Selmer and fine Selmer groups, along with explicit twisting and splitting conditions, to produce isomorphisms between Selmer groups of $\mathsf{E}$ and its quadratic twist $\mathsf{E}^K$ over the cyclotomic extension, hence equal algebraic $\lambda$-invariants and controlled rank behavior. Additionally, the paper derives density results for the twists and fields satisfying the required hypotheses, giving explicit formulas and positive-proportion outcomes in the large-$p$ regime, thereby enriching the understanding of rank growth and $p$-adic invariants in twist families.

Abstract

In this article, we use two different approaches -- one algebraic and the other analytic -- to study the variation of Iwasawa invariants of rational elliptic curves in some quadratic twist families. The analytic approach involves a thorough investigation of half-integral weight modular forms. On the other hand, the algebraic proof requires studying the BDP-Selmer groups and the fine Selmer groups.

Theorems & Definitions (55)

• Theorem A
• Theorem B
• Definition 2.1
• Conjecture 2.2: Greenberg--Iwasawa Main conjecture
• Theorem 2.3
• Definition 2.4
• Theorem 2.5
• proof
• Theorem 3.1
• proof