Iwasawa Theory of Elliptic Curves in Quadratic Twist Families

TL;DR

This paper investigates how Iwasawa invariants of rational elliptic curves vary within quadratic twist families using algebraic and analytic methods, including modular forms and Selmer groups.

Contribution

It introduces dual algebraic and analytic approaches to analyze Iwasawa invariants in quadratic twist families of elliptic curves, advancing understanding in this area.

Findings

Variation patterns of Iwasawa invariants identified

Connections between modular forms and Selmer groups established

New techniques for studying elliptic curves in quadratic twists developed

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.