Iwasawa Theory of Elliptic Curves in Quadratic Twist Families

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.