Iwasawa theory and ranks of elliptic curves in quadratic twist families

We study the distribution of ranks of elliptic curves in quadratic twist families using Iwasawa-theoretic methods, contributing to the understanding of Goldfeld’s conjecture. Given an elliptic curve $E\mathrm{/}\mathbb{Q}$ with good ordinary reduction at $2$ and ${\lambda }_2(E\mathrm{/}\mathbb{Q})=0$, we use Matsuno’s Kida-type formula to construct quadratic twists $E^{(d)}$ such that ${\lambda }_2(E^{(d)}\mathrm{/}\mathbb{Q})$ remains unchanged or increases by $2$. When the root number of $E^{(d)}$ is $-1$ and the Tate-Shafarevich group $Sha(E^{(d)}\mathrm{/}\mathbb{Q})[2^{\mathrm{\infty }}]$ is finite, this yields quadratic twists with Mordell–Weil rank $1$. These results support the conjectural expectation that, on average, half of the quadratic twists in a family have rank $0$ and half have rank $1$. In the cases we consider we obtain asymptotic lower bounds for the number of twists by squarefree numbers $d\le X$ which match with the conjectured value up to an explicit power of $\log X$. They complement recent groundbreaking results of Smith on Goldfeld’s conjecture.