Iwasawa theory and ranks of elliptic curves in quadratic twist families

TL;DR

This paper uses Iwasawa theory to analyze the distribution of ranks in quadratic twist families of elliptic curves, providing new constructions and asymptotic bounds that support Goldfeld's conjecture.

Contribution

It introduces Iwasawa-theoretic methods and Kida-type formulas to construct quadratic twists with controlled ranks, advancing understanding of rank distribution in elliptic curve families.

Findings

Constructs quadratic twists with rank 0 or 1 using Iwasawa theory.

Provides asymptotic lower bounds for the number of twists with certain ranks.

Supports the conjecture that half of the twists have rank 0 and half have rank 1.

Abstract

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/\mathbb{Q} $ with good ordinary reduction at $ 2 $ and $ \lambda_2(E/\mathbb{Q}) = 0 $, we use Matsuno's Kida-type formula to construct quadratic twists $ E^{(d)} $ such that $ \lambda_2(E^{(d)}/\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)}/\mathbb{Q})[2^\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 $d\leq X$ 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.