Factorization of measures and applications to the weak Goldfeld conjecture

TL;DR

This paper generalizes a measure factorization result to all primes and characters, then applies it to estimate the distribution of ranks of quadratic and cubic twists of elliptic curves over quadratic fields, supporting the weak Goldfeld conjecture.

Contribution

It extends Gross's measure factorization to all primes and characters, enabling new bounds on ranks of elliptic curve twists over quadratic fields.

Findings

Lower bounds on the proportion of quadratic fields with rank 1 elliptic curve twists.

Bounds on the proportion of twists with rank 0 or 1 over $Q$.

Application of measure factorization to support the weak Goldfeld conjecture.

Abstract

Extending Gross's result, we prove that a certain factorizaton of measures holds for all $p$ and any finite even Dirichlet character $\chi$ of any conductor, rather than only for split $p$ and $\chi$ with conductor a power of $p$. Using this generalization, we find lower bounds on the proportion of imaginary quadratic fields $K$ for which (under certain assumptions on the elliptic curve) a chosen quadratic twist of an elliptic curve $E$ over $K$ has rank $1$. We also find lower and upper bounds for the proportion of quadratic twists with rank $1$ when we vary $D$, the factor we twist by, under the assumption that $\omega$ (the prime factor counting function) is sufficiently close to a Gaussian distribution, as described by Erd\"os-Kac. We apply similar methods to cubic twists, and then derive analogous lower bounds for the proportion of imaginary quadratic fields for which a sextic twist has rank $1$. Lastly, for elliptic curves over $\mathbb{Q}$ satisfying certain assumptions, we find positive lower bounds on the proportion of quadratic twists (over $\mathbb{Q}$) which have rank $0$ and rank $1$, which yields examples of elliptic curves satisfying the weak Goldfeld conjecture.