Growth of the analytic rank of modular elliptic curves over quintic extensions

TL;DR

This paper demonstrates that for certain modular elliptic curves over totally real fields, the number of quintic extensions causing an increase in the analytic rank grows linearly with the discriminant bound, indicating rank growth is common.

Contribution

It establishes that the analytic rank of specific modular elliptic curves tends to grow over a positive proportion of quintic extensions, a phenomenon previously not well understood.

Findings

Number of quintic extensions with rank growth is proportional to X

Rank growth occurs frequently over quintic extensions

Results apply to elliptic curves with odd conductor and multiplicative reduction

Abstract

Given $F$ a totally real field and $E_{/F}$ a modular elliptic curve, we denote by $G_5(E_{/F};X)$ the number of quintic extensions $K$ of $F$ such that the norm of the relative discriminant is at most $X$ and the analytic rank of $E$ grows over $K$, i.e., $r_\mathrm{an}(E/K)>r_\mathrm{an}(E/F)$. We show that $G_5(E_{/F};X)\asymp_{+\infty} X$ when the elliptic curve $E_{/F}$ has odd conductor and at least one prime of multiplicative reduction. As Bhargava, Shankar and Wang \cite{BSW} showed that the number of quintic extensions of $F$ with norm of the relative discriminant at most $X$ is asymptotic to $c_{5,F} X$ for some positive constant $c_{5,F}$, our result exposes the growth of the analytic rank as a very common circumstance over quintic extensions.