Growth of the analytic rank of rational elliptic curves over quintic fields

Michele Fornea

Published 2018-02-20Version 1

Given a rational elliptic curve $E_{/\mathbb{Q}}$, we denote by $G_5(E;X)$ the number of quintic fields of absolute discriminant at most $X$ such that the analytic rank of $E$ grows over $K$, i.e., $\mathrm{r}_\mathrm{an}(E/K)>\mathrm{r}_\mathrm{an}(E/\mathbb{Q})$. We show that $G_5(E;X)\asymp_{+\infty} X$ when the elliptic curve $E$ has odd conductor and at least one prime of multiplicative reduction. As Bhargava showed that the number of quintic fields of absolute discriminant at most $X$ is asymptotic to $c_5 X$, for some positve constant $c_5>0$, our result exposes the growth of the analytic rank as a very common circumstance over quintic fields.