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

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.