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

Given $F$ a totally real number 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., $r_{\mathrm{an}}(E / K) \gt r_{\mathrm{an}}(E / F)$. 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 [1] 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.

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Received 21 March 2018

Accepted 10 February 2019

Published 6 March 2020