English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

Given $F$ a totally real field and $E_{/F}$ a modular elliptic curve, wedenote by $G_5(E_{/F};X)$ the number of quintic extensions $K$ of $F$ such thatthe 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 oddconductor 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 theanalytic rank as a very common circumstance over quintic extensions.

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