Bielliptic smooth plane curves and quadratic points

Abstract

Let $C_k$ be a smooth projective curve over a global field $k$, which is neither rational nor elliptic. Harris-Silverman, when $p=0$, and Schweizer, when $p>0$ together with an extra condition on the Jacobian variety $\operatorname{Jac}(C_k)$ arising from Mordell's conjecture, showed that $C$ has infinitely many quadratic points over some finite field extension $L/k$ inside $\overline{k}$ (a fixed algebraic closure of $k$) if and only if $C$ is hyperelliptic or bielliptic. Now, let $C_k$ be a smooth plane curve of a fixed degree $d\geq4$ with $p=0$ or $p>(d-1)(d-2)+1$ (up to an extra condition on $\operatorname{Jac}(C_k)$ in positive characteristic). Then, we prove that $C_k$ admits always finitely many quadratic points unless $d=4$. A so-called \emph{geometrically complete families} for the different strata of smooth bielliptic plane quartic curves by their automorphism groups, are given. Interestingly, we show (in a very simple way) that there are only finitely many quadratic extensions $k(\sqrt{D})$ of a fixed number field $k$, in which we may have more solutions to the Fermat's and the Klein's equations of degree $d\geq5$; $X^d+Y^d-Z^d=0$ and $X^{d-1}Y+Y^{d-1}Z+Z^{d-1}X=0$ respectively, than these over $k$ (the same holds for any non-singular projective plane equation of degree $d\geq 5$ over $k$, and also in general when $k$ is a global field after imposing an extra condition on $\operatorname{Jac}(C_k)$).