Bielliptic smooth plane curves and quadratic points

TL;DR

This paper investigates the finiteness of quadratic points on smooth plane curves of degree at least four, showing finiteness for most cases except degree four, and explores solutions to Fermat and Klein equations over quadratic extensions.

Contribution

It establishes new finiteness results for quadratic points on smooth plane curves of degree ≥4, especially highlighting the special case of degree 4 and providing explicit families for bielliptic quartics.

Findings

Finiteness of quadratic points for degree ≥5 curves over global fields.

Existence of infinitely many quadratic points only for degree 4 bielliptic plane quartics.

Limited quadratic extensions admit more solutions to Fermat and Klein equations than base fields.

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 $\operatorname{Jac}(C_k)$).