Bielliptic smooth plane curves and quadratic points
Eslam Badr, Francesc Bars

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 be a smooth projective curve over a global field , which is neither rational nor elliptic. Harris-Silverman, when , and Schweizer, when together with an extra condition on the Jacobian variety arising from Mordell's conjecture, showed that has infinitely many quadratic points over some finite field extension inside (a fixed algebraic closure of ) if and only if is hyperelliptic or bielliptic. Now, let be a smooth plane curve of a fixed degree with or (up to an extra condition on in positive characteristic). Then, we prove that admits always finitely many quadratic points unless . A so-called \emph{geometrically complete families} for the different strata of smooth bielliptic plane quartic curves by their automorphism groups, are…
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