Bielliptic smooth plane curves and quadratic points

Abstract

Let [Formula: see text] be a smooth plane curve of degree [Formula: see text] defined over a global field [Formula: see text] of characteristic [Formula: see text] or [Formula: see text] (up to an extra condition on [Formula: see text]). Unless the curve is bielliptic of degree four, we observe that it always admits finitely many quadratic points. We further show that there are only finitely many quadratic extensions [Formula: see text] when [Formula: see text] is a number field, in which we may have more points of [Formula: see text] than these over [Formula: see text]. In particular, we have this asymptotic phenomenon valid for Fermat’s and Klein’s equations. Second, we conjecture that there are two infinite sets [Formula: see text] and [Formula: see text] of isomorphism classes of smooth projective plane quartic curves over [Formula: see text] with a prescribed automorphism group, such that all members of [Formula: see text] (respectively [Formula: see text]) are bielliptic and have finitely (respectively infinitely) many quadratic points over a number field [Formula: see text]. We verify the conjecture over [Formula: see text] for [Formula: see text] and [Formula: see text]. The analog of the conjecture over global fields with [Formula: see text] is also considered.