Bielliptic smooth plane curves and quadratic points

Let C be a smooth plane curve of degree d ≥ 4 defined over a global field k of characteristic p = 0 or p > (d-1)(d-2)/2 (up to an extra condition on Jac(C)). 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 k(√D) when k is a number field, in which we may have more points of C than these over k. In particular, we have this asymptotic phenomenon valid for Fermat’s and Klein’s equations. Second, we conjecture that there are two infinite sets E and D of isomorphism classes of smooth projective plane quartic curves over k with a prescribed automorphism group, such that all members of E (respectively D) are bielliptic and have finitely (respectively infinitely) many quadratic points over a number field k. We verify the conjecture over k = Q for G = Z/6Z and GAP(16, 13). The analog of the conjecture over global fields with p > 0 is also considered.