TY - JOUR

T1 - Bielliptic smooth plane curves and quadratic points

AU - Badr, Eslam

AU - Bars, Francesc

N1 - Publisher Copyright:
© World Scientific Publishing Company.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

KW - Bielliptic curves

KW - Plane curves

KW - Quadratic points

KW - Twist of curves

UR - http://www.scopus.com/inward/record.url?scp=85093509538&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/34190ee8-194d-364d-a09a-9a03189218a9/

U2 - 10.1142/S1793042121500238

DO - 10.1142/S1793042121500238

M3 - Article

SN - 1793-0421

VL - 17

JO - International Journal of Number Theory

JF - International Journal of Number Theory

ER -