Bielliptic smooth plane curves and quadratic points
Badr, Eslam (Cairo University. Department of Mathematics (Egypt))
Bars Cortina, Francesc (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Date:	2021
Abstract:	Let Ck 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(Ck)). 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 Ck 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.
Grants:	Ministerio de Economía y Competitividad MTM2016-75980-P
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Language:	Anglès
Document:	Article ; recerca ; Versió sotmesa a revisió
Subject:	Plane curves ; Bielliptic curves ; Automorphism group ; Twist
Published in:	International Journal of Number Theory, Vol. 17, Issue 4 (May 2021) , p. 1047-1066, ISSN 1793-0421

DOI: 10.1142/S1793042121500238

Preprint 14 p, 588.1 KB

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