Comptes Rendus
no. 9
Number theory
ABC and the Hasse principle for quadratic twists of hyperelliptic curves
[ABC et le principe de Hasse pour les tordues de courbes hyperelliptiques]

Présenté par : the Editorial Board

Pete L. Clark1  ; Lori D. Watson1
1 Department of Mathematics, University of Georgia, Athens, GA 30606, United States
Comptes Rendus. Mathématique, Volume 356 (2018) no. 9, pp. 911-915.

En supposant la conjecture ABC, nous utilisons un travail de Granville pour montrer qu'une courbe hyperelliptique C/Q de genre au moins trois a une infinité de tordues quadratiques, qui violent le principe de Hasse si et seulement si elle n'a pas de point de branchement hyperelliptique rationnel sur Q.

Conditionally on the ABC conjecture, we apply work of Granville to show that a hyperelliptic curve C/Q of genus at least three has infinitely many quadratic twists that violate the Hasse Principle iff it has no Q-rational hyperelliptic branch points.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2018.07.007
Pete L. Clark 1 ; Lori D. Watson 1

1 Department of Mathematics, University of Georgia, Athens, GA 30606, United States 
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Pete L. Clark; Lori D. Watson. ABC and the Hasse principle for quadratic twists of hyperelliptic curves. Comptes Rendus. Mathématique, Volume 356 (2018) no. 9, pp. 911-915. doi : 10.1016/j.crma.2018.07.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.07.007/

[1] M. Bhargava; B.H. Gross; X. Wang A positive proportion of locally soluble hyperelliptic curves over Q have no point over any odd degree extension. With an appendix by Tim Dokchitser and Vladimir Dokchitser, J. Amer. Math. Soc., Volume 30 (2017), pp. 451-493

[2] P.L. Clark An “Anti-Hasse Principle” for prime twists, Int. J. Number Theory, Volume 4 (2008), pp. 627-637

[3] P.L. Clark Curves over global fields violating the Hasse Principle | arXiv

[4] P.L. Clark; J. Stankewicz Hasse Principle violations for Atkin–Lehner twists of Shimura curves, Proc. Amer. Math. Soc., Volume 146 (2018), pp. 2839-2851

[5] A. Granville Rational and integral points on quadratic twists of a given hyperelliptic curve, Int. Math. Res. Not. IMRN, Volume 8 (2007) (24 pp)

[6] Q. Liu Algebraic geometry and arithmetic curves. Translated from the French by Reinie Erné, Oxford Graduate Texts in Mathematics, Oxford Science Publications, vol. 6, Oxford University Press, Oxford, UK, 2002

[7] E. Ozman Points on quadratic twists of X0(N), Acta Arith., Volume 152 (2012), pp. 323-348

[8] M. Sadek On quadratic twists of hyperelliptic curves, Rocky Mountain J. Math., Volume 44 (2014), pp. 1015-1026

[9] J.-P. Serre Divisibilité de certained fonctions arithmétiques, Enseign. Math. (2), Volume 22 (1976), pp. 227-260

[10] P. Stevenhagen; H.W. Lenstra Chebotarëv and his density theorem, Math. Intell., Volume 18 (1996), pp. 26-37

[11] V. Vatsal Rank-one twists of a certain elliptic curve, Math. Ann., Volume 311 (1998), pp. 791-794

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