Number theory
ABC and the Hasse principle for quadratic twists of hyperelliptic curves
Comptes Rendus. Mathématique, Volume 356 (2018) no. 9, pp. 911-915.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2018.07.007
Clark, Pete L. 1; Watson, Lori D. 1

1 Department of Mathematics, University of Georgia, Athens, GA 30606, United States
@article{CRMATH_2018__356_9_911_0,
     author = {Clark, Pete L. and Watson, Lori D.},
     title = {ABC and the {Hasse} principle for quadratic twists of hyperelliptic curves},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {911--915},
     publisher = {Elsevier},
     volume = {356},
     number = {9},
     year = {2018},
     doi = {10.1016/j.crma.2018.07.007},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2018.07.007/}
}
TY  - JOUR
AU  - Clark, Pete L.
AU  - Watson, Lori D.
TI  - ABC and the Hasse principle for quadratic twists of hyperelliptic curves
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 911
EP  - 915
VL  - 356
IS  - 9
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2018.07.007/
DO  - 10.1016/j.crma.2018.07.007
LA  - en
ID  - CRMATH_2018__356_9_911_0
ER  - 
%0 Journal Article
%A Clark, Pete L.
%A Watson, Lori D.
%T ABC and the Hasse principle for quadratic twists of hyperelliptic curves
%J Comptes Rendus. Mathématique
%D 2018
%P 911-915
%V 356
%N 9
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2018.07.007/
%R 10.1016/j.crma.2018.07.007
%G en
%F CRMATH_2018__356_9_911_0
Clark, Pete L.; Watson, Lori D. 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. http://www.numdam.org/articles/10.1016/j.crma.2018.07.007/

[1] Bhargava, M.; Gross, B.H.; Wang, X. 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] Clark, P.L. An “Anti-Hasse Principle” for prime twists, Int. J. Number Theory, Volume 4 (2008), pp. 627-637

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

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

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

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

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

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

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

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

Cited by Sources: