ABC and the Hasse principle for quadratic twists of hyperelliptic curves

Pete L. Clark; Lori D. Watson

doi:10.1016/j.crma.2018.07.007

Number theory

ABC and the Hasse principle for quadratic twists of hyperelliptic curves

Presented by: the Editorial Board

Pete L. Clark ¹; Lori D. Watson ¹

¹ Department of Mathematics, University of Georgia, Athens, GA 30606, United States

Comptes Rendus. Mathématique, Volume 356 (2018) no. 9, pp. 911-915.

Abstract
Résumé

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: 2018-05-08
Accepted: 2018-07-13
Published online: 2018-08-03

DOI: 10.1016/j.crma.2018.07.007

Author's affiliations:

Pete L. Clark ¹; Lori D. Watson ¹

¹ Department of Mathematics, University of Georgia, Athens, GA 30606, United States

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     pages = {911--915},
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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/

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