Growth of points on hyperelliptic curves over number fields

Christopher Keyes

doi:10.5802/jtnb.1201

Christopher Keyes ¹

¹ Emory University Department of Mathematics 400 Dowman Dr., Atlanta, GA 30223, USA

Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 1, pp. 271-294.

Abstract
Résumé

Fix a hyperelliptic curve $C / ℚ$ of genus $g$ , and consider the number fields $K / ℚ$ generated by the algebraic points of $C$ . In this paper, we study the number of such extensions with fixed degree $n$ and discriminant bounded by $X$ . We show that when $g \geq 1$ and $n$ is sufficiently large relative to the degree of $C$ , with $n$ even if $deg C$ is even, there are $≫ X^{c_{n}}$ such extensions, where $c_{n}$ is a positive constant depending on $g$ which tends to $1 / 4$ as $n \to \infty$ . This result builds on work of Lemke Oliver and Thorne who, in the case where $C$ is an elliptic curve, put lower bounds on the number of extensions with fixed degree and bounded discriminant over which the rank of $C$ grows with specified root number.

Choisissons une courbe hyperelliptique $C / ℚ$ de genre $g$ et considérons les corps de nombres $K / ℚ$ engendrés par les points algébriques de $C$ . Dans cet article, nous étudions le nombre de telles extensions de degré fixé $n$ et de discriminant inférieur ou égal à $X$ . Nous montrons que lorsque $g \geq 1$ et $n$ est suffisamment grand par rapport au degré de $C$ (en supposant que $n$ est pair si $deg C$ est un nombre pair), ce nombre est $≫ X^{c_{n}}$ , où $c_{n}$ est une constante positive dépendant de $g,$ qui tend vers $1 / 4$ lorsque $n \to \infty$ . Ce résultat s’appuie sur le travail de Lemke Oliver et Thorne qui, dans le cas où $C$ est une courbe elliptique, donnent une minoration pour le nombre d’extensions de degré fixé et de discriminant borné sur lesquelles le rang de $C$ augmente avec une constante locale spécifiée.

Received: 2020-09-25
Revised: 2021-02-07
Accepted: 2022-05-20
Published online: 2022-07-07

DOI: 10.5802/jtnb.1201

Classification: 11G30, 12F05, 12E05
Mots-clés : Arithmetic statistics, hyperelliptic curves, Diophantine stability

Author's affiliations:

Christopher Keyes ¹

¹ Emory University Department of Mathematics 400 Dowman Dr., Atlanta, GA 30223, USA

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Growth of points on hyperelliptic curves over number fields},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {271--294},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Christopher Keyes. Growth of points on hyperelliptic curves over number fields. Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 1, pp. 271-294. doi : 10.5802/jtnb.1201. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1201/

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