Growth of points on hyperelliptic curves over number fields
Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 1, pp. 271-294.

Fix a hyperelliptic curve C/ of genus g, and consider the number fields K/ 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 g1 and n is sufficiently large relative to the degree of C, with n even if degC 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. 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/ de genre g et considérons les corps de nombres K/ 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 g1 et n est suffisamment grand par rapport au degré de C (en supposant que n est pair si degC 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. 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.

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DOI: 10.5802/jtnb.1201
Classification: 11G30,  12F05,  12E05
Keywords: Arithmetic statistics, hyperelliptic curves, Diophantine stability
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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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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