Growth of points on hyperelliptic curves over number fields

Keyes, Christopher

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

Choisissons une courbe hyperelliptique $C/\mathbb{Q}$ de genre $g$ et considérons les corps de nombres $K/\mathbb{Q}$ 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 $g\ge 1$ 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 $\gg X^{c_n}$, 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.

Fix a hyperelliptic curve $C/\mathbb{Q}$ of genus $g$, and consider the number fields $K/\mathbb{Q}$ 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 $g\ge 1$ and $n$ is sufficiently large relative to the degree of $C$, with $n$ even if $degC$ is even, there are $\gg X^{c_n}$ 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.

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

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