The rank of hyperelliptic Jacobians in families of quadratic twists

Petersen, Sebastian

Journal de théorie des nombres de Bordeaux, Tome 18 (2006) no. 3, pp. 653-676.

La variation du rang des courbes elliptiques sur $\mathbb{Q}$ dans des familles de “twists” quadratiques a été étudiée de façon détaillée par Gouvêa, Mazur, Stewart, Top, Rubin et Silverberg. On sait par exemple que chaque courbe elliptique sur $\mathbb{Q}$ admet une infinité de twists quadratiques de rang au moins $1$. Presque toutes les courbes elliptiques admettent même une infinité de twists de rang $\ge 2$ et on connaît des exemples pour lesquels on trouve une infinité de twists ayant rang $\ge 4$. On dispose pareillement de quelques résultats de densité. Cet article étudie la variation du rang des jacobiennes hyperelliptiques dans des familles de twists quadratiques, d’une manière analogue.

The variation of the rank of elliptic curves over $\mathbb{Q}$ in families of quadratic twists has been extensively studied by Gouvêa, Mazur, Stewart, Top, Rubin and Silverberg. It is known, for example, that any elliptic curve over $\mathbb{Q}$ admits infinitely many quadratic twists of rank $\ge 1$. Most elliptic curves have even infinitely many twists of rank $\ge 2$ and examples of elliptic curves with infinitely many twists of rank $\ge 4$ are known. There are also certain density results. This paper studies the variation of the rank of hyperelliptic Jacobian varieties in families of quadratic twists in an analogous way.

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