Bounds of the rank of the Mordell–Weil group of Jacobians of Hyperelliptic Curves

Daniels, Harris B.; Lozano-Robledo, Álvaro; Wallace, Erik

Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 1, pp. 231-258.

Dans cet article, nous étendons les travaux de Shanks et Washington sur les extensions cycliques et les courbes elliptiques associées aux corps cubiques les plus simples. En particulier, nous donnons des familles d’exemples de courbes hyperelliptiques $C:y^2=f\left(x\right)$ définies sur $\mathbb{Q}$, avec $f\left(x\right)$ de degré $p$, où $p$ est un nombre premier de Sophie Germain, telles que le rang du groupe de Mordell–Weil de la jacobienne $J/\mathbb{Q}$ de $C$ est borné par le genre de $C$ et le rang de la $2$-torsion du groupe des classes d’idéaux du corps (cyclique) défini par $f\left(x\right)$, et présentons des exemples où cette borne est optimale.

In this article we extend work of Shanks and Washington on cyclic extensions, and elliptic curves associated to the simplest cubic fields. In particular, we give families of examples of hyperelliptic curves $C:y^2=f\left(x\right)$ defined over $\mathbb{Q}$, with $f\left(x\right)$ of degree $p$, where $p$ is a Sophie Germain prime, such that the rank of the Mordell–Weil group of the jacobian $J/\mathbb{Q}$ of $C$ is bounded by the genus of $C$ and the $2$-rank of the class group of the (cyclic) field defined by $f\left(x\right)$, and exhibit examples where this bound is sharp.

Classification : 11G10, 14K15
Mots clés : Jacobian, hyperelliptic curve, Mordell–Weil, rank, Selmer, descent

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