A remark on the rank of Jacobians of hyperelliptic curves over \({\mathbb{Q}}\) over certain elementary abelian 2-extensions
The author proves the following theorem: Let \(f\in {\mathbb{Z}}[X]\) be a separable polynomial of odd degree \(\geq 3\). Let \({\mathcal C}\) be a smooth model of the curve \(y^ 2=f(x)\) and let \({\mathcal J}\) be the jacobian of \({\mathcal C}\). Then for every \(m\geq 1\) one can explicitly construct infinitely many extensions of \({\mathbb{Q}}\) of the form \(K={\mathbb{Q}}(\sqrt{d_ 1},...,\sqrt{d_ m})\) for which \(rank({\mathcal J}(K))\geq rank({\mathcal J}({\mathbb{Q}}))+m.\) For the proof the author proves the following two statements: Let \({\mathcal C}\to {\mathbb{P}}^ 1({\mathbb{Q}})\) be the associated two-sheeted covering. Then there are infinitely many ways to choose points \(x_ 1,...,x_ m\in {\mathbb{P}}^ 1({\mathbb{Q}})\), such that: (1) the fields \({\mathbb{Q}}(\sqrt{f(x_ i)})\) are linearly disjoint, and (2) the points in \({\mathcal J}({\mathcal C})({\bar {\mathbb{Q}}})\) associated to \(x_ 1,...,x_ m\) (with a fixed point \(x_{\infty}\) at infinity) are non-torsion.
scientific article; zbMATH DE number 4127387