论文标题
沿其2扭曲的属1和2的粘合曲线
Gluing curves of genus 1 and 2 along their 2-torsion
论文作者
论文摘要
令$ x $（分别$ y $）为基本场$ k $属1（分别2）的曲线，其特征不等于2。我们给出了curve $ z $ y twist twist（2,2,2,2）的曲线$ z $的标准。此外，我们给出了算法来构建曲线$ z $，一旦给出了$ x $和$ y $的方程式。其中的第一个涉及使用$ y $的Kummer品种的超平面部分，其降低为$ x $，而第二个是基于涉及$ \ Mathbb {c} $数值结果的插值方法，这些方法被证明是正确的。作为一个应用程序，我们在$ \ mathbb {q} $上找到了jacobian的扭曲，该扭曲是一个理性的70扭转点。
Let $X$ (resp. $Y$) be a curve of genus 1 (resp. 2) over a base field $k$ whose characteristic does not equal 2. We give criteria for the existence of a curve $Z$ over $k$ whose Jacobian is up to twist (2,2,2)-isogenous to the products of the Jacobians of $X$ and $Y$. Moreover, we give algorithms to construct the curve $Z$ once equations for $X$ and $Y$ are given. The first of these involves the use of hyperplane sections of the Kummer variety of $Y$ whose desingularization is isomorphic to $X$, whereas the second is based on interpolation methods involving numerical results over $\mathbb{C}$ that are proved to be correct over general fields a posteriori. As an application, we find a twist of a Jacobian over $\mathbb{Q}$ that admits a rational 70-torsion point.
