论文标题
属2曲线的显式两次下降
Explicit two-cover descent for genus 2 curves
论文作者
论文摘要
鉴于一个属$ 2 $曲线$ c $，在一个数字字段中定义了一个理性的Weierstrass点，我们构建了一个属属$ 5 $曲线的家族，这些曲线通过最大的未对$ c $的最大未熟悉的Abelian两次覆盖，并描述了雅各布人的同类型类别的明确模型，以限制其椭圆形的curves curves curves的规模。本文的所有构造都伴随着明确的公式，并在岩浆和/或sagemath中实现。我们将这些算法与椭圆chabauty结合使用，以$ \ m \ mathbb {q} $ $ \ mathbb {q} $的数据集$ \ mathbb {q} $，Mordell-Weil等级$ 2 $ 2 $或$ 3 $以前没有被证明是未经证明的。我们分析了此方法在计算一组理性点以及在某些情况下导致其失败的频率。
Given a genus $2$ curve $C$ with a rational Weierstrass point defined over a number field, we construct a family of genus $5$ curves that realize descent by maximal unramified abelian two-covers of $C$, and describe explicit models of the isogeny classes of their Jacobians as restrictions of scalars of elliptic curves. All the constructions of this paper are accompanied by explicit formulas and implemented in Magma and/or SageMath. We apply these algorithms in combination with elliptic Chabauty to a dataset of 7692 genus $2$ quintic curves over $\mathbb{Q}$ of Mordell-Weil rank $2$ or $3$ whose sets of rational points have not previously been provably computed. We analyze how often this method succeeds in computing the set of rational points and what obstacles lead it to fail in some cases.
