An explicit theory of heights for hyperelliptic Jacobians

超椭圆雅可比行列式的显式高度理论

• 批准号: 372107645
• 负责人: Professor Dr. Jan Steffen Müller
• 金额: --
• 依托单位: Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2023-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/372107645?language=en
• 关键词: explicit; theory; heights; hyperelliptic; Jacobians

The ultimate goal of this project is to improve and extend methods for solving diophantine equations of the form y^2 = f(x), where f is a polynomial, in integers or rational numbers. Equivalently, we are interested in the integral or rational points on the curve defined by the equation. Curves of this type are said to be hyperelliptic. Most of the available methods make use of the fact that the curve can be embedded into its Jacobian variety. This is an abelian variety of dimension equal to the genus of the curve (in our case, the genus is roughly half the degree of f) and thus carries the helpful structure of a group. To make use of this embedding, we need to know enough about the group of rational points on the Jacobian variety, which can be described by specifying finitely many generators.The theory of canonical heights is an indispensable tool when studying abelian varieties defined over number fields. Besides numerous theoretical applications, one needs to be able to compute the canonical height of a given rational point and to enumerate the set of all rational points of bounded canonical height in order to compute generators for the group of rational points of a given abelian variety. This is one of the fundamental tasks in the algorithmic theory of abelian varieties and is required, for instance, to numerically verify the celebrated conjecture of Birch and Swinnerton-Dyer for concrete examples. Such generators are especially interesting when the abelian variety in question is the Jacobian variety of a hyperelliptic curve. If, in this situation, we have generators for the group of rational points available, then there are efficient algorithms to compute the rational points on the curve with height below a prescribed bound, and the full set of integral points.Thus far, the explicit theory of (canonical) heights on Jacobians of hyperelliptic curves has been mostly restricted to curves of genus 2 or 3. One reason for the restriction to small genus is that one first needs an explicit theory for the so-called Kummer variety of the Jacobian, which at the moment is only available for genus at most 3.In the proposed project, we will extend the known results for genus 2 and 3 to larger genus, starting with an explicit theory of the Kummer variety. On the one hand, this will yield explicit formulas and efficient algorithms for genus at least up to 5, which should be essentially optimal for genus up to 3 and possibly beyond. We will implement these algorithms, thereby making the efficient computation of generators in moderate genus possible, with applications as discussed above. On the other hand, we expect that these explicit formulas will suggest generalizations to arbitrary genus (and possibly to non-hyperelliptic curves), which we will then attempt to prove. To this end, we will conceptualize some of the explicit results and proofs for genus 2 and 3, which will also lead to a deeper understanding of the theory of canonical heights.

该项目的最终目标是改进和扩展求解 y^2 = f(x) 形式的丢番图方程的方法，其中 f 是整数或有理数的多项式。同样，我们感兴趣的是方程定义的曲线上的积分点或有理点。这种类型的曲线被称为超椭圆曲线。大多数可用的方法都利用了曲线可以嵌入其雅可比变体这一事实。这是一个阿贝尔变体，其维数等于曲线的亏格（在我们的例子中，亏格大约是 f 的阶次的一半），因此具有群的有用结构。为了利用这种嵌入，我们需要充分了解雅可比簇上的有理点群，这可以通过指定有限多个生成元来描述。规范高度理论是研究数域上定义的阿贝尔簇时不可或缺的工具。除了大量的理论应用之外，我们还需要能够计算给定有理点的规范高度，并枚举有界规范高度的所有有理点的集合，以便计算给定阿贝尔簇的有理点组的生成元。这是阿贝尔簇算法理论的基本任务之一，例如，需要通过具体例子来数值验证 Birch 和 Swinnerton-Dyer 的著名猜想。当所讨论的阿贝尔簇是超椭圆曲线的雅可比簇时，这种生成器特别有趣。在这种情况下，如果我们有可用的有理点组的生成器，那么就有有效的算法来计算曲线上高度低于规定界限的有理点以及完整的积分点。到目前为止，超椭圆曲线雅可比行列式的（规范）高度的显式理论主要局限于 2 或 3 的曲线。限制为小亏格的一个原因是，首先需要 雅可比行列式所谓 Kummer 变体的显式理论，目前仅适用于最多 3 属。在拟议的项目中，我们将从 Kummer 变体的显式理论开始，将 2 和 3 属的已知结果扩展到更大的属。一方面，这将为至少最多 5 个属生成明确的公式和有效算法，这对于最多 3 个甚至可能更多的属来说应该是最佳的。我们将实现这些算法，从而使中等属中的生成器的有效计算成为可能，并具有如上所述的应用程序。另一方面，我们期望这些显式公式将建议对任意属（并且可能对非超椭圆曲线）的概括，然后我们将尝试证明这一点。为此，我们将对属2和属3的一些显式结果和证明进行概念化，这也将导致对规范高度理论的更深入的理解。