An explicit theory of heights for hyperelliptic Jacobians

这个项目的最终目标是改进和扩展求解形式为y^2=f(X)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，我们对方程所定义的曲线上的积分点或有理点感兴趣。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这样一个事实，即曲线可以嵌入到其雅可比变体中。这是一种维度等于曲线亏格的阿贝尔变种(在我们的例子中，亏格大约是f的一半)，因此承载着一个群的有用结构。要利用这种嵌入，我们需要对雅可比簇上的有理点群有足够的了解，它可以通过指定有限多个生成元来描述，典范高度理论是研究数域上定义的阿贝尔簇不可缺少的工具。除了大量的理论应用外，人们还需要能够计算给定有理点的标准高度，并且能够列举有界标准高度的所有有理点的集合，以便计算给定阿贝尔变种的有理点群的生成元。这是阿贝尔变种算法理论的基本任务之一，例如，对于具体的例子，需要用数字来验证著名的Birch和Swinnerton-Dyer猜想。当所讨论的阿贝尔变种是超椭圆曲线的雅可比变种时，这种生成器尤其有趣。在这种情况下，如果我们有一组有理点的生成器可用，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和全集的积分点。到目前为止，超椭圆曲线的雅可比上的(正则)高度的显式理论大多限于亏格2或3的曲线。限制在小亏格的一个原因是，人们首先需要所谓的雅可比的Kummer簇的显式理论，目前仅适用于亏格3。在提议的项目中，我们将把关于亏格2和亏格3的已知结果推广到更大的亏格，从Kummer簇的显式理论开始。一方面，这将产生至少5个亏格的显式公式和有效的算法，这对于亏格3甚至更大的亏格基本上应该是最优的。我们将实现这些算法，从而使中等亏格的生成元的有效计算成为可能，应用如上所述。另一方面，我们期望这些显式公式可以推广到任意亏格(也可能是非超椭圆曲线)，然后我们将尝试证明这一点。为此，我们将把亏格2和亏格3的一些明确的结果和证明概念化，这也将导致对正则高度理论的更深层次的理解。

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.