Algorithmic and Experimental Arithmetic Geometry

这项建议包含了两个项目，它们以不同的方式与代数簇的运算有关，特别是与它们的有理点的研究有关。第一个项目的目标是设计和实现一种算法，为给定的变量生成一组很好的方程；这对于用它进行进一步的计算很重要。在许多情况下，如果没有这样一个好的模型，进一步的计算是不可能的。第二个方案更具体地涉及高亏格曲线上的有理点。众所周知，在每条这样的曲线上只能有有限多个有理点，但到目前为止还没有明确确定这一集的算法。我们将改进和扩展现有的算法，涵盖某些情况。然后，我们将使用它们从许多曲线中收集统计信息，以更准确地了解它们的有理点的行为。另一方面，我们计划研究一种通用算法的可能方法，该算法可以在任何给定曲线上找到有理点集。

This proposal contains two projects related in various ways to the arithmetic of algebraic varieties and to the study of their rational points in particular. The aim of the first project is to devise and implement an algorithm that produces a nice set of equations for a given variety; this is important for doing further computations with it. In many cases, further computations would be infeasible without such a nice model. The second project is more specifically concerned with rational points on curves of higher genus. It is known that on each such curve there can be only finitely many rational points, but so far there is no algorithm that determines this set explicitly. We will improve and extend existing algorithms covering certain cases. We will then use them to gather statistical information from many curves to get some more precise idea on the behavior of their rational points. On the other hand, we plan to study possible approaches to a general algorithm that finds the set of rational points on any given curve.