Rational Points on Curves and Iterated p-adic Integrals

曲线上有理点的现代研究起源于古希腊人的数学，他们研究多项式方程的整数解。这些方程可能会变得非常棘手，有时会抵抗几个世纪的努力。 曲线的一个数值不变量--亏格--可以揭示相当多的关于它的有理点集的基本性质。 事实上，在1922年，莫德尔证明，一套合理的点曲线属至少2是有限的。这一点在近60年后被法尔明斯证明。然而，Faltings的证明并没有明确地构造这些曲线的有理点的有限集。产生一个构造性的，算法版本的法尔明斯定理仍然是一个具有挑战性的开放问题在数论。这个项目的主要目的是解决这个问题，通过迭代p-adic积分中的工具，给出新的算法来显式地找到某些高亏格曲线上的有理点。 PI将结合联合收割机的理论和明确的计算技术，以证明非阿贝尔Chabauty程序所带来的结果。 具体来说，PI将研究方法定位在一个稍大的有限集的合理点的p-adic点来自nonabelian Chabauty，并会给出实用的算法计算的p-adic函数切断这些p-adic点。为了促进这一点，PI将使用p-adic上同调的思想来产生迭代p-adic积分的快速算法。

The modern study of rational points on curves has its origins in the mathematics of the ancient Greeks, who studied whole number solutions to polynomial equations. These equations can turn out to be tremendously tricky to solve, sometimes resisting efforts for centuries. A numerical invariant -- the genus -- of the curve can reveal quite a bit about the fundamental nature of its set of rational points. Indeed, in 1922, Mordell conjectured that the set of rational points on curves of genus at least 2 is finite. This was proved nearly 60 years later by Faltings. However, Faltings' proof does not explicitly construct the finite set of rational points for these curves. Producing a constructive, algorithmic version of Faltings' theorem remains a challenging open problem in number theory. The main aim of this project is to address this problem by giving new algorithms for explicitly finding rational points on certain higher genus curves via tools in iterated p-adic integration. The PI will combine theoretical machinery and explicit computational techniques to prove results conjectured by the nonabelian Chabauty program. Specifically, the PI will study methods situating the finite set of rational points in a slightly larger finite set of p-adic points coming from nonabelian Chabauty and will give practical algorithms for computing the p-adic functions cutting out these p-adic points. To facilitate this, the PI will use ideas from p-adic cohomology to produce fast algorithms for iterated p-adic integration.