Quadratic Chabauty for integral points

在有理数上定义的至少两个属的代数曲线上的有理点或积分点的显式计算有许多重要的应用，但通常是一个众所周知的难题。Chabauty的方法经常成功地利用p进分析确定有理点，但它仅限于雅可比矩阵的莫德尔-韦尔秩严格小于格的曲线。为了克服这个问题，Kim提出了一个基于p-adic Hodge理论的扩展Chabauty方法的框架。他的理论预言，有理点（或至少是积分点）应该是迭代p进积分组合的零点。然而，将Kim的方法直接用于显式计算似乎非常困难。在Kim的哲学精神下，申请人在早期与J. Balakrishnan和A. Besser的联合工作中使用p进高度明确地写下了当秩等于属且曲线为奇次超椭圆时消失在积分点上的积分。在本课题中，我们将此技术推广到偶数次的超椭圆曲线、超椭圆曲线以及雅可比矩阵秩等于属的光滑平面四分体。我们还将该技术与基于p进高度的一种新的但相关的方法结合起来，这将简化该方法并使其适用于一些更大秩的示例。在发展了必要的理论之后，我们将实现计算这类曲线上积分点的完整算法。

The explicit computation of rational or integral points on algebraic curves of genus at least two defined over the rationals has many important applications, but is a notoriously difficult problem in general. Chabauty's method often succeeds in determining the rational points using p-adic analysis, but it is restricted to curves whose Jacobians have Mordell-Weil rank strictly less than the genus. To overcome this issue, M. Kim proposed a framework for extending Chabauty's method based on p-adic Hodge theory. His theory predicts that rational (or at least integral) points should be zeros of combinations of iterated p-adic integrals. However, it seems very difficult to use Kim's approach directly for explicit computations.In the spirit of Kim's philosophy, the applicant used p-adic heights in earlier joint work with J. Balakrishnan and A. Besser to explicitly write down such integrals vanishing in integral points when the rank equals the genus and the curve is hyperelliptic of odd degree. In the proposed project, we will extend this technique to hyperelliptic curves of even degree, superelliptic curves and smooth plane quartics whose Jacobians have rank equal to the genus. We will also combine the technique with a new, but related approach also based on p-adic heights, which will simplify the method and make it applicable for some larger rank examples as well.After developing the necessary theory, we will implement complete algorithms for the computation of integral points on such curves.