Quadratic Chabauty for integral points

有理数域上亏格至少为2的代数曲线上有理点或整点的显式计算有许多重要的应用，但通常是一个非常困难的问题。Chabauty的方法通常成功地确定合理的点使用p-adic分析，但它仅限于曲线的雅可比矩阵具有严格小于亏格的Mordell-Weil秩。为了解决这个问题，M。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（或至少是积分）点应该是迭代p-adic积分组合的零点。本着Kim的哲学精神，申请人在与J. Balakrishnan和A. Besser给出了当秩等于亏格且曲线为奇次超椭圆曲线时，积分在积分点处消失的表达式。在这个项目中，我们将把这种方法推广到偶数次超椭圆曲线、超椭圆曲线和光滑平面四次曲线，它们的雅可比行列式的秩等于亏格。我们也将联合收割机的技术与一个新的，但相关的方法也基于p-adic高度，这将简化的方法，使其适用于一些更大的秩的例子，以及发展必要的理论后，我们将实现完整的算法计算积分点的曲线。

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.