Explicit Chabauty-Kim theory for the thrice punctured line

设X表示三次穿孔线，设S表示有限质数集。那么关于X的s积分点的集合早就知道是有限的，但是找到所有点的算法还没有被开发出来。由Minhyong Kim开创的一个新理论可能是开发在各种情况下寻找积分点的算法的关键。我与Stefan Wewers的合作使我们离实现x的目标更近了一步。我们提出的项目将致力于构建这样一个算法。先验地，我们的算法将只提供点数量的上界，但Kim的猜想表明，实际上这个上界将是尖锐的。我们将使用我们的算法来计算许多例子。反过来，我们的计算将有助于完善猜想，并提供支持或反对其合理性的证据。

Let X denotes the thrice punctured line, and let S denote a finite set of prime numbers. Then the set of S-integral points of X has been known for a long time to be finite, but an algorithm for finding all points has yet to be developed. A new theory pioneered by Minhyong Kim may hold the key to developing algorithms for finding integral points in a wide range of situations. My collaborative work with Stefan Wewers has brought us closer to achieving this goal for our X. The project being proposed will be devoted to constructing such an algorithm. A priori, our algorithm will only provide an upper bound for the number of points, but Kim's conjecture implies that in fact this bound will be sharp. We will use our algorithms to compute many examples. In turn, our computations will help refine the conjecture and give evidence for or against its plausibility.