Explicit Chabauty-Kim theory for the thrice punctured line

设X表示三次穿孔直线，S表示有限素数集。那么X的S积分点的集合是有限的早已为人所知，但求所有点的算法还没有发展出来。由Minhyong Kim开创的一项新理论可能掌握着开发在广泛情况下寻找积分点的算法的关键。我与Stefan Wewers的合作使我们更接近于为我们的X实现这一目标。正在提议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的上界，但Kim的猜想暗示，实际上这个界将是尖锐的。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，并提供证据支持或反对其合理性。

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.