Explicit Chabauty-Kim theory for the thrice punctured line

三次穿刺线的显式 Chabauty-Kim 理论

• 批准号: 239470564
• 负责人: Dr. Ishai Dan-Cohen
• 金额: --
• 依托单位: Department of Mathematics and Statistics
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2013
• 资助国家: 德国
• 起止时间: 2012-12-31 至 2014-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/239470564?language=en
• 关键词: Explicit; Chabauty; Kim; theory; thrice

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.

设X表示三次穿孔线，S表示有限素数集。那么X的S-整点的集合在很长一段时间内都是有限的，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的合作使我们更接近实现X的这一目标。正在提出的项目将致力于构建这样一个算法。先验，我们的算法将只提供一个上限的点的数量，但金的猜想意味着，事实上，这个界限将是尖锐的。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可验证性的证据。