Explicit Chabauty-Kim theory for the thrice punctured line

• Applicant: Dr. Ishai Dan-Cohen
• Subject Area: Mathematics
• Term: from 2013 to 2015
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 239470564

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.

• DFG Programme: Priority Programmes
• Subproject of: SPP 1489: Algorithmic and Experimental Methods in Algebra, Geometry and Number Theory
• International Connection: USA
• Participating Person: Professorin Dr. Jennifer S. Balakrishnan