The S-unit equation and non-abelian Chabauty in depth 2

Abstract: The S-unit equation is a classical and well-studied Diophantine equation, with numerous connections to other Diophantine problems. Recent work of Kim and refinements due to Betts-Dogra have suggested new cohomological strategies to find rational and integral points on curves, based on but massively extending the classical method of Chabauty. At present, these methods are only conjecturally guaranteed to succeed in general, but they promise several applications in arithmetic geometry if they could be proved to always work. In order to better understand the conjectures of Kim that suggest that this method should work, we consider the case of the thrice punctured projective line, in "depth 2", the "smallest" non-trivial extension of the classical method. In doing so we get very explicit results for some S-unit equations, demonstrating the usability of the aforementioned cohomological methods in this setting. To do this we determine explicitly equations for (maps between) the (refined) Selmer schemes defined by Kim, and Betts-Dogra, which turn out to have some particularly simple forms. This is joint work with Alexander Betts, Theresa Kumpitsch, Martin Lüdtke, Angus McAndrew, Lie Qian, Elie Studnia, and Yujie Xu .

number theory

Audience: researchers in the topic

London number theory seminar

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