The S-unit equation and non-abelian Chabauty in depth 2
Linfoot Number Theory Seminar
16th June 2021, 11:00 am – 12:00 pm
Fry Building, Online
The S-unit equation is a classical and well-studied Diophantine equation, with numerous connections to other Diophantine problems.
Recently work of Kim and refinements due to Betts-Dogra, have suggested new cohomological strategies to find rational and integral points on curves, based on the classical method of Chabauty. In most cases, these methods are only conjecturally guaranteed to work, but they promise several applications in arithmetic geometry if they could be proved to always succeed.
In order to better understand the conjectures of Kim that suggest that this method should succeed, we look at 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. 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