Arithmetic Geometry and Number Theory RTG Seminar: Polylogarithms, Chabauty's Method, and the S-Unit Equation
Seminar | October 29 | 3:10-5 p.m. | 748 Evans Hall
David Corwin, UC Berkeley
In the first half, we will review Chabauty's and Skolem's methods. We will then explain how these can be generalized to the non-abelian Chabauty's method of Minhyong Kim. If time allows, I will also mention polylogarithms.
In the second half, we will describe work by the speaker and Ishai Dan-Cohen, building on previous work of Brown, Dan-Cohen, and Wewers, that computes with Kim's method in the case of the S-Unit equation, geometrically the projective line minus three points. For computational purposes, it is best to replace an abstract Galois group by an algebraic Tannakian Galois group, whose category of representations is equivalent to the relevant category of Galois representations. Being an algebraic group, this Tannakian Galois group is described by its Hopf algebra of regular functions. Elements of this Hopf algebra turn out to be motivic versions of special values of polylogarithms, and in the end, most of the explicit computations have to do with these motivic polylogarithms and their coproducts.