Selmer schemes seminar

Schedule:

22/11: Alex Betts (Harvard)

Title: Weight filtrations on Selmer schemes and effective non-abelian Chabauty

Abstract: One well-known deficiency of non-abelian group cohomology is that it produces relatively structureless objects: pointed sets, instead of the abelian groups one obtains from abelian group cohomology. Accordingly, one would expect that Selmer schemes, and cohomology schemes in general, are similarly unstructured. The aim of this talk is to present a countervailing view: that Selmer schemes are in fact highly structured objects, and moreover that this structure is of relevance in the non-abelian Chabauty method. The particular structure we will discuss takes the form of a "weight filtration" on the affine rings of Selmer schemes, which is induced from the weight filtration on the fundamental group. Time permitting, we will discuss an application of this idea to bounding the number of solutions to the S-unit equation.

29/11: Ishai Dan-Cohen (Ben-Gurion University of the Negev)

Title: Towards higher Mordellic obstruction devices

Abstract: The "motivic Sullivan models" considered by I. Iwanari, may be used to construct a new gadget which extracts concrete information from the rational motivic homotopy type beyond pi_1. Examples suggest that this gadget may eventually be used for bounding sets of integral points as in the method of Chabauty and Kim when pi_1 fails to provide enough information.

5/11: Noam Kantor

Title: Selmer Stacks and the Method of Lawrence and Venkatesh: Chabauty-Kim with the Relative Completion

Abstract: In this talk, I will explain how to go beyond the unipotent theory of the Chabauty-Kim method by allowing for a reductive portion of the etale fundamental group. The constructions I describe unify the Chabauty-Kim method with the Lawrence-Venkatesh approach to rational points. On top of that, our constructions give a tower of moduli spaces that refine those of Lawrence-Venkatesh, and provide a surprising solution to their problem with "the centralizer of Frobenius." This all begins with the definition of a Selmer stack, a moduli space of non-abelian Galois representations that combines Kim's Selmer schemes and a moduli space of usual linear Galois representations.

12/11: Jonathan Pridham (Edinburgh)

Title: Some theory behind homotopical obstructions in the Chabauty-Kim method

Abstract: Koszul duality in its commutative-Lie incarnation gives a systematic way to relate cohomology and homotopy groups, and indeed Sullivan and Quillen rational homotopy types, while avoiding the arbitrary choices common in rational homotopy theory.

When applied to etale, crystalline, or even motivic cohomology theories, we can use it to construct various obstruction towers, with explicit cohomology groups receiving obstructions for rational points to exist, or for adelic points to be rational. I might rely on the audience to decide the level of generality they would like to see at this point.

26/11: David Jarossay (Ben-Gurion University of the Negev)

Title : Towards the explicit motivic Chabauty-Kim method for M_0,n - joint work with Ishai Dan-Cohen

Abstract : Ishai Dan-Cohen, Stefan Wewers and David Corwin have developed an explicit motivic version of the Chabauty-Kim method for P^1 - {0,1,\infty}. We are extending it to M_0,n.

3/12: Martin Lüdtke (Goethe University Frankfurt)

Title: Refined Selmer equations for the thrice-punctured line in depth 2

Abstract: Betts--Dogra have proposed a refinement of the Chabauty--Kim method to find S-integral points of a curve by including in the definition of the Selmer scheme local conditions on the primes contained in S. We are applying the theory in the case of the thrice-punctured line. We determine explicit equations for the refined Chabauty--Kim sets in depth 2 for #S=2 and show that they are satisfied precisely by the solutions of the S-unit equation in many cases. The results were obtained by this year's Arizona Winter School project group under the guidance of Alex Betts. This is joint work with him and Alex Best, Theresa Kumpitsch, Angus McAndrew, Lie Qian, Elie Studnia, and Yujie Xu.

17/12: Minhyong Kim (Warwick)

Title: Selmer schemes, reciprocity laws, and defining equations

Abstract: This will be a very speculative talk about the possibility of finding defining equations for Selmer schemes by way of explicit reciprocity laws for Galois cohomology classes. Serious mathematicians are invited to tune out.