Non-abelian Chabauty method for motivic and rational approximation

The Chabauty-Coleman method

This lecture will discuss Chabauty’s theorem and Coleman’s formulation of the argument in terms of abelian integrals. We will then examine Kim’s reinterpretation of the Chabauty-Coleman method using the Tate module of the Jacobian and Kummer maps. Finally, we will introduce groupoids with Galois action, non-abelian cohomology, and non-abelian Kummer maps.

The Tannakian formalism

This lecture will give an overview of the Tannakian formalism, which associates a “fundamental group” to categories resembling categories of local systems on a space. We will briefly introduce pro-unipotent groups and their corresponding Tannakian categories, followed by examples of unipotent Tannakian categories leading to the definition of pro-unipotent étale, de Rham, and rigid fundamental groupoids. Finally, we will discuss Deligne’s motivic philosophy for motivic fundamental groups, with a possible introduction to the mixed Tate motivic fundamental groupoid.

Selmer Schemes and the Chabauty-Kim Locus

This lecture will give a brief overview of Bloch-Kato Selmer groups and introduce local and global Selmer schemes relative to systems of local Selmer conditions. We will prove their representability and then construct the Chabauty-Kim square and define the Chabauty-Kim locus. Finally, we will sketch the proof of the finiteness criterion.

Iterated Coleman Integration

This lecture will begin with a brief overview of the challenges of integration in the p-adic setting and Coleman’s solution using analytic continuation along Frobenius. We will then discuss integrals and iterated integrals, and their relation to parallel transport in unipotent vector bundles with integrable connection. Next, we will introduce Besser’s Tannakian approach to iterated Coleman integration and sketch the proof that the local non-abelian Kummer map is given by iterated Coleman integrals.

Computations of Rational and Integral Points

Material to be shared at a later date.

Computations of Rational and Integral Points

This lecture will discuss the computation of the Z[1/2]-integral points on the thrice-punctured line using non-abelian Chabauty. We will then give an overview of quadratic Chabauty and illustrate the method through an example.

Computing Chabauty-Kim loci in SageMath

To be announced.

References

• D. Corwin, From Chabauty's method to Kim's non-abelian Chabauty's method (unpublished notes)
• L. A. Betts, Foundations of non-abelian Chabauty (unpublished notes)
• W. McCallum and B. Poonen, The method of Chabauty and Coleman
• M. Kim, The motivic fundamental group of the projective line minus three points and the theorem of Siegel
• M. Kim, The unipotent Albanese map and Selmer varieties for curves
• G. Faltings, Mathematics around Kim's new proof of Siegel's Theorem
• P. Deligne, Le Groupe Fondamental de la Droite Projective Moins Trois Points
• M. Hadian, Motivic fundamental groups and integral points

• S. Bloch and K. Kato, L-functions and Tamagawa Numbers of Motives
• J. Bellaïche, Introduction to the conjecture of Bloch and Kato (notes from the Clay Mathematical Institute Summer School, Honolulu, Hawaii, 2009)
• L. A. Betts, Weight filtrations on Selmer schemes and the effective Chabauty-Kim method
• A. Besser, Coleman integration via the Tannakian formalism
• I. Dan-Cohen and S. Wewers, Explicit Chabauty-Kim theory for the thrice punctured line in depth two
• J. S. Balakrishnan and N. Dogra, Quadratic Chabauty and rational points I: p-adic heights

Lectures - Inter-universal Teichmüller principles in arithmetic geometry

Scientific committee: A. Betts (Cornell University, US), B. Collas (RIMS Kyoto University, JP), M. Kim (ICMS Edinburgh, UK), Y. Yatagawa (IS Tokyo, JP)

Org.: B. Collas (RIMS Kyoto, JP), T. Holzschuh (IHES, FR), S. Kelly (Tokyo Univ., JP)