Alexander Betts' webpage

Math 283Z: Foundations of non-abelian Chabauty

Harvard Spring 2023

The non-abelian Chabauty method of Minhyong Kim is one of the premier methods for computing rational or integral points on curves in practice. This course will give a thorough description of the foundations of the method and the theoretical ingredients it combines, and will conclude with an overview of some computations of rational or integral points. A course summary can be found here.

Lecture notes

These are copies of the notes I used for the lectures. Please feel free to contact me via email if you have any comments, queries or corrections. The eventual aim is to write these up properly into a book -- watch this space!

• Lecture 1: Descent and Chabauty--Coleman.
• Lecture 2: Groupoids and the profinite etale fundamental group.
• Lecture 3: Unipotent and pro-unipotent groups.
• Lecture 4: The pro-unipotent etale fundamental group I: Malcev completion.
• Lecture 5: The Tannakian formalism.
• Lecture 6: The pro-unipotent etale fundamental group II: unipotent Q_p-local systems.
• Lecture 7: The Galois action on the etale fundamental group.
• Lecture 8: Non-abelian cohomology.
• Lecture 9: Cohomology of pro-unipotent groups.
• Lecture 10: Bloch--Kato Selmer schemes.
• Lecture 11: The non-abelian Chabauty method, I.

Problems sheets

• Sheet 1: Groupoids and profinite etale fundamental groups.
• Sheet 2: Pro-categories, descending central series and unipotent groups.
• Sheet 3: Malcev completion and the Tannakian formalism.
• Sheet 4: Galois action and non-abelian cohomology.
• Sheet 5: Cohomology schemes and the Bloch--Kato exponential.
• Sheet 6: The Chabauty--Kim criterion.
• Final Exam. | Solutions.