Abelian and nonabelian Chabauty: reading course

This is the home page for a reading course to be held at UC San Diego during the fall 2019 and winter 2020 quarters, in preparation for the 2020 Arizona Winter School. The topic is the abelian (Chabauty-Coleman) and nonabelian (Chabauty-Kim, Balakrishnan-Dogra) methods for rigorously finding all rational points on suitable curves over number fields.

During the winter quarter, the seminar will meet Fridays 9:30-11:00 in APM 7218.

• January 10: Tannakian categories (Thomas Grubb). READING: Deligne and Milne, Tannakian categories (pdf).
• January 17: NO MEETING.
• January 24: Finiteness theorems via Kim's method (Baiming Qiao). READING: Kim, The motivic fundamental group of the projective line minus three points and the theorem of Siegel (pdf); Kim, The unipotent Albanese map and the Selmer varieties for curves (pdf).
• January 31: Poincaré bundles and quadratic Chabauty (Peter Wear). READING: Mumford, Abelian Varieties, chapter 1; Edixhoven and Lido, Geometric quadratic Chabauty (pdf).
• February 7: p-adic heights (Mingjie Chen). READING: Coleman and Gross, p-adic heights on curves (pdf); Balakrishnan and Besser, Coleman-Gross heights and the p-adic sigma function (pdf). See also this video by Balakrishnan.
• February 14: quadratic Chabauty for modular curves (Thomas Grubb). READING: Siksek, Quadratic Chabauty for modular curves (pdf); Dogra and Le Fourn, Quadratic Chabauty for modular curves and modular forms of rank one (pdf).
• February 21: quadratic Chabauty vs. Lawrence-Venkatesh (Peter Wear). READING: Balakrishnan et al, Two recent p-adic approaches towards the (effective) Mordell conjecture (pdf).
• February 28: explicit nonabelian Chabauty via motives (guest lecture by David Corwin). READING: Dan-Cohen and Wewers, Mixed Tate motives and the unit equation (pdf); Corwin, From Chabauty's method to Kim's non-abelian Chabauty's method (pdf).
• No meetings March 6 or March 13.

During the fall quarter, the seminar met Fridays 10:30-12:00 in APM 5829.

• October 11: overview of abelian Chabauty (Kiran Kedlaya). READING: W. McCallum and B. Poonen, The method of Chabauty and Coleman (pdf).
• October 18: the Mordell-Weil sieve (Harry Smit). READING: S. Siksek, Chabauty and the Mordell-Weil sieve, section 16 (pdf).
• October 25: Coleman integration (Mingjie Chen). READINGS: R. Coleman, Torsion points on curves and p-adic Abelian integrals (pdf); R. Coleman, Effective Chabauty (pdf).
• November 1: more examples of Chabauty (interactive session led by Harry Smit; notes on CoCalc). If possible, please bring a laptop with access to Magma (UCSD students: see sysadmins for a free copy).
• November 8: the Poonen-Stoll theorem (Thomas Grubb). READING: B. Poonen and M. Stoll, Most odd degree hyperelliptic curves have only one rational point (pdf); see also this video by Poonen.
• November 15: overview of nonabelian Chabauty (Baiming Qiao). READING: D. Corwin, From Chabauty's method to Kim's non-abelian Chabauty method (pdf).
• November 22: Chabauty at a prime of bad reduction (Peter Wear). READING: E. Katz, J. Rabinoff, and D. Zureick-Brown, Diophantine and tropical geometry, and uniformity of rational points on curves (pdf).
• December 6: continuation of Nov 22.
• On hiatus until January.

Possible additional topics and suggested readings:

• Chabauty for modular curves; Baker's reinterpretation of Kamienny's criterion. READING: M. Baker, Kamienny's criterion and the method of Chabauty and Coleman (pdf).
• Numerical methods for computing Coleman integrals; use of Frobenius lifting. READING: J. Balakrishnan, R. Bradshaw, and K.S. Kedlaya, Explicit Coleman integration for hyperelliptic curves (pdf).
• Reinterpretation via Grothendieck's section conjecture; segue to Kim's construction.
• Kim's definition of Selmer varieties.
• Quadratic Chabauty. READING: J. Balakrishnan and N. Dogra, Quadratic Chabauty I: p-adic heights (pdf); same authors, Quadratic Chabauty II: Generalised height functions on Selmer varieties (pdf); same authors, An effective Chabauty-Kim theorem (pdf).
• READING: J. Balakrishnan, N. Dogra, J.S. Müller, J. Tuitman, and J. Vonk, Explicit Chabauty-Kim for the split Cartan modular curve of level 13 (pdf).
• READING: Bresciani-Demeio-Lido-Lombardo nonabelian Chabauty study group (pdf).
• Lawrence's approach to the Mordell conjecture.