Chabauty–Coleman's method

Organized by Vidhu Adhihetty and Matthew Hase-Liu

For a nice curve defined over a number field with genus at least two, understanding its set of rational points is a fundamental problem in arithmetic geometry. Faltings famously proved that this set is finite, but his proof is ineffective, i.e. he did not provide an algorithm to compute the rational points. This learning seminar will focus on the ideas surrounding the method of Chabauty and Coleman, which gives an explicit strategy for computing rational points on curves whose Jacobian has low rank. We will also discuss variants and generalizations, such as non-abelian Chabauty, quadratic Chabauty, and Chabauty–Coleman for surfaces, as well as applications to arithmetic statistics (e.g. the work of Poonen–Stoll) and Mazur’s Program B.

We will follow parts of "p-adic approaches to rational and integral points on curves" by Poonen, "The method of Chabauty and Coleman" by McCallum–Poonen, "Abelian Chabauty" by Zureick-Brown, Kim's lecture notes on the foundations of non-abelian Chabauty, "Most odd degree hyperelliptic curves have only one rational point" by Poonen–Stoll, "Geometric quadratic Chabauty" by Edixhoven–Lido, "Computational tools for quadratic Chabauty" by Balakrishnan–Müller (many references from the corresponding Arizona Winter School), and "A Chabauty-Coleman bound for surfaces" by Caro–Pasten.

References

Here is a list of the references used in the seminar:

Schedule

We meet on Wednesday from 4:30 to 5:30 in Room 622.

Date Speaker Abstract References
09/10 Matthew Hase-Liu Overview of seminar and introduction to Chabauty–Coleman: This will mainly be an organizational meeting where we list out the topics and distribute talks. Time-permitting, I will begin explaining the overall strategy. Notes
09/17 Vidhu Adhihetty Coleman Integration: This week, we will develop the theory of Coleman integration on rigid varieties and see how it helps us prove finiteness statements for rational points on curves. Notes
09/24 Vidhu Adhihetty Explicit examples of effective point-counts via Chabauty-Coleman: We will wrap up our discussion of the Coleman integral and how it relates to and improves upon Chabauty's original idea. Afterwards, we will apply these ideas to get at effective point counts for explicit curves Notes
10/01 Matthew Hase-Liu Most odd degree hyperelliptic curves have only one rational point: We will discuss Poonen-Stoll's theorem that a positive fraction of odd degree hyperelliptic curves have only one rational point, and moreover that this fraction tends to 1 as the genus tends to infinity. Notes
10/08 Ethan Bottomley-Mason Nonabelian Chabauty: We understand the Chabauty method intrinsically in terms of the curve itself rather than the Jacobian. This extracts the essential information that the Jacobian provided and allows for generalizations of the method. We will see this via embeddings into semiabelian varieties and by utilizing non-abelian pro-p quotients of the fundamental group. Notes