Wenhan Dai

On Lawrence–Venkatesh’s proof of the Mordell conjecture

The purpose of this weekly seminar is to study Lawrence–Venkatesh’s proof of the Mordell conjecture. The main reference is [LV], and the survey [Po] gives a quick sketch for beginners. All talks will be given by participants, and each meeting will be about 90 minutes.

Organizer: Xinyi Yuan.
Time: 10:30–12:00 Wednesdays (starting on March 1).
Venue: Jia-yi-bing Big Lecture Hall, Jingchun Yuan.

Schedule

Part A

The following is about the proof argument of Mordell’s conjecture.

(3/1) Xinyi Yuan: Introduction and organization. I will review Faltings’ proof and sketch the main idea of [LV]’s proof. (notes)
(3/8) Han Hu: Review of the p-adic comparison theorems (étale / de Rham / crystalline). This is not in the paper, need an expert to do it. (my addendum notes (I))
(3/15) Han Hu: Continued with the introduction to p-adic comparison theorems. (my addendum notes (II))
(3/22) Shiquan Li: §3 of [LV]. Application of the comparison theorems to the Diophantine situation. (slides)
(3/29) Xingzhu Fang: §4 of [LV]. Prove the S-unit theorem. This is a baby case prior to the Mordell conjecture.
(4/5) Haoda Li: §5 of [LV]. Outline the proof of Mordell. Give a proof of [LV, Lemma 2.3] following [Wu, Proposition 2.7].
(4/12) Wenhan Dai: (To replace Xiangqian Yang) §6 of [LV]. This is to prove the technical Proposition 5.3 in §5. (notes)
(4/19) Binggang Qu: §7 of [LV]. Introduce the Kodaira–Parshin construction. This is purely geometric. (notes)
(No talk on PKU spring holidays.)
(5/24) Liang Xiao: §8 of [LV]. Compute monodromy of the Kodaira–Parshin construction. This is done over complex field by topological methods. Sketch the proof. (notes)

Part B

Up to now, the proof of Mordell’s conjecture is complete. The following is about families of hypersurfaces.

(Canceled) §9 of [LV]. The Bakker–Tsimerman theorem on on the period map.
(Canceled) §10 of [LV]. Proof of the main result on families of hypersurfaces.
(Canceled) §11 of [LV]. Proof of Proposition 10.6, which is mainly combinatorics and linear algebra.

References

[LV] Brian Lawrence, Akshay Venkatesh, Diophantine problems and p-adic period mappings. Invent. Math. 221 (2020), 893–999.
[Po] Bjorn Poonen, A p-adic approach to rational points on curves. Bull. Amer. Math. Soc. (N.S.) 58 (2021), no. 1, 45–56.
[Wu] G. Wüstholz, The finiteness theorems of Faltings. In: Rational points (Bonn, 1983/1984), Aspects Math., E6, pp.154–202. Friedr. Vieweg, Braunschweig (1984).