Wednesdays 4 pm, Ningzhai 宁斋 203, Tsinghua.
Organized by
Yichao Tian (MCM)
Yihang Zhu (YMSC)
The Mordell Conjecture was one of the greatest achievements in number theory in 20th century. It states that a projective smooth curve of genus at least 2 defined over a number field has only finitely many rational points. This conjecture was proposed by Mordell in 1922, and first proved by Faltings in 1983. Faltings’ approach uses deep results on abelian varieties, Galois representations and Arakelov geometry, and it reduces the proof to Tate conjecture and Shafarevich conjecture for abelian varieties, which also have great importance in number theory. Later on, Vojta gave a new proof of the Mordell Conjecture in 1989 using Diophantine approximation, and Lawrence and Venkatesh gave in 2019 a third proof using p-adic Hodge theory. Nowadays, topics related to further generalization and improvements of Mordell’s conjecture are still very active in the research of number theory.
This semester, we will continue from the previous semester and discuss the orginal proof by Faltings. We will still follow the outline in Lecture 1 of last semester. Notes.
Lecture 11, Oct 16: Theorem D' - Finiteness among isogenies induced by a p-divisible group. (朱艺航) notes
Ref: Rational Points Chapter IV, Bhatt-Snowden notes section 7.
Lecture 12, Oct 23: Theorems E,F,G - Semi-simplicity, Tate conjecture, and finiteness of isogeny classes. (李钒祎)
Ref: Rational Points IV, V.2; Bhatt-Snowden notes section 8.
Lecture 13, Oct 30: Theorem D - Finiteness within an isogeny class. (辜开源)
Ref: Rational Points V.3, Lipnowski notes. See also Bhatt-Snowden notes section 9 (for proof of Raynaud's theorem).
Lecture 14, Nov 7: Raynaud's theorem (李亦洲)
Lecture 15, Nov 21: The Kodaira-Parshin construction (扶磊) notes
Ref: Rational Points pp.183-197, Milne's notes IV.4, IV.5.
Lecture 16, Nov 27: Introduction to the Mordell Conjecture (袁新意) (Joint with TOPS.)
Xinyi Yuan's seminar program (for 2018)
Séminaire sur les pinceaux arithmétiques : la conjecture de Mordell
Abelian Varieties (Book draft by Edixhoven, van der Geer, Moonen)
Yihang Zhu's course notes on abelian varieties and Shimura varieties.
* For notes from the last semester, see the bottom part of this page.
Thursdays 2 pm. Location to be announced each time.
Organized by
Yichao Tian (MCM)
Yihang Zhu (YMSC)
The Mordell Conjecture was one of the greatest achievements in number theory in 20th century. It states that a projective smooth curve of genus at least 2 defined over a number field has only finitely many rational points. This conjecture was proposed by Mordell in 1922, and first proved by Faltings in 1983. Faltings’ approach uses deep results on abelian varieties, Galois representations and Arakelov geometry, and it reduces the proof to Tate conjecture and Shafarevich conjecture for abelian varieties, which also have great importance in number theory. Later on, Vojta gave a new proof of the Mordell Conjecture in 1989 using Diophantine approximation, and Lawrence and Venkatesh gave in 2019 a third proof using p-adic Hodge theory. Nowadays, topics related to further generalization and improvements of Mordell’s conjecture are still very active in the research of number theory.
During this semester and possibly continuing to the next, our goal is to go over Faltings’ original proof of Mordell’s conjecture, and if time allows, we may also discuss Lawrence--Venkatesh’s more recent approach. The aimed participants are advanced undergraduate students and beginning graduate students. We will ask professors from both Morningside Center and Tsinghua to give talks whenever suitable, while the students are also encouraged to volunteer to give talks.
*To Qiuzhen students: If your want to choose Number Theory as your future research direction, you are expected to participate in one of several number theory activities in order to have a solid background. The above seminar is one of the main activities we organize with the goal of preparing potential PhD students in number theory. The seminar will continue in the future semesters as well, and we hope to make it a platform for potential students and PhD advisors (from YMSC and MCM) to interact. More information on choosing number theory advisors will be announced later.
Lecture 1, Mar 14: Overview of the proof. (田一超) Notes.
Lecture 2, Mar 21: Abelian varieties (I). (朱艺航) Notes (handwritten) Latex version (感谢吴相东，张艺馨同学。经朱艺航修改。）
Lecture 3, Mar 28: Abelian varieties (II). (朱艺航) Latex notes (感谢吴相东，张艺馨同学。经朱艺航修改并补充。）
Lecture 4, Apr 11: Abelian varieties over the complex numbers. (谈夏羽)
Ref: Arithmetic Geometry Chapter IV.
Lecture 5, Apr 21: Abelian varieties over finite fields and Tate's theorem. (张驰)
Ref: Milne's notes Chapter IV, Bhatt-Snowden notes Section 2.
Lecture 6, Apr 25: Theory of heights. (黄治中 AMSS) notes by speaker
Ref: Rational Points Chapter II.
Lecture 7, May 9: Siegel modular varieties. (申旭 MCM) notes taken by Yihang
Lecture 8, May 23: Faltings height and Northcott property. (田一超 MCM) notes by speaker
Lecture 9, May 30: Finite group schemes and p-divisible groups (I). (许大昕 MCM) notes taken by Yihang
Lecture 10, Jun 13: Finite group schemes and p-divisible groups (II). (许大昕 MCM) notes by speaker
Ref for last two talks: Rational Points Chapter III.