Working seminar on rational points on curves (and higher dimensional varieties)

Organiser: Adrian Iovita, Jackson Morrow, Marc-Hubert Nicole, and Henri Darmon.

Time. Monday, 9:30-11:30 AM in room 6214 Aisensatdt Pavillion.

The goal of this seminar is to tie up the loose ends from the ill-fated Lawrence-Venkatesh seminar of Spring 2020 which was rudely interrupted by the onset of the pandemic.

The theme of this seminar is to discuss the various approaches to studying rational points on curves and higher dimensional varieties, with an emphasis on their common unifying features and on the recent new approach of Lawrence and Venkatesh.

The seminar is aimed at local Montreal participants who are encouraged to attend in person if at all possible. However, there will also be an on-line component for those who are unable to grace the CRM with their physical presence.

Participants

Patrick Allen
David Ayotte
Angelica Babei
Lea Beneish
Henri Darmon
Oriol Velasco Falguera
Francesc Gispert-Sanchez (remotely)
Ting-Han Huang
Adrian Iovita
Wanlin Li
David Lilienfeldt
Mike Lipnowski
Jackson Morrow
Oriol Navarro
Isabella Negrini
Marc-Hubert Nicole
Alice Pozzi (TBC)
James Rickards
Marti Roset Julia
Giovanni Rosso.
Ju-Feng Wu (remotely)
Peter Xu

Reading list . Here are a few references, in roughly increasing order of sophistication.

A rather pedestrian expository article aimed at beginning graduate students, based on a Clay summer school that took place in Gottingen during the Dark Ages when Chabauty-Kim and Lawrence-Venkatesh had not yet been discovered.

An article of Bjorn Poonen which compares and contrasts the Chabauty—Kim and Lawrence—Venkatesh methods.

A seven author paper by mathematicians ranging from Balakrishnan to Vonk, which presents the Chabauty—Kim and Lawrence—Venkatesh methods with a greater focus on computational aspects and questions of effectivity.

Last but certainly not least, the original article of Lawrence and Venkatesh.

Program

Week of January 11-15.
Speakers: Jackson and Henri
Jackson's notes.
Overview of the broad strategy of proof of Faltings, and of the proof of Lawrence-Venkatesh, stressing their common features as well as their differences. One possible approach to this would be to update the obsolete Clay summer school notes by fitting LV into a similar framework.

Week of January 18-22.
Speaker: Lea.
The Shafarevich problem and the Kodaira-Parshin construction reducing Mordell to Shafarevich.

Week of January 25-29.
Speaker: Angelica.
Galois representations.
Proof of Faltings' finiteness theorem for rational $\ell$-adic Galois representations with fixed Serre weights and bounded ramification.
References: [Dar, 2.5 -- 2.7]

Week of February 1-5.
Speaker: Alice.
Explanation, following Tate and Faltings, of how the semisimplicity of Galois representations, and the Tate conjectures, are obtained from the finiteness of isogeny classes over a field k.

Week of February 8-12.
Speaker: Francesc.
Algebraic de Rham cohomology and the Gauss-Manin connection.
Define hypercohomology and algebraic de Rham cohomology, and use GAGA to prove that for a smooth projective complex variety it agrees with classical de Rham cohomology (Grothendieck, first two pages). Recall the definition of connection, explain the algebraic definition of the Gauss-Manin connection in terms of differentiating k-forms (Katz-Oda up to Section 2)

Week of February 15-19.
Speaker: Adrian.
Period mappings and Galois representations- I.
§3.4 Begin with briefly recalling some basic facts about crystalline cohomology and discuss the commutative diagram (3.9). In the rest of the talk, focus on the complex and p-adic period mapping, and prove Lemmas 3.1 and 3.2.

Week of February 22-27.
Speaker: Angelica.
Period mappings and Galois representations- II.
Lemma 2.3 and §3.5 Recall the notion of a crystalline representation and the crystalline comparison theorem of Faltings. Recall the commutative diagram (3.9), and what it implies for the fully faithful embedding (3.12). In the second half of the talk, discuss in detail the proof of Proposition 3.4. The style of argument is repeated several times in the article and hence is crucial. You can assume the statement of Lemma 2.3 or outline a proof depending on the constraint on time.

Week of March 1-5.
Speaker: Marti.
Reference: Siegel's theorem via the Lawrence-Venkatesh method.
The S-unit equation-I.
§4.1-4.2 Discuss in detail how to obtain the desired finiteness from Lemma 4.2. After this, introduce the modified Legendre family which will be the target of the afore- mentioned Proposition 3.4. On an unrelated note, state and prove Lemma 2.1 as a warm-up for the kind of linear algebra involved in this set-up. This will be used in the next talk.

Week of March 8-12.
Speaker: Jackson.
The S-unit equation-II.
Secs. 4.3 – 4.4 Assuming the statement of Lemmas 4.3 (big monodromy) and 4.4 (generic simplicity), prove Lemma 4.2.
References: TBD.

Week of March 15-19.
Speaker: Wanlin Li.
Proof of Faltings’s Theorem.
Cover the whole of [LV, §5]. In particular, state Proposition 5.3 without proof and give an outlook on the key properties of the Kodaira-Parshin family (which will be introduced in a later talk). Explain how Theorem 5.4 (Falting’s theorem) is obtained by applying Proposition 5.3 to the Kodaira-Parshin family.
References: TBD.

Week of March 22-26.
Speaker: Daniel Barrera.
Topic: Rational points on the base of an abelian-by-finite family, I.
The goal of the next two talks is to prove [LV, Proposition 5.3]. This takes up the whole of [LV18, §6]. State and motivate the definition of an abelian-by-finite family (see Definition 5.1). Discuss what it means for such a family to have full monodromy. Begin by discussing the restrictions on the p-adic period mapping at a base point of an abelian-by-finite family. In the next part of the talk discuss the Equations (6.1)-(6.7) explicitly, and show that Proposition 5.3 can be deduced from Lemmas 2.3, 6.1 and 6.2.

Week of March 29- April 2.
Speaker: TBD.
Rational points on the base of an abelian-by-finite family, II.
Begin by recalling the statements of Lemmas 2.1 and 3.2. Also recall the fully faithful embedding (3.12) and the Crystalline comparision theorem, and apply it to the situation in hand. Complete the proof of Lemma 6.2. Next shift the attention to proving Lemma 6.1 (generic simplicity). Begin by stating Lemma 2.10 and if possible indicate a proof. Use this to prove the Sublemma. Recall the equation (6.7) to conclude the proof of Lemma 6.1 modulo the general position Lemmas 6.3 and 6.4, which can be stated without proof.
References: TBD.

Week of April 5-9.
Speaker: Marc-Hubert.
Kodaira-Parshin Family.
In this talk our aim will be to construct the Kodaira-Parshin family which is an analogue of the modified Legendre family from a previous talk. State the definition of a Prym variety and its variant the reduced Prym variety (see Section 7.2). Note the dimension of the reduced Prym variety. Show that the reduced Prym variety can also be described using (7.2), and thus has a natural relative variant. Construct the Kodaira-Parshin family assuming Proposition 7.1. Conclude by stating the key properties (i)-(iii) satisfied by the Kodaira-Parshin covers as stated in Section 5.
References: TBD.

Week of April 12-16.
Speaker: Peter Xu.
The monodromy of Kodaira-Parshin families, I.
This talk and the next are purely topological. The goal is to show the last missing ingredient in the proof of Faltings’ theorem: the Kodaira-Parshin family satisfies property (i) in the list after Proposition 5.3, i.e. it has full monodromy. Cover the first half of [LV, §8], i.e. §8.1-§8.4. Aside from 8.2.3, the main result should be Proposition 8.5. Recall background on mapping class groups and Dehn twists along the way.
References: TBD.

Week of April 19-23.
Speaker: TBD.
The monodromy of Kodaira-Parshin families, II.
Cover the second half of [LV, §8], i.e. §8.5-§8.6. Prove as much as you can.
References: TBD.