Scientific context
In this workshop we are interested in the recent developments of techniques coming from Arakelov theory, diophantine geometry or p-adic methods like Chabauty-Coleman or quadratic Chabauty, in view of studying rational points on curves, especially Shimura curves and modular curves.
Invited speakers
- Jennifer
S. Balakrishnan (Boston, USA)
Quadratic Chabauty IV - Computational aspects and examples
We continue the sequence of talks on quadratic Chabauty and describe various computational aspects of the method, which uses p-adic heights to find rational points on curves. Our main example will be the split (and non-split) Cartan curve of level 13 and the computation of its rational points.
- Netan
Dogra (Imperial college London)
Quadratic Chabauty II & III - The Chabauty-Kim method: global and local aspects
Following from the previous talk, we will explain some techniques for proving that the sets of p-adic points on a curve produced by Kim's theory are finite. We then introduce the twisting construction which may be used to relate this to p-adic heights. We will then describe the local aspects of the theory, including some background on nonabelian p-adic Hodge theory, with an emphasis on how to make these things explicit.
- Sebastián
Herrero M. (Gothenburg, Sweden)
p-adic distribution of Hecke and CM points
The aim of this talk is to describe the asymptotic distribution of Hecke and CM points on the moduli space of elliptic curves over the p-adic complex numbers. These are non-Archimedean analogues of classical results by Duke and Clozel-Ullmo. If time permits, we will present an application to the finitness of singular moduli that are S-units, extending a result of Habegger. This is joint work with Ricardo Menares and Juan Rivera-Letelier.
- Samuel Le
Fourn (Lyon, France) Around the classical Chabauty-Coleman method
In this introductory talk, I will explain how the classical Chabauty-Coleman method works. I will prove the original Chabauty's result, then the more explicit version established by Coleman based on p-adic integrals, and finally mention some interesting particular cases improving Coleman's bounds or method, while keeping a focus on how to make all of this effective.
- Jan Steffen
Müller (Groningen, Netherlands)
Quadratic Chabauty I - Setup and p-adic heights
We discuss how to make Chabauty-Kim theory explicit for a curve over the rationals of genus g>1 whose Jacobian has Mordell-Weil rank equal to g and Picard number >1, and which satisfies some additional conditions. This enables us to compute the rational points on such curves using p-adic heights. The main example we will discuss is the case of the split (and non-split) Cartan curve of level 13; this is joint work with J. Tuitman and J. Vonk and completes the classification of non-CM elliptic curves over Q with split Cartan level structure due to Bilu-Parent and Bilu-Parent-Rebolledo.
- Héctor
H. Pastén Vásquez (Harvard, USA) Shimura curves and the abc conjecture
I'm going to present some new unconditional results towards the abc conjecture. This is achieved by introducing new methods in the context of Shimura curves, in the aspects of Arakelov theory and automorphic forms. A main difference with the existing literature on modular forms and the abc conjecture is that for our approach it is not required to bound the degree of modular parametrizations of elliptic curves, but instead, we need to bound the variation of such degree for a fixed target elliptic curve as we choose among various available parametrizations coming from different Shimura curves. If time permits, I will also discuss more efficient ways to effectively bound modular degrees.
- Pierre
Parent (Bordeaux, France)
Stable models for modular curves in prime level
We describe stable models for modular curves in prime level, including the new case of non-split Cartan curves.
- Samir Siksek
(Warwick, UK) Quadratic Points on Some Modular Curves
We explain how the classical Chabauty can be extended to apply to symmetric powers of a curve under a suitable assumption on the Mordell-Weil rank. We also report on joint work (in progress) with Ekin Ozman aimed at determining the quadratic points on the classical modular curves X_0(N) of genera 3, 4, 5.
- Christian
Wutrich (Nottingham, UK) On the Mordell-Weil group as a Galois module
Let E be an elliptic curve over a number field k and let K/k be a finite Galois extension with group G. I propose to look at a few concrete cases and some general results on the structure of E(K) as a Z[G]-module, or more precisely at E(K) ox Z_p as a Z_p[G]-module. This links to questions about the equivariant Birch and Swinnerton-Dyer conjecture. Also we can also look at the case when k=Q_p.
Schedule
- Wednesday 17 January 2018
- 10:00 Opening (with coffee and (chocolate) croissants)
- 10:30 - 11:30 S. Le Fourn
- 13:45 - 14h45 J.S. Müller
- 15:00 - 16:00 S. Herrero M.
- Thursday 18 January 2018
- 9:00 - 10:00 P. Parent
- Coffee break
- 10:30 - 11:30 N. Dogra
- 13:45 - 14:45 H. Pastén
- 15:00 - 16:00 N. Dogra
- Coffee break
- 16:30 - 17:30 C. Wutrich
- Friday 19 January 2018
- 09:00 - 10:00 S. Siksek
- Coffee break
- 10:30 - 11:30 P. Parent
- 14:00 - 15:00 H. Pastén
- 15:15 - 16:15 J. Balakrishnan
Place of the meeting
The talks will be located at the Blaise Pascal mathematical laboratory of the university Clermont Auvergne (Clermont-Ferrand). Room 2222 (2nd floor).Participants (24)
Samuele Anni, Cécile Armana, Pascal Autissier, Jennifer Balakhrishnan, Nicolas Billerey, José Manuel Rodriguez Caballero, Marteen Derickx, Netan Dogra, Stevan Gajovic, Éric Gaudron, Sebastián Herrero M., Enis Kaya, Samuel Le Fourn, Davide Lombardo, Ricardo Menares, Jan Steffen Müller, Pierre Parent, Héctor Pastén, Fabien Pazuki, Marusia Rebolledo, Gaël Rémond, Samir Siksek, Andrea Surroca, Christian Wutrich.
Practical informations
Lunch : Cafétéria côté court
Restaurants for dinner : Bistro Vénitien (Pizzeria), Bougnat Burger (Local and/or organic products in a premium quality hamburger), La cassolette (simple fare, good quality-price ratio), La Gourmandine (for wine lovers), Un grain de saveur (fine cuisine at an affordable price), Lard et la Manière, La régalade, Le chardonnay (typical French bar), Le sisisi (traditional French food), L'instantané (very good, not too expensive).