Scientific context

L'objectif de cet atelier est de stimuler les collaborations scientifiques autour de quelques thèmes de géométrie diophantienne. Le nombre d'exposés est volontairement restreint pour laisser du temps aux échanges mathématiques.

The objective of this workshop is to stimulate scientific collaborations around some themes of Diophantine geometry. The number of presentations is deliberately limited to allow time for mathematical discussions.

Scientific committee : Richard Griffon et Fabien Pazuki.

Invited speakers

• Vincent Bosser (Caen, France)      Invariants modulaires singuliers de Drinfeld, hyperboles et unités.

  Bilu, Masser et Zannier ont démontré en 2013 que l'hyperbole XY=1 ne contient pas de points (j₁,j₂) où j₁ et j₂ sont des invariants modulaires singuliers, c'est-à-dire des invariants modulaires de courbes elliptiques CM. Habegger puis Bilu, Habegger, Kühne ont ensuite généralisé ce résultat et démontré qu'un invariant modulaire singulier ne peut jamais être une unité algébrique. Dans cet exposé, nous nous intéresserons à ces deux questions pour des invariants modulaires singuliers de Drinfeld, c'est-à-dire des invariants modulaires de modules de Drinfeld CM de rang 2. Nous verrons en particulier que l'hyperbole XY=1 ne contient pas de points dont les coordonnées sont des invariants modulaires singuliers de Drinfeld, et qu'il n'y a au plus qu'un nombre fini de tels invariants singuliers qui soient des unités algébriques. C'est un travail en commun avec Bruno Anglès, Cécile Armana et Fabien Pazuki.

• Guy Fowler (Hanover, Germany)      Multiplicative relations among differences of singular moduli

  Let n be a positive integer. We prove that there exist a finite set S ⊂ Cⁿ⁺¹ and finitely many algebraic curves T₁, …, T_k ⊂ Cⁿ⁺¹ with the following property. If (x₁, …, x_n, y) is an (n+1)-tuple of pairwise distinct singular moduli such that ∏ⁿ_i=1(x_i - y)^a_i = 1 for some nonzero integers a₁, …, a_n, then (x₁, …, x_n, y) ∈ S ∪ T₁ ∪ …∪ T_k. Further, the curves T₁, …, T_k may be described explicitly. This is joint work with Vahagn Aslanyan and Sebastian Eterović.

• Lorenzo Furio (Pisa, Italy)     Serre's Uniformity Question and proper subgroups of the normaliser of the non-split Cartan

  In 1972 Serre proved his celebrated Open Image Theorem, stating that for every rational elliptic curve E without complex multiplication there exists an integer N such that, for every prime p>N, the Galois representation on the p-torsion of E is surjective. In the same article, he asked whether the constant N can be taken to be independent of the curve, and this became known as Serre's Uniformity Question. In this talk, I will discuss the current progress towards an answer to this question, in particular the Runge method for modular curves developed by Bilu and Parent and the recent improvements obtained via this method by Le Fourn and Lemos, as well as explain how to solve some of the questions left open by the latter result. This is joint work with Davide Lombardo.

• Desirée Gijón Gómez (Copenhagen, Denmark)      Explicit bounds on the coefficients of modular polynomials

  We give explicit upper and lower bounds on the size of the coefficients of the modular polynomials Φ_n for the j-function. These bounds make explicit the best previously known asymptotic bounds. This is joint work with Fabien Pazuki and Florian Breuer.

• Nuno Hultberg (Copenhagen, Denmark)      Fields with few small points

  There are uncountably many algebraic extensions of a number field containing only finitely many special moduli. This relies on a quantitative notion of an algebraic field having few small points, weighted Northcott numbers. I will then present weighted Northcott numbers, their basic properties, and further applications.

• Samuel Le Fourn (Grenoble, France)      Bounds on the height of CM jacobians

  The problem of singular moduli admits several generalisations to higher-dimension, for example the following one. For a real quadratic field F and K/F a CM extension which is a quartic cyclic extension of Q, consider the jacobians of curves of genus 2 with CM by $\mathcal{O}_K$ such that the curves themselves have potentially good reduction everywhere. Habegger and Pazuki proved that their height can be bounded (ineffectively) only in terms of F. In the present work (joint with Linda Frey and Elisa Lorenzo-Garcia), we provide (under GRH) an explicit bound in F. In this talk, I will explain the main difficulties to be overcome to obtain explicit bounds, and the somewhat surprising link with questions about Minkowski's bound that arise.

• Pietro Mercuri (Rome, Italy)      Asymptotic self-intersection of the Arakelov canonical sheaf of X₀(N)

  We recall some basic notions and the general ideas of Arakelov theory and we sketch the proof that the self-intersection of the Arakelov canonical sheaf of the classical modular curves X₀(N) is asymptotic to 3glog N, for N→+∞ and coprime with 6, where g is the genus of X₀(N).

• Fabien Pazuki (Copenhagen, Denmark)      Cycloalkanes and elliptic curves

  The aim of the talk is to explain an unexpected link between a class a molecules composed of carbon and hydrogen atoms, and the theory of elliptic curves over finite fields. The correspondance is of topological nature and doesn't include, so far, any of the crucial geometric features of the cycloalkanes. We will nevertheless explain how modular curves help making this connection, the role of modular polynomials, and give details about explicit computations we performed. The talk is based on joint work with Henry Bambury and Francesco Campagna.

• Emanuele Tron (Sorbonne Université, France)      Some problems on elliptic curves and GCDs

  Arithmetical properties of the reductions of an algebraic group translate into results on so-called "GCD problems" associated to it. We give a detailed account of this correspondence and recent progress in case the group has an elliptic curve as a direct factor. We then focus on more exotic problems, in particular the following question about Frobenius traces: can GCDs detect isogeny classes?

Color code for titles: Blackboard, video projector and blackboard & video projector.

Schedule

• Lundi 4 décembre
  • 10h   Reception (and coffee/tea)
  • 10h30 - 11h30   Fabien Pazuki
  • 13h30 - 14h30   Pietro Mercuri


• Mardi 5 décembre
  • 10h30 - 11h30   Lorenzo Furio
  • 13h30 - 14h30   Samuel Le Fourn


• Mercredi 6 décembre
• 10h30 - 11h30   Emanuele Tron
• 13h30 - 14h30   Desirée Gijón Gómez


• Jeudi 7 décembre
• 10h30 - 11h30   Vincent Bosser
• 13h30 - 14h30   Guy Fowler


• Thursday 7 at 7:30 pm: conference dinner at the restaurant Le Devant


• Vendredi 8 décembre
• 10h30 - 11h30   Nuno Hultberg

Place of the meeting

The talks will be located at the Blaise Pascal mathematical laboratory of the university Clermont Auvergne (Clermont-Ferrand). Room 218 (2nd floor).

Participants (22)

Cécile Armana, Martin Azón, Henry Bambury, Nicolas Billerey, Vincent Bosser, Francesco Campagna, Gabriel Dospinescu, Guy Fowler, Lorenzo Furio, Éric Gaudron, Desirée Gijón Gómez, Samuel Le Fourn, Richard Griffon, Nuno Hultberg, François Martin, Pietro Mercuri, Fabien Pazuki, Pierre Parent, Marusia Rebolledo, Gaël Rémond, Florian Tilliet, Emanuele Tron

Practical informations

Take care to keep all your restaurant or travel invoices, so that we can reimburse you later !

Lunch : Les Hauts de l'Artière

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).

Sponsor

This meeting was made possible thanks to the support of the ANR project J-invariant and the IRN Mathematics in Denmark and France (MaDeF).