Team seminars on Antoine Leudière

From elliptic curves to Drinfeld modules

Tue, 09 Dec 2025 00:00:00 +0000

Elliptic curves play a critical role in certain practical applications: cryptography, computer algebra. They are one of the main tools in number theory and for the arithmetic of characteristic zero objects like number fields. A class of objects was invented specifically to play the role of elliptic curves in the arithmetic of positive characteristic and function fields: that of Drinfeld modules.

Our goal is to present the main ideas of the theory Drinfeld modules, the analogies with elliptic curves, and to motivate Drinfeld modules as credible alternatives to elliptic curves for certain applications: coding theory and computer algebra.

From elliptic curves to Drinfeld modules

Tue, 02 Dec 2025 00:00:00 +0000

Point counting without points (again)

Wed, 26 Nov 2025 00:00:00 +0000

Drinfeld modules are the analogues of elliptic curves in positive characteristic. They are essential objects in number theory for studying function fields. They do not have points, in the traditional sense—we’re going to count them anyway! The first methods achieving this were inspired by classical elliptic curve results; we will instead explore an algorithm based on so-called Anderson motives that achieves greater generality. Joint work with Xavier Caruso.

Point counting on Drinfeld modules

Tue, 08 Jul 2025 00:00:00 +0000

We present explicit methods for point counting on objects known as Drinfeld modules. These can be seen as function field analogue of elliptic curves; they were introduced in the 1970s to provide function fields with an explicit class field theory, including a theory of complex multiplication. They played a key role in the resolution of some cases of the geometrical Langlands program by Laurent Lafforgue, who received the Fields medal for his work. More recently, they were used for state-of-the-art polynomial factorization over finite fields (Doliskani-Narayanan-Schost).

A computation on Drinfeld modules

Thu, 29 May 2025 00:00:00 +0000

The development of algebraic geometry has shed light on deep similarities between the classical number theory (characteristic zero, number fields), and its positive characteristic analogue (centered on curves and function fields). The latter turned out easier to work with: from a theoretical point of view, some results are unconditional (e.g. Riemann hypothesis for function fields); from a computational point of view, a lot of elementary procedures can be performed efficiently (e.g. polynomial factorization, as opposed to integer factorization).

Computations in positive and zero characteristic

Sat, 03 May 2025 00:00:00 +0000

Number theory is often seen as the science of integer numbers. Yet, arithmetical similarities between the ring of integers, and that of polynomials over a finite field, have led to establishing a “number theory” in positive characteristic that strikingly resembles its classical analogue.

It is, however, easier to work in positive characteristic: function fields come from algebraic varieties, which give new tools for their study. For example, in positive characteristic, the Riemann hypothesis as well as some cases of the Langlands program are actually theorems. The lack of geometrical interpretation is a fundamental obstruction to our understanding of the classical case.

Modules de Drinfeld: action de groupe de classe explicite et implémentation

Fri, 23 Sep 2022 00:00:00 +0000

Motivés par la cryptographie des isogénies, nous parlerons de l’algorithmique des modules de Drinfeld et leurs isogénies.

Les modules de Drinfeld ont été introduits dans les années 1970 pour construire une théorie du corps de classe des corps de fonctions explicite, comme celle-ci peut l’être pour les corps de nombres : le corps de classes de Hilbert d’un corps quadratique imaginaire est engendré par les j-invariants des courbes elliptiques ayant multiplication complexe dans ce corps. En ce sens, la théorie modules de Drinfeld fournit un « analogue corps de fonctions » des courbes elliptiques.

An explicit CRS-like action with Drinfeld modules

Tue, 14 Jun 2022 00:00:00 +0000

L’une des pierres angulaires de la cryptographie des isogénies est l’action (dite CRS), simplement transitive, du groupe des classes d’un ordre d’un corps quadratique imaginaire, sur un certain ensemble de classes d’isomorphismes de courbes elliptiques ordinaires.

L’échange de clé non-interactif basé sur cette action (espace homogène difficile) est relativement lent (de Feo, Kieffer, Smith, 2019) ; la structure du groupe (Beullens, Kleinjung, Vercauteren, 2019) est difficile à calculer. Pour palier à cela, nous décrivons une action, simplement transitive, de la jacobienne d’une courbe hyperelliptique imaginaire, sur un certain ensemble de classes d’isomorphismes de modules de Drinfeld.

An explicit CRS-like action with Drinfeld modules

Mon, 30 May 2022 00:00:00 +0000

L’échange de clé non-interactif basé sur cette action est relativement lent (de Feo, Kieffer, Smith, 2019) ; la structure du groupe sous-jacent (Beullens, Kleinjung, Vercauteren, 2019) est particulièrement difficile à calculer. Cela nous incite à adapter cette construction à d’autres objets mathématiques. Dans ce contexte, nous décrivons une action, simplement transitive, de la jacobienne d’une courbe hyperelliptique imaginaire, sur un certain ensemble de classes d’isomorphismes de modules de Drinfeld.