Generators of the group of modular units for Γ 1 (N) over the rationals
[Générateurs du groupe des unités modulaires pour Γ 1 (N) sur les rationnels]
Streng, Marco1
1Universiteit Leiden, Niels Bohrweg 1, 2333 CA Leiden (The Netherlands)
Annales Henri Lebesgue, Tome 6 (2023), pp. 95-116

We give two explicit sets of generators of the group of invertible regular functions over Q on the modular curve Y 1 (N).

The first set of generators is very surprising. It is essentially the set of defining equations of Y 1 (k) for k≤N/2 when all these modular curves are simultaneously embedded into the affine plane, and this proves a conjecture of Derickx and Van Hoeij [DvH14]. This set of generators is an elliptic divisibility sequence in the sense that it satisfies the same recurrence relation as the elliptic division polynomials.

The second set of generators is explicit in terms of classical analytic functions known as Siegel functions. This is both a generalization and a converse of a result of Yang [Yan04, Yan09].

Nous donnons deux ensembles explicites de générateurs du groupe des fonctions régulières inversibles à coefficients rationnels sur la courbe modulaire Y 1 (N).

Le premier ensemble de générateurs est très surprenant. C’est essentiellement l’ensemble des équations qui définissent Y 1 (k) pour k≤N/2 quand toutes ces courbes modulaires sont plongées simultanément dans le plan affine, ce qui prouve une conjecture de Derickx et Van Hoeij [DvH14]. Cet ensemble de générateurs est une suite de divisibilité elliptique dans le sens où il satisfait la même relation de récurrence que les polynômes de division elliptiques.

Le second ensemble de générateurs est explicite en termes de fonctions analytiques classiques appelées fonctions de Siegel. C’est à la fois une généralisation et une réciproque d’un résultat de Yang [Yan04, Yan09].

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DOI : 10.5802/ahl.160
Classification : 11G16, 11B37, 11B39, 11F03, 14H52
Keywords: modular units, modular functions, elliptic divisibility sequences, divsion polynomials

Streng, Marco  1

1 Universiteit Leiden, Niels Bohrweg 1, 2333 CA Leiden (The Netherlands)
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Streng, Marco. Generators of the group of modular units for $\Gamma ^1(N)$ over the rationals. Annales Henri Lebesgue, Tome 6 (2023), pp. 95-116. doi: 10.5802/ahl.160

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