Generators of the group of modular units for Γ 1 (N) over the rationals
Annales Henri Lebesgue, Volume 6 (2023), pp. 95-116.


Keywords modular units, modular functions, elliptic divisibility sequences, divsion polynomials


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 kN/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].


[dL10] de Looij, Rutger Elliptic divisibility sequences, Masters thesis, Mathematical Sciences, Universiteit Utrecht, Netherlands (2010) (written under the supervision of Gunther Cornelissen,

[Dri73] Drinfeld, Vladimir G. Two theorems on modular curves, Funkts. Anal. Prilozh., Volume 7 (1973) no. 2, pp. 83-84 | MR | Zbl

[DvH14] Derickx, Maarten; van Hoeij, Mark Gonality of the modular curve X 1 (N), J. Algebra, Volume 417 (2014), pp. 52-71 | DOI | MR | Zbl

[Fri11] Fricke, Robert Die elliptischen Funktionen und ihre Anwendungen. Erster Teil. Die funktionentheoretischen und analytischen Grundlagen, Springer, 2011 (Reprint of the 1916 original) | MR | Zbl

[IMS + 12] Ingram, Patrick; Mahé, Valéry; Silverman, Joseph H.; Stange, Katherine E.; Streng, Marco Algebraic divisibility sequences over function fields, J. Aust. Math. Soc., Volume 92 (2012) no. 1, pp. 99-126 | DOI | MR | Zbl

[Jin13] Jin, Jinbi Homogeneous division polynomials for Weierstrass elliptic curves (2013) (

[KL75] Kubert, Daniel S.; Lang, Serge Units in the modular function field. II. A full set of units, Math. Ann., Volume 218 (1975) no. 2, pp. 175-189 | DOI | MR | Zbl

[KL77] Kubert, Daniel S.; Lang, Serge Units in the modular function field. IV. The Siegel functions are generators, Math. Ann., Volume 227 (1977) no. 3, pp. 223-242 | DOI | MR | Zbl

[KL81] Kubert, Daniel S.; Lang, Serge Modular units, Grundlehren der Mathematischen Wissenschaften, 244, Springer, 1981 | DOI | MR | Zbl

[Kub81] Kubert, Daniel S. The square root of the Siegel group, Proc. Lond. Math. Soc., Volume 43 (1981) no. 2, pp. 193-226 | DOI | MR | Zbl

[KY17] Koo, Ja Kyung; Yoon, Dong Sung Generators of the ring of weakly holomorphic modular functions for Γ 1 (N), Ramanujan J., Volume 42 (2017) no. 3, pp. 583-599 | DOI | MR | Zbl

[Lab18] Labrande, Hugo Computing Jacobi’s theta in quasi-linear time, Math. Comput., Volume 87 (2018) no. 311, pp. 1479-1508 | DOI | MR | Zbl

[Man72] Manin, Yuri I. Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 36 (1972), pp. 19-66 | MR | Zbl

[Mar67] Markushevich, Alekseĭ I. Theory of Functions of a Complex Variable. Vol. III, Selected Russian Publications in the Mathematical Sciences, Chelsea Publishing; Prentice Hall, 1967 (revised English edition translated and edited by Richard A. Silverman) | Zbl

[Nas16] Naskręcki, Bartosz Divisibility sequences of polynomials and heights estimates, New York J. Math., Volume 22 (2016), pp. 989-1020 | MR | Zbl

[SageMath14] The Sage Developers SageMath, the Sage Mathematics Software System (Version 6.2), 2014 (

[Sil86] Silverman, Joseph H. The arithmetic of elliptic curves, Graduate Texts in Mathematics, 106, Springer, 1986 | DOI | MR | Zbl

[Yan04] Yang, Yifan Transformation Formulas for Generalized Dedekind Eta Functions, Bull. Lond. Math. Soc., Volume 36 (2004) no. 5, pp. 671-682 | DOI | MR | Zbl

[Yan09] Yang, Yifan Modular units and cuspidal divisor class groups of X 1 (N), J. Algebra, Volume 322 (2009) no. 2, pp. 514-553 | DOI | MR | Zbl