Package: drinfeld-modular-forms
Drinfeld Modular FormsThis SageMath package provides an implementation for computing with Drinfeld modular forms for the full modular group.InstallationThis package has been tested on SageMath version 9.8 and higher. It is
not guaranteed to work on previous versions.Install from PyPIThe easiest way to install this package is via PyPI. You simply have to run SageMath first and then type the following commandsage: pip install drinfeld-modular-formsInstall from source codeYou can also install this package by cloning the source code from theGithub repo.Next, you have to runmake installinside the project's folder. You can also run the following command:sage -pip install --upgrade --no-index -v .If there is any changes to the current repo, you will then simply need to pull the changes and run the above command again.UsageAfter running SageMath, you can import the functionalities of this package by typing the following command:sage: from drinfeld_modular_forms import *DocumentationThe documentation is available at this address:https://davidayotte.github.io/drinfeld_modular_formsExamplesOne may create the ring of Drinfeld modular forms:sage: from drinfeld_modular_forms import DrinfeldModularFormsRing
    sage: A = GF(3)['T']; K = Frac(A); T = K.gen()
    sage: M = DrinfeldModularFormsRing(K, 2)
    sage: M.ngens()  # number of generators
    2The elements of this ring are viewed as multivariate polynomials in a choice of generators for the ring. The current implemented generators are the coefficient forms of a universal Drinfeld module over the Drinfeld period domain (see theorem 17.5 in [1]). In the computation below, the formsg1andg2corresponds to the weightq - 1Eisenstein series and the Drinfeld modular discriminant of weightq^2 - 1respectively.sage: M.inject_variables()
    Defining g1, g2
    sage: F = (g1 + g2)*g1; F
    g1*g2 + g1^2Note that elements formed with polynomial relationsg1andg2may not be homogeneous in the weight and may not define a Drinfeld modular form. We will call elements of this ringgraded Drinfeld modular forms.In the case of rank 2, one can compute the expansion at infinity of any graded form:sage: g1.expansion()
    1 + ((2*T^3+T)*t^2) + O(t^7)
    sage: g2.exansion()
    t^2 + 2*t^6 + O(t^8)
    sage: ((g1 + g2)*g2).expansion()
    1 + ((T^3+2*T+1)*t^2) + ((T^6+T^4+2*T^3+T^2+T)*t^4) + 2*t^6 + O(t^7)This is achieved via theA-expansion theory developed by López-Petrov in [3] and [4]. We note that the returned expansion is a lazy power series. This means that it will compute on demands any coefficient up to any precision:sage: g2[600]  # 600-th coefficient
    T^297 + 2*T^279 + T^273 + T^271 + T^261 + 2*T^253 + T^249 + 2*T^243 + 2*T^171 + T^163 + T^153 + 2*T^147 + 2*T^145 + T^139 + T^135 + T^129 + 2*T^123 + 2*T^121 + T^117 + T^115 + T^111 + 2*T^109 + T^105 + 2*T^99 + 2*T^97 + T^93 + T^91 + T^87 + 2*T^85 + T^81 + 2*T^75 + T^69 + T^67 + T^63 + 2*T^61 + 2*T^51 + 2*T^45 + T^43 + T^39 + T^29 + T^27 + 2*T^21 + T^19 + T^13 + 2*T^11 + T^9 + T^7 + 2*T^3 + 2*TIn rank 2, it is also possible to compute the normalized Eisenstein series of weightq^k - 1(see (6.9) in [2]):sage: from drinfeld_modular_forms import DrinfeldModularFormsRing
    sage: q = 3
    sage: A = GF(q)['T']; K = Frac(A); T = K.gen()
    sage: M = DrinfeldModularFormsRing(K, 2)
    sage: M.eisenstein_series(q^3 - 1)  # weight q^3 - 1
    g1^13 + (-T^9 + T)*g1*g2^3NotesThis package is based on the intial implementation of Alex Petrov.Drinfeld modules are currently being implemented in SageMath. Seehttps://github.com/sagemath/sage/pull/350263. As of March 2023, this PR is merged in the current latest development version of SageMath.Further DevelopmentsAdd Hecke operators computations.Add general Goss polynomialsReferences[1] Basson D., Breuer F., Pink R., Drinfeld modular forms of arbitrary rank, Part III: Examples,https://arxiv.org/abs/1805.12339[2] Gekeler, E.-U., On the coefficients of Drinfelʹd modular forms. Invent. Math. 93 (1988), no. 3, 667–700[3] López, B. A non-standard Fourier expansion for the Drinfeld discriminant function. Arch. Math. 95, 143–150 (2010).https://doi.org/10.1007/s00013-010-0148-7[4] Petrov A., A-expansions of Drinfeld modular forms. J. Number Theory 133 (2013), no. 7, 2247–2266