drinfeld-modular-forms

Drinfeld Modular Forms

This SageMath package provides an implementation for computing with Drinfeld modular forms for the full modular group.

Installation

This package has been tested on SageMath version 9.8 and higher. It is not guaranteed to work on previous versions.

Install from PyPI

The easiest way to install this package is via PyPI. You simply have to run SageMath first and then type the following command

sage: pip install drinfeld-modular-forms

Install from source code

You can also install this package by cloning the source code from the Github repo.

Next, you have to run make install inside 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.

Usage

After running SageMath, you can import the functionalities of this package by typing the following command:

sage: from drinfeld_modular_forms import *

Documentation

The documentation is available at this address:

https://davidayotte.github.io/drinfeld_modular_forms

Examples

One 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
    2

The 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 forms g1 and g2 corresponds to the weight q - 1 Eisenstein series and the Drinfeld modular discriminant of weight q^2 - 1 respectively.

    sage: M.inject_variables()
    Defining g1, g2
    sage: F = (g1 + g2)*g1; F
    g1*g2 + g1^2

Note that elements formed with polynomial relations g1 and g2 may not be homogeneous in the weight and may not define a Drinfeld modular form. We will call elements of this ring graded 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 the A-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*T

In rank 2, it is also possible to compute the normalized Eisenstein series of weight q^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*g2^3

Notes

This package is based on the intial implementation of Alex Petrov.

Drinfeld modules are currently being implemented in SageMath. See https://github.com/sagemath/sage/pull/350263. As of March 2023, this PR is merged in the current latest development version of SageMath.

Further Developments

• Add Hecke operators computations.
• Add general Goss polynomials

References

• [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