معرفی شرکت ها

تبلیغات ما

مشتریان به طور فزاینده ای آنلاین هستند. تبلیغات می تواند به آنها کمک کند تا کسب و کار شما را پیدا کنند.

مشاهده بیشتر

تبلیغات ما

مشتریان به طور فزاینده ای آنلاین هستند. تبلیغات می تواند به آنها کمک کند تا کسب و کار شما را پیدا کنند.

مشاهده بیشتر

تبلیغات ما

مشتریان به طور فزاینده ای آنلاین هستند. تبلیغات می تواند به آنها کمک کند تا کسب و کار شما را پیدا کنند.

مشاهده بیشتر

تبلیغات ما

مشتریان به طور فزاینده ای آنلاین هستند. تبلیغات می تواند به آنها کمک کند تا کسب و کار شما را پیدا کنند.

مشاهده بیشتر

تبلیغات ما

مشتریان به طور فزاینده ای آنلاین هستند. تبلیغات می تواند به آنها کمک کند تا کسب و کار شما را پیدا کنند.

مشاهده بیشتر

توضیحات

SageMath implementation of Drinfeld modular forms

ویژگی	مقدار
سیستم عامل	-
نام فایل	drinfeld-modular-forms-0.0.2
نام	drinfeld-modular-forms
نسخه کتابخانه	0.0.2
نگهدارنده	[]
ایمیل نگهدارنده	[]
نویسنده	David Ayotte
ایمیل نویسنده	davidayotte94@outlook.com
آدرس صفحه اصلی	https://github.com/DavidAyotte/drinfeld_modular_forms
آدرس اینترنتی	https://pypi.org/project/drinfeld-modular-forms/
مجوز	-

# 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](https://github.com/DavidAyotte/drinfeld_modular_forms). 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

نیازمندی

مقدار	نام
>=2	sphinx

نحوه نصب

نصب پکیج whl drinfeld-modular-forms-0.0.2:

pip install drinfeld-modular-forms-0.0.2.whl

نصب پکیج tar.gz drinfeld-modular-forms-0.0.2:

pip install drinfeld-modular-forms-0.0.2.tar.gz