On the computation of modular forms on noncongruence subgroups

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Mathematics of Computation Pub Date : 2023-10-30 DOI:10.1090/mcom/3903

David Berghaus, Hartmut Monien, Danylo Radchenko

引用次数: 1

Abstract

We present two approaches that can be used to compute modular forms on noncongruence subgroups. The first approach uses Hejhal’s method for which we improve the arbitrary precision solving techniques so that the algorithm becomes about up to two orders of magnitude faster in practical computations. This allows us to obtain high precision numerical estimates of the Fourier coefficients from which the algebraic expressions can be identified using the LLL algorithm. The second approach is restricted to genus zero subgroups and uses efficient methods to compute the Belyi map from which the modular forms can be constructed.

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非同余子群上模形式的计算

我们提出了两种可用于计算非同余子群上模形式的方法。第一种方法使用了Hejhal的方法，我们改进了任意精度求解技术，使算法在实际计算中速度提高了两个数量级。这使我们能够获得傅立叶系数的高精度数值估计，从中可以使用LLL算法识别代数表达式。第二种方法被限制为零属子群，并使用有效的方法来计算Belyi映射，从Belyi映射可以构造模形式。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematics of Computation 数学-应用数学

CiteScore

3.90

自引率

5.00%

发文量

审稿时长

7.0 months

期刊介绍： All articles submitted to this journal are peer-reviewed. The AMS has a single blind peer-review process in which the reviewers know who the authors of the manuscript are, but the authors do not have access to the information on who the peer reviewers are. This journal is devoted to research articles of the highest quality in computational mathematics. Areas covered include numerical analysis, computational discrete mathematics, including number theory, algebra and combinatorics, and related fields such as stochastic numerical methods. Articles must be of significant computational interest and contain original and substantial mathematical analysis or development of computational methodology.