Computational aspects of Maass cusp forms

马斯尖点形式的计算方面

• 批准号: 2271345
• 金额: --
• 依托单位: University of Bristol
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2019
• 资助国家: 英国
• 起止时间: 2019 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2271345
• 关键词: Computational; aspects; Maass; cusp; forms

Modular forms are certain holomorphic complex-valued functions with many symmetry properties. Some notable applications include monstrous moonshine and string theory (work that earned Borcherds the fields medal in 1998), Wiles' solution of Fermat's last theorem, and recent novel applications to sphere packing by Viazovska et al. Maass forms are closely related to modular forms, however certain conditions like holomorphy are no longer required which then makes them harder to compute. A notable application of Maass forms is in the solution to the question of what integers can be represented as the sum of three squares.Hejhal introduced an algorithm to find Maass cusp forms in 1990's, however this relies on a heuristic argument and has not been proven rigourously to converge in general for congruence subgroups. Some advances from a decade ago demonstrate that rigorous computation of Maass forms is possible from a different method using the Selberg trace formula, however this has only been carried out in the simplest case of the modular group.An effort to compute a large, rigorous, database of holomorphic modular forms for the L-functions and modular forms database (LMFDB) was completed in 2016, and already has several citations. The main aim for this project is provide a method to compute and certify a large database of Maass forms for arbitrary level and character. The project will explore some novel computation and certification methods based on the Selberg trace formula, namely with relation to Hecke operators. All relevant data will be published in the LMFDB.

模形式是一类具有许多对称性质的全纯复值函数。一些值得注意的应用包括巨大的月光和弦理论（工作赢得Borcherds领域奖章在1998年），怀尔斯的解决方案费马的最后定理，以及最近的新应用领域包装维亚佐夫斯卡等人。Maass形式的一个值得注意的应用是解决什么整数可以表示为三个平方和的问题。Hejhal在20世纪90年代介绍了一种算法来寻找Maass尖点形式，但是这依赖于启发式论证，并且没有被严格证明在一般情况下对同余子群收敛。十年前的一些进展表明，使用Selberg迹公式可以通过不同的方法严格计算Maass形式，但这只在模群的最简单情况下进行。2016年，为L-函数和模形式数据库（LMFDB）计算一个大型的、严格的全纯模形式数据库的工作已经完成，并且已经有了一些引用。本课题的主要目的是提供一种方法来计算和验证一个大型的任意级别和字符的Maass格式数据库。该项目将探索基于Selberg迹公式的一些新的计算和证明方法，即与Hecke算子的关系。所有相关数据将在LMFDB中公布。