Computational aspects of Maass cusp forms

模形式是一类具有多种对称性的全纯复值函数。一些值得注意的应用包括可怕的月光和弦理论(这项工作为BorCherds赢得了1998年的菲尔兹奖)，Wiles的费马大定理的解，以及Viazovska等人最近在球面包装方面的新应用。Maass形式与模形式密切相关，但是不再需要像全纯这样的某些条件，这使得它们更难计算。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1990年的‘S’中提出了一个寻找Maass尖点形式的算法，但是这个算法依赖于启发式的论证，并且没有被严格地证明对于同余子群一般是收敛的。十年前的一些进展表明，使用塞尔伯格迹公式的不同方法可以严格计算Maass形式，但这只在最简单的模群情况下执行。为L函数与模形式数据库计算大型、严格的全纯模形式数据库的努力于2016年完成，并已被多次引用。这个项目的主要目的是提供一种计算和认证任意级别和字符的Maass表格的大型数据库的方法。该项目将探索一些基于Selberg迹公式的新的计算和验证方法，即与Hecke算子有关的方法。所有相关数据将在LMFDB上公布。

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.