Harmonic Maass Forms

调和弱Maass形式是当前研究的热点。权为k的调和弱Maass形式是关于模群的子群变换的上半平面的光滑函数，它被权为k的双曲拉普拉斯函数所零化，并且在尖点至多具有线性指数增长。这种形式分解为全纯部分和非全纯部分。非全纯部分与椭圆尖点形式的周期积分有关。全纯部分，有时被称为模拟模形式，出现在各种数学上下文中。在20世纪70年代，Zagier证明了Hurwitz类数的生成函数是一种模拟的模形式。最近，Bringmann、Ono和Zwegers证明了Ramanujan的模拟theta函数是模拟的模块形式。后来的工作将调和弱Maass形式理论应用于几何、数学物理和概率论，以及与数论更密切相关的主题，如分拆和Q级数、椭圆模形式、奇异模的迹、Borcherds积和模L函数。在这些联系的启发下，这个建议解决了调和弱Maass形式的全纯部分的问题。特别是，PI的目的是研究调和弱Maass形式和椭圆模形式之间需要相互作用的问题。PI的研究兴趣是数论。本提案中的主题源于二十世纪初传奇数学家斯里尼瓦萨·拉马努詹的工作。拉马努扬遗产的一个重要部分与他发现的大量美学公式有关，这些公式涉及被称为Q系列的函数。在他生命的末期，他发现了一个特殊的Q系列家族，他称之为“模拟的theta函数”。虽然这些函数与模形式有许多共同的性质，模形式是现代数论的一类核心函数，但模拟theta函数不是模形式。Bringmann，Ono和Zwegers最近的工作最终解决了这个问题，即模拟theta函数和自同构形式理论之间的确切关系，自同构形式理论包含了模形式理论。这个问题已经悬而未决大约80年了。我们现在知道，一个模拟的theta函数是称为调和弱Maass形式的某一自同构形的“部分”或“一半”。这一发现激发了对一类更广泛的函数的研究，称为模拟模形式，它作为调和弱Maass形式的“部分”出现。这些函数不仅适用于数论，还适用于其他数学领域，如几何、拓扑和概率，以及数学物理。

Harmonic weak Maass forms are the subject of much current interest. A harmonic weak Maass form of weight k is a smooth function of the upper half-plane which transforms with respect to a subgroup of the modular group, is annhilated by the hyperbolic Laplacian in weight k, and has at most linear exponential growth at cusps. Such forms decompose into holomorphic and non-holomorphic parts. Non-holomorphic parts are related to period integrals of elliptic cusp forms. The holomorphic parts, sometimes called mock modular forms, appear in a variety of mathematical contexts. In the 1970s, Zagier showed that the generating function for Hurwitz class numbers is a mock modular form. More recently,Bringmann, Ono, and Zwegers showed that Ramanujan's mock theta functions are mock modular forms. Subsequent works applied the theory of harmonic weak Maass forms to geometry, mathematical physics, and probability theory, and to topics more closely allied to number theory such as partitions and q-series, elliptic modular forms, traces of singular moduli, Borcherds products, and modular L-functions. Motivated by these connections, this proposal addresses questions concerning the holomorphic parts of harmonic weak Maass forms. In particular, the PI intends to studyproblems requiring interplay between harmonic weak Maass forms and elliptic modular forms.The PI's research interest is in number theory. The topics in this proposal originate in work of the legendary mathematician Srinivasa Ramanujan in the early twentieth century. An important part of Ramanujan's legacy concerns the large quantity of aesthetic formulas he discovered involving functions called q-series. Toward the end of his life, he discovered a particular family of q-series which he called "mock theta functions". Though these functions share many properties with modular forms, a class of functions central to modern number theory, the mock theta functions are not modular forms. Recent works of Bringmann, Ono, and Zwegers finally settled the question, as to the precise relationship between the mock theta functions and the theory of automorphic forms, a theory which subsumes the theory of modular forms. This question had been outstanding for roughly eighty years. We now know that a mock theta function is "part" or "half" of a certain automorphic form called a harmonic weak Maass form. This discovery motivates the study of a broader class of functions, called mock modular forms, which arise as "parts" of harmonic weak Maass forms. These functions have applications not only to number theory, but to other areas of mathematics such as geometry, topology, and probability, and to mathematical physics.