Arithmetic properties of automorphic forms-Bounds on Fourier coefficients and the interplay between hypergeometric series and automorphic forms

首席研究员凯瑟琳·布林曼（Kathrin Bringmann）提出了对自守形式的以下算术性质的深入研究：傅立叶系数的界限以及自守形式和超几何级数的联系。 的PI 在估计自守形式的系数方面有一个活跃的研究计划。特别是她计划研究系数的不同种类的自守形式的小重量，包括半整数重量模形式和西格尔模形式。改进现有的估计有着广泛的应用，包括物理学，椭圆曲线，表示论，代数几何和二次型。这个建议的第二个重点是研究超几何级数和自守形式之间的联系，特别是经典的模形式，弱马斯形式和马斯尖点形式。 关于超几何级数是模的例子的文献是广泛的，进一步追求这些和它们的解释是一个活跃的研究领域 因为它们应用于数学和物理学的许多领域。 然而，在这方面， 这些分散的结果的证明远远达不到一个全面的理论来描述超几何级数和自守形式之间的相互作用。 这种情况进一步复杂的模拟θ函数，收集22个q系列定义的拉马努金在他的最后一封信哈代。 虽然它们类似于模q-级数，但这些函数并不是模形式的傅立叶展开式的微小修改。尽管如此，它们拥有许多惊人的特性，并且一直是数量惊人的重要作品的主题。最近，许多光已经阐明了拉马努金的模拟θ函数的性质。通过PI，Ono和Zwegers的工作，现在已知这些函数是重量为1/2的全纯部分 弱Maass形式，并且开始出现模块形式和Maass形式出现的更清晰的画面 基本超几何级数 由于很难描述马斯形式的傅立叶展开式，PI计划超越现有的有限列表，并开发一般定理，说明基本超几何级数和自守形式之间的精确相互作用。 由于这些在数学和物理的不同领域发挥着重要作用，PI希望新开发的理论将在那里得到应用。

The Principal Investigator, Kathrin Bringmann, proposes an intense study of the following arithmetic properties of automorphic form: bounds for Fourier coefficients and the connection of automorphic forms and hypergeometric series. The PI has an active research program on estimating coefficients of automorphic forms. In particular she plans to study coefficients of different kinds of automorphic forms of small weight including half-integer weight modular forms and Siegel modular forms. Improving existing estimates has a wide range of applications to many areas including physics, elliptic curves, representation theory, algebraic geometry, and quadratic forms. The second emphasis of this proposal is to investigate the connection between hypergeometric series and automorphic forms, in particular classical modular forms, weak Maass forms, and Maass cusp forms. The literature on examples of hypergeometric series that are modular is extensive, and the pursuit of further of these and their interpretation is an active area of research due to their applications to many areas of mathematics and to physics. However, the proofs of these scattered results fall far short of a comprehensive theory to describe the interplay between hypergeometric series and automorphic forms. The situation is further complicated by the mock theta functions, a collection of 22 q-series defined by Ramanujan in his last letter to Hardy. Though they resemble modular q-series, these functions do not arise as minor modifications of the Fourier expansions of modular forms. Nevertheless they possess many striking properties, and have been the subject of an astonishing number of important works. Recently, much light has been shed on the nature of Ramanujan's mock theta functions. By work of the PI, Ono, and Zwegers it is now known that these functions are the holomorphic parts of weight 1/2 weak Maass forms, and a clearer picture is beginning to emerge of which modular forms and Maass forms arise from basic hypergeometric series. Since it is very difficult to describe Fourier expansions of Maass forms, the PI plans to go beyond the finite list of tantalizing examples that exist and develop general theorems which illustrate the precise interplay between basic hypergeometric series and automorphic forms. Since these play an important role in different areas of mathematics and physics, and the PI expects that the newly developed theories will have applications there.