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

首席研究员Kathrin Bringmann建议深入研究自同构形式的以下算术性质：傅里叶系数的界限以及自同构形式与超几何级数的联系。PI在自同构形式的系数估计方面有一个活跃的研究项目。她特别计划研究各种小权值自同构形式的系数，包括半整数权值模形式和西格尔模形式。改进现有的估计在许多领域有广泛的应用，包括物理、椭圆曲线、表示理论、代数几何和二次型。本文的第二个重点是研究超几何级数与自同构形式之间的联系，特别是经典模形式、弱质量形式和质量尖形式。关于模的超几何级数的例子的文献是广泛的，由于它们在数学和物理的许多领域的应用，对这些和它们的解释的进一步追求是一个活跃的研究领域。然而，对这些零散结果的证明远远不能提供一个全面的理论来描述超几何级数和自同构形式之间的相互作用。由于Ramanujan在他给Hardy的最后一封信中定义了22个q级数的模拟函数，情况变得更加复杂。虽然它们类似于模q级数，但这些函数并不是模形式的傅里叶展开的微小修改。然而，它们具有许多惊人的特性，并已成为数量惊人的重要作品的主题。最近，人们对拉马努金模拟函数的性质有了更多的了解。通过PI， Ono和Zwegers的工作，现在我们知道这些函数是1/2弱质量形式的全纯部分，并且一个更清晰的图像开始出现，模形式和质量形式来自基本的超几何级数。由于描述质量形式的傅里叶展开是非常困难的，PI计划超越现有的有限诱人的例子列表，并发展一般定理，以说明基本超几何级数和自同构形式之间的精确相互作用。由于它们在数学和物理的不同领域发挥着重要作用，PI期望新发展的理论将在这些领域得到应用。

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.