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建议对自同构形式的下列算术性质进行深入研究：傅立叶系数的界以及自同构形式与超几何级数的联系。PI在估计自同构形式的系数方面有一个积极的研究计划。特别是，她计划研究不同类型的小重量自同构形式的系数，包括半整数权重模形式和西格尔模形式。改进现有的估计在许多领域都有广泛的应用，包括物理、椭圆曲线、表示理论、代数几何和二次型。第二个重点是研究超几何级数和自同构形之间的联系，特别是经典的模形式、弱Maass形式和Maass尖点形式。关于模超几何级数的例子的文献很多，由于它们在数学和物理的许多领域中的应用，对这些例子及其解释的进一步追求是一个活跃的研究领域。然而，这些零散结果的证明远远没有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。模拟的theta函数使情况变得更加复杂，这是Ramanujan在他给Hardy的最后一封信中定义的22个Q系列的集合。虽然它们类似于模Q级数，但这些函数并不是作为模形式的傅里叶展开的微小修改而出现的。尽管如此，它们拥有许多引人注目的特性，并已成为数量惊人的重要作品的主题。最近，人们对Ramanujan的模拟theta函数的性质有了很多了解。通过Pi、Ono和Zweger的工作，现在已经知道这些函数是权1/2弱Maass形式的全纯部分，关于哪些模形式和Maass形式产生于基本超几何级数的更清晰的图景开始出现。由于很难描述Maass形式的傅里叶展开，PI计划超越现有的诱人例子的有限列表，并发展说明基本超几何级数和自同构形式之间精确相互作用的一般定理。由于这些理论在数学和物理的不同领域发挥着重要作用，国际和平研究所预计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.