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.
