Automorphic Forms on GL(2) and Their Behaviors at Finite Cusps develops a systematic study of Fourier expansions at arbitrary cusps, combining classical modular form theory with modern adelic and representation-theoretic methods. While the classical theory often emphasizes expansions at infinity, this book places finite cusps at the center of the arithmetic theory.The text presents both classical and adelic approaches to Fourier expansions, emphasizing the role of local Whittaker models in understanding the behavior of Fourier coefficients at finite cusps. It further develops the metaplectic and Weil representation framework for half-integral weight modular forms and theta functions, providing a unified perspective on transformation laws and twisted theta series. These methods are applied to problems and conjectures concerning finite-cusp expansions, including work of Goldfeld and Gunnells.Designed for graduate students and researchers, the book provides a detailed introduction to automorphic forms on GL(2), together with modern tools from harmonic analysis and representation theory.
