Hecke's Theory of Modular Forms and Dirichlet Series (2nd Printing and Revisions) (Monographs in Number Theory)

In 1938, at the Institute for Advanced Study, E Hecke gave a series of lectures on his theory of correspondence between modular forms and Dirichlet series. Since then, the Hecke correspondence has remained an active feature of number theory and, indeed, it is more important today than it was in 1936 when Hecke published his original papers. This book is an amplified and up-to-date version of the former author's lectures at the University of Illinois at Urbana-Champaign, based on Hecke's notes. Providing many details omitted from Hecke's notes, it includes various new and important developments in recent years. In particular, several generalizations and analogues of the original Hecke theory are briefly described in this concise volume. Drawing upon Hecke's notes and his own lectures, Berndt (mathematics, U. of Illinois at Urbana-Champaign) and co-author Knopp (Mathematics, Temple U.) present new and important developments of recent years in the theory of correspondence. They begin by describing Hecke's main correspondence theorem and establish a couple of useful lemmas, then work through a fundamental region. They closely work through three cases, in which the lemma is greater than two, less than two, and equal to two, and describe Bochner's generalization of the main correspondence theorem of Hecke with related results and present identities equivalent to the functional equation and to the modular relation. They provide a comprehensive bibliography and useful index. Annotation ©2008 Book News, Inc., Portland, OR (booknews.com) This book is an amplified and up-to-date version of the first named author's lectures at the University of Illinois at Urbana-Champaign, based on Hecke's notes. Providing many details omitted from Hecke's notes, it includes various new and important developments in recent years. In particular, several generalizations and analogues of the original Hecke theory are briefly described in this concise volume