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Hecke's Theory of Modular Forms and Dirichlet Series

Bruce C. Berndt and Marvin I. Knopp
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
Michael Berg
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I had the great good fortune to receive my graduate education from Audrey Terras at UC San Diego in the early 1980s. UCSD was something of a hotbed for research in modular forms, with Terras, Harold Stark, and Ron Evans on the permanent faculty and a continuous stream of wonderful visitors in evidence all the time. Happily Evans, Stark, and Terras are still going strong at my alma mater and I often try to steer my best undergraduate students toward La Jolla, given, too, that UCSD’s Department of Mathematics can boast of so many strengths besides number theory.

In any event, back in those halcyon days when I started my PhD with Audrey she presented me right off the bat with a photocopy of Erich Hecke’s 1938 lecture notes from the Institute of Advanced Study, “…[on] Dirichlet series, modular forms, and quadratic forms.” I was to study these notes carefully as they contained the master’s own presentation of his (i.e. the Hecke) correspondence; I ended up writing my thesis on later related work by André Weil.

Roughly speaking, the Hecke Correspondence entails a dictionary between certain modular forms of a given type (weight, level, character, &c.) and Dirichlet series obeying an corresponding functional equation; going back and forth between these two kinds of analytic objects is facilitated by the formalism of the Mellin (integral) transform and its inverse. In this context one also meets the classical version of the Hecke operators; the modular forms that get to play have to be simultaneous eigenfunctions for this family of operators. This obviously augurs suggestive connections with unitary representation theory, and the game is afoot: a treasure trove of gorgeous mathematics.

Given the role that modular forms play in number theory these days (with Weil championing them in the 1960s and fostering their dramatic resurgence, and with their lead role in Andrew Wiles’ work on the Shimura-Taniyama-Weil conjecture and Fermat’s Last Theorem), and noting also that Dirichlet series (or L-functions) are arguably the lifeblood of analytic number theory, the book under review, Hecke’s Theory of Modular Forms and Dirichlet Series, has far more than historical significance. The authors, Bruce Berndt and Marvin Knopp, need no introduction, of course, and their mastery of the subject is abundantly evident on every page of the book. It is also worth noting that Berndt and Knopp include in their discussion a treatment of the interesting generalization of Hecke’s work given in the 1950s by S. Bôchner. This only increases the great worth of Hecke’s Theory of Modular Forms and Dirichlet Series as a gateway into research in the theory of modular forms. Indeed, in the Preface, in Knopp’s section, we find the passage: “For the past 35 years I have employed both sets of notes [i.e. Hecke’s original IAS notes and a 1970 note set by Berndt based on the former] to introduce graduate students to the Hecke theory and the broader theory of modular/automorphic forms.” And rightly so!

On a personal note, seeing that my copy of Hecke’s IAS lecture notes is in a pitiable state, I am very happy to meet this beloved material again twenty five years later, presented in this beautiful book. Hecke’s Theory of Modular Forms and Dirichlet Series will be of huge importance to fledgling number theorists working on modular forms — in fact, given Hecke’s exalted place in this part of the firmament, the book is really indispensable.

Michael Berg is Professor of Mathematics at Loyola Marymount University.

  • The Main Correspondence Theorem
  • A Fundamental Region
  • The Case λ > 2
  • The Case λ < 2
  • The Case λ = 2
  • Bochner's Generalization of the Main Correspondence Theorem of Hecke and Related Results
  • Identities Equivalent to the Functional Equation and to the Modular Relation