You are here

Six Short Chapters on Automorphic Forms and L-functions

Ze-Li Dou and Qiao Zhang
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
, on

The first sentence of the preface is what grabbed my attention: “This modest tract offers a little light reading on a heavy subject.” Very nice, and much appreciated.

What we have here is exactly what the title says: there are six chapters, and they are all related in one way or another to automorphic forms. They derive from expository talks given by the authors in many different settings to audiences who they describe by saying that “Although all were interested in the topics in general, many were not experts in the field.” My guess is that this means that they were active mathematicians whose research work came close to the field of modular and automorphic forms, but who may not have been experts in the particular topics at hand. The book would be very slow going for someone totally ignorant of automorphic things.

The dominant influence is clearly that of Goro Shimura, and that is by no means a bad thing. The first chapter is probably the most accessible, since it begins with some basic definitions and motivation for considering classical modular forms. The second chapter begins by summarizing the basic definitions in two pages and then launches right into more technical stuff on periods of automorphic forms. Things get even more technical in the later chapters. Along the way, we meet many important theorems and conjectures, from the Shimura-Taniyama Conjecture (now a theorem, of course) to the authors’ own work.

It would be easy to put down the book by arguing that it is too technical for the broader mathematical community and not technical enough for those doing research in the field. But I don't think that would be correct. In fact, the book may well serve as light reading for the modular/automorphic community, and could be of great value to graduate students beginning to try to understand the field. 

Fernando Q. Gouvêa is probably still a member of the modular/automorphic community. 

Modular forms and the Shimura-Taniyama Conjecture

Periods of automorphic forms

Lifting of automorphic forms

Zeros of L-functions

Special Values of L-functions

Theta lifts and periods with respect to a quadratic extension.