A Short Course in Automorphic Functions

This nice little book was originally published in 1966 in the famous “Athena Series” of short mathematical monographs. It offers a very clear, if somewhat old-fashioned, introduction to the classical theory of discontinuous groups and automorphic functions.

All holomorphic automorphisms of the complex sphere are Möbius transformations, which Lehner calls “linear transformations”: \[ z \mapsto \frac{az+b}{cz+d}, \] where \(a,b,c,d\) are complex numbers and \( ad-bc\neq 0\). The set of all such transformations is a group, and the main object of study in the first chapter of this book are its discrete subgroups, called “discontinuous groups of linear transformations.”

Lehner quickly restricts himself to transformations where \(a,b,c,d\in\mathbb{R}\) and \(ad-bc=1\). These are the transformations that map the complex upper half-plane to itself. The most interesting groups of transformations are those that are discrete. The most famous of these are the modular group (require \(a,b,c,d\in\mathbb{Z}\)) and its subgroups, but Lehner’s treatment is general.

The goal of the first chapter is to describe how these groups act on the upper half-plane, proving in particular that one can always find a nice fundamental domain. Along the way Lehner shows how to turn the upper half plane into a model of plane hyperbolic geometry, which allows him to reinterpret real Möbius transformations as isometries. The discussion is very interesting and not too hard to follow.

In the second chapter, Lehner introduces automorphic forms and functions, constructing them via Poincaré series. This chapter is heavy on analysis, since one has to prove that various series and integrals converge and are invariant (or transform nicely) under the group action. It culminates with the definition of the Petersson inner product. A long appended “note” gives a brisk summary of the theory of Hecke operators and the link between modular forms and Dirichlet series. The half-century distance between then and now becomes particularly visible in this chapter, for example when Lehner talks of “automorphic forms of dimension \(-2m\)” where we would say “modular forms of weight \(2m\).”

In the third and last chapter, Lehner considers the quotient of the (extended) upper half-plane by the group action and shows how to make it into a Riemann surface. Once that is done, there is some discussion of how to transfer theorems back and forth. Using the Riemann-Roch theorem, the dimensions of various spaces of automorphic forms are calculated. This is the least self-contained of the chapters, with frequent references to Springer’s famous Introduction to Riemann Surfaces.

Lehner says that the book assumes only “the usual first courses in complex analysis, topology, and algebra.” In fact, most of the topology is done from scratch (especially in chapter 3), and the algebra amounts to knowing some basic group theory. The analytic prerequisites, however, are quite serious, certainly more than one would learn (today, at least) in an undergraduate course.

There are two ways to write a brief book. One can limit the topics discussed or one can be ruthlessly concise. Lehner does both, but one must add that he is also incredibly clear. The result is a book that one can read with huge profit: many sections need unpacking, but the effort is rewarded. It is also an easy book to skim: Lehner usually tells his reader what is going on, so one can easily leave the details of an argument for later consideration and still follow the overall thread.

This book should not be confused with another small book by the same author, Lectures on Modular Forms, published in 1969 by the National Bureau of Standards. That is a very different book, a bit more modern, paying much more attention to Hecke operators and the coefficients of the \(q\)-expansion of modular forms. Like this book, however, it is short and very clear, a pleasure to read. Lehner clearly knew what he was doing.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College. He likes modular forms, good writing, and old books.