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Analytic Functions of Several Complex Variables

Kendrick Press
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I don’t remember if I was a senior or a first-year graduate student when I had the great good fortune of first encountering the work of Carl Ludwig Siegel in a marvelous seminar on transcendental number theory. We studied Siegel’s famous monograph, Transcendental Numbers, which appeared as the sixteenth volume in the Annals of Mathematics Studies series. In point of fact, Transcendental Numbers had been prepared by Richard Bellman from his notes to “a course of lectures given by Professor Carl Ludwig Siegel at Princeton during the spring of 1946.”

What struck me (very forcefully) about Transcendental Numbers already then, at a rather tender mathematical age, was that the style or way of doing mathematics favored by Siegel, or perhaps it’s better to say, exemplified by him, was something quite different. The results were obviously magnificent and modern (we were headed for the Siegel-Shidlovsky theorem), but both the methods used, and certainly the presentation, appeared to be almost anachronistically classical, for lack of a better word. A few years later I found the same evaluation could be applied to Siegel’s book, Symplectic Geometry.

Over the years since graduate school I have come to find out all sorts of marvelous things about the man Siegel, biographical data (see not only Constance Reid’s Hilbert and Courant in Göttingenand New York but also Weil’s Apprenticeship of a Mathematician and Georgiadou’s Constantin Carethéodory) as well as abundant anecdotal material (see, e.g., Krantz’s Mathematical Apocrypha and Yandell's The Honors Class ). The picture that emerges is that of an iconoclastic and even reactionary scholar of the highest order, given to a certain impishness, who allegedly preferred to compute a logarithm by hand to looking it up in a table. (How Siegel would have despised today’s ubiquitous calculators and computers!) And it is perhaps Siegel’s reactionary preference for things classical, doubtless bred into him in the Göttingen mileu of Hilbert and Minkowski (though he was Landau’s student), that accounts for his idiosyncratic style.

Indeed, given that Edmund Landau, Siegel’s thesis advisor, was truly one of the grandmasters of the application of analytic methods to number theoretic problems, it comes as no surprise that classical complex function theory should feature so prominently in so many of Siegel’s own works. And in the book under review, Analytic Functions of Several Complex Variables, Siegel focuses on the analytic methods required to treat functions possessing 2n independent periods and, subsequently, automorphic functions with bounded (fundamental) domain. This is obviously part of Siegel’s own development of the theory of higher dimensional modular forms that have come to bear his name, a subject of considerable current interest. To see the subject treated in Siegel’s own meticulous style, with classical methods dominating, is a marvelous experience in itself as well as terrific training in the techniques of applying analytic methods to multiply periodic functions.

In the preface to the book Siegel notes that his goal in these lectures (now with Paul T. Bateman playing the role of scribe, just as Richard Bellman had done a couple of years earlier) is twofold, namely, to explicate the theory of automorphic functions of n complex variables in two parts: first the focus falls on meromorphic functions with 2n periods and Riemann’s theta theorem; second Siegel deals with automorphic functions with bounded fundamental domain “under adequate assumptions for the underlying group.”

This terse description of the book’s content all but belies what ensues, in that the amount of deep and beautiful mathematics Siegel develops in the pages that follow seems disproportionate to such a compactly phrased objective. But this misconception is dispelled by recalling how much is required even to develop the theory of elliptic modular forms properly, where a single complex variable suffices. For what Siegel has in mind the notoriously more intricate subject of several complex variables is required, and in Analytic Functions of Several Complex Variables we find it developed beautifully, from the Weierstrass preparation theorem, through Cousin’s Problem(s) and Carathéodory’s lemma, to the great classical field of abelian functions (which in fact constitute a field in the algebraic sense). Siegel then proceeds to automorphic functions, as such, after discussing Riemann surfaces and topological groups; he closes with a treatment of the themes of the existence of discontinuous groups and the modular group of degree p.

The book contains a few misprints, really of no consequence: they do not detract from the very high quality of the mathematics presented here, and the masterful exposition Siegel gives of it. Analytic Functions of Several Complex Variables is a true gem of analytic number theory and the general theory of automorphic functions and modular forms.

Michael Berg is Professor of Mathematics at Loyola Marymount University.

Date Received: 
Thursday, April 24, 2008
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Carl L. Siegel
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Michael Berg
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Monday, June 9, 2008