The theory of modular forms is without question among the most popular and most lucrative areas of research in modern number theory, perhaps the most popular of all.
The huge role modular forms played in the proof of the Shimura-Taniyama-Weil conjecture (“every rational elliptic curve is modular”) and its titanic corollary, Fermat’s Last Theorem (“Gerhard Frey’s elliptic curve doesn’t exist”) at the hands of Andrew Wiles in the 1990s has a lot to do with this current vogue, but there’s a lot more to it than that. It is in fact possible to trace the current flourishing of the subject back some forty years ago, to the work of such figures as André Weil, Gôro Shimura, Hans Maaß, and Martin Eichler (with my apologies for any pioneer I’m overlooking). And, behind the (blessedly erstwhile) Iron Curtain we should of course mention the Gel’fand school, including the recently deceased Ilya Piatetskii-Shapiro, and, in Leningrad, the author of the book under review, Anatoli Andrianov.
By the way, speaking of pioneers, it’s impossible to resist parroting Eichler’s wonderful quip, cited by Peter Sarnak in “The Proof,” doubtless the best NOVA program of all time (dealing with Wiles achievement): There are five elementary arithmetical operations: addition, subtraction, multiplication, division, and… modular forms!
But this recent and ongoing flurry of activity in modular forms is really its second flowering. Its first Spring occurred as early as the famous competition between Felix Klein and Henri Poincaré concerning automorphic functions, the subsequent (or even contemporaneous) work of Robert Fricke, and then the corresponding activity of the Hilbert School, especially Erich Hecke and his student Hans Petersson.
Arguably it was Hecke’s work on the relationship between modular forms and L-functions that claims the lion’s share of the credit for giving shape to the modern theory. Indeed, this is where we first encounter what is now often called the Hecke correspondence, matching modular forms whose Fourier coefficients are eigenvalues of an algebra of linear operators (the Hecke operators, of course, populating the archetypal Hecke algebra) with Dirichlet series obeying a certain functional equation. Of course, in a dramatic fashion, this hearkens back to nothing less than Riemann’s 1859 landmark paper, Ueber die Anzahl der Primzahlen unterhalb einer gegebenen Größe, dealing with his zeta function and the inner life of the primes, where two proofs are given for the zeta functional equation, the second of which is obtained by transforming the zeta function, via an integral transform, into a theta function — and a theta function is a modular form of half-integral weight.
Well, Hecke was Hilbert’s student, and Carl Ludwig Siegel was Landau’s student, all during what has been called the Golden Age of the Georg August Universität at Göttingen, with Klein still at the helm as far as the department’s administration was concerned. With these trailblazers present, making for a terrific critical mass, it comes as no surprise that marvelous evolutions should occur in the theory of modular forms, and so it came to pass that the classical theory was supplemented by such important generalizations as the theory of Hilbert modular forms and (considerably later?) the theory of Siegel modular forms. It is the latter that forms the focus of the book at hand.
Classically, an elliptic modular form is characterized as a complex function by being invariant under the action of a suitable subgroup Γ of SL(2,Z), modulo a critically important weight factor, or multiplier. And in light of this group action (by fractional linear transformations) a modular form can be realized as living on the complex (or Poincaré) upper half-plane as well as on the fundamental domain obtained by dividing out by the underlying action of Γ.
Analogously (at least to an extent) a Siegel modular form lives on the Cartesian product of n copies of the complex upper half plane divided out by an action of the modular symplectic group Sp(n,Z), a.k.a. the Siegel modular group of genus n, and is itself invariant (modulo a factor of huge importance) under the action of a discrete group K, commensurable with Sp(n,Z) in the ambient group Sp(n,R).
Furthermore, here too there is a fundamental domain in the game, but its construction is a comparatively much more tricky business involving the Minkowski theory of positive definite quadratic forms. Here we also encounter the famous question of whether such a fundamental domain has finite volume: for Siegel modular forms it does. Additionally, it is the case that when calculating with modular forms their expandability into Fourier series is of paramount importance, and this feature is present for both classical and Siegel modular forms. The fact that Andrianov gets to this theorem on already on p. 22 says a good deal about the pace he sets in this book.
So it is that Introduction to Siegel Modular Forms and Dirichlet Series is a compact but masterful presentation of this important generalization of the classical theory, and a good deal more. Andrianov describes the book as “a concise but basically complete and self-contained introduction to the multiplicative theory of Siegel modular forms, Hecke operators, and zeta functions, including the classical case of modular forms of one variable.” And all the bases are covered: “radial Dirichlet series attached to modular forms of genera 1 and 2,… Hecke-Shimura rings for symplectic and related groups… multiplicative properties of Fourier coefficients of modular forms and the related Euler product factorization of radial Dirichlet series attached to eigenfunctions [of a Hecke algebra],” the latter being a very famous classical theme, as already hinted at above. Andrianov goes on to state that “[t]his leads us to Hecke zeta functions of modular forms of one variable [again a classical theme] and to spinor (or Andrianov [!]) zeta functions of Siegel modular forms of genus two…”
With the book coming in at 174 pages this makes for a pretty dense presentation of very serious and deep mathematics. Andrianov says in the Preface that “[n]o special knowledge is presupposed for the reading of this book beyond standard courses in algebra and calculus (one and several variables), although some skill in working with mathematical texts would be helpful.” Granting that these traits were necessary for a reader of the book, are they also sufficient? I think not. To be sure everything is done in great detail in this book, but it’s a tall order to require of the reader that he should pick up facility and comfort with Fourier series, large chunks of complex analysis and complex function theory, rings (or algebras) of operators, infinite products, etc., “working with mathematical texts.” Manifestly the likely beneficiaries of this wonderful book are the obvious candidates: students of number theory with their qualifying examinations behind them, or very gifted undergraduates who have already learned group theory and complex analysis, some topology, some rings and fields, and so on. Additionally, it seems to me that what Andrianov does in the book under review were most highly appreciated by some one who already possesses a good deal of knowledge of the classical case. (I recommend R. C. Gunning’s Lectures on Modular Forms, and the indicated part of Serre’s Cours d’Arithmétique.) This having been said, Introduction to Siegel Modular Forms and Dirichlet Series should perhaps be compared to Siegel’s own Symplectic Geometry, at least as far as the geometrical part of the discussion is concerned, and the first observation to be made is that Andrianov is far more readable.
If I may be forgiven a personal aside, I have in fact always found Siegel difficult to read, be it in a geometric connection as is the case here, in connection with his analytic theory of quadratic forms (with none other than André Weil as his interpreter and expositor: “Sur certains groupes d’opérateurs unitaires,” Acta Math. 111 (1964)), or in connection with transcendental numbers. I should like to think it’s largely a question of style, and, by contrast, Andrianov’s (modern) style is excellent.
Having introduced a personal note, let me finish on another such. I first came across Andrianov’s name en passant when I was a graduate student in the early 1980s. I was working on Weil’s converse Hecke correspondence at the time, looking for representation theoretic and, by-and-by, functional analytic connections at the time: up to my eyeballs in the one-variable case, with no occasion (or time) to study anything else really, and certainly not Andrianov. I’m happy to get to look at it now: it’s fantastic!
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.
Preface.- Introduction: The Two Features of Arithmetical Zeta Functions.- Modular Forms.- Dirichlet Series of Modular Forms.- Hecke-Shimura Rings of Double Cosets.- Hecke Operators.- Euler Factorization of Radial Series.- Conclusion: Other Groups, Other Horizons.- Notes.- Short Bibliography.-