omplex multiplication of elliptic or elliptic modular functions” and “[a]pplications of Hecke operators to the zeta functions of algebraic curves and abelian varieties,” and, beyond this, or preliminary to it (in the book’s first few chapters), to give “an introductory account of the theory of automorphic functions of one complex variable, along with the fundamentals of Hecke operators.”</p>
<p class="Body">It was in connection with the latter that I, in my university student days, first made the acquaintance not only of the book under review but of Shimura as an author. Many margins, as well as a few of the book’s front or back blank pages, are filled with my scribbles and revisiting the book again now highlights its great utility. When I first used it it served beautifully. Now, about three decades later, I can better appreciate both the depth and elegance of this book.</p>
<p class="Body">So it is that even as he presents an exceptionally thorough discussion of automorphic functions and Hecke operators, Shimura aims at bigger game: he goes on to deal with CM (in the adelic dialect) in two ways (see below); he applies Hecke operators as indicated in order to explicate some work by Martin Eichler done in 1954, the focus falling on Hasse-Weil.</p>
<p class="Body">It should be evident even to outsiders that Shimura’s mathematical taste is superb and his instinct for what matters is equally marvelous. This is borne out is spades, of course, by the role played by the Shimura-Taniyama-Weil conjecture in Andrew Wiles’ conquest of Fermat’s Last Theorem. And one notes that FLT is really only ancillary to a much more sweeping theorem, said conjecture to the effect that all rational elliptic curves are modular, which circumstance only underscores Shimura’s prescience. (It is of course impossible to fail to call to mind Shimura’s filmed comment — at the end of <em>The Proof</em> — conveying his reaction to Wiles’ achievement: “I told you so…”)</p>
<p class="Body">Proceeding to the material substance of <em>Introduction to the Arithmetic Theory of Automorphic Functions</em>, regarding CM of elliptic modular functions, Shimura focuses, as mentioned, on two themes: “[o]ne is concerned with the behavior of an elliptic curve and its points of finite order under automorphisms of the number field in question … [while t]he other is closely connected to the field <em>F</em> of all modular forms of all levels whose Fourier coefficients belong to cyclotomic fields.” And then we get the following tantalization: “It will be shown that the group of all automorphisms of <em>F </em>is isomorphic to the adelization of GL(2, <strong>Q</strong>) modulo rational scalar matrices and the archimedean part. Then the reciprocity law in the maximal abelian extension of an imaginary quadratic field is given as a certain commutativity of the action of the adèles with the specialization of the functions of <em>F</em>.”</p>
<p class="Body">Next, regarding Eichler’s result, Shimura states that “[first] Hasse-Weil will be verified for the algebraic curves uniformized by modular functions.” And then comes the <em>pièce de résistance</em>: “Further we shall show that if a cusp form of weight 2 is a common eigenfunction of the Hecke operators… the product of several Dirichlet series associated with it coincides, up to finitely many Euler factors, with the zeta function of a certain abelian variety…”</p>
<p class="Body">Thus, in a few well-chosen words, Shimura stakes out the territory he proposes to survey in the book, including parts of class-field theory, the theory of elliptic curves, abelian varieties (at least to some extent), and the theory of modular forms and automorphic functions, of course. This is very serious business, if also the quintessence of modern number theory, obviously still very much the Queen of Mathematics.</p>
<p class="Body">Shimura goes on to offer as “an application” of the result a discussion and treatment of the proposition that the classical class field theoretic question, “Can one construct abelian extensions of a real quadratic field by analytic means?”, can be given “a positive, if not complete, answer…” Wonderful icing on the cake.</p>
<p class="Body">The first three chapters of <em>Introduction to the Arithmetic Theory of Automorphic Functions</em> present a course on automorphic functions and modular forms of unsurpassed quality; there is probably no better treatment available of the subject of Hecke operators than Chapter 3. Then we get to elliptic curves, abelian extensions of imaginary quadratic fields and CM (for elliptic curves), after which it is on to higher level modular forms. The book finishes with a long and manifestly weighty chapter on zeta functions for curves and abelian varieties, followed by a treatment of the cohomology groups attached to cusp forms.</p>
<p class="Body">The reader should be forewarned that Shimura’s style is quite compact: he is not one to waste words. But, to be sure, the book is full of wonderful arguments and complete treatments of the themes indicated above.</p>
<p>There is little more to be said. Shimura is a master. The book is a masterpiece.</p>
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<p>Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.</p>
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