Studies in the Development of Modern Mathematics, 2. New York etc.: Gordon and Breach Science Publishers. x, 411 p. $ 90.00; £46.00 (1991).
The arithmetical theory of modular functions has been a central theme in the 19th century mathematics. Many outstanding mathematicians have contributed a wealth of ideas and results to this fascinating area, and it has taken more then 150 years for this theory to reach its contemporary, highly developed state. At present, the theory of modular functions enjoys a new period of great interest, vast activities, and spectacular applications, particularly in algebraic geometry, arithmetical geometry, algebraic number theory, and - recently - in the mathematical framework of quantum field theory.
One of the main sources of the theory of modular functions is the theory of complex multiplication of elliptic functions, whose origin can be traced back to earlier papers by Gauss, Abel, and Eisenstein. However, the theory of complex multiplication is mainly associated with the name of Leopold Kronecker, who not only made decisive contributions to it, but also inspired (and challenged) the following generations of mathematicians by a general problem which he called the most beloved dream of his youth (“liebster Jugendtraum”). This dream of his was to see the formulation of a complete theory of complex multiplication in full comprehension.
In the present book, the author studies Kronecker’s “Jugendtraum” from its genesis, i.e., from the development of elliptic function theory, up to the present state of modular function theory, including its most recent achievements and applications. According to this pretentious programme, the book is divided into three parts.
Part I is of historical nature and describes the development of complex multiplication theory in its relationship to the arithmetical theory of modular functions. Chapter I is devoted to Kronecker’s biography, comments on his main works, and the lasting impact of his ingenious ideas. Chapter II sketches the development of various topics in mathematics (such as elliptic functions, modular equations, theta- functions, elliptic curves, the origins of algebraic number theory from Gauss to Kummer, etc.) that finally led to questions related to Kronecker’s “Jugendtraum”. Chapter III discusses the works of Kronecker’s predecessors with regard to complex multiplication theory, namely the papers of Gauss, Abel, and Eisenstein.
Kronecker’s synthesis, i.e., his papers devoted to his “Jugendtraum”, is analyzed in Chapters IV, whilst Chapter V explains the development of complex multiplication theory after Kronecker, i.e., from the research at the end of the 19th century, via M. Deuring’s algebraic approach, up to its multidimensional generalization formulated by Weil, Shimura and Taniyama between 1950 and 1960. In an Appendix A, at the end of Part I, the author recalls some definitions and results from algebraic number theory, elliptic function theory, and the theory of elliptic curves, which are helpful to the reader of this historical part of the book.
Part II is just a reprint of L. Kronecker’s original and very important paper “Zur Theorie der elliptischen Funktionen. XI” published in Berl. Ber. 1886, 701-780 (1986; JFM 18.0396.04), which contains many crucial ideas of the arithmetical theory of modular functions, viewed by Kronecker himself as the fundamentals for his “Jugendtraum”.
Part III, which takes up nearly one half of the book, is devoted to the current state and various applications of the theory. Chapter I discusses three main approaches to modular function theory from the contemporary point of view, namely the viewpoint of function theory (modular functions and modular forms), the viewpoint of algebraic geometry (modular curves), and the viewpoint of representation theory (automorphic representations). Some topics from the arithmetic theory of modular forms are treated in Chapter II. Here the author focusses on the Eichler-Shimura relation, whose prototype was discovered by Kronecker in his paper reprinted in Part II of the book, and its applications to zeta-functions of modular curves, cusp forms and -adic representations, and the modern theory of complex multiplication.
Chapter III deals with a very recent development in the theory, namely with the finite-characteristic analogue in the function field case, the so-called Drinfeld theory of modular functions. This includes Drinfeld’s theory of elliptic modules over a field and their morphisms, the theory of nonarchimedean modular forms on rigid analytic spaces, the Tate algebra, moduli schemes for elliptic modules and their compactifications, and some fragments from the Langlands program.
The final Chapter IV is devoted to five selected, utmost brilliant and celebrated applications of the theory of modular functions. The first four of them deal with rather subtle problems in number theory, namely with effective methods in the arithmetic of complex quadratic fields, bounds for division points on elliptic curves, the main conjecture in the Iwasawa theory and its proof (by Mazur and Wiles) via the theory of abelian extensions of and their class fields, and the very recently discovered link between modular function theory and Fermat’s Last Theorem (via the Taniyama-Weil conjecture). The fifth, and concluding, application consists in explaining Goppa’s construction of error-correcting codes from algebraic curves over finite fields and its application to different classes of modular curves. This leads to important new results concerning the parameters for error-correcting codes of large length. Chapter III ends with an Appendix B, in which the author recalls the basic facts from the arithmetic theory of elliptic curves, as they are used through the entire Chapter III. The whole treatise is enhanced by some bibliographic remarks to Part III (i.e., to the recent developments), and by a carefully selected bibliography, ranging from the origins to the most recent papers.
Altogether, this book is a brilliant piece of mathematical culture, an invaluable guide and reference book of encyclopedic character, and a welcome source of inspiration for any interested reader, including specialists in the field, students, and historians specialized in the 19th century mathematics. The excellent style of writing and the many (complete or sketched) proofs, the introductory remarks, and the added appendices make this book even largely self-contained, at least for the first, informative reading. The author really succeeded in producing a masterpiece of mathematical writing!