For most elliptic curves, the only endomorphisms are multiplication by integers. But for a select class of elliptic curves there are some extra endomorphisms. These are the *complex multiplication elliptic curves* usually called CM elliptic curves or elliptic curves of CM type. If E is a CM elliptic curve, tensoring its endomorphism ring End(E) with the field of rational numbers **Q** one gets an imaginary quadratic field extension of **Q** and End(E) is an order in that extension. Elliptic curves are one-dimensional Abelian varieties, and there is a similar theory of complex multiplication for those varieties.

Sometimes having extra endomorphisms is an advantage. For example, it was for these curves that the Shimura-Taniyama conjecture was first proved. It is also for CM elliptic curves that the Birch and Swinnerton-Dyer conjecture is known to be true. But at other times CM curves and/or CM Abelian varieties are the hard case. This seems to be the case for the Hodge conjecture for Abelian varieties of CM type.

The classical theory of elliptic curves of CM type has an illustrious history. It starts with the pioneering work of Kronecker and Weber in the late 19th century, to Takagi, Fricke and Fueter in the early 20th century, up to Hasse, Deuring, Eichler and Shimura in the mid 20th century, to name a few mathematicians that have made fundamental contributions to the arithmetic of CM elliptic curves. As can be gathered from the previous roll call, elliptic curves of CM type come with a lot of arithmetic baggage. The first important result in this context is analogous to the Kronecker-Weber theorem: All Abelian extensions of the field of rational numbers are generated by the values of the exponential function at the torsion points of the unit circle. Kronecker conjectured (actually, “dreamt”) that for imaginary quadratic fields there is a corresponding analogous result: The maximal abelian extension of an imaginary quadratic field is generated by the *j*-invariant of a CM elliptic curve and its torsion points.

The book under review is a comprehensive account of the classical theory of complex multiplication for elliptic curves, with some concessions to the current trends in applications, including a chapter on elliptic curve cryptography.

The exposition attempts to be self-contained, and in the first three chapters we find the basic facts on elliptic functions (lattices, double periodic functions, the Weierstrass Ã-function, Eisenstein series, q-expansions and torsion points), modular functions (congruence subgroups of the modular group, modular forms and functions, the j-invariant), basic facts from algebraic number theory (for quadratic number fields and their orders, including some results on Dirichlet densitiy for prime ideals in a quadratic number field and a brief review of class-field theory: The Artin map, ray class-fields, the Hilbert class-field and the conductor-discriminant formula for imaginary quadratic fields).

The middle part of the book, chapters 4, 5 and 6, are devoted to the main results for the arithmetic of CM elliptic curves. Chapter four starts by showing that modular functions of level N have algebraic values when calculated at quadratic imaginary numbers, and some factorization properties of these singular values. These singular values are put to use in chapter five to obtain abelian extensions of quadratic number fields. In chapter six, using the singular values of the j-modular function and Weber’s τ function the corresponding ring class-fields and ray class-fields are constructed. One highlight of chapter 6 is Heegner’s proof that there are only nine imaginary quadratic fields of class-number one. Further arithmetic applications are given in chapters 11 and 12: Class number formulas, analogues to the class number formula for cyclotomic fields, and arithmetic interpretations of these formulas.

After obtaining integer bases for ray class-fields in chapter 7, in the next chapter the Galois module structure of certain integral objects is given. These are relatively new results, basically obtained in 1980s and 1990s after the initial input by Leopoldt in the 1960s. For their use in the applications chapter, the last sections of chapter 8 are devoted to study explicit models of elliptic curves by Weierstrass, Fueter and Deuring. These models are put to work on cryptographic applications on chapter 10.

Before the publication of this book there were no monographs totally aimed to the classical theory of elliptic curves of CM type, only chapters in some books on the arithmetic of elliptic curves, e.g., Chapter 2 of the second volume of Silverman's treatise on the arithmetic of elliptic curves. The publication of the book under review fills this void and prepares the interested reader for the more abstract theory of complex multiplication for Abelian varieties of arbitrary dimension.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is [email protected].