1.1. From the Publisher.

The study of abelian manifolds forms a natural generalisation of the theory of elliptic functions, that is, of doubly periodic functions of one complex variable. When an abelian manifold is embedded in a projective space it is termed an abelian variety in an algebraic geometrical sense. This introduction presupposes little more than a basic course in complex variables.

1.2. Review by: Helmut Klingen.
Mathematical Reviews MR0366934 (51 #3180).

This book is a concise and easily readable introduction to the theory of abelian functions. The first chapter presents some preliminaries on compact Riemann surfaces, especially the Riemann-Roch theorem and Abel's theorem, as well as a short survey of elliptic functions and some background on functions of several complex variables. In Chapter 2 an elementary approach to Weil's proof of the existence of theta-functions for an arbitrary positive divisor on a complex torus is given. Then the author proves the classical theorem that a non-trivial theta-function exists if and only if there is a non-trivial positive semi-definite Riemann form on the given torus. Frobenius' formula for the dimension of the space of theta-functions of a given type is derived. Consequences are concerned with the structure of the field of abelian functions and the projective embedding of an abelian variety. The last chapter deals with morphisms, polarisations and the duality theory of abelian varieties; especially Poincaré's complete reducibility theorem for abelian varieties is proved and the ring of endomorphisms of an abelian variety is studied.

This book can be recommended to beginning graduate students and presupposes not much more than a basic complex variable course.

2.1. From the Publisher.

This is a 2001 account of Algebraic Number Theory, a field which has grown to touch many other areas of pure mathematics. It is written primarily for beginning graduate students in pure mathematics, and encompasses everything that most such students are likely to need; others who need the material will also find it accessible. It assumes no prior knowledge of the subject, but a firm basis in the theory of field extensions at an undergraduate level is required, and an appendix covers other prerequisites. The book covers the two basic methods of approaching Algebraic Number Theory, using ideals and valuations, and includes material on the most usual kinds of algebraic number field, the functional equation of the zeta function and a substantial digression on the classical approach to Fermat's Last Theorem, as well as a comprehensive account of class field theory. Many exercises and an annotated reading list are also included.

- By a world class researcher.
- Based on a graduate course given at the University of Cambridge.
- Much broader coverage than other similar books.

2.2. From the Preface.

This book is intended both for number theorists and more generally for working algebraists, though some sections (notably §15) are likely to be of interest only to the former. It is largely an account of mainstream theory; but for example Chapter 3 and §20 should be seen as illustrative applications.

An algebraic number field is by definition a finite extension of $\mathbb{Q}$, and algebraic number theory was initially defined as the study of the properties of algebraic number fields. Like any empire, its borders have subsequently grown. The higher reaches of algebraic number theory are now one of the crown jewels of mathematics. But algebraic number theory is not merely interesting in itself. It has become an important tool over a wide range of pure mathematics; and many of the ideas involved generalise, for example to algebraic geometry. Some applications to Diophantine equations can be found among the exercises, but there has not been room for other applications.

Algebraic number theory was originally developed to attack Fermat's Last Theorem - the assertion that $x^{n} + y^{n} = z^{n}$ has no non-trivial integer solutions for $n > 2$. It provided proofs that many values of $n$ are impossible; some of the simpler arguments are in §13. But it did not provide a proof for all $n$, though recently the theorem has been proved by Andrew Wiles, assisted by Richard Taylor, by much more sophisticated methods (which still use a great deal of algebraic number theory). There are still respectable mathematicians seeking a more elementary proof, and this is not a ridiculous quest; but even if a more elementary proof is found, it is almost bound to be highly sophisticated.

There are two obvious ways of approaching algebraic number theory, one by means of ideals and the other by means of valuations. Each has its advantages, and it is desirable to be familiar with both. They are covered in Chapters 1 and 2 respectively. In this. book I have chosen to put the main emphasis on ideals, but properties which really relate to local fields (whether or not the latter are made explicit) are usually best handled by means of valuations. Chapter 3 then applies the general theory to particular kinds of number field. The first two chapters (perhaps omitting §9), together with the easier parts of Chapter 3 and the first half of the Appendix, would form a satisfactory and self-contained one-term graduate course.

Though §9 is more advanced than the rest of Chapter 2, its logical home is there; it is needed in Chapters 4 and 5, and introduces language which is widely used across number theory. The somewhat peripheral §12 depends on the results stated in §14 and proved in §15 of Chapter 4, as do parts of §13.1, and thus they are not in the correct logical order; but there are advantages in collecting all the information on special kinds of number field in a single chapter.

There are important results which, though not in appearance analytic, can as far as we know only be proved by analytic methods. Indeed it has been said: 'The zeta function knows everything about the number field; we just have to prevail on it to tell us.' Some of what it has already told us can be found in Chapter 4.

The more advanced parts of the algebraic theory are generally known as class field theory; most of the proofs involve Galois cohomology, either openly or in disguise. Anyone who writes a book on algebraic number theory is faced with a dilemma when he comes to class field theory. Most authors stop short of it; but working algebraists ought to know the main results of class field theory, though few of them need to understand the rather convoluted proofs. I would think it wrong to make no mention of class field theory; but to have included the proofs and the necessary background material would have doubled the length of the book without doubling its value. In consequence §§17 and 18 present an exposition of class field theory without proofs. In §19 we deduce the general reciprocity theorems, which are the simplest major applications of class field theory. In addition, §20 contains a proof of the Kronecker-Weber Theorem that every abelian extension of the rationals is cyclotomic; it is this result which made the general structure of classical class field theory plausible long before it was proved. The proof of the Kronecker-Weber Theorem is also rather convoluted, but it illustrates most of the ideas in the first two chapters.

The reader needs to know the standard results about field extensions of finite degree, including the relevant Galois theory. The properties of finitely generated abelian groups and lattices, and of norms and traces, are described in §A1.1 and §A1.2. Most readers will already know these results, but those who do not will need to start by reading these two subsections. The existence of Haar measure and the Haar integral (described without proofs in §A1.3) is a fact which all working mathematicians should know, though again they have no need to study the proofs. Indeed, the main use of the general theory is to provide motivation and guidance; in any particular case one can expect to be able to define explicitly an integral having the required properties, and thereby evade any appeal to the general theory. The status of §A2 is rather different. The Galois theory of infinite extensions is not actually needed anywhere in this book; but anyone who uses the results in Chapter 5 may need to consider field extensions of infinite degree. The remaining subsections of §A2 cover (without proofs) characters, duality and Fourier transforms on locally compact abelian groups; these are prerequisites for §15, but also for much of advanced number theory.

The book concludes with a substantial collection of exercises. Others can be found in the text; see the index. The latter are results which are too peripheral to justify the provision of a detailed proof but which may be interesting or useful to the reader. Each of them is provided with 'stepping-stones': intermediate results which are individually not too difficult and which should enable the reader to construct a complete proof.

2.3. Review by: Władysław Narkiewicz.
Mathematical Reviews MR1826558 (2002a:11117).

The book contains an introduction to the theory of algebraic numbers. It contains 5 chapters. The first presents the principal theorems of the theory (using the ideal-theoretic approach): unique ideal factorisation, Dirichlet's unit theorem and the finiteness of the class number. In the case of a normal extension the first three Hilbert groups are introduced. The second chapter introduces valuations and completions and uses them to prove the different theorem as well as the discriminant theorem. It concludes with the definition and principal properties of idèles and adèles. The theory is illustrated in the next chapter by quadratic, pure cubic, biquadratic and cyclotomic fields. As an application a proof of Fermat's Last Theorem for regular prime exponents is given (using certain facts from class field theory, which are presented without proofs in Chapter 5). Chapter 4 is devoted to analytic methods: Dedekind's zeta-function and Hecke's $L$-functions are introduced and the functional equation for them is established, following the steps of Tate's thesis. The presentation of class field theory in the last chapter is done both in the classical and modern way and is applied to give proofs of the quadratic reciprocity law for algebraic number fields and of the Kronecker–Weber theorem. There are 8 appendices explaining various algebraic and analytic topics used in the main body of the book.

This book seems to be ideally suited for non-specialists who would like to know quickly what algebraic number theory is about but who do not wish to study thick volumes dealing with that subject.