One of the most interesting themes in modern number theory is the surprising and deep connection between the arithmetic properties of algebraic objects and special values of their associated L-functions. For instance, the Birch and Swinnerton-Dyer conjecture asserts that the order of vanishing at s = 1 of the Hasse-Weil L-function of an elliptic curve E equals the free rank of the group of rational points on E. Much more generally, the Bloch-Kato conjecture relates the values of motivic L-functions with the order of Tate-Shafarevich groups that are defined cohomologically. This book is, in part, an introduction to the work of Beilinson and Kato, who discovered the connection in certain settings between critical values of L-functions and certain cohomology classes (the Kato-Beilinson zeta-elements). We will summarize the contents of the book below.

There has been an incredible number of developments in this area in recent times; hundreds of research articles on these topics are published every year. However, there is an underwhelming number of expository articles and books that attempt to make recent work accessible to a larger audience. Delbourgo’s book is a very welcome contribution in this department, as it covers much material that is well-known to the experts but difficult (or impossible) to find in print unless the researcher patiently wades through a myriad very technical research papers. It is fantastically convenient to have all these results summarized in one single volume with a uniform approach, notation and goals. Moreover, this book does a very good job at motivating every step and putting results into perspective, so that the reader is aware at all times of what has been accomplished so far and what remains ahead.

A word of caution, however, is in order: this is an extremely technical area, and this is an extremely technical book. The reader will need a very solid background before being able to parse even Chapter 2. To my surprise, the author writes in the introduction that “The reader who has done a graduate-level course in algebraic number theory, should have no trouble at all in understanding most of the material.” This is perplexing, since (in my opinion) the reader of this book certainly needs a solid background on algebraic number theory, but in an extremely broad sense: from a complete treatment of number fields and local fields to the theory of elliptic curves and modular forms (with an approach heavy in cohomology), passing by p-adic analysis and K-theory. I don’t know of any graduate-level course in algebraic number theory that would come close to cover even one of those topics. In any case, as I mentioned above, the book should be very useful for graduate students and young researchers who do have a solid background in algebraic number theory, very broadly construed.

Here is a summary of the contents of the book. Chapter 1 consists of a very brief review (24 pages!) of some of the background material that the reader needs to be familiar with before diving into the rest of the book: elliptic curves, Tate modules and their associated Galois representations, Hasse-Weil L-functions, complex multiplication (CM), the Mordell-Weil, Selmer and Tate-Shafarevich group, the Birch and Swinnerton-Dyer conjecture (BSD), modular forms and Hecke algebras.

In Chapter 2, the author introduces p-adic L-functions through the language of measure theory. The goal here is to state non-archimedean versions of the BSD conjecture. The local Iwasawa machinery of Perrin-Riou is also introduced to replace analytic p-adic L-functions with other objects of more algebraic flavor. Then the text develops the theory of Kato’s p-adic zeta-elements, which combines the K-theory approach of Beilinson and the work of Coates and Wiles that relates the L-function of a CM elliptic curve with a certain Euler system of elliptic units.

All these elements are used in Chapter 3 to develop a new way to construct p-adic L-functions, which assigns a modular symbol to each Euler system. The author explains how these symbols (M-symbols) can be deformed along a cyclotomic variable. Chapter 4 is a very useful “user’s guide to Hida theory”. In Hida theory the weight is considered as a new variable and the modular symbols can be deformed once again along this weight variable.

In the rest of the book, the author develops the theory of two-variable Euler systems and their deformations (which allow a formulation of a Tamagawa number conjecture for the universal nearly-ordinary Galois representations) and concentrates on the study of the arithmetic of p-ordinary families. In the last chapter of the book, the whole theory comes together to state a two-variable main conjecture of Iwasawa theory of elliptic curves without error terms.

Álvaro Lozano-Robledo is Assistant Professor of Mathematics and Associate Director of the Q Center at the University of Connecticut.