Theta constants, Riemann surfaces and the modular group. An introduction with applications to uniformization theorems, partition identities and combinatorial number theory (Q2756813)

The theory of theta functions is rather like the planet Venus -- it has two aspects, like the Evening and Morning Star. One of these is the role that theta functions play in the function theory and in particular the general theory of uniformization, the other is a large number of beautiful formulae which these relate functions to one another and which often have remarkable number-theoretic, representation-theoretic or combinatorial interpretations. Although both aspects are both beautiful and of great significance they are, like the Evening and Morning Star rarely to be seen at the same time -- here one should note that it does happen that Venus passes to the North of the Sun so that in Northern latitudes it passes from being the Evening Star; this phenomenon will be visible in nothern Norway in May, 2002. In this beautiful book the authors unite these two aspects. The basis of their treatment is function-theoretic and they describe the theory of the theta functions with characteristics and theta constants much more carefully than the standard texts do; whereas this theory is ``well-known'' it is often very difficult to find statements one needs and the authors point out that one of their motivations was just this problem. Having built up the function-theoretic basis in the first two chapters they then study many special cases of the function theory of modular varieties using theta constants to provide enough functions and forms. In the fourth chapter the emphasis changes to identities between theta constants. In the final three chapters it is the applications to combinatorial and arithmetical questions that comes into the centre of the considerations. There are many identities and congruences given here that are new, at least to the reviewer. In proving them the authors have made use of modern computer-based methods to both find and prove identities. This is not only heuristically and technically useful, it also gives one confidence in the truth of the identities, for many developments since the time of Ramanujan need more than human patience to verify. It is a real pleasure to read this book. In the introduction to his ``A brief introduction to theta functions''; Holt, 1961, \textit{R. Bellman} writes, ``The theory of elliptic functions is the fairyland of mathematics. The mathematician who once gazes upon this enchanting and wondrous domain crowed with the most beautiful relations and concepts is forever captivated.'' With this book the modern mathematician has a contemporary path into this world.