Ever since Euler’s pathbreaking discovery of the marvelous identity \[ \prod^\infty_{n=1} (1- q^n)= \sum^{+\infty}_{m=-\infty}(-1)^m q^{m(3m- 1)/2}, \] theta functions and in particular the eta function never ceased to be a source of amazing mathematical findings.
In the book under review mainly a highly interesting special class of theta functions is investigated, the class of Hecke theta series for quadratic number fields. These series provide a pleasant method to construct modular forms on subgroups of \(\text{SL}_2(\mathbb Z)\), e.g. in the case of weight  1 where other methods fail. An extra bonus of Hecke theta series for quadratic number fields is that these functions are automatically Hecke eigenforms. The author makes use of this fact as follows: Modular forms on subgroups of \(\text{SL}_2(\mathbb Z)\) may also be constructed by linear combinations of quotients of eta functions (with suitable arguments). It may be tedious to show that such a form is in fact a Hecke eigenform. But once one can show that the form is equal to a Hecke theta series, one has not only proved some usually remarkable theta series identity but also the bonus result that the form is a Hecke eigenform.
In the course of roughly 25 years the author collected a huge number of examples on the approach indicated above. Among these are some 150 most remarkable special examples of theta series of weight one on three distinct quadratic number fields (two of them imaginary quadratic, the third of them real quadratic) where the theta series coincide with one and the same linear combination of eta products. The functions \(\eta^2(z)\), \(\eta(z)\eta(2z)\), \(\eta(z)\eta(5z)\), \(\eta(z)\eta(7z)\) are known example for this strange phenomenon.
The present monograph splits into two parts of quite different length. Part I comprises roughly 100 pages and contains the theoretical background. Besides the relevant facts on e.g. the eta function, Eisenstein series, Fricke groups and Hecke eigenforms the author thoroughly investigates the problem as to when an “eta product” \[ \prod_{m\mid N}\eta(mz)^{a_m} \] (where the \(a_m\) are integers, possibly zero or negative) is holomorphic at all cusps of the group \(\Gamma_0(N)\). This problem is transformed into a system of linear inequalities for the exponents \(a_m\) with rational coefficients. These inequalities describe a cone with vertex at the origin in a space of dimension \(\sigma_0(N)\), the number of divisors of \(N\). The exponents \(a_m\) of holomorphic eta products are the coordinates of the lattice points in this cone. Lattice points in the interior yield cusp forms whereas boundary points correspond to non-cuspidal forms. The lattice points for an eta product of (integral or half-integral) weight \(k> 0\) correspond to the intersection of the cone at hand with the hyperplane \(\sum_{m\mid N} a_m= 2k\). This intersection is a compact simplex of dimension \(\sigma_0(N)- 1\). For any fixed weight \(k\) and arbitrary level \(N\) there exist only finitely many holomorphic eta products of level \(N\) and weight \(k\).
In order to provide a means to determine all holomorphic eta products of a given level \(N\) and weight \(k\) explicitly, the author develops an algorithm which lists all lattice points in a rationally defined simplex. This makes it possible to construct linear combinations of the corresponding eta products whose sufficiently long initial segments of Fourier coefficients coincide with those of suitable Hecke theta series. Consequently the functions under consideration coincide and are Hecke eigenforms. In order to facilitate the computations with Hecke theta series the author represents ideals by ideal numbers. The tools for the computations are developed with great care.
Part II comprising roughly 500 pages contains a wealth of explicit examples. Since any attempt to give a reasonably complete survey of the rich contents of this part must fail we can only advise everybody who needs modular forms of low weights for the Hecke or Fricke groups to consult the monograph under review and (s)he will most probably find useful information. In Sect. 8 identities for eta products of weights \({1\over 2}\) and \({3\over 2}\) are collected. The remaining Sections 9–31 deal with eta products of small integral weight, predominantly of weight 1. Most of the identities in the later sections of this monograph are supposed to be new.
Clearly this is a most valuable addition to the literature on modular forms, and the modular forms people must be most grateful to the author for his fine achievement.