Minking Eie’s Topics in Number Theory appears as the second volume in the Monographs in Number Theory series, published by World Scientific Publishers. The first volume in the series is an offering by P. T. Bateman and Harold Diamond, Analytic Number Theory — An Introductory Course. It strikes me that there is a pleasing continuity in the game here: it bodes well for the whole series as a valuable resource in number theory, particularly in view of the exciting things available in the interface between analytic methods and modular forms (which arguably covers just about all of the latter, if one interprets the attendant representation theory in the style of, say, Gel’fand, Graev, and Piatetskii-Shapiro). Eie’s book explicitly goes at some of this avant garde material in the sense that his focus falls on, among other things, Jacobi forms, first introduced by Zagier and Eichler in 1985 (in a beautiful Birkhäuser publication, The Theory of Jacobi Forms), and subsequently on Eie’s own development of Jacobi Forms over the Cayley numbers in the 1990s. So this book is really quite unusual among works in analytic number theory since it provides a rather quick line of ascent to something not only pretty exotic and exciting but also absent from the usual repertoire of a practitioner of the art.
Topics in Number Theory is split into two parts, “Theory of Modular Forms of One Variable,” and (not surprisingly, given part one), “Theory of Modular Forms of Several Variables.” These headings cover huge expanses of mathematics, of course, and so it is unavoidable that Eie’s own predilections take center stage before too long. For example, the functional equation for Riemann’s zeta function appears already on p. 29, soon to be followed by a chapter titled, “Zeta functions of modular forms,” which features theta series and Hecke theory (hear hear!). Then Eie goes on to Dedekind eta functions, Selberg’s trace formula, and Euler sums, including coverage of recent developments. All this takes place in about 120 pages. Dense stuff, but Eie has good mathematical taste, as well as a pretty decent pedagogical sense, in that he ends each (dense) chapter with an exercise set that will bring an appropriate amount of sweat to the reader’s brow.
The latter half of the book, the part concerned with modular forms of many variables, starts off with symplectic geometry in the sense of C. L. Siegel, going all the way up to Fourier analysis in this context and the attendant more sophisticated theta series — all this in only 40 pages (so gird your loins). Thereafter, it’s Jacobi forms, replete with Hecke operators and a Selberg trace formula in this context, as well as Jacobi-Eisenstein series, and Jacobi cusp forms. Manifestly, given the above remarks, this contains a good deal of Eie’s own work of no more than 20 years’ vintage.
Finally, the book comes equipped with three appendices, on things p-adic (always welcome), weighted sum formulas of multiple zeta functions (very viable stuff these days), and on the density of quadratic forms over the Cayley numbers (tantalizing if exotic). Eie is certainly keen on being his presentation being self-contained, at least to a reasonable degree, but nevertheless I don’t think this book is good fare for neophytes.
Topics in Number Theory is a very interesting book indeed. If I may end on a personal note, I hope to have occasion to use it in my own work in the not too distant future.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.