The name of J. W. S.Cassels will always be associated in my mind with a particularly happy and formative event near the close of my undergraduate years at UCLA. In those halcyon times (or so they certainly seem to me now, many years later, despite objective evidence to the contrary), number theory seminars dominated my schedule — indeed, this had been the case for several years already, almost always to the detriment of my proper progress in required or recommended courses. But UCLA’s number theory group was very friendly, informal, but also phenomenally active and inspiring. My contact with them galvanized my desire to be a number theorist myself; I certainly haven’t regretted it for a moment.
The glamorous beauty of class field theory worked on me like a magnet at that time, so it was a cause for celebration for me to discover (as a primary reference for the aforementioned seminar) the famous Algebraic Number Theory, edited by Cassels and Albrecht Fröhlich; this work was, and is, almost always referred to simply as “Cassels-Fröhlich.”
In this wonderful book Fröhlich’s first chapter, on local fields, is followed by Cassels’ second chapter, on local fields, and then it’s off to the races: Birch on Kummer theory, Atiyah and Wall on group cohomology, Serre on local class field theory, Tate on global class field theory, Roquette on class field towers, Serre again on complex multiplication — and more. There’s also a history offered by Helmut Hasse himself, and the closing chapter is nothing less than Tate’s famous thesis. It’s a terrific book, if remarkably compact, and should adorn every arithmetician’s shelf if there’s any justice in the world.
Already in Cassels-Fröhlich one encounters Cassels’ elegant and precise writing style, both in the preface and the second chapter. His style is also droll and humorous; for example, regarding the discrete topology on a global field he says, “It is impossible to conceive of any other uniquely defined topology in k. This metamathematical reason is more persuasive than the argument that follows!” He then goes on to give a formal proof, of course. Such asides certainly make for a very enjoyable reading experience: solid and serious mathematics presented meticulously and rigorously but with the author’s infectious joy seeping through.
It is exactly the same story for the book under review, Cassels’ Rational Quadratic Forms. This book appeared originally in the London Mathematical Society’s Distinguished Monographs series going on thirty years ago, and is now reissued in paperback by Dover Publications.
Cassels starts off the excursion (in the Preface) with a pair of quotes by H. Brandt and by B. L. van der Waerden. Brandt claims (in Hochdeutsch, which Cassels doesn’t translate; I’m afraid I like that sort of thing myself) that “whereas the subject of rational quadratic forms certainly belongs to the most rarified realms of number theory, it is only in the area of binary quadratic forms that we encounter a somewhat complete theory. Beyond this one finds nothing but an unsatisfactorily chaotic situation ...” (my free-ish translation). And thereupon van der Waerden is made to weigh in with (also im deutsch) “… in Ganzen ist die Theorie nach wie vor, wie Brandt ganz richtig sagt, in einem chaotischen Zustand,” largely echoing Brandt. But then Cassels enters the scene: “The above quotations may give a false impression.” What a marvelous hook, no?
And, to be sure, what follows is worth it, and then some: “The material of the book is largely nineteenth century but the treatment is structured by two twentieth century insights. The first, which seems to have come to its full recognition by the work of Hasse and Witt, is that the theory of forms over fields is logically simpler and more complete than that over rings. It is therefore appropriate, contrary to what seemed natural earlier, to study forms with rational coefficients and under rational equivalence before attacking integral forms and integral equivalence.” So there, Brandt!
He goes on to say, “[t]he second major insight, due to Hensel and Hasse, is the perspective introduced by the p-adic view-point.” And then (the date of this writing being 1978): “... as the p-adic numbers are not as yet as well-known as they should be to the broad audience to the broad audience to which this book is addressed, no knowledge of them has been presupposed.” (So there’s a pedagogical dividend to be had, too.)
More from the eminently informative Preface: “After the introduction there is a section of forms over fields in general (characteristic ≠ 2) including the vital ‘Witt’s Lemma.’ There follow forms over the p-adic field Qp and then forms over… Q. Here the relation with the Qp is very tight. Things happen over Q if and only if they happen over all the Qp… Then the same sequence is repeated for integral forms. First the theory over a general PID, then over the p-adic integers, and finally over the rational integers. For integral forms the relation between the local and the global is not quite so tight and the notion of ‘genus’ measures the discrepancy. Two forms are in the same genus if it cannot be shown by purely local [i.e. p-adic] methods that they are not integrally equivalent.”
Cassels continues, in the Preface as well as in the book, with spinor genera, which come up in connection with the spin group, “the two-sheeted simply connected covering group of the orthogonal group,” and we’re suddenly in the vicinity of the analytic theory of quadratic forms of Carl Ludwig Siegel and the work of André Weil of the middle 1960s. But this is not touched on in Cassels’ book, whose orientation is, for lack of a better word, more algebraic and arithmetic.
Then we get the reduction theory of definite forms, “the theory introduced by Hermite and in a better form by Minkowski,” followed by the composition theory for binary forms referred to so lovingly by Brandt and van der Waerden, and finally two appendices, “Definite forms” (“In this appendix we discuss aspects of the theory of positive definite integral forms which do not appropriately fit into the body of the book...”), and “Analytic methods.” The second appendix contains coverage of some work by Siegel as well as a brief discussion of modular forms.
Each chapter of the book is capped off by a section titled “Examples,” which include sundry exercises often having to do with comparatively concrete cases of what came just before; there are also more autonomous exercise to be had here.
Cassels’ Rational Quadratic Forms is obviously a wonderful contribution to the genre by a master of form and composition, if I may be excused an(other) egregious pun. It’s well worth reading… slowly and attentively: there’s an awful lot there in the book’s almost 400 pages.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.