The book under review deals with one of those themes in number theory that are utterly irresistible to every practitioner of the art: Ramanujan’s magic. The focus falls on the theory of elliptic functions, and this is certainly the sexiest arithmetical thing going these days. It is a prelude to the theory of modular forms and this is in turn one of the pillars of the proof of Fermat’s Last Theorem. I guess only the Ricci flow and Poincaré-3 can compete for sheer seductive power with the theory of elliptic functions.
Make no mistake, sexy can be profound. This is a true treasure trove of wonderful mathematics, as one can observe a priori from the fact that the subject has captivated titans such as Abel, Eisenstein, Weierstrass, Jacobi, and, indeed, the redoubtable Srinivasa Ramanujan. The present 6th volume in the “Monographs in Number Theory” series is a paean addressed to Ramanujan by the late K. Venkatachaliengar; the present edition comes edited and revised by Shaun Cooper. In the Preface to the revised edition, Cooper cites a letter from André Weil to Venkatachaliengar relating to the latter’s original project: “I can well appreciate the difficulty and value of writing such a book, and I am sure the mathematical world will be grateful to you for having written it.” And then Bruce Berndt’s review is quoted: “The author has studied Ramanujan’s papers and notebooks over a period of several decades. His keen insights, beautiful new theorems, and elegant proofs presented in this monograph will enrich readers…” Thus, we have here a work of loving scholarship imbued with both historical virtues and mathematical originality: a true multiple threat (and treat).
Venkatachaliengar himself starts off his monograph with a Preface noting, right off the bat, that “Ramanujan had not seen any standard book on elliptic functions before he went to England … [but] he proves his basic identity … , modestly claims that the[se] results … really do belong to the theory of elliptic functions, and draws the interest of the reader to the simplicity of his proofs.” Moreover, in the 1916 paper, “On Certain Arithmetical Functions,” containing his celebrated identity, he also provides a similar identity that is equivalent to the differential equation satisfied by Weierstass’ elliptic function. Venkatachaliengar notes that “these are connected with several interesting results in his notebooks [and] from these identities the basic properties of elliptic functions can be derived in a purely algebraic way without making any use of the Cauchy-Liouville methods of [the] theory of functions.” And then: “Ramanujan remarks in a very modest way: ‘The elementary proof of these formulæ given in the preceding sections seems to be of some interest in itself.’” Evidently we have here the understatement of the day, be it Ramanujan’s or today.
But this is just the start of the game. To be sure, Venkatachaliengar’s book starts off with the indicated explication of Ramanujan’s basic identity, but he goes on after that to the attendant differential equations, the Jordan-Kronecker formula, the theory of and formulas on partitions (p. 50), classical material on the hypergeometric function, connections with Weierstrass and Jacobi (covering more than two chapters), and finally an exploration of themes surrounding the Legendre modular function (p. 127 ff.). The appendices deal with, respectively, singular moduli, the identity of the quintuple product, and (yes!) the addition theorem for elliptic integrals — we come full circle, classically (up to homotopy, anyway).
It is obvious that every arithmetician should want to own a copy of this book, and every modular former should put it on his “to be handled-with-loving-care-shelf.” As for myself, I will place my copy in the same connected component in which Weil’s Elliptic Functions According to Eisenstein and Kronecker resides, besides lot of other truly wonderful books on the subject.
The only possible caveat is that the reader of Venkatachaliengar’s fine, fine book should be willing to enter into that part of the mathematical world where Euler, Jacobi, and Ramanujan live: beautiful formulas everywhere, innumerable computations with infinite series, and striking manouevres with infinite products. But I guess that this kind of predilection, non-negotiable for number theorists, is pretty wide-spread across the mathematical spectrum: Venkatachaliengar’s book (reissued) should be a smash hit!
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.