Elliptic and Modular Functions from Gauss to Dedekind to Hecke

Everyone really should see the wonderful documentary “The Proof,” dealing with Andrew Wiles’ monumental work on Fermat’s Last Theorem. To be precise, in the 1990s Wiles proved the Shimura-Taniyama-Weil conjecture, to the effect that every rational elliptic curve is modular. (Wiles originally proved the conjecture in most cases, and the proof was then completed by Breuil, Conrad, Diamond, and Taylor.) Courtesy of work by, e.g., Serre and Ribet, Wiles’s result suffices to yield the Fermat Theorem. Truly gorgeous and dramatic work! The video wonderfully conveys both aspects of the story, the great mathematical elegance of Wiles’s work and the personal element of what Wiles went through as he carried out these labors of love that came with high highs and low lows. At some point in the film (at around 13:05, actually), Wiles quotes Martin Eichler’s aphorism that there are five, not four, arithmetic operations, namely, addition, subtraction, multiplication, division, and … modular forms. So, given that none other than Carl Friedrich Gauss advised us that arithmetic (or number theory) is the Queen of Mathematics, whence it follows that we should all learn it (yeah, right …), we should all seriously study modular forms. And then, indeed, Wiles’s irresistible work on Shimura-Taniyama-Weil (+Fermat) need no longer be resisted (working modulo Galois representations and elliptic curves as things one needs to know). All right, to be fair, number theory is a huge subject and, even though it is unparalleled in its beauty, really getting off the ground in this subject is a highly non-trivial affair, but certainly modular forms are a major theme, and that brings us to the present book.

Roy sets himself some very clear boundaries: “from Gauss to Dedekind to Hecke.” I think this is an interesting approach, given the phenomenally rich history of the subject, the depth and beauty of the work done by 19th century scholars working in this area (add Abel, Jacobi, Eisenstein, Kronecker, Hurwitz, Hermite, Weierstrass, and Dirichlet to the three mentioned above), and Olympian extent of the field. In a very natural way, Hecke qualifies as a transitional figure in that one might argue that the subject took off in a different (modern) direction due to his work (in the early 20th century), with a great deal of beautiful and deep material coming before as well as after.

When I studied this subject in graduate school in the early 1980s, I was given to read (and dissect!) Erich Hecke’s 1938 Institute for Advanced Study lecture notes, which really do amount to the start of a revolution: he plays modular forms and Dirichlet series off against each other via the use of the Mellin and inverse Mellin transform. In so doing he builds (the start) of a dictionary between these two sets of number theoretic objects; this is what is meant by the Hecke correspondence. It is worth noting that in 1966 André Weil provided a very far-reaching extension of what Hecke had done, and that representation theory entered the game in a major way. It is perhaps reasonable to characterize these developments as essentially modern, in contrast to what came before.

But it is absolutely critical to give the “earlier” material its due, seeing that it is still a very fertile area for research and study, and also that the corresponding classical results are not to be missed. Although Roy advertises what he does in perhaps more prosaic terms, what his book offers is nothing less than a careful and detailed treatment of this classical material in all its beauty. Says Roy: “The purpose of this work is to present the fundamental results of modular function theory as developed during the 19th and early 20th centuries, focusing particularly on those interesting methods and techniques that appear to have been overlooked or are not generally well-known.” I would suggest that in all fairness they should be well-known, simply because of their beauty (Gauss knew what he was talking about) and depth, and the evident pedigree of the mathematicians who worked in the subject. I’d like to stipulate, also, that if today’s fledgling number theorists would actually read what is probably at this point an all but forgotten work, namely, G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers (see in particular the first line of the fourth paragraph of Stenger’s review), Roy’s evaluation that the results he deals with are to a large extent “overlooked or … not generally well-known” would cease to apply.

All right, then, what are some of these 19th and 20th century gems Roy presents us with? Well, here are six: Gauss on elliptic functions as well as theta functions and modular forms; Jacobi’s theory of theta functions; Eisenstein’s theory of elliptic functions (and let me plug another fantastic book: Weil’s Elliptic Functions According to Eisenstein and Kronecker); Dedekind’s \(\eta\)-function; Ramanujan’s \(\tau\)-function; Dirichlet series and modular forms (Ch. 14); and “The Hecke Operators” (Chapter 16).

So, as the book draws to a close (the sixteenth chapter is the last one), Roy presents what I characterized above as material one might regard as setting the stage for a transition to modern themes. However, the preceding chapters are clear testimony to the profound influence of 19th century titans.

Finally, it needs to be stressed that Roy does much more than present these mathematical works as museum pieces. He takes pains to tie them in to modern work when reasonable and appropriate, and that of course just adds to the quality of his work. I am very excited to have a copy of this wonderful book in my possession.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.