When you go to Jacquet’s webpage at Columbia you come across a list of papers placed online for the convenience of the discerning visitor. This list reads like a roadmap of much of modern number theory: “Jacquet-Langlands” is there, of course, as are papers on the Selberg trace formula, Rankin-Selberg stuff, Whittaker models, automorphic functions on GL(2) and GL(3), Euler products, (automorphic) representations, base change for GL(3), and various things Kloosterman. What everybody knows comes through clear as a bell: Hervé Jacquet is a major (and now senior) figure in the huge arithmetical revolution wrought in the early 1960s with, to pick an obvious landmark, André Weil’s revamping and expansion of Erich Hecke’s ideas of the earlier parts of the century. Even before I did my graduate work on modular forms and Dirichlet series in the early 1980s I had heard about the magnificently influential 1970 opus mentioned first above, always just called “Jacquet-Langlands,” appearing at that time as a Springer Lecture Notes affair and weighing in at well over 500 pages. It is a wonderful thing that this still critical text, whose actual title is Automorphic Functions on GL(2), is now online as a polished production of a mere 300 pages or so. (This text is actually only Part I of two: Part II (same title) was authored by Jacquet alone.)
The salient point is that a modern number theorist cannot help but encounter Jacquet’s work, and, indeed, should be happy to do so: this is beautiful stuff. Recall the marvelous bit in the wonderful documentary, “The Proof,” about Andrew Wiles’ conquest of FLT (actually Shimura-Taniyama-Weil, which implies Fermat), where Wiles himself jokes that Martin Eichler stated that there five arithmetical operations: addition, subtraction, multiplication, division, and … modular forms. The things Jacquet does, largely centered on things automorphic and modular, are accordingly of central importance to the entire modern enterprise.
The book under review contains a selecta of Jacquet’s papers, and every single choice (there are fifteen, a small representative subset of the whole) is superb. Edited by Dorian Goldfeld, these Collected Works contain, to name only three of these gems, Jacquet’s gorgeous paper with Shalika (r.i.p.: he passed away last year) titled, “A non-vanishing theorem for zeta-functions of GL(n)”; the paper with Gelbart on connecting automorphic representations of GL(2) with such for GL(3); and Jacquet’s solo paper of 2001, “Factorization of period integrals,” whose abstract reads: “We show that for certain quadratic extensions E/F of number fields the period integral of a cusp form on GL(3,E) over the unitary group Ho in three variables is a product of local linear factors.”
The latter paper is something of a model of both (very) serious and solid mathematizing and restraint and balance: in the section “Concluding Remarks,” Jacquet notes that “The same techniques can be used to prove the conjecture [What conjecture? See p. 429…] for other representations. As a matter of fact this is done in [a paper by Lapid and Rogawski]. However, it is difficult to prove the conjecture for all representations [of GL(3)] … thus we prefer to limit ourselves to the above cases.” Fair enough, it’s a wonderful paper as it stands. But then Jacquet goes on to a wonderful discussion, the first (telling) sentence of which is: “One expects the above results to generalize in a straightforward way to the groups GL(n) with n odd …” Then turn the page and read: “Now we discuss the local situation for n even …” To be sure, Jacquet is properly careful, but the closing section to this paper is not only full of good stuff (including a hard-core lemma on linear independence of Bessel distributions), it ends, in its own right, with a conjecture pointing toward a future exploration. This all adds up to a splendid example of mathematical mastery coupled with insightful commentary and prescience.
And the latter phrase describes Jacquet’s work across the spectrum, so to speak, and this volume of his Collected Works is a wonderful contribution to the literature in modern number theory, where we might define “modern” as meaning something like post-‘Jacquet-Langlands’. All modern arithmeticians, in particular the members of the everywhere dense subset of modular formers, should properly covet this book. I am happy I have my copy…
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.