Notes on Jacquet-Langlands' Theory

Automorphic Forms on \(GL(2)\), by Hervé Jacquet and Robert P. Langlands, is a foundational document in the Langlands program connecting automorphic representations and Galois representations. First published in 1970, the book was found difficult to read and digest by many. Roger Godement’s Notes on Jacquet-Langlands is the record of the lectures Godement gave at the Institute for Advanced Study to help people understand and read the book. The introduction begins:

The purpose of these notes would have been better explained if we had chosen another title, namely Jacquet-Langlands’ Theory Made Easy; it occurred to us at the last moment that a more pedestrian choice would be more prudent, since after all the author is in a rather bad position to judge…

Godement gives a no-nonsense account of a difficult theory, of course, but every so often one sees a flash of wit or a note of sarcasm much like the last phrase in that sentence. These are most visible in the footnotes. My favorite is attached to an exercise on page 115 of this edition: “We don’t know how to solve it, of course!”

I can’t resist one more quote, from the beginning of Chapter 3, on “The Global Theory.” The issue is how to denote a non-archimedean absolute value on a global field. Godement writes \(|x|_{\mathfrak p}\) and adds (p. 98; a rare transcription error appears here in this edition: \(\nu\) instead of \(v\).):

In Weil’s Basic Number Theory and in Jacquet and Langlands they write \(v\) instead of \(\mathfrak p\). We prefer our \(\mathfrak p\)’s; they remind us of the Great Dynasty-- Gauss, Jacobi, Dirichlet, Riemann, Dedekind, Kronecker, Hilbert, Minkowski, Hecke Artin, Hasse, etc. … -- Salvation through Zahlentheorie.

When I first worked with these notes in graduate school, they were typed with mathematical symbols added in by hand, which makes the remark about \(\mathfrak p\) even funnier. A scan of the original notes is available online at Bill Casselman’s Classic Lecture Notes site. Here they are presented in typeset form in a nice binding, with introductory notes from Jacquet and Langlands themselves.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.