The question is in the title: The book

Godement-Jacquet "Zeta functions of simple algebras"

is from 1971. Has there ever been a textbook introduction to this material, or at least part of it? (but beyond just Tate's thesis).

  • $\begingroup$ Are you interested in GL(n), or division algebras, or the general case of CSAs? $\endgroup$
    – Kimball
    Mar 14 at 0:14

As you mention in your question, for $\mathrm{GL}_1$, this is just Tate's thesis revisited.

The only exposition on this that I know of occurs in "Automorphic Representations and $L$-Functions for the General Linear Group" by Dorian Goldfeld and Joseph Hundley. In particular, Chapter 11 of volume 1 covers the theory of the Godement-Jacquet zeta integral for $\mathrm{GL}_2(\mathbb{A}_{\mathbb{Q}})$ in some detail. Chapter 15 of volume 2 covers the same theory for $\mathrm{GL}_n(\mathbb{A}_{\mathbb{Q}})$ with $n$ arbitrary.

More generally, the two volumes are a detailed introduction to the theory of automorphic representations of $\mathrm{GL}_n(\mathbb{A}_{\mathbb{Q}})$ at a reasonable pace. There are other books that cover the $\mathrm{GL}_2$ theory at a little higher level (for example, "Automorphic Forms and Representations" by Daniel Bump or the original, "Automorphic Forms on $\mathrm{GL}(2)$" by Hervé Jacquet and Robert Langlands) or at a quicker pace ("Automorphic Forms on Adele Groups" by Stephen Gelbart). To my knowledge, however, Goldfeld-Hundley is the only textbook that covers the Godement-Jacquet zeta integral in detail.

I should also mention that Hervé Jacquet has also written two short surveys on the Godement-Jacquet theory: "Principal $L$-Functions of the Linear Group", which appears in the famous Corvallis proceedings, and "Principal $L$-Functions for $\mathrm{GL}(n)$".

Edit: per the review of Goldfeld-Hundley by Ramin Takloo-Bighash in the Bulletin of the American Mathematical Society, "The book under review is the first book since the appearance of [Godement-Jacquet] in 1972 that gives a thorough treatment of the principal $L$-function using matrix coefficients."

Second edit: Google Scholar has informed me that there is a new book coming out soon, The Genesis of the Langlands Program, that includes a long chapter by Goldfeld and Jacquet called "Automorphic Representations and $L$-Functions for $\mathrm{GL}(n)$", which seems to be a detailed discussion of the theory of the Godement-Jacquet zeta integral. You can read some of this chapter on Google Books.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.