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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).

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

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