Zeta Functions of Simple Algebras
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260 Roger Godement Universite de Paris, Paris/France
Herve Jacquet The City University of New York, New York, NY/USA
Zeta Functions of Simple Algebras
Spri nger-Verlag Berlin· Heidelberg· New York 1972
AMS Subject Classifications (1970): 10 D20, 12 A 70, 12 B 35, 22 E 50
ISBN 3-540-05797-8 Springer-Verlag Berlin' Heidelberg· New York ISBN 0-387-05797-8 Springer-Verlag New York· Heidelberg· Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation) reprinting, re-use of illustrations) broadcJsting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin' Heidelberg 1972. Library of Congress Catalog Card Number 72-76391. Offsetdruck: Julius Beltz, HemsbachlBcrgstr.
rnt roduction Two of the best known achievements of Hecke are his theory of L-functions with grossencharacter and his theory of the Dirichlet series associated to automorphic forms on an automorphic form on
GL(l)
GL(2)
Since a grossencharacter is
and Hecke's results on L-functions can be
best proved by using Tate's technique, it is natural to ask if it is possible to extend Tate's technique to
GL(2), or, more generally, to the
multiplicative group of an arbitrary simple algebra.
This is, of course,
not a new question and many authors have given partial answers. The purpose of this set of notes is to give an affirmative and, in some sense, complete answer.
Of course, complete is a relative word.
A complete treatment of the question would imply at the very least a complete knowledge of the representations of the local groups. not available at the moment.
This is
Actually the existence of the "absolutely
cuspidal representations" is not even proved in this set of notes.
*
When a complete list of the irreducible representations of the local groups becomes available, it will presumably be relatively easy to compute the factors attached in this paper to such a representation. Also in the global theory, I have restricted myself to the case of cusp forms.
If my understanding of the theory is correct, the other
forms should not give essentially new Euler products.
However, this is
not to say that it would be without interest to consider other forms as well.
*In
particular, we do not prove the existence of nontrivial functions in the space of Lemma 5.3.
IV Since this is a joint work and I alone had responsibility for writing the final version of these notes, deciding on the plan and the contents, and making mistakes, it is perhaps best to explain the genesis of this work.
In 1967, R. Godement gave a series of lectures in Princeton
and later on, in Tokyo, on the global theory.
Chapter II is largely, if
not totally, based on the notes he made available to me. only the local problems remaine
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