Automorphic Forms on GL (2)
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114 H. Jacquet University of Maryland, College Park
R. P. Lang lands Yale University, New Haven
Automorphic Forms on GL (2)
Springer-Verlag Berlin· Heidelberg · New York 1970
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e
by Springer-Verlag Berlin' Heidelberg 1970. Library of Congress Catalog Card Number Title No. 3270.
Printed in Germany.
Table of Contents Chapter I : § 1.
§ § § § § § §
2.
3. 4.
5. 6.
7. 8.
Theory
We11 representations
•..•••••••.•••..•••.••••••••••••••••
1
Representatlons of GL(2,F) ln the non-archlmedean case •• 23 The prlncipal serles for flelds •••••••••• 92 Examples of absolutely cuspldal representatlons ••••••••••124 Representatlons of GL(2,1K) •••••••••••••••••••••••••••• 153 Representatlons of GL(2,t •••••••••••••••••••••••••••• 220 Cllaracters
• . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 243
Odds and ends
Chapter II :
• • • • • • • • • • . • • • • • • • • • • • • • . • • • • • • • • • • • • • • • •• 277
Global Theory
§ 9. The global Heeke algebra ••••••••••••••••••••••••••••••• § 10. Automorphlc forms •••..••.••••••••.•••••••••••••.•••.••• § 11. Heeke theory •.............................. . . . . . . . . . . .. § 12. Some extraordlnary representatlons •••••••••••••••••••••
Chapter III
.
301 324 349
392
Quaternlon Algebras
§ 13. Zeta-functions for M(2, F) •••••••••••••••••••••••••••• 417 § 14. Automorphlc forms and quaternlon algebras ••••••••••••••• 456 § 15. Some orthogonality relatlons ••••••••••••••••••••••••••• 471 § 16. An application of tne Selberg trace formula •••••••••••• 494
Introduction Two of the best known of Hecke's achievements are his theory of
" L - functions with grossencharakter, which are
which can be represented by
products, and his theory of the
products associated to automorphlc forms on
GL(2).
grgssencharakter ls an automorphic form on ask if the
serles
Since a
one is tempted to
products associated to automorphlc forms on
a role in the theory of numbers similar to that h
L - functions With grossencharakter.
In
GL(2) by the
do they bear the
same relation to the Artln
L - functions associated to two-dimensional
representations of a
group as the Hecke
the Artin
L - functions bear to
L - functions associated to one-dimensional representatlons?
Although we cannot answer the question definitively one of the purposes of these notes is to provide some evidence that the answer is affirmative. The evidence is presented in S12. along Hecke.
It comes from reexamining,
suggested by a recent paper of Anything
the original work of
novel in our reexa
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