Modular Functions of One Variable VI Proceedings International Confe
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627 Modular Functions of One Variable VI Proceedings International Conference, University of Bonn, Sonderforschungsbereich Theoretische Mathematik July 2-14, 1976
Edited by J.P. Serre and D. B. Zagier
Springer-Verlag Berlin Heidelberg NewYork 1977
Editors
Jean-Pierre Serre College de France 75231 Paris Cedex 05/France Don Bernard Zagier Mathematisches Institut der Universitat Bonn Wegelerstr.10 53 Bonn/BRD
AMS Subject Classifications (1970): lOC15, 10005, 10025, 12A99, 14H45, 14K22, 14K25 ISBN 3-540-08530-0 ISBN 0-387-08530-0
Springer-Verlag Berlin Heidelberg New York Springer-Verlag New York Heidelberg Berlin
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© by Springer-Verlag Berlin Heidelberg 1977 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2140/3140-543210
PREFACE This is the second and final volume of proceedings for the Conference on Modular Forms held in Bonn in July 1976. The first volume appeared as Lecture Notes nO 601, under the title "Modular Functions of One Variable V" (cf. Lecture Notes nO 320, 349, 350 and 476).
Jean- Pierre Serre
Don Zagier
CONTENTS
M. RAZAR,
Values of Dirichlet series at integers in the
1
critical strip C. MORENO,
Analytic properties of Euler products of automorphic representations
11
J-P. SERRE, H.M. STARK,
Modular forms of weight
27
H. COHEN, J. OESTERLE,
Dimension des espaces de formes modulaires
69
M-F.. VIGNE'RAS"
Facteurs gamma et equations fonctionnelles
79
D. ZAGIER,
Modular forms whose Fourier coefficients
1/2
involve zeta-functions of quadratic fields D. ZAGIER,
Correction to "The Eichler-Selberg trace formula on
H. COHEN,
105 171
SL (;z ) II 2
A lifting of modular forms in one variable to Hilbert modular forms in two variables
175
M. EICHLER,
Theta functions over
197
M-F. VIGNERAS,
Series theta des formes quadratiques
and over
indefinies
227
S. GELBART,
Automorphic forms and Artin's conjecture
241
S. KUDLA,
Relations between automorphic forms produced by theta-functions
F. HIRZEBRUCH,
The ring of Hilbert modular forms for real quadratic fields of small discriminant
A.N. ANDRIANOV,
277
287
On zeta-functions of Rankin type associated with Siegel modular forms
List of adresses of authors
325 339
Raz-1
Values of Dirichlet Series at Integers in the Critical Strip by Michael J. Razar
§l. Introduction This note is primarily a summary of some recent work on the values at integer points in the critical strip of Dirichlet series associated to newforms on
(N).
The first such results seem to be due
to Shimura [9J, who derived them for the Dirichlet series a
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