Modular Functions of One Variable III Proceedings International Summ
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350 Modular Functions of One Variable III Proceedings International Summer School University of Antwerp, RUCA July 17 -August 3, 1972
Edited by W Kuijk and J-P. Serre
S pringer-Verlag Berlin Heidelberg New York Tokyo
Editors
Willem Kuijk Rijksuniversitair Centrum Antwerpen, Leerstoel Algebra Groenenborgerlaan 171, 2020 Antwerpen, Belgium Jean-Pierre Serre College de France, 11, pl. Marcenn Berthelot 75231 Paris Cedex 05, France
1st Edition 1973 2nd Corrected Printing 1986
Mathematics Subject Classification (1970): 10005, 10025, 10C 15, 14K22, 14K25 ISBN 3-540-06483-4 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-06483-4 Springer-Verlag New York Heidelberg Berlin Tokyo
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© by Springer-Verlag Berlin Heidelberg 1973 Printed in Germany Printing and binding: Beltz Offsetdruck, HemsbachlBergstr. 2146/3140-543210
Preface
This is Volume 3 of the Proceedings of the International Summer School on "Modular functions of one variable and arithmetical applications" which took place at RUCA, Antwerp University, from July 17 to August 3, 1972. It contains papers by P.Cartier-¥.Roy, B.Dwork, N.Katz, J-P.Serre and H.P.F.Swinnerton-Dyer on congruence properties of modular forms, l-adic representations, p-adic modular forms and p-adic zeta functions.
W.Kuyk
J-P.Serre
CONTENTS
H.P.F. SWINNERTON-DYER
B. DWORK
On l-adic representations and congruences for coefficients of modular forms
1
The Up operator of Atkin on modular functions of level 2 with growth conditions
57
N. KATZ
p-adic properties of modular schemes and modular forms
69
J-P. SERRE
Formes modulaires et fonctions zeta p-adiques
191
P. CARTIER-Y. ROY
Certains calculs numeriques relatifs a l'interpolation p-adique des series de Dirichlet
269
Mailing addresses of authors
350
He~~n
e.L.
Siegel gewidmet
ON l-ADIC REPRESENTATIONS AND CONGRUENCES FOR COEFFICIENTS OF MODULAR FORMS
BY H.P.F. SWINNERTON-DYER
International Summer School on Modular Functions Antwerp 1972
SwD-2
2
CONTENTS
1.
Introduction.
p.3
2.
The possible images of Pl.
p.l0
3.
Modular forms mod l.
p.1S
4.
The exceptional primes.
p.26
5.
Congruences modulo powers of l.
p.36
Appendix
p.43
References
p.ss
3
SwD-3
ON l-ADIC REPRESENTATIONS AND CONGRUENCES FOR COEFFICIENTS OF MODULAR FORMS •
1.
Introduction.
The work I shall describe in these lectures has two themes, a classical one going back to Ramanujan (8J and a modern one initiated by Serre and Deligne [3].
(9)
To describe the classical theme, let the unique cusp
form of weight 12 for the full modular group be written (1)
and note that the associated Dirichlet s
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