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Analytic continuation of overconvergent eigenforms
Author(s):
Kevin
Buzzard
Journal:
J. Amer. Math. Soc.
16
(2003),
29-55.
MSC (2000):
Primary 11F80, 11F33;
Secondary 11G18, 14G22, 14G35
Posted:
September 19, 2002
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Abstract:
Let  be an overconvergent  -adic eigenform of level  ,  , with non-zero  -eigenvalue. We show how  may be analytically continued to a subset of  containing, for example, all the supersingular locus. Using these results we extend the main theorem of our earlier work with R. Taylor to many ramified cases.
References:
- [Be]
-
P. Berthelot, Cohomologie rigide et
cohomologie rigide à supports propres,
preprint available at http://www.maths.univ-rennes1.fr/~berthelo/.
- [Bu]
-
J. P. Buhler, Icosahedral Galois representations,
Springer-Verlag, Berlin, 1978, Lecture Notes in
Mathematics, Vol. 654.
MR
58:22019
- [BGR]
-
S. Bosch, U. Güntzer, R. Remmert,
Non-Archimedean Analysis,
Grundlehren der mathematischen
Wissenschaften 261, Springer, 1984. MR
86b:32031
- [BDST]
-
K. Buzzard, M. Dickinson, N. Shepherd-Barron,
R. Taylor,
On Icosahedral Galois Representations,
Duke
Math. Journal 109 (2001), 283-318.
- [BT]
-
K. Buzzard and R. Taylor, Companion
forms and weight
1 forms, Annals of Mathematics 149
(1999), 905-919. MR
2000j:11062
- [Col1]
-
R. Coleman, Reciprocity laws on curves,
Compos. Math. 72
(1989), 205-235. MR
91c:14028
- [Col2]
-
R. Coleman, The Monodromy Pairing,
Asian J.
Math. 4 (2000), 315-330. MR
2001k:14083
- [Con]
-
B. Conrad, Modular curves and rigid
analytic
spaces, pre-print.
- [D]
-
F. Diamond, On deformation rings and
Hecke rings,
Ann. Math. 144 (1996), 137-166. MR
92d:11172
- [DR]
-
P. Deligne and M. Rapoport, Les schémas
de modules
de courbes elliptiques, LNM 349, Springer-Verlag
(1971), 123-165. MR
49:2762
- [E]
-
S. Edixhoven, Minimal resolution and
stable reduction of
 , Ann. Inst. Fourier
40(1) (1990), 31-67.
MR
92f:11080
- [G]
-
F. Gouvêa, Arithmetic of
 -adic modular forms,
Springer LNM
1304 (1988) MR
91e:11056
- [K]
-
N. Katz,
 -adic properties of modular schemes
and
modular forms, LNM 350 (1973), 69-190.
MR
56:5434
- [Kö]
-
U. Köpf, Über eigentliche
Familien algebraischer
Varietäten über affinoiden Räumen,
Schriftenreihe des Mathematischen
Instituts der Universität Münster,
2 Serie, Heft 7 (1974). MR
54:10657
- [KM]
-
N. Katz and B. Mazur, Arithmetic moduli
of elliptic
curves, Annals of Math. Stud. 108,
Princeton University Press (1985).
MR
86i:11024
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Additional Information:
Kevin
Buzzard
Affiliation:
Department of Mathematics, Imperial College, Huxley Building, 180 Queen's Gate, London SW7 2B2, England
Email:
buzzard@ic.ac.uk
DOI:
10.1090/S0894-0347-02-00405-8
PII:
S 0894-0347(02)00405-8
Keywords:
Galois representations,
$p$-adic modular forms
Received by editor(s):
September 24, 2001
Posted:
September 19, 2002
Additional Notes:
The author would like to thank the Miller Institute and UC Berkeley for the financial support and hospitality they offered him whilst he was obtaining the majority of these results. The write-up was done over a period of several years, in Rennes, the IHP in Paris, Cambridge UK, and Imperial College London, and the author would also like to thank these institutions for their hospitality. He would also like to thank the referee for several helpful remarks
Copyright of article:
Copyright
2002,
American Mathematical Society
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