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Analytic continuation of overconvergent eigenforms


Author: Kevin Buzzard
Journal: J. Amer. Math. Soc. 16 (2003), 29-55
MSC (2000): Primary 11F80, 11F33; Secondary 11G18, 14G22, 14G35
Published electronically: September 19, 2002
MathSciNet review: 1937198
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Abstract: Let $f$ be an overconvergent $p$-adic eigenform of level $Np^r$, $r\geq1$, with non-zero $U_p$-eigenvalue. We show how $f$ may be analytically continued to a subset of $X_1(Np^r)^{{an}}$ 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.


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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: https://doi.org/10.1090/S0894-0347-02-00405-8
Keywords: Galois representations, $p$-adic modular forms
Received by editor(s): September 24, 2001
Published electronically: 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
Article copyright: © Copyright 2002 American Mathematical Society