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Slopes of the U7 operator acting on a space of overconvergent modular forms

Part of: Arithmetic problems. Diophantine geometry Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 May 2012

L. J. P. Kilford  and
Ken McMurdy
L. J. P. Kilford
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, United Kingdom (email: [email protected])
Ken McMurdy
Affiliation:
Department of Mathematics (TAS), Ramapo College of New Jersey, 505 Ramapo Valley Rd, Mahwah NJ 07430, USA (email: [email protected])

Abstract

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Let χ be the primitive Dirichlet character of conductor 49 defined by χ(3)=ζ for ζ a primitive 42nd root of unity. We explicitly compute the slopes of the U7 operator acting on the space of overconvergent modular forms on X1(49) with weight k and character χ7k−6 or χ8−7k, depending on the embedding of ℚ(ζ) into ℂ7. By applying results of Coleman and of Cohen and Oesterlé, we are then able to deduce the slopes of U7 acting on all classical Hecke newforms of the same weight and character.

MSC classification

Secondary: 11F11: Holomorphic modular forms of integral weight 11F33: Congruences for modular and $p$-adic modular forms 14G35: Modular and Shimura varieties 14G22: Rigid analytic geometry
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 15 , May 2012 , pp. 113 - 139
Copyright
Copyright © London Mathematical Society 2012

References

[1]Buzzard, K., ‘Analytic continuation of overconvergent eigenforms’, J. Amer. Math. Soc. 16 (2003) 29–55.CrossRefGoogle Scholar
[2]Cohen, H. and Oesterlé, J., ‘Dimensions des espaces de formes modulaires’, Modular functions of one variable VI (Bonn, Germany 1976), Lecture Notes in Mathematics 627 (Springer, Berlin, 1977) 69–78.CrossRefGoogle Scholar
[3]Coleman, R., ‘Classical and overconvergent modular forms’, Invent. Math. 124 (1996) 215–241.CrossRefGoogle Scholar
[4]Coleman, R., ‘Classical and overconvergent modular forms of higher level’, J. Théor. Nombres Bordeaux 9 (1997) 395–403.CrossRefGoogle Scholar
[5]Coleman, R., ‘p-adic Banach spaces and families of modular forms’, Invent. Math. 127 (1997) 417–479.CrossRefGoogle Scholar
[6]Coleman, R. and McMurdy, K., ‘Stable reduction of X 0(p 3)’, Algebra Number Theory 4 (2010) 357–431.Google Scholar
[7]Deligne, P. and Rapoport, M., ‘Les schémas de modules de courbes elliptiques’, Modular functions of one variable II (Antwerp, Belgium 1972), Lecture Notes in Mathematics 349 (Springer, Berlin, 1973) 143–316.CrossRefGoogle Scholar
[8]Diamond, F. and Im, John, ‘Modular forms and modular curves’, Seminar on Fermat’s Last Theorem, CMS Conference Proceedings 17 (American Mathematical Society, Providence, RI, 1995) 39–133.Google Scholar
[9]Gross, B., ‘A tameness criterion for Galois representations associated to modular forms (mod p)’, Duke Math. J. 61 (1990) 445–517.CrossRefGoogle Scholar
[10]Katz, N., ‘ p-adic properties of modular forms and modular curves’, Modular functions of one variable III (Antwerp, Belgium 1972), Lecture Notes in Mathematics 350 (1973) 69–190.CrossRefGoogle Scholar
[11]Kilford, L. J. P., ‘On the slopes of the U 5 operator acting on overconvergent modular forms’, J. Théor. Nombres Bordeaux 20 (2008) 165–182.CrossRefGoogle Scholar
[12]Loeffler, D., ‘Spectral expansions of overconvergent modular functions’, Int. Math. Res. Not. IMRN 2007 (2007) Art. ID rnm050.Google Scholar
[13]McMurdy, K., ‘Explicit parametrizations of ordinary and supersingular regions of X0(pn)’, Modular curves and abelian varieties, Progress in Mathematics 224 (Birkhäuser, Basel, 2004) 165–179.CrossRefGoogle Scholar
[14]Stein, W. A.et al., ‘Sage mathematics software, version 4.6’, 2010, http://www.sagemath.org.Google Scholar
[15]Stein, W. A., ‘The Modular Forms Database’, http://modular.math.washington.edu/Tables/.Google Scholar
[16]Shimura, G., Introduction to the arithmetic theory of automorphic functions (Princeton University Press, Princeton, NJ, 1971).Google Scholar