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Geometric level raising for p-adic automorphic forms

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  05 August 2010

James Newton
James Newton*
Affiliation:
Department of Mathematics, South Kensington Campus, Imperial College London, SW7 2AZ, London, UK (email: james.newton07@imperial.ac.uk)
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Abstract

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We present a level-raising result for families of p-adic automorphic forms for a definite quaternion algebra D over ℚ. The main theorem is an analogue of a theorem for classical automorphic forms due to Diamond and Taylor. We show that certain families of forms old at a prime l intersect with families of l-new forms (at a non-classical point). One of the ingredients in the proof of Diamond and Taylor’s theorem (which also played a role in earlier work of Taylor) is the definition of a suitable pairing on the space of automorphic forms. In our situation one cannot define such a pairing on the infinite dimensional space of p-adic automorphic forms, so instead we introduce a space defined with respect to a dual coefficient system and work with a pairing between the usual forms and the dual space. A key ingredient is an analogue of Ihara’s lemma which shows an interesting asymmetry between the usual and the dual spaces.

Keywords

p-adic modular formslevel raising

MSC classification

Secondary: 11F33: Congruences for modular and $p$-adic modular forms
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 2 , March 2011 , pp. 335 - 354
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Buzzard, K., On p-adic families of automorphic forms, in Modular curves and abelian varieties, Progress in Mathematics, vol. 224 (Birkhäuser, Basel, 2004).Google Scholar
[2]Buzzard, K., Eigenvarieties, in L-functions and Galois representations, London Mathematical Society Lecture Notes Series, vol. 320 (Cambridge University Press, Cambridge, 2007), 59–120.CrossRefGoogle Scholar
[3]Chenevier, G., Familles p-adiques de formes automorphes pour GLn, J. Reine Angew. Math. 570 (2004), 143–217.Google Scholar
[4]Chenevier, G., Une correspondance de Jacquet–Langlands p-adique, Duke Math. J. 126 (2005), 161–194.CrossRefGoogle Scholar
[5]Coleman, R. and Mazur, B., The eigencurve, in Galois representations in arithmetic algebraic geometry (Durham, 1996), London Mathematical Society Lecture Note Series, vol. 254 (Cambridge University Press, Cambridge, 1998), 1–113.Google Scholar
[6]Diamond, F. and Taylor, R., Non-optimal levels of mod l modular representations, Invent. Math. 115 (1994), 435–462.CrossRefGoogle Scholar
[7]Emerton, M., On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms, Invent. Math. 164 (2006), 1–84.CrossRefGoogle Scholar
[8]Ihara, Y., On modular curves over finite fields, in Int. Colloq. on discrete subgroups of Lie groups and applications to moduli, Bombay, 1973 (Oxford University Press, Oxford, 1975), 161–202.Google Scholar
[9]Kisin, M., Overconvergent modular forms and the Fontaine–Mazur conjecture, Invent. Math. 153 (2003), 373–454.CrossRefGoogle Scholar
[10]Mazur, B., Modular curves and the Eisenstein ideal, Publ. Math. Inst. Hautes Études Sci. 47 (1977), 33–186.CrossRefGoogle Scholar
[11]Newton, J., Level raising and completed cohomology (2010), Preliminary version at http://www.ma.ic.ac.uk/∼jjmn07/.CrossRefGoogle Scholar
[12]Paulin, A., Local to global compatibility on the eigencurve (l not equal p), arXiv:0709.4190v1.Google Scholar
[13]Paulin, A., Failure of the local to global principle in the eigencurve, arXiv:1001.2051v1.Google Scholar
[14]Ribet, K., Congruence relations between modular forms, in Proc. Int. Congress of Mathematicians, Warsaw, 16–24 August 1983 (PWN, Warsaw and North Holland, Amsterdam, 1984), 503–514.Google Scholar
[15]Schneider, P., Nonarchimedian function analysis (Springer, Berlin, 2002).CrossRefGoogle Scholar
[16]Serre, J.-P., Trees, (Springer, Berlin, 1980), translated from the French by John Stillwell.Google Scholar
[17]Stevens, G., Rigidanalyticmodularsymbols (1994), http://math.bu.edu/people/ghs/research.d/.Google Scholar
[18]Taylor, R., On Galois representations associated to Hilbert modular forms, Invent. Math. 98 (1989), 265–280.CrossRefGoogle Scholar