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Weights of Modular Forms on so+(2, l) and Congruences Between Eisenstein Series and Cusp forms of Half-Integral Weight on SL2

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  21 December 2009

Richard Hill
Richard Hill
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, England.
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Abstract

Let E be a level 1, vector valued Eisenstein series of half-integral weight, normalized so that the coefficients are all in ℤ. It is shown that there is a level one vector valued cusp form f with the same weight as E and with coefficients in ℤ, which is congruent to E modulo the constant term of E.

MSC classification

Secondary: 11F33: Congruences for modular and $p$-adic modular forms 11F37: Forms of half-integer weight; nonholomorphic modular forms
Type
Research Article
Information
Mathematika , Volume 54 , Issue 1-2 , December 2007 , pp. 47 - 58
Copyright
Copyright © University College London 2007

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References

1Antoniadis, J. A. and Kohnen, W., Congruences between cusp forms and Eisenstein series of half-integral weight. Abh. Math. Sem. Univ. Hamburg 57 (1987), 157–164.CrossRefGoogle Scholar
2Borcherds, R. E., Automorphic forms with singularities on Grassmanians. Invent. Math. 132 (1998), 491–562.CrossRefGoogle Scholar
3Borcherds, R. E., The Gross-Kohnen-Zagier theorem in higher dimensions. Duke Math. J. 97 (1999), 219–233.CrossRefGoogle Scholar
4Brown, J., Saito-Kurokawa lifts and applications to the Bloch-Kato conjecture. Compositio Math. (to appear).Google Scholar
5Bruinier, J. H., Borcherds Products on O(2, l) and Chern Classes of Heegner Divisors Springer Lecture Notes in Mathematics 1780 (2002).CrossRefGoogle Scholar
6Bruinier, J. H. and Funke, J., On two geometric theta lifts. Duke Math. J. 125 (2004), 45–90.CrossRefGoogle Scholar
7Bruinier, J. H. and Kuss, M., Eisenstein series attached to lattices and modular forms on orthogonal groups. Manuscripta Math. 106 (2001), 443–459.CrossRefGoogle Scholar
8Deligne, P., Extensions centrales non résiduellement finies de groupes arithmetiques. C. R. Acad. Sci. Paris 287 (1978), 203–208.Google Scholar
9Hahn, A. J. and O'Meara, O. T., The Classical Groups and K-Theory. Grundlehren der Mathematischen Wissenschaften 291, Springer-Verlag (Berlin, 1989).CrossRefGoogle Scholar
10Hill, R., Fractional weights and non-congruence subgroups. In Automorphic Forms and Representations of Algebraic Groups over Local Fields (ed. Saito, H. and Takahashi, T.), Surikenkoukyuroku Series 1338 (2003), 71–80.Google Scholar
11Kneser, M., Normalteiler ganzzahliger Spingruppen. J. Reine Angew. Math. 311/312 (1979), 191–214.Google Scholar
12Koblitz, N., p-adic congruences and modular forms of half integer weight. Math. Ann. 274 (1986), 199–220.CrossRefGoogle Scholar
13McGraw, W. J., The rationality of vector valued modular forms associated with the Weil representation. Math. Ann. 326 (2003), 105–122.CrossRefGoogle Scholar
14Serre, J.-P. and Stark, H. M., Modular forms of weight 1/2. Modular Functions of One Variable, VI (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), Lecture Notes in Math. 627, Springer (Berlin, 1977), 27–67.CrossRefGoogle Scholar
15Shintani, T., On construction of holomorphic cusp forms of half integral weight. Nagoya Math. J. 58 (1975), 83–126.CrossRefGoogle Scholar
16Weil, A., Sur certains groupes d'opérateurs unitaires. Acta Math. 111 (1964), 143–211.CrossRefGoogle Scholar