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On the mass formula of supersingular abelian varieties with real multiplications

Part of: Forms and linear algebraic groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  09 April 2009

Chia-Fu Yu
Chia-Fu Yu
Affiliation:
Department of MathematicsColumbia UniversityNew York, NY 10027USA e-mail: [email protected]
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Abstract

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A geometric mass concerning supersingular abelian varieties with real multiplications is formulated and related to an arithmetic mass. We determine the exact geometric mass formula for superspecial abelian varieties of Hubert-Blumenthal type. As an application, we compute the number of the irreducible components of the supersingular locus of some Hubert-Blumenthal varieties in terms of special values of the zeta function.

Keywords

mass formulasupersingular locussuperspecial abelian varietiesHilbert-Blumenthal varieties

MSC classification

Secondary: 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11E41: Class numbers of quadratic and Hermitian forms
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 78 , Issue 3 , June 2005 , pp. 373 - 392
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Bachmat, E. and Goren, E. Z., ‘On the non-ordinary locus in Hillbet-Blumenthal surfaces’, Math. Ann. 313 (1999), 475–506.CrossRefGoogle Scholar
[2]Deligne, P. and Pappas, G., ‘Singularités des espaces de modules de Hillbert, en les caractéristiques divisant le discriminant’, Compositio Math. 90 (1994), 59–79.Google Scholar
[3]Geer, G. van der, ‘Cycles on the moduli space of abelian varieties’, in: Moduli of curves and abelian varieties, Aspects Math. E33 (Vieweg, Braunschweig, 1999) pp. 65–89.CrossRefGoogle Scholar
[4]Gross, B. H., ‘Algebraic modular forms’, Israel J. Math. 113 (1999), 61–93.CrossRefGoogle Scholar
[5]Kottwitz, R. E., ‘Points on some Shimura varieties over finite fields’, J. Amer. Math. Soc. 5 (1992), 373–444.CrossRefGoogle Scholar
[6]Li, K.-Z. and Oort, F., Moduli of supersingular abelian varieties, Lecture Notes in Math. 1680 (Springer, New York, 1998).CrossRefGoogle Scholar
[7]Rapoport, M., ‘Compactifications de l'espaces de modules de Hilbert-Blumenthal’, Compositio Math. 36 (1978), 255–335.Google Scholar
[8]Serre, J.-P., ‘Two letters of quaternnions and modular forms (mod p)’, Israel J. Math. 95 (1996), 281–299.CrossRefGoogle Scholar
[9]Shimura, G., ‘Some exact formulas for quaternion unitary groups’, J. Reine Angew. Math. 509 (1999), 67–102.CrossRefGoogle Scholar
[10]Yu, C.-F., ‘On reduction of Hilbert-Blumenthal varieties’, Ann. Inst. Fourier 53 (2003), 2105–2154.CrossRefGoogle Scholar
[11]Yu, C.-F., ‘On the supersingular locus in Hilbert-Blumenthal 40folds’, J. Algebraic Geom. 12 (2003), 653–698.CrossRefGoogle Scholar
[12]Zink, T., ‘Isogenieklassen von Pubkten von Shimuramannigfaltigkeiten mit Werten in einem endlichen Körper’, Math. Nachr. 112 (1983), 103–124.CrossRefGoogle Scholar