Algebra & Number Theory

An analogue of Sturm's theorem for Hilbert modular forms

Yuuki Takai

Full-text: Open access

Abstract

In this paper, we consider congruences of Hilbert modular forms. Sturm showed that mod elliptic modular forms of weight k and level Γ1(N) are determined by the first (k12)[Γ1(1):Γ1(N)] mod Fourier coefficients. We prove an analogue of Sturm’s result for Hilbert modular forms associated to totally real number fields. The proof uses the positivity of ample line bundles on toroidal compactifications of Hilbert modular varieties.

Article information

Source
Algebra Number Theory, Volume 7, Number 4 (2013), 1001-1018.

Dates
Received: 22 November 2011
Revised: 30 August 2012
Accepted: 4 September 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513729989

Digital Object Identifier
doi:10.2140/ant.2013.7.1001

Mathematical Reviews number (MathSciNet)
MR3095234

Zentralblatt MATH identifier
1330.11030

Subjects
Primary: 11F41: Automorphic forms on GL(2); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces [See also 14J20]
Secondary: 11F30: Fourier coefficients of automorphic forms 11F33: Congruences for modular and $p$-adic modular forms [See also 14G20, 22E50] 14C17: Intersection theory, characteristic classes, intersection multiplicities [See also 13H15]

Keywords
Hilbert modular forms and varieties congruences of modular forms Sturm's theorem toroidal and minimal compactifications intersection numbers

Citation

Takai, Yuuki. An analogue of Sturm's theorem for Hilbert modular forms. Algebra Number Theory 7 (2013), no. 4, 1001--1018. doi:10.2140/ant.2013.7.1001. https://projecteuclid.org/euclid.ant/1513729989


Export citation

References

  • A. Ash, D. Mumford, M. Rapoport, and Y. Tai, Smooth compactification of locally symmetric varieties, Lie Groups: History, Frontiers and Applications 4, Math. Sci. Press, Brookline, MA, 1975.
  • S. Baba, K. Chakraborty, and Y. N. Petridis, “On the number of Fourier coefficients that determine a Hilbert modular form”, Proc. Amer. Math. Soc. 130:9 (2002), 2497–2502.
  • C.-L. Chai, “Arithmetic minimal compactification of the Hilbert–Blumenthal moduli spaces”, Ann. of Math. $(2)$ 131:3 (1990), 541–554.
  • P. Deligne and M. Rapoport, “Les schémas de modules de courbes elliptiques”, pp. 143–316 in Modular functions of one variable (Antwerp, 1972), vol. 2, edited by P. Deligne and W. Kuyk, Lecture Notes in Mathematics 349, Springer, Berlin, 1973.
  • L. Dieulefait, A. Pacetti, and M. Schuett, “Modularity of the Consani–Scholten quintic”, preprint, 2010.
  • M. Dimitrov, “Compactifications arithmétiques des variétés de Hilbert et formes modulaires de Hilbert pour $\Gamma\sb 1(\mathfrak{c},\mathfrak{n})$”, pp. 527–554 in Geometric aspects of Dwork theory, edited by A. Adolphson et al., de Gruyter, Berlin, 2004.
  • M. Dimitrov and J. Tilouine, “Variétés et formes modulaires de Hilbert arithmétiques pour $\Gamma\sb 1(\mathfrak{c},\mathfrak{n})$”, pp. 555–614 in Geometric aspects of Dwork theory, edited by A. Adolphson et al., de Gruyter, Berlin, 2004.
  • K. Doi and M. Ohta, “On some congruences between cusp forms on $\Gamma \sb{0}(N)$”, pp. 91–105 in Modular functions of one variable (Bonn, 1976), vol. 5, Lecture Notes in Mathematics 601, Springer, Berlin, 1977.
  • T. Ekedahl, “Canonical models of surfaces of general type in positive characteristic”, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 97–144.
  • G. Faltings and C.-L. Chai, Degeneration of abelian varieties, Ergeb. Math. Grenzgeb. (3) 22, Springer, Berlin, 1990.
  • W. Fulton, Intersection theory, 2nd ed., Ergeb. Math. Grenzgeb. (3) 2, Springer, Berlin, 1998.
  • G. van der Geer, Hilbert modular surfaces, Ergeb. Math. Grenzgeb. (3) 16, Springer, Berlin, 1988.
  • R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics 52, Springer, New York, 1977.
  • H. Hida, $p$-adic automorphic forms on Shimura varieties, Springer, New York, 2004.
  • N. M. Katz, “$p$-adic $L$-functions for CM fields”, Invent. Math. 49:3 (1978), 199–297.
  • R. Lazarsfeld, Positivity in algebraic geometry, I: Classical setting, line bundles and linear series, Ergeb. Math. Grenzgeb. (3) 48, Springer, Berlin, 2004.
  • T. Miyake, Modular forms, Springer, Berlin, 1989.
  • L. Moret-Bailly, Pinceaux de variétés abéliennes, Astérisque 129, Société Mathématique de France, Paris, 1985.
  • D. Mumford, “Hirzebruch's proportionality theorem in the noncompact case”, Invent. Math. 42 (1977), 239–272.
  • M. Rapoport, “Compactifications de l'espace de modules de Hilbert–Blumenthal”, Compositio Math. 36:3 (1978), 255–335.
  • J. Sturm, “On the congruence of modular forms”, pp. 275–280 in Number theory (New York, 1984–1985), edited by D. V. Chudnovsky et al., Lecture Notes in Mathematics 1240, Springer, Berlin, 1987.
  • Y. Takai, An analogy of Sturm's theorem for real quadratic fields, Ph.D. thesis, Nagoya University, 2010.