Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

Request Permissions   Purchase Content 


Hecke and Sturm bounds for Hilbert modular forms over real quadratic fields

Authors: Jose Ignacio Burgos Gil and Ariel Pacetti
Journal: Math. Comp. 86 (2017), 1949-1978
MSC (2010): Primary 11F41
Published electronically: November 8, 2016
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ K$ be a real quadratic field and $ \mathscr {O}_K$ its ring of integers. Let $ \Gamma $ be a congruence subgroup of $ \mathrm {SL}_2(\mathscr {O}_K)$ and $ M_{(k_1,k_2)}(\Gamma )$ be the finite dimensional space of Hilbert modular forms of weight $ (k_1,k_2)$ for $ \Gamma $. Given a form $ f(z) \in M_{(k_1,k_2)}(\Gamma )$, how many Fourier coefficients determine it uniquely in such space? This problem was solved by Hecke for classical forms, and Sturm proved its analogue for congruences modulo a prime ideal. The present article solves the same problem for Hilbert modular forms over $ K$. We construct a finite set of indices (which depends on the cusps desingularization of the modular surface attached to $ \Gamma $) such that the Fourier coefficients of any form in such set determines it uniquely.

References [Enhancements On Off] (What's this?)

  • [BCP02] Srinath Baba, Kalyan Chakraborty, and Yiannis N. Petridis, On the number of Fourier coefficients that determine a Hilbert modular form, Proc. Amer. Math. Soc. 130 (2002), no. 9, 2497-2502 (electronic). MR 1900854,
  • [Cha90] C.-L. Chai, Arithmetic minimal compactification of the Hilbert-Blumenthal moduli spaces, Ann. of Math. (2) 131 (1990), no. 3, 541-554. MR 1053489,
  • [Coh93] Henri Cohen, A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR 1228206
  • [Cox89] David A. Cox, Primes of the Form $ x^2 + ny^2$: Fermat, Class Field Theory and Complex Multiplication, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1989. MR 1028322
  • [DPS12] Luis Dieulefait, Ariel Pacetti, and Matthias Schütt, Modularity of the Consani-Scholten quintic, Doc. Math. 17 (2012), 953-987. With an appendix by José Burgos Gil and Pacetti. MR 3007681
  • [Fre03] E. Freitag, Modular embeddings of hilbert modular surfaces,, 2003.
  • [Gar90] Paul B. Garrett, Holomorphic Hilbert Modular Forms, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1990. MR 1008244
  • [Gor02] Eyal Z. Goren, Lectures on Hilbert Modular Varieties and Modular Forms, CRM Monograph Series, vol. 14, American Mathematical Society, Providence, RI, 2002. With the assistance of Marc-Hubert Nicole. MR 1863355
  • [Hec70] Erich Hecke, Mathematische Werke, Vandenhoeck&Ruprecht, Göttingen, 1970 (German). Mit einer Vorbemerkung von B. Schoenberg, einer Anmerkung von Carl Ludwig Siegel, und einer Todesanzeige von Jakob Nielsen; Zweite durchgesehene Auflage. MR 0371577
  • [Her87] Carl Friedrich Hermann, Thetareihen und modulare Spitzenformen zu den Hilbertschen Modulgruppen reell-quadratischer Körper, Math. Ann. 277 (1987), no. 2, 327-344 (German). MR 886425,
  • [Her89] Carl Friedrich Hermann, Thetareihen und modulare Spitzenformen zu den Hilbertschen Modulgruppen reell-quadratischer Körper. II, Math. Ann. 283 (1989), no. 4, 689-700 (German). MR 990596,
  • [Kat73] Nicholas M. Katz, $ p$-adic properties of modular schemes and modular forms, Modular Functions of One Variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Mathematics, vol. 350, Springer, Berlin, 1973, pp. 69-190. MR 0447119
  • [KU85] Toshiyuki Katsura and Kenji Ueno, On elliptic surfaces in characteristic $ p$, Math. Ann. 272 (1985), no. 3, 291-330. MR 799664,
  • [Pap95] Georgios Pappas, Arithmetic models for Hilbert modular varieties, Compositio Math. 98 (1995), no. 1, 43-76. MR 1353285
  • [PAR12] The PARI Group, Bordeaux,
    PARI/GP, version 2.6.0, 2012.
    available from
  • [Rap78] M. Rapoport, Compactifications de l'espace de modules de Hilbert-Blumenthal, Compositio Math. 36 (1978), no. 3, 255-335 (French). MR 515050
  • [Shi94] Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Publications of the Mathematical Society of Japan, vol. 11, Princeton University Press, Princeton, NJ, 1994. Reprint of the 1971 original; Kanô Memorial Lectures, 1. MR 1291394
  • [vdG78] G. van der Geer, Hilbert modular forms for the field $ {\bf Q}(\surd 6)$, Math. Ann. 233 (1978), no. 2, 163-179. MR 0491516
  • [vdG88] Gerard van der Geer, Hilbert Modular Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 16, Springer-Verlag, Berlin, 1988. MR 930101
  • [vdGZ77] G. van der Geer and D. Zagier, The Hilbert modular group for the field $ {\bf Q}(\surd 13)$, Invent. Math. 42 (1977), 93-133. MR 0485704

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 11F41

Retrieve articles in all journals with MSC (2010): 11F41

Additional Information

Jose Ignacio Burgos Gil
Affiliation: ICMAT (CSIC-UAM-UCM-UC3), C/ Nicolás Cabrera 13-15, 28049 Madrid, Spain

Ariel Pacetti
Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires and IMAS, CONICET, Argentina

Received by editor(s): July 1, 2015
Received by editor(s) in revised form: August 9, 2015, and January 19, 2016
Published electronically: November 8, 2016
Additional Notes: The first author was partially supported by grant MTM2013-42135-P
The second author was partially supported by CONICET PIP 2010-2012 11220090100801, ANPCyT PICT-2013-0294 and UBACyT 2014-2017-20020130100143BA
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society