Abstract:
Let K be a real quadratic field and OK its ring of integers. Let Γ be a congruence subgroup of SL2(OK) and M(k1,k2)(Γ) be the finite dimensional space of Hilbert modular forms of weight (k1, k2) for Γ. Given a form f(z) ∈ M(k1,k2)(Γ), 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 Γ) such that the Fourier coefficients of any form in such set determines it uniquely. © 2016 American Mathematical Society.
Registro:
Documento: |
Artículo
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Título: | Hecke and sturm bounds for Hilbert modular forms over real quadratic fields |
Autor: | Gil, J.I.B.; Pacetti, A. |
Filiación: | ICMAT (CSIC-UAM-UCM-UC3), C/ Nicolás Cabrera 13-15, Madrid, 28049, Spain Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires and IMAS, CONICET, Argentina
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Año: | 2017
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Volumen: | 86
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Número: | 306
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Página de inicio: | 1949
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Página de fin: | 1978
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DOI: |
http://dx.doi.org/10.1090/mcom/3187 |
Título revista: | Mathematics of Computation
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Título revista abreviado: | Math. Comput.
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ISSN: | 00255718
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Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00255718_v86_n306_p1949_Gil |
Referencias:
- Baba, S., Chakraborty, K., Petridis, Y.N., On the number of Fourier coefficients that determine a Hilbert modular form (2002) Proc. Amer. Math. Soc, 130 (9), pp. 2497-2502. , (electronic). MR1900854
- Chai, C.-L., Arithmetic minimal compactification of the Hilbert-Blumenthal moduli spaces (1990) Ann. of Math. (2), 131 (3), pp. 541-554. , MR1053489
- Cohen, H., (1993) A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics, 138. , Springer-Verlag, Berlin,. MR1228206
- Cox, D.A., (1989) Primes of the Form x2 + ny2: Fermat, Class Field Theory and Complex Multiplication, , A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York. MR1028322
- Dieulefait, L., Pacetti, A., Schütt, M., Modularity of the Consani-Scholten quintic (2012) Doc. Math, 17, pp. 953-987. , With an appendix by José Burgos Gil and Pacetti. MR3007681
- Freitag, E., (2003) Modular embeddings of hilbert modular surfaces, , http://www.rzuser.uni-heidelberg.de/~t91/index4.html
- Garrett, P.B., (1990) Holomorphic Hilbert Modular Forms, , The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA. MR1008244
- Goren, E.Z., (2002) Lectures on Hilbert Modular Varieties and Modular Forms, CRM Monograph Series, 14. , American Mathematical Society, Providence, RI,. With the assistance of Marc-Hubert Nicole. MR1863355
- Hecke, E., (1970) Mathematische Werke (German), Vandenhoeck&Ruprecht, Göttingen, , Mit einer Vorbemerkung von B. Schoenberg, einer Anmerkung von Carl Ludwig Siegel, und einer Todesanzeige von Jakob Nielsen; Zweite durchgesehene Auflage. MR0371577
- Hermann, C.F., Thetareihen und modulare Spitzenformen zu den Hilbertschen Modulgruppen reell-quadratischer Körper (German) (1987) Math. Ann, 277 (2), pp. 327-344. , MR886425
- Hermann, C.F., Thetareihen und modulare Spitzenformen zu den Hilbertschen Modulgruppen reell-quadratischer Körper. II (German) (1989) Math. Ann, 283 (4), pp. 689-700. , MR990596
- Katz, N.M., p-adic properties of modular schemes and modular forms, Modular Functions of One Variable, III (1972) Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 350, pp. 69-190. , Lecture Notes in Mathematics, Springer, Berlin. MR0447119 1973
- Katsura, T., Ueno, K., On elliptic surfaces in characteristic p (1985) Math. Ann, 272 (3), pp. 291-330. , MR799664
- Pappas, G., Arithmetic models for Hilbert modular varieties (1995) Compositio Math, 98 (1), pp. 43-76. , MR1353285
- (2012) Bordeaux, PARI/GP, version 2.6.0, , http://pari.math.u-bordeaux.fr/
- Rapoport, M., Compactifications de l'espace de modules de Hilbert-Blumenthal (French) (1978) Compositio Math, 36 (3), pp. 255-335. , MR515050
- Shimura, G., (1994) Introduction to the Arithmetic Theory of Automorphic Functions, Publications of the Mathematical Society of Japan, 11. , Princeton University Press, Princeton, NJ,. Reprint of the 1971 original; Kanô Memorial Lectures, 1. MR1291394
- van der Geer, G., Hilbert modular forms for the field Q(v6) (1978) Math. Ann, 233 (2), pp. 163-179. , MR0491516
- van der Geer, G., (1988) Hilbert Modular Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 16. , Springer-Verlag, Berlin,. MR930101
- van der Geer, G., Zagier, D., The Hilbert modular group for the field Q(v13) (1977) Invent. Math, 42, pp. 93-133. , MR0485704
Citas:
---------- APA ----------
Gil, J.I.B. & Pacetti, A.
(2017)
. Hecke and sturm bounds for Hilbert modular forms over real quadratic fields. Mathematics of Computation, 86(306), 1949-1978.
http://dx.doi.org/10.1090/mcom/3187---------- CHICAGO ----------
Gil, J.I.B., Pacetti, A.
"Hecke and sturm bounds for Hilbert modular forms over real quadratic fields"
. Mathematics of Computation 86, no. 306
(2017) : 1949-1978.
http://dx.doi.org/10.1090/mcom/3187---------- MLA ----------
Gil, J.I.B., Pacetti, A.
"Hecke and sturm bounds for Hilbert modular forms over real quadratic fields"
. Mathematics of Computation, vol. 86, no. 306, 2017, pp. 1949-1978.
http://dx.doi.org/10.1090/mcom/3187---------- VANCOUVER ----------
Gil, J.I.B., Pacetti, A. Hecke and sturm bounds for Hilbert modular forms over real quadratic fields. Math. Comput. 2017;86(306):1949-1978.
http://dx.doi.org/10.1090/mcom/3187