Abstract:
We prove that the Consani-Scholten quintic, a Calabi-Yau threefold over Q, is Hilbert modular. For this, we refine several techniques known from the context of modular forms. Most notably, we extend the Faltings-Serre-Livné method to induced fourdimensional Galois representations over Q. We also need a Sturm bound for Hilbert modular forms; this is developed in an appendix by José Burgos Gil and the second author.
Registro:
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Artículo
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Título: | Modularity of the Consani-Scholten quintic |
Autor: | Gil, J.B.; Dieulefait, L.; Pacetti, A.; Schütt, M. |
Filiación: | ICMAT-CSIC, Nicolas Cabrera 13-15, Madrid, 28049, Spain Departament d'Àlgebra i Geometria, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via de les Corts Catalanes, 585, Barcelona, 08007, Spain Departamento de Matemática, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria, Buenos Aires, C.P:1428, Argentina Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany
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Palabras clave: | Consani-Scholten quintic; Faltings-Serre-Livné method; Hilbert modular form; Sturm bound |
Año: | 2012
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Volumen: | 17
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Número: | 2012
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Página de inicio: | 953
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Página de fin: | 987
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Título revista: | Documenta Mathematica
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Título revista abreviado: | Doc. Math.
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ISSN: | 14310635
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Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_14310635_v17_n2012_p953_Gil |
Referencias:
- Bruinier, J.H., Gil, J.I.B., Kühn, U., Borcherds products and arithmetic intersection theory on Hilbert modular surfaces (2007) Duke Math. J., 139 (1), pp. 1-88
- Buzzard, K., Diamond, F., Jarvis, F., On Serre's conjecture for mod ℓ Galois representations over totally real fields (2010) Duke Math. J., 155 (1), pp. 105-161
- Chai, C.-L., Arithmetic minimal compactification of the Hilbert-Blumenthal moduli spaces (1990) Ann. of Math. (2), 131 (3), pp. 541-554
- Cohen, H., (2000) Advanced topics in computational number theory, volume 193 of Graduate Texts in Mathematics, , Springer-Verlag New York
- Consani, C., Scholten, J., Arithmetic on a quintic threefold (2001) Internat. J. Math., 12 (8), pp. 943-972
- Dieulefait, L., Dimitrov, M., Explicit determination of images of Galois representations attached to Hilbert modular forms (2006) J. Number Theory, 117 (2), pp. 397-405
- Dieulefait, L., Guerberoff, L., Pacetti, A., Proving modularity for a given elliptic curve over an imaginary quadratic field (2010) Math. Comp., 79 (270), pp. 1145-1170
- Dimitrov, M., On Ihara's lemma for Hilbert modular varieties (2009) Compos. Math., 145 (5), pp. 1114-1146
- Dieulefait, M., Manoharmayum, J., Modularity of rigid Calabi-Yau threefolds over Q (2003) Calabi-Yau varieties and mirror symmetry (Toronto, ON, 2001), 38, pp. 159-166. , Fields Inst. Com-mun. Amer. Math. Soc., Providence, RI
- Dieulefait, L., Pacetti, L., Schütt, M., http://www.iag.uni-hannover.de/~schuett/publik_en.html, Table of eigenvalues of a hilbert modular form; Deligne, P., Rapoport, M., Les schémas de modules de courbes elliptiques (1973) Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), 349, pp. 143-316. , Lecture Notes in Math. Springer Berlin
- Goren, E.Z., (2002) Lectures on Hilbert modular varieties and modular forms, volume 14 of CRM Monograph Series, , American Mathematical Society, Providence RI With the assistance of Marc-Hubert Nicole
- Hulek, K., Verrill, H., On the modularity of Calabi-Yau threefolds containing elliptic ruled surfaces (2006) Mirror symmetry. V AMS/IP Stud. Adv. Math., 38, pp. 19-34. , Amer. Math. Soc., Providence, RI With an appendix by Luis V. Dieulefait
- Khare, C., Wintenberger, J.-P., Serre's modularity conjecture (2009) I. Invent. Math., 178 (3), pp. 485-504
- Khare, C., Wintenberger, J.-P., Serre's modularity conjecture. II. (2009) Invent. Math., 178 (3), pp. 505-586
- Livné, R., Cubic exponential sums and Galois representations (1987) Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985) Contemp. Math, 67, pp. 247-261. , Amer. Math. Soc., Providence, RI
- PARI/GP, version 2.4.3 (2008), http://pari.math.u-bordeaux.fr/; Rapoport, M., Compactifications de l'espace de modules de Hilbert-Blumenthal (1978) Compositio Math, 36 (3), pp. 255-335
- Shimura, G., The special values of the zeta functions associated with Hilbert modular forms (1978) Duke Math. J., 45 (3), pp. 637-679
- Taylor, R., On Galois representations associated to Hilbert modular forms (1989) Invent. Math., 98 (2), pp. 265-280
- van der Geer, G., (1988) Hilbert modular surfaces Ergeb-nisse der Mathematik und ihrer Grenzgebiete, 16 (3). , [Results in Math-ematics and Related Areas (3)] Springer-Verlag Berlin
Citas:
---------- APA ----------
Gil, J.B., Dieulefait, L., Pacetti, A. & Schütt, M.
(2012)
. Modularity of the Consani-Scholten quintic. Documenta Mathematica, 17(2012), 953-987.
Recuperado de https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_14310635_v17_n2012_p953_Gil [ ]
---------- CHICAGO ----------
Gil, J.B., Dieulefait, L., Pacetti, A., Schütt, M.
"Modularity of the Consani-Scholten quintic"
. Documenta Mathematica 17, no. 2012
(2012) : 953-987.
Recuperado de https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_14310635_v17_n2012_p953_Gil [ ]
---------- MLA ----------
Gil, J.B., Dieulefait, L., Pacetti, A., Schütt, M.
"Modularity of the Consani-Scholten quintic"
. Documenta Mathematica, vol. 17, no. 2012, 2012, pp. 953-987.
Recuperado de https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_14310635_v17_n2012_p953_Gil [ ]
---------- VANCOUVER ----------
Gil, J.B., Dieulefait, L., Pacetti, A., Schütt, M. Modularity of the Consani-Scholten quintic. Doc. Math. 2012;17(2012):953-987.
Available from: https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_14310635_v17_n2012_p953_Gil [ ]