Icosahedral invariants and Shimura curves. (English. French summary) Zbl 1386.14139
Shimura curves are moduli spaces of abelian surfaces with quaternion multiplication, but it is a non-trivial problem to find explicit models of Shimura curves. This paper obtains explicit models of Shimura curves for quaternion algebra with small discriminant. Explicit defining equations of the Shimura curves are obtained in the weighted projective space \({\mathbb{P}}(1:3:5)=\text{{Proj}}({\mathbb{C}}[A,B,C])\) where \(A,B,C\) are the Klein icosahedral invariants of weight \(1,3\) and \(5\), respectively.
The Klein invariants \(A, B\) and \(C\) give the Hilbert modular forms associated to \(\sqrt{5}\) via the period map for a family of \(K3\) surfaces. Using the period map for several families of \(K3\) surfaces, explicit models of Shimura curves with small discriminants are produced.
The Klein invariants \(A, B\) and \(C\) give the Hilbert modular forms associated to \(\sqrt{5}\) via the period map for a family of \(K3\) surfaces. Using the period map for several families of \(K3\) surfaces, explicit models of Shimura curves with small discriminants are produced.
Reviewer: Noriko Yui (Kingston)
MSC:
| 14J28 | \(K3\) surfaces and Enriques surfaces |
| 14G35 | Modular and Shimura varieties |
| 11G18 | Arithmetic aspects of modular and Shimura varieties |
Keywords:
\(K3\) surfaces; abelian surfaces; Shimura curves; Hilbert modular functions; quaternion algebraReferences:
| [1] | An, Sang Yook; Kim, Seog Young; Marshall, David C.; Marshall, Susan H.; McCallum, William G.; Perlis, Alexander R., Jacobians of Genus One Curves, J. Number Theory, 90, 2, 304-315 (2001) · Zbl 1066.14035 · doi:10.1006/jnth.2000.2632 |
| [2] | Besser, Amnon, Elliptic fibrations of \(K3\) surfaces and QM Kummer surfaces, Math. Z., 288, 2, 283-308 (1998) · Zbl 0936.14027 · doi:10.1007/PL00004613 |
| [3] | Bonfanti, Matteo Alfonso; Van Geemen, Bert, Abelian surfaces with an automorphism and quaternionic multiplication, Can. J. Math., 68, 1, 24-43 (2016) · Zbl 1337.14035 · doi:10.4153/CJM-2014-045-4 |
| [4] | Clingher, Adrian; Doran, Charles F., Lattice polarized \(K3\) surfaces and Siegel modular forms, Adv. Math., 231, 1, 172-212 (2012) · Zbl 1245.14036 · doi:10.1016/j.aim.2012.05.001 |
| [5] | Cox, David; Little, John; O’Shea, Donal, Using algebraic geometry, 185, xii+499 p. pp. (1998), Springer · Zbl 0920.13026 |
| [6] | Dolgachev, Igor V., Mirror symmetry for lattice polarized \(K3\) surfaces, J. Math. Sci., New York, 81, 3, 2599-2630 (1996) · Zbl 0890.14024 · doi:10.1007/BF02362332 |
| [7] | Elkies, Noam, Shimura curve computations, Algorithmic number theory. 3rd international symposium, 1423, 1-47 (1998), Springer · Zbl 1010.11030 |
| [8] | Elkies, Noam, Shimura curve computations via \(K3\) surfaces of Néron-Severi rank at least \(19\), Algorithmic number theory. 8th international symposium, 5011, 196-211 (2008), Springer · Zbl 1205.11069 |
| [9] | Elkies, Noam; Kumar, Abhinav, \(K3\) Surfaces and equations for Hilbert modular surfaces, Algebra Number Theory, 8, 10, 2297-2411 (2014) · Zbl 1317.11048 · doi:10.2140/ant.2014.8.2297 |
| [10] | van der Geer, Gerard, Hilbert modular surfaces, 16, ix+291 p. pp. (1988), Springer · Zbl 0634.14022 |
| [11] | Hashimoto, Kenji; Nagano, Atsuhira; Ueda, Kazushi, Modular surfaces associated with toric \(K3\) surfaces (2014) |
| [12] | Hashimoto, Ki-Ichiro, Explicit form of quaternion modular embeddings, Osaka J. Math., 32, 3, 533-546 (1995) · Zbl 0851.11034 |
| [13] | Hashimoto, Ki-Ichiro; Murabayashi, Naoki, Shimura curves as intersections of Humbert surfaces and defining equations of QM-curves of genus two, Tôhoku Math. J., 47, 2, 271-296 (1995) · Zbl 0838.11044 · doi:10.2748/tmj/1178225596 |
| [14] | Hirzebruch, Friedrich, The ring of Hilbert modular forms for real quadratic fields of small discriminant, Modular Funct. of one Var. VI, Proc. int. Conf., Bonn 1976, 627, 287-323 (1977), Springer · Zbl 0369.10017 |
| [15] | Humbert, Georges, Oeuvres de G. Humbert 2, pub. par les soins de Pierre Humbert et de Gaston Julia, Sur les fonctions abéliennes singulières, 297-401 (1936), Gauthier-Villars · JFM 62.1039.01 |
| [16] | Klein, Felix, Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade (1884), Tauber · JFM 16.0061.01 |
| [17] | Kohel, David R.; Verrill, Helena A., Fundamental Domains for Shimura Curves, J. Théor. Nombres Bordx, 15, 1, 205-222 (2003) · Zbl 1044.11052 · doi:10.5802/jtnb.398 |
| [18] | Kumar, Abhinav, \(K3\) surfaces associated to curves of genus two, Int. Math. Res. Not., 16 (2008) · Zbl 1145.14029 |
| [19] | Kurihara, Akira, On some examples of equations defining Shimura curves and the Mumford uniformization, J. Fac. Sci., Univ. Tokyo, 25, 277-300 (1978) · Zbl 0428.14012 |
| [20] | Müller, Rolf, Hilbertsche Modulformen und Modulfunktionen zu \(\mathbb{Q}(\sqrt{5})\), Arch. Math., 45, 239-251 (1985) · Zbl 0552.10016 · doi:10.1007/BF01275576 |
| [21] | Nagano, Atsuhira, Period differential equations for the families of \(K3\) surfaces with two parameters derived from the reflexive polytopes, Kyushu J. Math., 66, 1, 193-244 (2012) · Zbl 1262.14047 · doi:10.2206/kyushujm.66.193 |
| [22] | Nagano, Atsuhira, A theta expression of the Hilbert modular functions for \(\sqrt{5}\) via period of \(K3\) surfaces, Kyoto J. Math., 53, 4, 815-843 (2013) · Zbl 1285.14042 · doi:10.1215/21562261-2366102 |
| [23] | Nagano, Atsuhira, Double integrals on a weighted projective plane and Hilbert modular functions for \(\mathbb{Q}(\sqrt{5})\), Acta Arith., 167, 4, 327-345 (2015) · Zbl 1394.11040 · doi:10.4064/aa167-4-2 |
| [24] | Nagano, Atsuhira, Icosahedral invariants and a construction of class fields via periods of \(K3\) surfaces (2017) · Zbl 1448.11120 |
| [25] | Nagano, Atsuhira; Shiga, Hironori, Modular map for the family of abelian surfaces via elliptic \(K3\) surfaces, Math. Nachr., 288, 1, 89-114 (2015) · Zbl 1307.32011 · doi:10.1002/mana.201300142 |
| [26] | Rotger, Victor, Modular Shimura varieties and forgetful maps, Trans. Am. Math. Soc., 356, 4, 1535-1550 (2004) · Zbl 1049.11061 · doi:10.1090/S0002-9947-03-03408-1 |
| [27] | Rotger, Victor, Modular curves and Abelian varieties. Based on lectures of the conference, Bellaterra, Barcelona, July 2002, 224, Shimura curves embedded in Igusa’s threefold, 263-276 (2004), Birkhäuser · Zbl 1071.11035 |
| [28] | Shimura, Goro, Construction of class fields and zeta functions of algebraic curves, Ann. Math., 85, 58-159 (1967) · Zbl 0204.07201 · doi:10.2307/1970526 |
| [29] | Shimura, Goro, On canonical models of arithmetic quotients of bounded symmetric domains, Ann. Math, 91, 144-222 (1970) · Zbl 0237.14009 · doi:10.2307/1970604 |
| [30] | Shimura, Goro, On the real points of an arithmetic quotient of a bounded symmetric domain, Math. Ann., 215, 135-164 (1975) · Zbl 0394.14007 · doi:10.1007/BF01432692 |
| [31] | Shimura, Goro, Abelian Varieties with Complex Multiplication and Modular Functions, 46, ix+217 p. pp. (1998), Princeton University Press · Zbl 0908.11023 |
| [32] | Vignéras, Marie-France, Arithmétiques des algèbres de quaternions, 80, vii+169 p. pp. (1980), Springer · Zbl 0422.12008 |
| [33] | Voight, John, Arithmetic geometry, 8, Shimura Curve Computations, 103-113 (2009), Clay Mathematics Institute · Zbl 1250.11063 |
| [34] | Yang, Yifan, Quaternionic loci in Siegel’s modular threefolds (2015) |
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