Algebra & Number Theory

K3 surfaces and equations for Hilbert modular surfaces

Noam Elkies and Abhinav Kumar

Full-text: Open access

Abstract

We outline a method to compute rational models for the Hilbert modular surfaces Y(D), which are coarse moduli spaces for principally polarized abelian surfaces with real multiplication by the ring of integers in (D), via moduli spaces of elliptic K3 surfaces with a Shioda–Inose structure. In particular, we compute equations for all thirty fundamental discriminants D with 1<D<100, and analyze rational points and curves on these Hilbert modular surfaces, producing examples of genus-2 curves over  whose Jacobians have real multiplication over .

Article information

Source
Algebra Number Theory, Volume 8, Number 10 (2014), 2297-2411.

Dates
Received: 22 January 2013
Revised: 26 August 2013
Accepted: 28 October 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513730328

Digital Object Identifier
doi:10.2140/ant.2014.8.2297

Mathematical Reviews number (MathSciNet)
MR3298543

Zentralblatt MATH identifier
1317.11048

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: 14G35: Modular and Shimura varieties [See also 11F41, 11F46, 11G18] 14J28: $K3$ surfaces and Enriques surfaces 14J27: Elliptic surfaces

Keywords
elliptic K3 surfaces moduli spaces Hilbert modular surfaces abelian surfaces real multiplication genus-2 curves

Citation

Elkies, Noam; Kumar, Abhinav. K3 surfaces and equations for Hilbert modular surfaces. Algebra Number Theory 8 (2014), no. 10, 2297--2411. doi:10.2140/ant.2014.8.2297. https://projecteuclid.org/euclid.ant/1513730328


Export citation

References

  • S. Y. An, S. Y. Kim, D. C. Marshall, S. H. Marshall, W. G. McCallum, and A. R. Perlis, “Jacobians of genus one curves”, J. Number Theory 90:2 (2001), 304–315.
  • C. Birkenhake and H. Lange, Complex abelian varieties, 2nd ed., Grundlehren der Mathematischen Wissenschaften 302, Springer, Berlin, 2004.
  • A. Brumer, “The rank of $J\sb 0(N)$”, pp. 3, 41–68 in Columbia University Number Theory Seminar (New York, 1992), Astérisque 228, Soc. Math. France, Paris, 1995.
  • A. Clingher and C. F. Doran, “Modular invariants for lattice polarized $K3$ surfaces”, Michigan Math. J. 55:2 (2007), 355–393.
  • P. Deligne and G. Pappas, “Singularités des espaces de modules de Hilbert, en les caractéristiques divisant le discriminant”, Compositio Math. 90:1 (1994), 59–79.
  • L. Dembélé and A. Kumar, “Examples of abelian surfaces with everywhere good reduction”, preprint, 2013.
  • I. V. Dolgachev, “Mirror symmetry for lattice polarized $K3$ surfaces”, J. Math. Sci. 81:3 (1996), 2599–2630.
  • N. D. Elkies, “Elliptic and modular curves over finite fields and related computational issues”, pp. 21–76 in Computational perspectives on number theory (Chicago, IL), edited by D. A. Buell and J. T. Teitelbaum, AMS/IP Stud. Adv. Math. 7, Amer. Math. Soc., Providence, RI, 1998.
  • N. D. Elkies, “Shimura curve computations via $K3$ surfaces of Néron–Severi rank at least 19”, pp. 196–211 in Algorithmic number theory, edited by A. J. van der Poorten and A. Stein, Lecture Notes in Comput. Sci. 5011, Springer, Berlin, 2008.
  • N. D. Elkies, “Elliptic curves of high rank over $\Q$ and $\Q(t)$”, In preparation.
  • R. Friedman, “A new proof of the global Torelli theorem for $K3$ surfaces”, Ann. of Math. $(2)$ 120:2 (1984), 237–269.
  • G. van der Geer, Hilbert modular surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 16, Springer, Berlin, 1988.
  • V. A. Gritsenko and V. V. Nikulin, “Siegel automorphic form corrections of some Lorentzian Kac–Moody Lie algebras”, Amer. J. Math. 119:1 (1997), 181–224.
  • D. Gruenewald, Explicit Algorithms for Humbert Surfaces, Ph.D. thesis, University of Sydney, 2008, http://echidna.maths.usyd.edu.au/~davidg/thesis.pdf.
  • W. Hausmann, “The fixed points of the symmetric Hilbert modular group of a real quadratic field with arbitrary discriminant”, Math. Ann. 260:1 (1982), 31–50.
  • F. E. P. Hirzebruch, “Hilbert modular surfaces”, Enseignement Math. $(2)$ 19 (1973), 183–281.
  • F. Hirzebruch and A. van de Ven, “Hilbert modular surfaces and the classification of algebraic surfaces”, Invent. Math. 23 (1974), 1–29.
  • F. Hirzebruch and G. van der Geer, Lectures on Hilbert modular surfaces, Séminaire de Mathématiques Supérieures 77, Presses de l'Université de Montréal, 1981.
  • F. Hirzebruch and D. Zagier, “Classification of Hilbert modular surfaces”, pp. 43–77 in Complex analysis and algebraic geometry, edited by J. W. L. Baily and T. Shioda, Iwanami Shoten, Tokyo, 1977.
  • R. W. H. T. Hudson, Kummer's quartic surface, Cambridge University Press, 1990.
  • J.-i. Igusa, “Arithmetic variety of moduli for genus two”, Ann. of Math. $(2)$ 72 (1960), 612–649.
  • H. Inose, “Defining equations of singular $K3$ surfaces and a notion of isogeny”, pp. 495–502 in Proceedings of the International Symposium on Algebraic Geometry (Kyoto University, 1977), edited by M. Nagata, Kinokuniya Book Store, Tokyo, 1978.
  • C. Khare and J.-P. Wintenberger, “Serre's modularity conjecture, I”, Invent. Math. 178:3 (2009), 485–504.
  • C. Khare and J.-P. Wintenberger, “Serre's modularity conjecture, II”, Invent. Math. 178:3 (2009), 505–586.
  • V. S. Kulikov, “Degenerations of $K3$ surfaces and Enriques surfaces”, Izv. Akad. Nauk SSSR Ser. Mat. 41:5 (1977), 1008–1042, 1199. In Russian; translated in Math. USSR-Izv. 11:5 (1977), 957–989.
  • A. Kumar, “$K3$ surfaces associated with curves of genus two”, Int. Math. Res. Not. 2008:6 (2008), Art. ID rnm165, 26.
  • A. Kumar, “Elliptic fibrations on a generic Jacobian Kummer surface”, J. Algebraic Geom. 23 (2014), 599–667.
  • M. Kuwata and T. Shioda, “Elliptic parameters and defining equations for elliptic fibrations on a Kummer surface”, pp. 177–215 in Algebraic geometry in East Asia (Hanoi, 2005), edited by K. Konno and V. Nguyen-Khac, Adv. Stud. Pure Math. 50, Math. Soc. Japan, Tokyo, 2008.
  • Q. Liu, D. Lorenzini, and M. Raynaud, “On the Brauer group of a surface”, Invent. Math. 159:3 (2005), 673–676.
  • R. van Luijk, “K3 surfaces with Picard number one and infinitely many rational points”, Algebra Number Theory 1:1 (2007), 1–15.
  • J.-F. Mestre, “Construction de courbes de genre $2$ à partir de leurs modules”, pp. 313–334 in Effective methods in algebraic geometry (Castiglioncello, 1990), edited by T. Mora and C. Traverso, Progr. Math. 94, Birkhäuser, Boston, 1991.
  • J. S. Milne, “On a conjecture of Artin and Tate”, Ann. of Math. $(2)$ 102:3 (1975), 517–533.
  • J. S. Milne, “On the conjecture of Artin and Tate”, 1975, http://jmilne.org/math/articles/add/1975a.pdf. Notes on Ann. of Math. (2) 102:3 (1975), 517–533.
  • D. R. Morrison, “On $K3$ surfaces with large Picard number”, Invent. Math. 75:1 (1984), 105–121.
  • V. V. Nikulin, “Finite groups of automorphisms of Kählerian $K3$ surfaces”, Trudy Moskov. Mat. Obshch. 38 (1979), 75–137. In Russian.
  • V. V. Nikulin, “Integer symmetric bilinear forms and some of their geometric applications”, Izv. Akad. Nauk SSSR Ser. Mat. 43:1 (1979), 111–177, 238. In Russian; translated in Math. USSR-Izv. 14:1 (1980), 103–167.
  • T. Oda, Periods of Hilbert modular surfaces, Progress in Mathematics 19, Birkhäuser, Boston, 1982.
  • U. Persson and H. Pinkham, “Degeneration of surfaces with trivial canonical bundle”, Ann. of Math. $(2)$ 113:1 (1981), 45–66.
  • I. Piatetski-Shapiro and I. R. Shafarevich, “Torelli's theorem for algebraic surfaces of type ${\rm K}3$”, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 530–572. In Russian; translated in Math. USSR-Izv. 5:3 (1971), 547–588.
  • M. Rapoport, “Compactifications de l'espace de modules de Hilbert–Blumenthal”, Compositio Math. 36:3 (1978), 255–335.
  • K. A. Ribet, “Abelian varieties over $\bf Q$ and modular forms”, pp. 241–261 in Modular curves and abelian varieties, edited by J. Cremona et al., Progr. Math. 224, Birkhäuser, Basel, 2004.
  • B. Runge, “Endomorphism rings of abelian surfaces and projective models of their moduli spaces”, Tohoku Math. J. $(2)$ 51:3 (1999), 283–303.
  • T. Shioda, “On elliptic modular surfaces”, J. Math. Soc. Japan 24 (1972), 20–59.
  • T. Shioda, “On the Mordell–Weil lattices”, Comment. Math. Univ. St. Paul. 39:2 (1990), 211–240.
  • T. Shioda, “Kummer sandwich theorem of certain elliptic $K3$ surfaces”, Proc. Japan Acad. Ser. A Math. Sci. 82:8 (2006), 137–140.
  • J. Tate, “Endomorphisms of abelian varieties over finite fields”, Invent. Math. 2 (1966), 134–144.
  • J. Tate, “On the conjectures of Birch and Swinnerton-Dyer and a geometric analog”, in Séminaire Bourbaki $1965/1966$ (Exposé 306), W. A. Benjamin, Amsterdam, 1966. Reprinted as pp. 415–440 in Séminaire Bourbaki 9, Soc. Math. France, Paris, 1995.
  • J. Tate, “Algorithm for determining the type of a singular fiber in an elliptic pencil”, pp. 33–52 in Modular functions of one variable, IV, edited by B. J. Birch and W. Kuyk, Lecture Notes in Math. 476, Springer, Berlin, 1975.
  • J. Wilson, Curves of genus $2$ with real multiplication by a square root of $5$, Ph.D. thesis, Oxford University, 1998, http://eprints.maths.ox.ac.uk/32/1/wilson.pdf.
  • J. Wilson, “Explicit moduli for curves of genus 2 with real multiplication by ${\bf Q}(\sqrt5)$”, Acta Arith. 93:2 (2000), 121–138.

Supplemental materials