Algebra & Number Theory

Arithmetic of singular Enriques surfaces

Klaus Hulek and Matthias Schütt

Full-text: Open access

Abstract

We study the arithmetic of Enriques surfaces whose universal covers are singular K3 surfaces. If a singular K3 surface X has discriminant d, then it has a model over the ring class field H(d). Our main theorem is that the same holds true for any Enriques quotient of X. It is based on a study of Néron–Severi groups of singular K3 surfaces. We also comment on Galois actions on divisors of Enriques surfaces.

Article information

Source
Algebra Number Theory, Volume 6, Number 2 (2012), 195-230.

Dates
Received: 13 March 2010
Revised: 28 November 2010
Accepted: 29 December 2010
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513729776

Digital Object Identifier
doi:10.2140/ant.2012.6.195

Mathematical Reviews number (MathSciNet)
MR2950152

Zentralblatt MATH identifier
1248.14043

Subjects
Primary: 14J28: $K3$ surfaces and Enriques surfaces
Secondary: 11E16: General binary quadratic forms 11G15: Complex multiplication and moduli of abelian varieties [See also 14K22] 11G35: Varieties over global fields [See also 14G25] 14J27: Elliptic surfaces

Keywords
Enriques surface singular K3 surface elliptic fibration Néron–Severi group Mordell–Weil group complex multiplication

Citation

Hulek, Klaus; Schütt, Matthias. Arithmetic of singular Enriques surfaces. Algebra Number Theory 6 (2012), no. 2, 195--230. doi:10.2140/ant.2012.6.195. https://projecteuclid.org/euclid.ant/1513729776


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References

  • W. Barth and C. Peters, “Automorphisms of Enriques surfaces”, Invent. Math. 73:3 (1983), 383–411.
  • W. P. Barth, K. Hulek, C. A. M. Peters, and A. Van de Ven, Compact complex surfaces, 2nd ed., Ergebnisse der Mathematik $(3)$ 4, Springer, Berlin, 2004.
  • A. Beauville, “On the Brauer group of Enriques surfaces”, Math. Res. Lett. 16:6 (2009), 927–934.
  • F. A. Bogomolov and Y. Tschinkel, “Density of rational points on Enriques surfaces”, Math. Res. Lett. 5:5 (1998), 623–628.
  • D. A. Cox, Primes of the form $x\sp 2 + ny\sp 2$: Fermat, class field theory and complex multiplication, Wiley, New York, 1989.
  • N. D. Elkies and M. Schütt, “K3 families of high Picard rank”, (2008).
  • N. D. Elkies and M. Schütt, “Modular forms and K3 surfaces”, 2008.
  • B. H. Gross, Arithmetic on elliptic curves with complex multiplication, Lecture Notes in Mathematics 776, Springer, Berlin, 1980.
  • K. Hulek and M. Schütt, “Enriques surfaces and Jacobian elliptic K3 surfaces”, Math. Z. 268:3-4 (2011), 1025–1056.
  • 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, 1977), edited by M. Nagata, Kinokuniya, Tokyo, 1978.
  • J. H. Keum, “Every algebraic Kummer surface is the K3-cover of an Enriques surface”, Nagoya Math. J. 118 (1990), 99–110.
  • S. Kondō, “Enriques surfaces with finite automorphism groups”, Japan. J. Math. $($N.S.$)$ 12:2 (1986), 191–282.
  • D. R. Morrison, “On K3 surfaces with large Picard number”, Invent. Math. 75:1 (1984), 105–121.
  • S. Mukai and Y. Namikawa, “Automorphisms of Enriques surfaces which act trivially on the cohomology groups”, Invent. Math. 77:3 (1984), 383–397.
  • V. Nikulin, “Integral symmetric bilinear forms and some of their applications”, Math. USSR, Izv. 14 (1980), 103–167.
  • K.-i. Nishiyama, “The Jacobian fibrations on some K3 surfaces and their Mordell–Weil groups”, Japan. J. Math. $($N.S.$)$ 22:2 (1996), 293–347.
  • H. Ohashi, “On the number of Enriques quotients of a K3 surface”, Publ. Res. Inst. Math. Sci. 43:1 (2007), 181–200.
  • I. 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.
  • R. Schertz, “Die singulären Werte der Weberschen Funktionen $\mathfrak f,\mathfrak f\sb{1},\mathfrak f\sb{2},$ $\gamma \sb{2},$ $\gamma \sb{3}$”, J. Reine Angew. Math. 286/287 (1976), 46–74.
  • M. Schütt, “Fields of definition of singular K3 surfaces”, Commun. Number Theory Phys. 1:2 (2007), 307–321.
  • M. Schütt, “Arithmetic of a singular K3 surface”, Michigan Math. J. 56:3 (2008), 513–527.
  • M. Schütt, “K3 surfaces with Picard rank 20”, Algebra Number Theory 4:3 (2010), 335–356.
  • M. Schütt and T. Shioda, “Elliptic surfaces”, pp. 51–160 in Algebraic geometry in East Asia (Seoul, 2008), edited by J. Keum et al., Adv. Stud. Pure Math. 60, Math. Soc. Japan, Tokyo, 2010.
  • A. S. Sert öz, “Which singular K3 surfaces cover an Enriques surface”, Proc. Amer. Math. Soc. 133:1 (2005), 43–50.
  • I. R. Shafarevich, “On the arithmetic of singular K3-surfaces”, pp. 103–108 in Algebra and analysis (Kazan, 1994), edited by M. M. Arslanov et al., de Gruyter, Berlin, 1996.
  • I. Shimada, “Transcendental lattices and supersingular reduction lattices of a singular K3 surface”, Trans. Amer. Math. Soc. 361 (2009), 909–949.
  • I. Shimada and D.-Q. Zhang, “Classification of extremal elliptic K3 surfaces and fundamental groups of open K3 surfaces”, Nagoya Math. J. 161 (2001), 23–54. http://www.ams.org/mathscinet-getitem?mr=2002d:14056MR 2002d:14056
  • G. Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan 11, Iwanami Shoten, Tokyo, 1971.
  • 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.
  • T. Shioda, “Correspondence of elliptic curves and Mordell–Weil lattices of certain elliptic K3's”, pp. 319–339 in Algebraic cycles and motives, vol. 2, edited by J. Nagel and C. Peters, London Math. Soc. Lecture Note Ser. 344, Cambridge Univ. Press, 2007.
  • T. Shioda and H. Inose, “On singular K3 surfaces”, pp. 119–136 in Complex analysis and algebraic geometry, edited by W. L. Baily, Jr. and T. Shioda, Iwanami Shoten, Tokyo, 1977.
  • T. Shioda and N. Mitani, “Singular abelian surfaces and binary quadratic forms”, pp. 259–287 in Classification of algebraic varieties and compact complex manifolds, edited by H. Popp, Lecture Notes in Mathematics 412, Springer, Berlin, 1974.
  • J. H. Silverman, Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics 151, Springer, New York, 1994.
  • H. Sterk, “Finiteness results for algebraic K3 surfaces”, Math. Z. 189:4 (1985), 507–513.
  • 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 (Antwerp, 1972), edited by B. J. Birch and W. Kuyk, Lecture Notes in Math. 476, Springer, Berlin, 1975.
  • P. J. Weinberger, “Exponents of the class groups of complex quadratic fields”, Acta Arith. 22 (1973), 117–124.