Journal of the Mathematical Society of Japan

Elliptic fibrations on K3 surfaces and Salem numbers of maximal degree

Xun YU

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Abstract

We study the maximal Salem degree of automorphisms of K3 surfaces via elliptic fibrations. In particular, we establish a characterization of such maximum in terms of elliptic fibrations with infinite automorphism groups. As an application, we show that any supersingular K3 surface in odd characteristic has an automorphism the entropy of which is the natural logarithm of a Salem number of degree 22.

Article information

Source
J. Math. Soc. Japan, Volume 70, Number 3 (2018), 1151-1163.

Dates
Received: 24 August 2016
Revised: 20 January 2017
First available in Project Euclid: 25 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1529892024

Digital Object Identifier
doi:10.2969/jmsj/75907590

Mathematical Reviews number (MathSciNet)
MR3830803

Zentralblatt MATH identifier
06966978

Subjects
Primary: 14J28: $K3$ surfaces and Enriques surfaces 14J50: Automorphisms of surfaces and higher-dimensional varieties
Secondary: 14D06: Fibrations, degenerations 14G99: None of the above, but in this section

Keywords
K3 surfaces automorphisms elliptic fibrations Salem numbers

Citation

YU, Xun. Elliptic fibrations on K3 surfaces and Salem numbers of maximal degree. J. Math. Soc. Japan 70 (2018), no. 3, 1151--1163. doi:10.2969/jmsj/75907590. https://projecteuclid.org/euclid.jmsj/1529892024


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References

  • M. Artin, Supersingular K3 surfaces, Ann. Sci. École Norm. Sup. (4), 7 (1974), 543–567.
  • J. Blanc and S. Cantat, Dynamical degrees of birational transformations of projective surfaces, J. Amer. Math. Soc., 29 (2016), 415–471.
  • S. Brandhorst, Dynamics on supersingular K3 surfaces and automorphisms of Salem degree 22, http://arxiv.org/pdf/1507.02092v1.pdf.
  • S. Brandhorst, Automorphisms of Salem degree 22 on supersingular K3 surfaces of higher Artin invariant–-a short note, https://arxiv.org/pdf/1609.02348v1.pdf.
  • S. Brandhorst and V. González-Alonso, Automorphisms of minimal entropy on supersingular K3 surfaces, https://arxiv.org/pdf/1609.02716v1.pdf.
  • H. Esnault and V. Srinivas, Algebraic versus topological entropy for surfaces over finite fields, Osaka J. Math., 50 (2013), 827–846.
  • H. Esnault and K. Oguiso, Non-liftability of automorphism groups of a K3 surface in positive characteristic, Math. Ann., 363 (2015), 1187–1206.
  • H. Esnault, K. Oguiso and X. Yu, Automorphisms of elliptic K3 surfaces and Salem numbers of maximal degree, Algebraic Geometry, 3 (2016), 496–507.
  • A. Kumar, Elliptic fibrations on a generic Jacobian Kummer surface, J. Algebraic Geometry, 23 (2014), 599–667.
  • M. Lieblich and D. Maulik, A note on the cone conjecture for K3 surfaces in positive characteristic, http://arxiv.org/pdf/1102.3377v3.pdf.
  • C. Liedtke, Supersingular K3 surfaces are unirational, Inventiones mathematicae, 200 (2015), 979–1014.
  • C. T. McMullen, Dynamics on K3 surfaces: Salem numbers and Siegel disks, J. Reine Angew. Math., 545 (2002), 201–233.
  • V. V. Nikulin, Factor groups of groups of automorphisms of hyperbolic forms with respect to subgroups generated by 2-reflections, Algebrogeometric applications, J. Math. Sci., 22 (1983), 1401–1475.
  • V. V. Nikulin, K3 surfaces with interesting groups of automorphisms, J. Math. Sci., 95 (1999), 2028–2048.
  • V. V. Nikulin, Elliptic fibrations on K3 surfaces, Proc. Edinb. Math. Soc. (2), 57 (2014), 253–267.
  • K. Oguiso, On Jacobian fibrations on the Kummer surfaces of the product of non-isogenous elliptic curves, J. Math. Soc. Japan, 41 (1989), 651–680.
  • K. Oguiso, Automorphisms of hyperkähler manifolds in the view of topological entropy, Algebraic geometry, Contemp. Math., 422, Amer. Math. Soc., Providence, RI, 2007, 173–185.
  • K. Oguiso, Mordell–Weil groups of a hyperkähler manifold–-a question of F. Campana, Publ. RIMS, 44 (2009), 495–506.
  • K. Oguiso, Salem polynomials and the bimeromorphic automorphism group of a hyper-Kähler manifold, Selected papers on analysis and differential equations, Amer. Math. Soc. Transl. Ser., 230 (2010), 201–227.
  • K. Oguiso, Pisot units, Salem numbers and higher dimensional projective manifolds with primitive automorphisms of positive entropy, https://arxiv.org/pdf/1608.03122v3.pdf.
  • A. Ogus, Supersingular K3 crystals, Journées de Géométrie Algébrique de Rennes, Astérisque, 64 (1979), 3–86.
  • A. Ogus, A crystalline Torelli theorem for supersingular K3 surfaces, Progr. Math., 36 (1983), 361–394.
  • I. I. Piatetsky-Shapiro and I. R. Shafarevich, A Torelli theorem for algebraic surfaces of type K3, Math. USSR-Izv., 5 (1971), 547–588.
  • A. N. Rudakov and I. R. Shafarevich, Supersingular K3 surfaces over fields of characteristic 2, Izv. Akad. Nauk SSSR, 42 (1978), 848–869, Math. USSR-Izv., 13 (1979), 147–165.
  • M. Schütt, Dynamics on supersingular K3 surfaces, http://arxiv.org/pdf/1502.06923v2.pdf.
  • I. Shimada, Automorphisms of supersingular K3 surfaces and Salem polynomials, Exp. Math., 25 (2016), 389–398.
  • T. Shioda, Algebraic cycles on certain K3 surfaces in characteristic $p$, Manifolds-Tokyo 1973, Proc. Internat. Conf., Tokyo, 1973, Univ. Tokyo Press, Tokyo, 1975, 357–364.
  • T. Shioda, On the Mordell–Weil lattices, Comm. Math. Univ. St. Paul, 39 (1990), 211–240.