\newcommand{\etalchar}[1]{$^{#1}$}
\begin{thebibliography}{KMPS10}

\bibitem[Art74]{Art}
M.~Artin.
\newblock Supersingular {$K3$} surfaces.
\newblock {\em Ann. Sci. \'Ecole Norm. Sup. (4)}, 7:543--567 (1975), 1974.

\bibitem[ASD73]{ASD}
M.~Artin and H.~P.~F. Swinnerton-Dyer.
\newblock The {S}hafarevich-{T}ate conjecture for pencils of elliptic curves on
  {$K3$} surfaces.
\newblock {\em Invent. Math.}, 20:249--266, 1973.

\bibitem[Ati57]{At}
M.~F. Atiyah.
\newblock Complex analytic connections in fibre bundles.
\newblock {\em Trans. Amer. Math. Soc.}, 85:181--207, 1957.

\bibitem[B{\etalchar{+}}85]{BeK3}
A.~Beauville et~al.
\newblock {\em G\'eom\'etrie des surfaces {$K3$}: modules et p\'eriodes}.
\newblock Soci\'et\'e Math\'ematique de France, Paris, 1985.
\newblock Papers from the seminar held in Palaiseau, October 1981--January
  1982, Ast\'erisque No. 126 (1985).

\bibitem[Bea78]{BeaBook}
Arnaud Beauville.
\newblock {\em Surfaces alg\'ebriques complexes}.
\newblock Soci\'et\'e Math\'ematique de France, Paris, 1978.
\newblock Avec une sommaire en anglais, Ast{\'e}risque, No. 54.

\bibitem[Bea99]{Bea}
Arnaud Beauville.
\newblock Counting rational curves on {$K3$} surfaces.
\newblock {\em Duke Math. J.}, 97(1):99--108, 1999.

\bibitem[BHT09]{BHT}
F.~A. Bogomolov, B.~Hassett, and Yu. Tschinkel.
\newblock Constructing rational curves on {$K3$} surfaces, 2009.
\newblock to appear in the {\em Duke Mathematical Journal}.

\bibitem[BL00]{BL}
Jim Bryan and Naichung~Conan Leung.
\newblock The enumerative geometry of {$K3$} surfaces and modular forms.
\newblock {\em J. Amer. Math. Soc.}, 13(2):371--410 (electronic), 2000.

\bibitem[Blo72]{Bloch}
Spencer Bloch.
\newblock Semi-regularity and de{R}ham cohomology.
\newblock {\em Invent. Math.}, 17:51--66, 1972.

\bibitem[BT00]{bt}
F.~A. Bogomolov and Yu. Tschinkel.
\newblock Density of rational points on elliptic {$K3$} surfaces.
\newblock {\em Asian J. Math.}, 4(2):351--368, 2000.

\bibitem[BT05a]{bt-IMRN}
Fedor Bogomolov and Yuri Tschinkel.
\newblock Curves in abelian varieties over finite fields.
\newblock {\em Int. Math. Res. Not.}, (4):233--238, 2005.

\bibitem[BT05b]{bt-K3}
Fedor Bogomolov and Yuri Tschinkel.
\newblock Rational curves and points on {$K3$} surfaces.
\newblock {\em Amer. J. Math.}, 127(4):825--835, 2005.

\bibitem[BZ09]{BZ}
F.~Bogomolov and Yu. Zarhin.
\newblock Ordinary reduction of {K}3 surfaces.
\newblock {\em Cent. Eur. J. Math.}, 7(2):206--213, 2009.

\bibitem[Che99]{XiChen}
Xi~Chen.
\newblock Rational curves on {$K3$} surfaces.
\newblock {\em J. Algebraic Geom.}, 8(2):245--278, 1999.

\bibitem[Che02]{XiChen2}
Xi~Chen.
\newblock A simple proof that rational curves on {$K3$} are nodal.
\newblock {\em Math. Ann.}, 324(1):71--104, 2002.

\bibitem[CL10]{ChenLewis}
Xi~Chen and James~D. Lewis.
\newblock Density of rational curves on {$K3$} surfaces, 2010.
\newblock arXiv:1004.5167v2.

\bibitem[Del72]{DeK3}
Pierre Deligne.
\newblock La conjecture de {W}eil pour les surfaces {$K3$}.
\newblock {\em Invent. Math.}, 15:206--226, 1972.

\bibitem[Del81]{DeLift}
P.~Deligne.
\newblock Rel\`evement des surfaces {$K3$} en caract\'eristique nulle.
\newblock In {\em Algebraic surfaces (Orsay, 1976--78)}, volume 868 of {\em
  Lecture Notes in Math.}, pages 58--79. Springer, Berlin, 1981.
\newblock Prepared for publication by Luc Illusie.

\bibitem[EJ08a]{EJ1}
Andreas-Stephan Elsenhans and J{\"o}rg Jahnel.
\newblock {$K3$} surfaces of {P}icard rank one and degree two.
\newblock In {\em Algorithmic number theory}, volume 5011 of {\em Lecture Notes
  in Comput. Sci.}, pages 212--225. Springer, Berlin, 2008.

\bibitem[EJ08b]{EJ2}
Andreas-Stephan Elsenhans and J{\"o}rg Jahnel.
\newblock {$K3$} surfaces of {P}icard rank one which are double covers of the
  projective plane.
\newblock In {\em Higher-dimensional geometry over finite fields}, volume~16 of
  {\em NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur.}, pages 63--77. IOS,
  Amsterdam, 2008.

\bibitem[EJ09a]{EJ3}
Andreas-Stephan Elsenhans and J{\"o}rg Jahnel.
\newblock On the computation of the {P}icard group for {$K3$} surfaces, 2009.
\newblock arXiv:1006.1724.

\bibitem[EJ09b]{EJ4}
Andreas-Stephan Elsenhans and J{\"o}rg Jahnel.
\newblock The {P}icard group of a {$K3$} surface and its reduction modulo $p$,
  2009.
\newblock arXiv:1006.1972.

\bibitem[Ell04]{El}
Jordan~S. Ellenberg.
\newblock {$K3$} surfaces over number fields with geometric {P}icard number
  one.
\newblock In {\em Arithmetic of higher-dimensional algebraic varieties (Palo
  Alto, CA, 2002)}, volume 226 of {\em Progr. Math.}, pages 135--140.
  Birkh\"auser Boston, Boston, MA, 2004.

\bibitem[GH78]{GH}
Phillip Griffiths and Joseph Harris.
\newblock {\em Principles of algebraic geometry}.
\newblock Wiley-Interscience [John Wiley \& Sons], New York, 1978.
\newblock Pure and Applied Mathematics.

\bibitem[JR01]{JR}
K.~Joshi and C.S. Rajan.
\newblock Frobenius splitting and ordinarity, 2001.
\newblock arXiv:math/0110070v1 (but not in the published version).

\bibitem[KMPS10]{KMPS}
A.~Klemm, D.~Maulik, R.~Pandharipande, and E.~Scheidegger.
\newblock {N}oether-{L}efschetz theory and the {Y}au-{Z}aslow conjecture.
\newblock {\em J. Amer. Math. Soc.}, 23:1013--1040, 2010.
\newblock arXiv:0807.2477v2.

\bibitem[KS59]{KS}
K.~Kodaira and D.~C. Spencer.
\newblock A theorem of completeness of characteristic systems of complete
  continuous systems.
\newblock {\em Amer. J. Math.}, 81:477--500, 1959.

\bibitem[KS67]{KugaSatake}
Michio Kuga and Ichir{\^o} Satake.
\newblock Abelian varieties attached to polarized {$K\sb{3}$}-surfaces.
\newblock {\em Math. Ann.}, 169:239--242, 1967.

\bibitem[LL05]{LL}
Junho Lee and Naichung~Conan Leung.
\newblock Yau-{Z}aslow formula on {$K3$} surfaces for non-primitive classes.
\newblock {\em Geom. Topol.}, 9:1977--2012 (electronic), 2005.

\bibitem[LN80]{LN}
William~E. Lang and Niels~O. Nygaard.
\newblock A short proof of the {R}udakov-\v {S}afarevi\v c theorem.
\newblock {\em Math. Ann.}, 251(2):171--173, 1980.

\bibitem[LP81]{LP}
Eduard Looijenga and Chris Peters.
\newblock Torelli theorems for {K}\"ahler {$K3$} surfaces.
\newblock {\em Compositio Math.}, 42(2):145--186, 1980/81.

\bibitem[LW06]{LW}
Jun Li and Baosen Wu.
\newblock Note on a conjecture of {G}opakumar-{V}afa.
\newblock {\em Chinese Ann. Math. Ser. B}, 27(2):219--242, 2006.

\bibitem[MM83]{mori-mukai}
Shigefumi Mori and Shigeru Mukai.
\newblock The uniruledness of the moduli space of curves of genus {$11$}.
\newblock In {\em Algebraic geometry ({T}okyo/{K}yoto, 1982)}, volume 1016 of
  {\em Lecture Notes in Math.}, pages 334--353. Springer, Berlin, 1983.

\bibitem[NO85]{NO}
Niels Nygaard and Arthur Ogus.
\newblock Tate's conjecture for {$K3$} surfaces of finite height.
\newblock {\em Ann. of Math. (2)}, 122(3):461--507, 1985.

\bibitem[Nyg79]{Ny}
Niels~O. Nygaard.
\newblock A {$p$}-adic proof of the nonexistence of vector fields on {$K3$}\
  surfaces.
\newblock {\em Ann. of Math. (2)}, 110(3):515--528, 1979.

\bibitem[Ogu79]{ogus}
Arthur Ogus.
\newblock Supersingular {$K3$} crystals.
\newblock In {\em Journ\'ees de G\'eom\'etrie Alg\'ebrique de Rennes (Rennes,
  1978), Vol. II}, volume~64 of {\em Ast\'erisque}, pages 3--86. Soc. Math.
  France, Paris, 1979.

\bibitem[P{\v{S}}{\v{S}}71]{SPS}
I.~I. Pjatecki{\u\i}-{\v{S}}apiro and I.~R. {\v{S}}afarevi{\v{c}}.
\newblock Torelli's theorem for algebraic surfaces of type {${\rm K}3$}.
\newblock {\em Izv. Akad. Nauk SSSR Ser. Mat.}, 35:530--572, 1971.

\bibitem[Ran95]{Ran}
Ziv Ran.
\newblock Hodge theory and deformations of maps.
\newblock {\em Compositio Math.}, 97(3):309--328, 1995.

\bibitem[R{\v{S}}78]{RS}
A.~N. Rudakov and I.~R. {\v{S}}afarevi{\v{c}}.
\newblock Vector fields on elliptic surfaces.
\newblock {\em Uspekhi Mat. Nauk}, 33(6(204)):231--232, 1978.

\bibitem[RS81]{RS2}
A.~N. Rudakov and I.~R. Shafarevich.
\newblock Surfaces of type {$K3$} over fields of finite characteristic.
\newblock In {\em Current problems in mathematics, {V}ol. 18}, pages 115--207.
  Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1981.
\newblock translation in Journal of Mathematical Sciences, Volume 22, Number 4,
  July, 1983, pages 1476--1533.

\bibitem[Sch68]{Sch}
Michael Schlessinger.
\newblock Functors of {A}rtin rings.
\newblock {\em Trans. Amer. Math. Soc.}, 130:208--222, 1968.

\bibitem[SD74]{SDK3}
B.~Saint-Donat.
\newblock Projective models of {$K-3$} surfaces.
\newblock {\em Amer. J. Math.}, 96:602--639, 1974.

\bibitem[Shi74]{Sh}
Tetsuji Shioda.
\newblock An example of unirational surfaces in characteristic {$p$}.
\newblock {\em Math. Ann.}, 211:233--236, 1974.

\bibitem[Shi77]{Shi77}
Tetsuji Shioda.
\newblock On unirationality of supersingular surfaces.
\newblock {\em Math. Ann.}, 225(2):155--159, 1977.

\bibitem[Shi04]{Shimada}
Ichiro Shimada.
\newblock Supersingular {$K3$} surfaces in characteristic 2 as double covers of
  a projective plane.
\newblock {\em Asian J. Math.}, 8(3):531--586, 2004.

\bibitem[Siu83]{Siu}
Y.~T. Siu.
\newblock Every {$K3$} surface is {K}\"ahler.
\newblock {\em Invent. Math.}, 73(1):139--150, 1983.

\bibitem[Tat65]{Tate}
John~T. Tate.
\newblock Algebraic cycles and poles of zeta functions.
\newblock In {\em Arithmetical {A}lgebraic {G}eometry ({P}roc. {C}onf. {P}urdue
  {U}niv., 1963)}, pages 93--110. Harper \& Row, New York, 1965.

\bibitem[Ter85]{Te}
Tomohide Terasoma.
\newblock Complete intersections with middle {P}icard number {$1$} defined over
  {${\bf Q}$}.
\newblock {\em Math. Z.}, 189(2):289--296, 1985.

\bibitem[vG00]{vG}
Bert van Geemen.
\newblock Kuga-{S}atake varieties and the {H}odge conjecture.
\newblock In {\em The arithmetic and geometry of algebraic cycles (Banff, AB,
  1998)}, volume 548 of {\em NATO Sci. Ser. C Math. Phys. Sci.}, pages 51--82.
  Kluwer Acad. Publ., Dordrecht, 2000.

\bibitem[vL07]{vL}
Ronald van Luijk.
\newblock K3 surfaces with {P}icard number one and infinitely many rational
  points, 2007.

\bibitem[Voi92]{Voisin}
Claire Voisin.
\newblock Sur la stabilit\'e des sous-vari\'et\'es lagrangiennes des
  vari\'et\'es symplectiques holomorphes.
\newblock In {\em Complex projective geometry ({T}rieste, 1989/{B}ergen,
  1989)}, volume 179 of {\em London Math. Soc. Lecture Note Ser.}, pages
  294--303. Cambridge Univ. Press, Cambridge, 1992.

\bibitem[Voi96]{Voi}
Claire Voisin.
\newblock Remarks on zero-cycles of self-products of varieties.
\newblock In {\em Moduli of vector bundles (Sanda, 1994; Kyoto, 1994)}, volume
  179 of {\em Lecture Notes in Pure and Appl. Math.}, pages 265--285. Dekker,
  New York, 1996.

\bibitem[Wu07]{Wu}
Baosen Wu.
\newblock The number of rational curves on {$K3$} surfaces.
\newblock {\em Asian J. Math.}, 11(4):635--650, 2007.

\bibitem[YZ96]{YZ}
Shing-Tung Yau and Eric Zaslow.
\newblock B{PS} states, string duality, and nodal curves on {$K3$}.
\newblock {\em Nuclear Phys. B}, 471(3):503--512, 1996.

\bibitem[Zar93]{Za}
Yuri~G. Zarhin.
\newblock Transcendental cycles on ordinary {$K3$} surfaces over finite fields.
\newblock {\em Duke Math. J.}, 72(1):65--83, 1993.

\end{thebibliography}