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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Transcendental lattices and supersingular reduction lattices of a singular $ K3$ surface


Author: Ichiro Shimada
Journal: Trans. Amer. Math. Soc. 361 (2009), 909-949
MSC (2000): Primary 14J28; Secondary 14J20, 14H52
Published electronically: July 30, 2008
MathSciNet review: 2452829
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Abstract: A $ K3$ surface $ X$ defined over a field $ k$ of characteristic 0 is called singular if the Néron-Severi lattice $ \mathrm{NS}(X)$ of $ X\otimes \overline{k}$ is of rank $ 20$. Let $ X$ be a singular $ K3$ surface defined over a number field $ F$. For each embedding $ \sigma: F\hookrightarrow \mathbb{C}$, we denote by $ T(X^\sigma)$ the transcendental lattice of the complex $ K3$ surface $ X^\sigma$ obtained from $ X$ by $ \sigma$. For each prime $ \mathfrak{p}$ of $ F$ at which $ X$ has a supersingular reduction $ X_{\mathfrak{p}}$, we define $ L(X, \mathfrak{p})$ to be the orthogonal complement of $ \mathrm{NS}(X)$ in $ \mathrm{NS}(X_{\mathfrak{p}})$. We investigate the relation between these lattices $ T(X\sp\sigma)$ and $ L(X,\mathfrak{p})$. As an application, we give a lower bound for the degree of a number field over which a singular $ K3$ surface with a given transcendental lattice can be defined.


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  • 1. M. Artin, Supersingular 𝐾3 surfaces, Ann. Sci. École Norm. Sup. (4) 7 (1974), 543–567 (1975). MR 0371899 (51 #8116)
  • 2. Théorie des intersections et théorème de Riemann-Roch, Lecture Notes in Mathematics, Vol. 225, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6); Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre. MR 0354655 (50 #7133)
  • 3. N. Bourbaki, Éléments de mathématique. 23. Première partie: Les structures fondamentales de l’analyse. Livre II: Algèbre. Chapitre 8: Modules et anneaux semi-simples, Actualités Sci. Ind. no. 1261, Hermann, Paris, 1958 (French). MR 0098114 (20 #4576)
  • 4. J. W. S. Cassels, Rational quadratic forms, London Mathematical Society Monographs, vol. 13, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1978. MR 522835 (80m:10019)
  • 5. Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR 1228206 (94i:11105)
  • 6. François R. Cossec and Igor V. Dolgachev, Enriques surfaces. I, Progress in Mathematics, vol. 76, Birkhäuser Boston, Inc., Boston, MA, 1989. MR 986969 (90h:14052)
  • 7. David A. Cox, Primes of the form 𝑥²+𝑛𝑦², A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1989. Fermat, class field theory and complex multiplication. MR 1028322 (90m:11016)
  • 8. Max Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197–272 (German). MR 0005125 (3,104f)
  • 9. David R. Dorman, Global orders in definite quaternion algebras as endomorphism rings for reduced CM elliptic curves, Théorie des nombres (Quebec, PQ, 1987) de Gruyter, Berlin, 1989, pp. 108–116. MR 1024555 (90j:11043)
  • 10. Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, and Angelo Vistoli, Fundamental algebraic geometry, Mathematical Surveys and Monographs, vol. 123, American Mathematical Society, Providence, RI, 2005. Grothendieck’s FGA explained. MR 2222646 (2007f:14001)
  • 11. William Fulton, Rational equivalence on singular varieties, Inst. Hautes Études Sci. Publ. Math. 45 (1975), 147–167. MR 0404257 (53 #8060)
  • 12. William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323 (99d:14003)
  • 13. Benedict H. Gross and Don B. Zagier, On singular moduli, J. Reine Angew. Math. 355 (1985), 191–220. MR 772491 (86j:11041)
  • 14. Alexander Grothendieck, Fondements de la géométrie algébrique. [Extraits du Séminaire Bourbaki, 1957–1962.], Secrétariat mathématique, Paris, 1962 (French). MR 0146040 (26 #3566)
  • 15. Alexander Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 4, Société Mathématique de France, Paris, 2005 (French). Séminaire de Géométrie Algébrique du Bois Marie, 1962; Augmenté d’un exposé de Michèle Raynaud. [With an exposé by Michèle Raynaud]; With a preface and edited by Yves Laszlo; Revised reprint of the 1968 French original. MR 2171939 (2006f:14004)
  • 16. Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157 (57 #3116)
  • 17. Hiroshi Inose, Defining equations of singular 𝐾3 surfaces and a notion of isogeny, Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977) Kinokuniya Book Store, Tokyo, 1978, pp. 495–502. MR 578868 (81h:14021)
  • 18. Serge Lang, Elliptic functions, 2nd ed., Graduate Texts in Mathematics, vol. 112, Springer-Verlag, New York, 1987. With an appendix by J. Tate. MR 890960 (88c:11028)
  • 19. David Mumford, Lectures on curves on an algebraic surface, With a section by G. M. Bergman. Annals of Mathematics Studies, No. 59, Princeton University Press, Princeton, N.J., 1966. MR 0209285 (35 #187)
  • 20. V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 1, 111–177, 238 (Russian). MR 525944 (80j:10031)
  • 21. I. I. Pjateckiĭ-Šapiro and I. R. Šafarevič.
    Torelli's theorem for algebraic surfaces of type $ K3$.
    Izv. Akad. Nauk SSSR Ser. Mat., 35:530-572, 1971.
    Reprinted in I. R. Shafarevich, Collected Mathematical Papers, Springer-Verlag, Berlin, 1989, pp. 516-557.
  • 22. I. Reiner, Maximal orders, London Mathematical Society Monographs. New Series, vol. 28, The Clarendon Press, Oxford University Press, Oxford, 2003. Corrected reprint of the 1975 original; With a foreword by M. J.\ Taylor. MR 1972204 (2004c:16026)
  • 23. A. N. Rudakov and I. R. Shafarevich, Surfaces of type 𝐾3 over fields of finite characteristic, Current problems in mathematics, Vol. 18, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1981, pp. 115–207 (Russian). MR 633161 (83c:14027)
  • 24. Matthias Schütt, Fields of definition of singular 𝐾3 surfaces, Commun. Number Theory Phys. 1 (2007), no. 2, 307–321. MR 2346573 (2008g:14060), http://dx.doi.org/10.4310/CNTP.2007.v1.n2.a2
  • 25. I. R. Shafarevich, On the arithmetic of singular 𝐾3-surfaces, Algebra and analysis (Kazan, 1994) de Gruyter, Berlin, 1996, pp. 103–108. MR 1465448 (98h:14041)
  • 26. Ichiro Shimada, On normal 𝐾3 surfaces, Michigan Math. J. 55 (2007), no. 2, 395–416. MR 2369942 (2008i:14056), http://dx.doi.org/10.1307/mmj/1187647000
  • 27. I. Shimada.
    On arithmetic Zariski pairs in degree $ 6$. Preprint, 2006. To appear in Adv. Geom.
    http://arxiv.org/abs/math.AG/0611596.
  • 28. I. Shimada.
    Non-homeomorphic conjugate complex varieties.
    Preprint, 2007. http://arxiv.org/abs/math.AG/0701115.
  • 29. I. Shimada and De-Qi Zhang.
    Dynkin diagrams of rank $ 20$ on supersingular $ K3$ surfaces.
    Preprint, 2005. http://www.math.sci.hokudai.ac.jp/~shimada/preprints.html.
  • 30. T. Shioda and H. Inose, On singular 𝐾3 surfaces, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 119–136. MR 0441982 (56 #371)
  • 31. T. Shioda.
    Correspondence of elliptic curves and Mordell-Weil lattices of certain $ K3$ surfaces.
    Preprint.
  • 32. Tetsuji Shioda, The elliptic 𝐾3 surfaces with with a maximal singular fibre, C. R. Math. Acad. Sci. Paris 337 (2003), no. 7, 461–466 (English, with English and French summaries). MR 2023754 (2004j:14046), http://dx.doi.org/10.1016/j.crma.2003.07.007
  • 33. Tetsuji Shioda and Naoki Mitani, Singular abelian surfaces and binary quadratic forms, Classification of algebraic varieties and compact complex manifolds, Springer, Berlin, 1974, pp. 259–287. Lecture Notes in Math., Vol. 412. MR 0382289 (52 #3174)
  • 34. Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210 (87g:11070)
  • 35. Joseph H. Silverman, Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. MR 1312368 (96b:11074)
  • 36. John Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1966), 134–144. MR 0206004 (34 #5829)
  • 37. André Weil, Variétés abéliennes et courbes algébriques, Actualités Sci. Ind., no. 1064 = Publ. Inst. Math. Univ. Strasbourg 8 (1946), Hermann & Cie., Paris, 1948 (French). MR 0029522 (10,621d)

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Additional Information

Ichiro Shimada
Affiliation: Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo 060-0810, Japan
Address at time of publication: Department of Mathematics, Graduate School of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, 739-8526 Japan
Email: shimada@math.sci.hokudai.ac.jp, shimada@math.sci.hiroshima-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04560-1
PII: S 0002-9947(08)04560-1
Received by editor(s): November 8, 2006
Received by editor(s) in revised form: April 16, 2007
Published electronically: July 30, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.