Skip to main content
Log in

Fake projective planes

  • Published:
Inventiones mathematicae Aims and scope

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Bombieri, E.: Canonical models of surfaces of general type. Publ. Math., Inst. Hautes Étud. Sci. 42, 171–220 (1972)

    MATH  Google Scholar 

  2. Borel, A., Prasad, G.: Finiteness theorems for discrete subgroups of bounded covolume in semisimple groups. Publ. Math., Inst. Hautes Étud. Sci. 69, 119–171 (1989)

    MATH  MathSciNet  Google Scholar 

  3. Borevich, Z.I., Shafarevich, I.R.: Number theory. Academic Press, New York (1966)

    MATH  Google Scholar 

  4. Friedman, E.: Analytic formulas for the regulator of a number field. Invent. Math. 98, 599–622 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  5. Holzapfel, R.-P.: Ball and surface arithmetics. Vieweg & Sohn, Wiesbaden (1998)

    MATH  Google Scholar 

  6. Ishida, M.-N., Kato, F.: The strong rigidity theorem for non-archimedean uniformization. Tohoku Math. J. 50, 537–555 (1998)

    MATH  MathSciNet  Google Scholar 

  7. Klingler, B.: Sur la rigidité de certains groupes fonndamentaux, l’arithméticité des réseaux hyperboliques complexes, et les ‘faux plans projectifs’. Invent. Math. 153, 105–143 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Kollár, J.: Shafarevich maps and automorphic forms. Princeton University Press, Princeton (1995)

    MATH  Google Scholar 

  9. Martinet, J.: Petits discriminants des corps de nombres. Number theory days, 1980 (Exeter, 1980), Lond. Math. Soc. Lect. Note Ser., vol. 56, pp. 151–193. Cambridge Univ. Press, Cambridge-New York, (1982)

  10. Mostow, G.D.: Strong rigidity of locally symmetric spaces. In: Ann. Math. Stud., vol. 78. Princeton University Press, Princeton (1973)

    MATH  Google Scholar 

  11. Mumford, D.: An algebraic surface with K ample, K 2=9, p g =q=0. Am. J. Math. 101, 233–244 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  12. Narkiewicz, W.: Elementary and analytic theory of algebraic numbers, 3rd edn. Springer, New York (2000)

    Google Scholar 

  13. Odlyzko, A.M.: Some analytic estimates of class numbers and discriminants. Invent. Math. 29, 275–286 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  14. Odlyzko, A.M.: Discriminant bounds. http://www.dtc.umn.edu/∼odlyzko/unpublished/index.html.

  15. Platonov, V.P., Rapinchuk, A.S.: Algebraic groups and Number theory. Academic Press, New York (1994)

    MATH  Google Scholar 

  16. Prasad, G.: Volumes of S-arithmetic quotients of semi-simple groups. Publ. Math., Inst. Hautes Étud. Sci. 69, 91–117 (1989)

    MATH  Google Scholar 

  17. Prasad, G., Rapinchuk, A.S.: Computation of the metaplectic kernel. Publ. Math., Inst. Hautes Étud. Sci. 84, 91–187 (1996)

    MATH  MathSciNet  Google Scholar 

  18. Prasad, G., Yu, J.-K.: On finite group actions on reductive groups and buildings. Invent. Math. 147, 545–560 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. Reider, I.: Vector bundles of rank 2 and linear systems on algebraic surfaces. Ann. Math. 127, 309–316 (1988)

    Article  MathSciNet  Google Scholar 

  20. Riehm, C.: The norm 1 group of \(\mathfrak{p}\)-adic division algebra. Am. J. Math. 92, 499–523 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  21. Rogawski, J.D.: Automorphic representations of unitary groups in three variables. In: Ann. Math. Stud., vol. 123. Princeton University Press, Princeton (1990)

    MATH  Google Scholar 

  22. Scharlau, W.: Quadratic and hermitian forms. Springer, New York (1985)

    MATH  Google Scholar 

  23. Serre, J.-P.: Cohomologie des groupes discrets. In: Ann. Math. Stud., vol. 70. Princeton University Press, Princeton (1971)

    Google Scholar 

  24. Serre, J.-P.: A course in arithmetic. Springer, New York (1973)

    MATH  Google Scholar 

  25. Serre, J.-P.: Galois cohomology. Springer, New York (1997)

    MATH  Google Scholar 

  26. Siegel, C.L.: Berechnung von Zetafunktionen an ganzzahligen Stellen. Nachr. Akad. Wiss. Gött., 87–102 (1969)

  27. Slavutskii, I.Sh.: On the Zimmert estimate for the regulator of an algebraic field. English translation of Mat. Zametki in Math. Notes 51, 531–532 (1992)

    Google Scholar 

  28. Tits, J.: Classification of algebraic semisimple groups. Algebraic Groups and Discontinuous Subgroups. Proc. Am. Math. Soc. Symp. Pure Math. 9, 33–62 (1966)

  29. Tits, J.: Reductive groups over local fields. Proc. Am. Math. Soc. Symp. Pure Math. 33(1), 29–69 (1979)

    MathSciNet  Google Scholar 

  30. Tsuyumine, S.: On values of L-functions of totally real algebraic number fields at integers. Acta Arith. 76(4), 359–392 (1996)

    MATH  MathSciNet  Google Scholar 

  31. Washington, L.C.: Introduction to cyclotomic fields, 2nd edn. In: Grad. Texts Math., vol. 83. Springer, New York, (1997)

    MATH  Google Scholar 

  32. Yeung, S.-K.: Integrality and arithmeticity of co-compact lattices corresponding to certain complex two-ball quotients of Picard number one. Asian J. Math. 8, 107–130 (2004)

    MATH  MathSciNet  Google Scholar 

  33. Zimmert, R.: Ideale kleiner Norm in Idealklassen und eine Regulatorabschätzung. Invent. Math. 62, 367–380 (1981)

    Article  MathSciNet  Google Scholar 

  34. The Bordeaux Database, Tables obtainable from: ftp://megrez.math.u-bordeaux.fr/pub/numberfields/

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Gopal Prasad.

Additional information

Dedicated to David Mumford

Rights and permissions

Reprints and permissions

About this article

Cite this article

Prasad, G., Yeung, SK. Fake projective planes. Invent. math. 168, 321–370 (2007). https://doi.org/10.1007/s00222-007-0034-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-007-0034-5

Keywords

Navigation