Japanese Journal of Mathematics

, Volume 11, Issue 1, pp 15–32 | Cite as

Perfectoid Shimura varieties

Takagi Lectures

Abstract

This note explains some of the author’s work on understanding the torsion appearing in the cohomology of locally symmetric spaces such as arithmetic hyperbolic 3-manifolds.

The key technical tool was a theory of Shimura varieties with infinite level at p: As p-adic analytic spaces, they are perfectoid, and admit a new kind of period map, called the Hodge–Tate period map, towards the flag variety. Moreover, the (semisimple) automorphic vector bundles come via pullback along the Hodge–Tate period map from the flag variety.

In the case of the Siegel moduli space, the situation is fully analyzed in [12]. We explain the conjectural picture for a general Shimura variety.

Mathematics Subject Classification (2010)

14G35 11F03 11F80 14G22 

Keywords and phrases

Shimura varieties Galois representations perfectoid spaces 

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References

  1. 1.
    A. Ash, Galois representations and cohomology of GL(n, Z), In: Séminaire de Théorie des Nombres, Paris, 1989–90, Progr. Math., 102, Birkhäuser Boston, Boston, MA, 1992, pp. 9–22.Google Scholar
  2. 2.
    W.L. Baily, Jr. and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2), 84 (1966), 442–528.Google Scholar
  3. 3.
    Bergeron N., Venkatesh A.: The asymptotic growth of torsion homology for arithmetic groups. J. Inst. Math. Jussieu 12, 391–447 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    F. Calegari and M. Emerton, Completed cohomology—a survey, In: Non-Abelian Fundamental Groups and Iwasawa Theory, London Math. Soc. Lecture Note Ser., 393, Cambridge Univ. Press, Cambridge, 2012, pp. 239–257.Google Scholar
  5. 5.
    P. Deligne, Travaux de Shimura, In: Séminaire Bourbaki, 23ème année (1970/71), Exp. No. 389, Lecture Notes in Math., 244, Springer-Verlag, 1971, pp. 123–165.Google Scholar
  6. 6.
    G. Faltings, Arithmetic varieties and rigidity, In: Seminar on Number Theory, Paris, 1982– 83, Progr. Math., 51, Birkhäuser Boston, Boston, MA, 1984, pp. 63–77.Google Scholar
  7. 7.
    G. Faltings, A relation between two moduli spaces studied by V.G. Drinfeld, In: Algebraic Number Theory and Algebraic Geometry, Contemp. Math., 300, Amer. Math. Soc., Providence, RI, 2002, pp. 115–129.Google Scholar
  8. 8.
    L. Fargues, A. Genestier and V. Lafforgue, L’isomorphisme entre les tours de Lubin–Tate et de Drinfeld, Progr. Math., 262, Birkhäuser Verlag, Basel, 2008.Google Scholar
  9. 9.
    R. Huber, Étale Cohomology of Rigid Analytic Varieties and Adic Spaces, Aspects Math., 30, Friedr. Vieweg & Sohn, Braunschweig, 1996.Google Scholar
  10. 10.
    Pfaff J.: Exponential growth of homological torsion for towers of congruence subgroups of Bianchi groups. Ann. Global Anal. Geom. 45, 267–285 (2014)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Scholze P.: Perfectoid Spaces. Publ. Math. Inst. Hautes Études Sci. 116, 245–313 (2012)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    P. Scholze, On torsion in the cohomology of locally symmetric varieties, preprint, arXiv:1306.2070.
  13. 13.
    P. Scholze, Perfectoid spaces: a survey, In: Current Developments in Mathematics 2012, Int. Press, Somerville, MA, 2013, pp. 193–227.Google Scholar
  14. 14.
    Scholze P., Weinstein J.: Moduli of p-divisible groups. Camb. J. Math. 1, 145–237 (2013)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Mathematical Society of Japan and Springer Japan 2016

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität BonnBonnDeutschland

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