Skip to main content

Perfectoid Shimura varieties


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.

This is a preview of subscription content, access via your institution.


  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.

  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.

  3. 3.

    Bergeron N., Venkatesh A.: The asymptotic growth of torsion homology for arithmetic groups. J. Inst. Math. Jussieu 12, 391–447 (2013)

    MathSciNet  Article  MATH  Google 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.

  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.

  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.

  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.

  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.

  9. 9.

    R. Huber, Étale Cohomology of Rigid Analytic Varieties and Adic Spaces, Aspects Math., 30, Friedr. Vieweg & Sohn, Braunschweig, 1996.

  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)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Scholze P.: Perfectoid Spaces. Publ. Math. Inst. Hautes Études Sci. 116, 245–313 (2012)

    MathSciNet  Article  MATH  Google 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.

  14. 14.

    Scholze P., Weinstein J.: Moduli of p-divisible groups. Camb. J. Math. 1, 145–237 (2013)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Peter Scholze.

Additional information

This article is based on the 14th Takagi Lectures that the author delivered at the University of Tokyo on November 15 and 16, 2014.

Communicated by: Takeshi Saito

About this article

Verify currency and authenticity via CrossMark

Cite this article

Scholze, P. Perfectoid Shimura varieties. Jpn. J. Math. 11, 15–32 (2016).

Download citation

Keywords and phrases

  • Shimura varieties
  • Galois representations
  • perfectoid spaces

Mathematics Subject Classification (2010)

  • 14G35
  • 11F03
  • 11F80
  • 14G22