Interpolating Hodge–Tate and de Rham periods

  • Shrenik Shah


We study the interpolation of Hodge–Tate and de Rham periods over rigid analytic families of Galois representations. Given a Galois representation on a coherent locally free sheaf over a reduced rigid space and a bounded range of weights, we obtain a stratification of this space by locally closed subvarieties where the Hodge–Tate and bounded de Rham periods (within this range) as well as 1-cocycles form locally free sheaves. We also prove strong vanishing results for higher cohomology. Together, these results give a simultaneous generalization of results of Sen, Kisin, and Berger–Colmez. The main result has been applied by Varma in her proof of geometricity of Harris–Lan–Taylor–Thorne Galois representations as well as in several works of Ding.



  1. 1.
    Bellaïche, J., Chenevier, G.: Families of Galois representations and Selmer groups. Astérisque 324 (2009)Google Scholar
  2. 2.
    Bellovin, R.: \(p\)-adic Hodge theory in rigid analytic families. Algebra Number Theory 9(2), 371–433 (2015)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Berger, L., Colmez, P.: Familles de représentations de de Rham et monodromie \(p\)-adique. Astérisque 319, 303–337 (2008)MATHGoogle Scholar
  4. 4.
    Bierstone, E., Milman, P.D.: Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant. Invent. Math. 128(2), 207–302 (1997)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bosch, S., Güntzer, U., Remmert, R.: Non-Archimedean analysis. Grundlehren der Mathematischen Wissenschaften, vol. 261. Springer, Berlin (1984)MATHGoogle Scholar
  6. 6.
    Bourbaki, N.: Elements of Mathematics. Commutative Algebra. Hermann, Paris (1972). (Translated from the French) MATHGoogle Scholar
  7. 7.
    Brinon, O., Conrad, B.: CMI summer school notes on \(p\)-adic Hodge theory (2009) (Preprint) Google Scholar
  8. 8.
    Chenevier, G., Harris, M.: Construction of automorphic Galois representations II. Camb. J. Math. 1, 53–73 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Conrad, B.: Irreducible components of rigid spaces. Ann. Inst. Fourier (Grenoble) 49(2), 473–541 (1999)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Diao, H., Liu, R.: The eigencurve is proper. Duke Math. J. 165(7), 1381–1395 (2016)MathSciNetMATHGoogle Scholar
  11. 11.
    Ding, Y.: Companion points and locally analytic socle for \({\rm GL} _2({L})\) (2016) (Preprint) Google Scholar
  12. 12.
    Ding, Y.: Formes modulaires \(p\)-adiques sur les courbes de Shimura unitaires et compatibilité local-global (2016) (Preprint) Google Scholar
  13. 13.
    Ding, Y.: \(\cal{L}\)-invariants, partially de Rham families, and local-global compatibility. Annales de l’Institut Fourier 67(4), 1457–1519 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Eilenberg, S., MacLane, S.: Cohomology theory in abstract groups I. Ann. Math. 2(48), 51–78 (1947)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Emerton, M.: Locally Analytic Vectors in Representations of Locally \(p\)-adic Analytic Groups, vol. 248, no. 1175. Memoirs of the American Mathematical Society, Providence (2017)Google Scholar
  16. 16.
    Fontaine, J.-M.: Le corps des périodes \(p\)-adiques. With an appendix by Pierre Colmez. Astérisque 223, 59–111 (1994)MATHGoogle Scholar
  17. 17.
    Harris, M., Lan, K.-W., Taylor, R., Thorne, J.: On the rigid cohomology of certain Shimura varieties. Res. Math. Sci. 3, 1–308 (2016)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Hellmann, E.: Families of \(p\)-adic Galois representations and \((\varphi,\Gamma )\)-modules. Comment. Math. Helv. 91(4), 721–749 (2016)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Iovita, A., Zaharescu, A.: Galois theory of \(B_{\rm dR}^+\). Compositio Math. 117(1), 1–31 (1999)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Iovita, A., Zaharescu, A.: Generating elements for \(B_{\rm dR}^+\). J. Math. Kyoto Univ. 39(2), 233–248 (1999)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Jorza, A.: \(p\)-adic families and Galois representations for \({\rm GSp}(4)\) and \({\rm GL}(2)\). Math. Res. Lett. 19(5), 987–996 (2012)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Kaplansky, I.: Projective modules. Ann. Math. 2(68), 372–377 (1958)CrossRefMATHGoogle Scholar
  23. 23.
    Kedlaya, K., Liu, R.: On families of (\(\varphi \), \(\Gamma \))-modules. Algebra Number Theory 4(7), 943–967 (2010)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Kedlaya, K., Pottharst, J., Xiao, L.: Cohomology of arithmetic families of \((\varphi,\Gamma )\)-modules. J. Am. Math. Soc. 27(4), 1043–1115 (2014)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Kisin, M.: Overconvergent modular forms and the Fontaine–Mazur conjecture. Invent. Math. 153(2), 373–454 (2003)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Liu, R.: Semistable periods of finite slope families. Algebra Number Theory 9(2), 435–458 (2015)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Liu, R.: Triangulation of refined families. Comment. Math. Helv. 90(4), 831–904 (2015)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Luu, M.: Deformation Theory and Local-Global Compatibility of Langlands Correspondences, vol. 238, no. 1123. Memoirs of the American Mathematical Society, Providence (2015)Google Scholar
  29. 29.
    Mazur, B., Wiles, A.: On \(p\)-adic analytic families of Galois representations. Compositio Math. 59(2), 231–264 (1986)MathSciNetMATHGoogle Scholar
  30. 30.
    Neukirch, J., Schmidt, A., Wingberg, K.: Cohomology of Number Fields. Grundlehren der Mathematischen Wissenschaften, vol. 323, 2nd edn. Springer, Berlin (2008)MATHGoogle Scholar
  31. 31.
    Pérez-García, C., Schikhof, W.H.: Locally Convex Spaces Over Non-Archimedean Valued Fields. Cambridge Studies in Advanced Mathematics, vol. 119. Cambridge University Press, Cambridge (2010)CrossRefMATHGoogle Scholar
  32. 32.
    Pottharst, J.: Analytic families of finite-slope Selmer groups. Algebra Number Theory 7(7), 1571–1612 (2013)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Schneider, P.: Non-Archimedean Functional Analysis. Springer Monographs in Mathematics. Springer, Berlin (2002)Google Scholar
  34. 34.
    Sen, S.: Continuous cohomology and \(p\)-adic Galois representations. Invent. Math. 62(1), 89–116 (1980)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Sen, S.: The analytic variation of \(p\)-adic Hodge structure. Ann. Math. (2) 127(3), 647–661 (1988)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Sen, S.: An infinite-dimensional Hodge–Tate theory. Bull. Soc. Math. Fr. 121(1), 13–34 (1993)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Shah, S.: \(p\)-adic approaches to the Langlands program. Ph.D. thesis (2014)Google Scholar
  38. 38.
    Skinner, C.: A note on the \(p\)-adic Galois representations attached to Hilbert modular forms. Doc. Math. 14, 241–258 (2009)MathSciNetMATHGoogle Scholar
  39. 39.
    Skinner, C., Urban, E.: The Iwasawa main conjectures for \({\rm GL}_2\). Invent. Math. 195(1), 1–277 (2014)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Tate, J.: \(p\)-divisible groups. In: Proceedings of a Conference on Local Fields (Driebergen, 1966), pp. 158–183. Springer, Berlin (1967)Google Scholar
  41. 41.
    Temkin, M.: Functorial desingularization of quasi-excellent schemes in characteristic zero: the nonembedded case. Duke Math. J. 161(11), 2207–2254 (2012)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Varma, I.: Crystallinity of Galois representations associated to regular algebraic cuspidal automorphic representations of \({\rm GL}_n\) (2018) (Preprint) Google Scholar

Copyright information

© SpringerNature 2018

Authors and Affiliations

  1. 1.Department of MathematicsColumbia UniversityNew YorkUSA

Personalised recommendations