Algebra & Number Theory

First covering of the Drinfel'd upper half-plane and Banach representations of $\mathrm{GL}_2(\mathbb{Q}_p)$

Lue Pan

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


For an odd prime p, we construct some admissible Banach representations of GL2(p) that conjecturally should correspond to some 2-dimensional tamely ramified, potentially Barsotti–Tate representations of Gal(p ̄p) via the p-adic local Langlands correspondence. To achieve this, we generalize Breuil’s work in the semistable case and work on the first covering of the Drinfel’d upper half-plane. Our main tool is an explicit semistable model of the first covering.

Article information

Algebra Number Theory, Volume 11, Number 2 (2017), 405-503.

Received: 12 October 2015
Revised: 17 June 2016
Accepted: 18 November 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11S37: Langlands-Weil conjectures, nonabelian class field theory [See also 11Fxx, 22E50]
Secondary: 22E50: Representations of Lie and linear algebraic groups over local fields [See also 20G05] 11F85: $p$-adic theory, local fields [See also 14G20, 22E50] 11G25: Varieties over finite and local fields [See also 14G15, 14G20]

Drinfel'd upper half-plane $p$-adic local Langlands correspondence of $\mathrm{GL}_2(\mathbb{Q}_p)$


Pan, Lue. First covering of the Drinfel'd upper half-plane and Banach representations of $\mathrm{GL}_2(\mathbb{Q}_p)$. Algebra Number Theory 11 (2017), no. 2, 405--503. doi:10.2140/ant.2017.11.405.

Export citation


  • P. Berthelot, “Géométrie rigide et cohomologie des variétés algébriques de caractéristique $p$”, pp. 3, 7–32 in Introductions aux cohomologies $p$-adiques (Luminy, 1984), Mém. Soc. Math. France (N.S.) 23, 1986.
  • S. Bosch, Lectures on formal and rigid geometry, Lecture Notes in Mathematics 2105, Springer, Cham, 2014.
  • J.-F. Boutot and H. Carayol, “Uniformisation $p$-adique des courbes de Shimura: les théorèmes de Čerednik et de Drinfel'd”, pp. 7, 45–158 in Courbes modulaires et courbes de Shimura (Orsay, 1987/1988), Astérisque 196-197, Société Mathématique de France, 1991.
  • C. Breuil, “Sur quelques représentations modulaires et $p$-adiques de $\GL_2(\Q_p)$, II”, J. Inst. Math. Jussieu 2:1 (2003), 23–58.
  • C. Breuil, “Invariant $\mathscr{L}$ et série spéciale $p$-adique”, Ann. Sci. École Norm. Sup. $(4)$ 37:4 (2004), 559–610.
  • C. Breuil, “Representations of Galois and of $\GL_2$ in characteristic $p$”, course notes, Columbia University, 2007, hook \posturlhook.
  • C. Breuil and A. Mézard, “Représentations semi-stables de ${\rm GL}_2(\mathbb Q_p)$, demi-plan $p$-adique et réduction modulo $p$”, pp. 117–178 in $p$-adic representations of $p$-adic groups, III : global and geometrical methods, edited by L. Berger et al., Astérisque 331, Société Mathématique de France, Paris, 2010.
  • C. J. Bushnell and G. Henniart, The local Langlands conjecture for $\GL(2)$, Grundlehren der Math. Wissenschaften 335, Springer, Berlin, 2006.
  • R. F. Coleman, “Reciprocity laws on curves”, Compositio Math. 72:2 (1989), 205–235.
  • R. Coleman and A. Iovita, “The Frobenius and monodromy operators for curves and abelian varieties”, Duke Math. J. 97:1 (1999), 171–215.
  • P. Colmez, “Une correspondance de Langlands locale $p$-adique pour les représentations semi-stables de dimension $2$”, preprint, 2004, hook \posturlhook.
  • P. Colmez, “Représentations de $\GL_2(\Q_p)$ et $(\phi,\Gamma)$-modules”, pp. 281–509 in Représentations $p$-adiques de groupes $p$-adiques, II : Représentations de $\GL_2(\Q_p)$ et $(\varphi,\Gamma)$-modules, edited by L. Berger et al., Astérisque 330, 2010.
  • P. Colmez, G. Dospinescu, and V. Paškūnas, “The $p$-adic local Langlands correspondence for ${\rm GL}_2(\Q_p)$”, Camb. J. Math. 2:1 (2014), 1–47.
  • P. Deligne and G. Lusztig, “Representations of reductive groups over finite fields”, Ann. of Math. $(2)$ 103:1 (1976), 103–161.
  • G. Dospinescu and A.-C. Le Bras, “Revêtements du demi-plan de Drinfeld et correspondance de Langlands $p$-adique”, preprint, 2015.
  • V. G. Drinfel'd, “Elliptic modules”, Mat. Sb. $($N.S.$)$ 94(136) (1974), 594–627, 656. In Russian; translated in Math. USSR-Sb. 23:4 (1974), 561–592.
  • V. G. Drinfel'd, “Coverings of $p$-adic symmetric domains”, Funkcional. Anal. i Priložen. 10:2 (1976), 29–40. In Russian; translated in Funct. Anal. Appl. 10:2 (1976), 107–115.
  • M. Emerton, “$p$-adic $L$-functions and unitary completions of representations of $p$-adic reductive groups”, Duke Math. J. 130:2 (2005), 353–392.
  • J.-M. Fontaine, “Représentations $l$-adiques potentiellement semi-stables”, pp. 321–347 in Périodes $p$-adiques (Bures-sur-Yvette, 1988), Astérisque 223, Société Mathématique de France, Paris, 1994.
  • H. Gillet and W. Messing, “Cycle classes and Riemann–Roch for crystalline cohomology”, Duke Math. J. 55:3 (1987), 501–538.
  • E. Große-Klönne, “Rigid analytic spaces with overconvergent structure sheaf”, J. Reine Angew. Math. 519 (2000), 73–95.
  • E. Große-Klönne, “Finiteness of de Rham cohomology in rigid analysis”, Duke Math. J. 113:1 (2002), 57–91.
  • E. Große-Klönne, “De Rham cohomology of rigid spaces”, Math. Z. 247:2 (2004), 223–240.
  • B. Haastert and J. C. Jantzen, “Filtrations of the discrete series of $\SL_2(q)$ via crystalline cohomology”, J. Algebra 132:1 (1990), 77–103.
  • N. M. Katz, “Crystalline cohomology, Dieudonné modules, and Jacobi sums”, pp. 165–246 in Automorphic forms, representation theory and arithmetic (Bombay, 1979), Tata Inst. Fund. Res. Studies in Math. 10, Tata Inst. Fundamental Res., Bombay, 1981.
  • N. M. Katz and W. Messing, “Some consequences of the Riemann hypothesis for varieties over finite fields”, Invent. Math. 23 (1974), 73–77.
  • R. Kiehl, “Theorem A und Theorem B in der nichtarchimedischen Funktionentheorie”, Invent. Math. 2 (1967), 256–273.
  • C. Perez-Garcia and W. H. Schikhof, Locally convex spaces over non-Archimedean valued fields, Cambridge Studies in Advanced Mathematics 119, Cambridge University Press, 2010.
  • M. Raynaud, “Schémas en groupes de type $(p,\dots, p)$”, Bull. Soc. Math. France 102 (1974), 241–280.
  • D. Savitt, “On a conjecture of Conrad, Diamond, and Taylor”, Duke Math. J. 128:1 (2005), 141–197.
  • P. Schneider, Nonarchimedean functional analysis, Springer, Berlin, 2002.
  • P. Schneider and U. Stuhler, “The cohomology of $p$-adic symmetric spaces”, Invent. Math. 105:1 (1991), 47–122.
  • P. Schneider and J. Teitelbaum, “Banach space representations and Iwasawa theory”, Israel J. Math. 127 (2002), 359–380.
  • J. Teitelbaum, “Geometry of an étale covering of the $p$-adic upper half plane”, Ann. Inst. Fourier $($Grenoble$)$ 40:1 (1990), 68–78.
  • J. T. Teitelbaum, “Modular representations of $\PGL_2$ and automorphic forms for Shimura curves”, Invent. Math. 113:3 (1993), 561–580.