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Journal of the American Mathematical Society

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ISSN 1088-6834 (online) ISSN 0894-0347 (print)

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Maximal varieties and the local Langlands correspondence for $GL(n)$
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by Mitya Boyarchenko and Jared Weinstein
J. Amer. Math. Soc. 29 (2016), 177-236
Published electronically: April 3, 2015


The cohomology of the Lubin-Tate tower is known to realize the local Langlands correspondence for $GL(n)$ over a nonarchimedean local field. In this article we make progress toward a purely local proof of this fact. To wit, we find a family of formal schemes $\mathcal {V}$ such that the generic fiber of $\mathcal {V}$ is isomorphic to an open subset of Lubin-Tate space at infinite level, and such that the middle cohomology of the special fiber of $\mathcal {V}$ realizes the local Langlands correspondence for a broad class of supercuspidals (those whose Weil parameters are induced from an unramified degree $n$ extension). The special fiber of $\mathcal {V}$ is related to an interesting variety $X$, defined over a finite field, which is “maximal” in the sense that the number of rational points of $X$ is the largest possible among varieties with the same Betti numbers as $X$. The variety $X$ is derived from a certain unipotent algebraic group, in an analogous manner as Deligne-Lusztig varieties are derived from reductive algebraic groups.
  • Théorie des topos et cohomologie étale des schémas. Tome 3, Lecture Notes in Mathematics, Vol. 305, Springer-Verlag, Berlin-New York, 1973 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4); Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de P. Deligne et B. Saint-Donat. MR 0354654
  • Mitya Boyarchenko, Deligne-Lusztig constructions for unipotent and $p$-adic groups (2012). arXiv:1207.5876.
  • Mitya Boyarchenko and Vladimir Drinfeld, A motivated introduction to character sheaves and the orbit method for unipotent groups in positive characteristic (2006).
  • Mitya Boyarchenko and Jared Weinstein, Geometric realization of special cases of local Langlands and Jacquet-Langlands correspondences (2013). arXiv:1303.5795.
  • A. J. de Jong, Crystalline Dieudonné module theory via formal and rigid geometry, Inst. Hautes Études Sci. Publ. Math. 87 (1998), 175., DOI 10.1007/BF02698863
  • P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, vol. 569, Springer-Verlag, Berlin, 1977 (French). Séminaire de géométrie algébrique du Bois-Marie SGA $4\frac {1}{2}$. MR 463174, DOI 10.1007/BFb0091526
  • P. Deligne, La conjecture de Weil II, Publ. Math. IHES 52 (1980), 137–252., DOI 10.1007/BF02684780
  • P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. (2) 103 (1976), no. 1, 103–161. MR 393266, DOI 10.2307/1971021
  • Laurent Fargues and Jean-Marc Fontaine, Courbes et fibrés vectoriels en théorie de Hodge $p$-adique, (2011).
  • M. J. Hopkins and B. H. Gross, Equivariant vector bundles on the Lubin-Tate moduli space, Topology and representation theory (Evanston, IL, 1992) Contemp. Math., vol. 158, Amer. Math. Soc., Providence, RI, 1994, pp. 23–88. MR 1263712, DOI 10.1090/conm/158/01453
  • Michael Harris and Richard Taylor, The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, vol. 151, Princeton University Press, Princeton, NJ, 2001. With an appendix by Vladimir G. Berkovich. MR 1876802
  • Guy Henniart, Correspondance de Langlands-Kazhdan explicite dans le cas non ramifié, Math. Nachr. 158 (1992), 7–26 (French, with French summary). MR 1235293, DOI 10.1002/mana.19921580102
  • Guy Henniart, Correspondance de Jacquet-Langlands explicite. I. Le cas modéré de degré premier, Séminaire de Théorie des Nombres, Paris, 1990–91, Progr. Math., vol. 108, Birkhäuser Boston, Boston, MA, 1993.
  • Roger E. Howe, Tamely ramified supercuspidal representations of $\textrm {Gl}_{n}$, Pacific J. Math. 73 (1977), no. 2, 437–460. MR 492087, DOI 10.2140/pjm.1977.73.437
  • R. Huber, A generalization of formal schemes and rigid analytic varieties, Math. Z. 217 (1994), no. 4, 513–551. MR 1306024, DOI 10.1007/BF02571959
  • David Kazhdan, On lifting, Lie group representations, II (College Park, Md., 1982/1983) Lecture Notes in Math., vol. 1041, Springer, Berlin, 1984, pp. 209–249. MR 748509, DOI 10.1007/BFb0073149
  • Jonathan Lubin and John Tate, Formal complex multiplication in local fields, Ann. of Math. (2) 81 (1965), 380–387. MR 172878, DOI 10.2307/1970622
  • Peter Scholze, Perfectoid spaces, Publ. Math. Inst. Hautes Études Sci. 116 (2012), 245–313. MR 3090258, DOI 10.1007/s10240-012-0042-x
  • Peter Scholze and Jared Weinstein, Moduli of $p$-divisible groups, Camb. J. Math. 1 (2013), no. 2, 145–237. MR 3272049, DOI 10.4310/CJM.2013.v1.n2.a1
  • J. Tate, Number theoretic background, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 3–26. MR 546607
  • Jared Weinstein, Good reduction of affinoids on the Lubin-Tate tower, Doc. Math. 15 (2010), 981–1007. MR 2745690
  • Jared Weinstein, Semistable models for modular curves of arbitrary level (2012).
  • Teruyoshi Yoshida, On non-abelian Lubin-Tate theory via vanishing cycles, Algebraic and arithmetic structures of moduli spaces (Sapporo 2007), Adv. Stud. Pure Math., vol. 58, Math. Soc. Japan, Tokyo, 2010, pp. 361–402. MR 2676163, DOI 10.2969/aspm/05810361
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Bibliographic Information
  • Mitya Boyarchenko
  • Affiliation: Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, Michigan 48109
  • MR Author ID: 710764
  • Email:
  • Jared Weinstein
  • Affiliation: Department of Mathematics and Statistics, Boston University, 111 Cummington Mall, Boston, Massachusetts 02215
  • MR Author ID: 867611
  • Email:
  • Received by editor(s): November 8, 2011
  • Received by editor(s) in revised form: July 15, 2013, and December 17, 2014
  • Published electronically: April 3, 2015
  • Additional Notes: The second author is supported by NSF Award DMS-1303312.
  • © Copyright 2015 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 29 (2016), 177-236
  • MSC (2010): Primary 11S37, 11G25, 14G22; Secondary 11G18
  • DOI:
  • MathSciNet review: 3402698