Duke Mathematical Journal

Local-global compatibility and the action of monodromy on nearby cycles

Ana Caraiani

Full-text: Access denied (no subscription detected)

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


We strengthen the local-global compatibility of Langlands correspondences for GLn in the case when n is even and lp. Let L be a CM field, and let Π be a cuspidal automorphic representation of GLn(AL) which is conjugate self-dual. Assume that Π is cohomological and not “slightly regular,” as defined by Shin. In this case, Chenevier and Harris constructed an l-adic Galois representation Rl(Π) and proved the local-global compatibility up to semisimplification at primes v not dividing l. We extend this compatibility by showing that the Frobenius semisimplification of the restriction of Rl(Π) to the decomposition group at v corresponds to the image of Πv via the local Langlands correspondence. We follow the strategy of Taylor and Yoshida, where it was assumed that Π is square-integrable at a finite place. To make the argument work, we study the action of the monodromy operator N on the complex of nearby cycles on a scheme which is locally étale over a product of strictly semistable schemes and we derive a generalization of the weight spectral sequence in this case. We also prove the Ramanujan–Petersson conjecture for Π as above.

Article information

Duke Math. J. Volume 161, Number 12 (2012), 2311-2413.

First available in Project Euclid: 6 September 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R39: Langlands-Weil conjectures, nonabelian class field theory [See also 11Fxx, 22E55]
Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11F80: Galois representations 14G35: Modular and Shimura varieties [See also 11F41, 11F46, 11G18]


Caraiani, Ana. Local-global compatibility and the action of monodromy on nearby cycles. Duke Math. J. 161 (2012), no. 12, 2311--2413. doi:10.1215/00127094-1723706. https://projecteuclid.org/euclid.dmj/1346936109

Export citation


  • [A] J. Arthur, The invariant trace formula, II: Global theory, J. Amer. Math. Soc. 1 (1988), 501–554.
  • [AC] J. Arthur and L. Clozel, Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, Ann. of Math. Stud. 120, Princeton Univ. Press, Princeton, 1989.
  • [AGV] M. Artin, A. Grothendieck, and J. L. Verdier, Théorie des topos et cohomologie étale des schémas, Séminaire de Géométrie Algébrique du Bois-Marie (SGA 4), Lecture Notes in Math. 305, Springer, Berlin, 1973.
  • [Bad] A. I. Badulescu, Jacquet-Langlands et unitarisabilité, J. Inst. Math. Jussieu 6 (2007), 349–379.
  • [B] A. A. Beĭlinson, “On the derived category of perverse sheaves” in $K$-theory, Arithmetic and Geometry (Moscow, 1984–1986), Lecture Notes in Math. 1289, Springer, Berlin, 1987, 27–41.
  • [BBD] A. A. Beĭlinson, J. Bernstein, and P. Deligne, “Faisceaux pervers” in Analyse et topologie sur les espaces singuliers, I (Luminy, 1981), Astérisque 100, Soc. Math. France, Paris, 1982, 5–171.
  • [BZ] I. N. Bernstein and A. V. Zelevinsky, Induced representations of reductive $p$-adic groups, I, Ann. Sci. Éc. Norm. Supér. (4) 10 (1977), 441–472.
  • [Bor] A. Borel, “Automorphic $L$-functions” in Automorphic Forms, Representations and $L$-functions (Corvallis, Ore., 1977), Proc. Sympos. Pure Math. 33, Amer. Math. Soc., Providence, 1979, 27–61.
  • [CH] G. Chenevier and M. Harris, Construction of automorphic Galois representations, II, preprint, http://people.math.jussieu.fr/~harris/ (accessed 24 July 2012).
  • [Cl1] L. Clozel, Représentations galoisiennes associées aux représentations automorphes autoduales de $\operatorname{GL}(n)$, Publ. Math. Inst. Hautes Études Sci. 73 (1991), 97–145.
  • [Cl2] Clozel, L., Purity reigns supreme, to appear in Int. Math. Res. Not. IMRN, preprint, http://fa.institut.math.jussieu.fr/node/45 (accessed 24 July 2012).
  • [D] V. G. Drinfeld, Elliptic modules (in Russian), Mat. Sb. (N.S.) 94(136), no. 4, 594–627; English translation in Math. USSR Sb. 23 (1974), 561–592.
  • [E] T. Ekedahl, “On the adic formalism” in The Grothendieck Festschrift, Vol. II, Progr. Math. 87, Birkhäuser, Boston, 1990, 197–218.
  • [GDK] A. Grothendieck, P. Deligne, and N. Katz, Groupes de monodromie en géométrie algébrique. I, Séminaire de Géométrie Algébrique du Bois-Marie (SGA 7 I), Lecture Notes in Math. 288, Springer, Berlin, 1972.
  • [GD1] A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique, III: Étude cohomologique des faisceaux cohérents, II, Publ. Math. Inst. Hautes Études Sci. 17 (1963).
  • [GD2] A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique, IV: Étude locale des schémas et des morphismes de schémas, IV, Publ. Math. Inst. Hautes Études Sci. 32 (1967).
  • [HT] M. Harris and R. Taylor, The Geometry and Cohomology of Some Simple Shimura Varieties, with an appendix by V. G. Berkovich, Ann. of Math. Stud. 151, Princeton Univ. Press, Princeton, 2001.
  • [I1] L. Illusie, “Autour du théorème de monodromie locale” in Périodes $p$-adiques (Bures-sur-Yvette, 1988), Astérisque 223, Soc. Math. France, Paris, 1994, 9–57.
  • [I2] L. Illusie, “An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic étale cohomology” in Cohomologies $p$-adiques et applications arithmétiques, II, Astérisque 279, Soc. Math. France, Paris, 2002, 271–322.
  • [K] K. Kato, “Logarithmic structures of Fontaine-Illusie” in Algebraic Analysis, Geometry, and Number Theory (Baltimore, 1988), Johns Hopkins Univ. Press, Baltimore, 1989.
  • [KN] K. Kato and C. Nakayama, Log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over $\mathbb{C}$, Kodai Math. J. 22 (1999), 161–186.
  • [KM] N. M. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves, Ann. of Math. Stud. 108, Princeton Univ. Press, Princeton, 1985.
  • [Ko1] R. E. Kottwitz, On the $\lambda$-adic representations associated to some simple Shimura varieties, Invent. Math. 108 (1992), 653–665.
  • [Ko2] R. E. Kottwitz, Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992), 373–444.
  • [L] K.-W. Lan, Arithmetic compactifications of PEL-type Shimura varieties, Ph.D. dissertation, Harvard University, Cambridge, Mass., 2008.
  • [Ma] E. Mantovan, On the cohomology of certain PEL-type Shimura varieties, Duke Math. J. 129 (2005), 573–610.
  • [Na] C. Nakayama, Nearby cycles for log smooth families, Compos. Math. 112 (1998), 45–75.
  • [O] A. Ogus, Relatively coherent log structures, preprint, 2009.
  • [RZ] M. Rapoport and Th. Zink, Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik, Invent. Math. 68 (1982), 21–101.
  • [Re] R. Reich, Notes on Beilinson’s “How to glue perverse sheaves, J. Singul. 1 (2010), 94–115.
  • [S] M. Saito, Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988), 849–995.
  • [Sa] T. Saito, Weight spectral sequences and independence of $l$, J. Inst. Math. Jussieu 2 (2003), 583–634.
  • [She] D. Shelstad, $L$-indistinguishability for real groups, Math. Ann. 259 (1982), 385–430.
  • [Sh1] S. W. Shin, Counting points on Igusa varieties, Duke Math. J. 146 (2009), 509–568.
  • [Sh2] S. W. Shin, A stable trace formula for Igusa varieties, J. Inst. Math. Jussieu 9 (2010), 847–895.
  • [Sh3] S. W. Shin, Galois representations arising from some compact Shimura varieties, Ann. of Math. (2) 173 (2011), 1645–1741.
  • [Tad] M. Tadic, Classification of unitary representations in irreducible representations of general linear group (non-Archimedean case), Ann. Sci. Éc. Norm. Supér. (4) 19 (1986), 335–382.
  • [Tat] J. Tate, “Classes d’isogenie des variétés abéliennes sur un corps fini” in Séminaire Bourbaki, 1968/1969, no. 352, Lecture Notes in Math. 179, Springer, Berlin, 1971.
  • [TY] R. Taylor and T. Yoshida, Compatibility of local and global Langlands correspondences, J. Amer. Math. Soc. 20 (2007), 467–493.
  • [Y] T. Yoshida, “On non-abelian Lubin-Tate theory via vanishing cycles” in Algebraic and Arithmetic Structures of Moduli Spaces (Sapporo, 2007), Adv. Stud. Pure Math. 58, Math. Soc. Japan, Tokyo, 2010, 361–402.