Abstract
Let \(\mathrm{F}\) be a non-Archimedean local field of residual characteristic p, and \(\ell \) be a prime number different from p. We consider the local Jacquet–Langlands correspondence between \(\ell \)-adic discrete series of \(\mathrm{GL}_n(\mathrm{F})\) and an inner form \(\mathrm{GL}_m(\mathrm{D})\). We show that it respects the relationship of congruence modulo \(\ell \). More precisely, we show that two integral \(\ell \)-adic discrete series of \(\mathrm{GL}_n(\mathrm{F})\) are congruent modulo \(\ell \) if and only if the same holds for their Jacquet–Langlands transfers to \(\mathrm{GL}_m(\mathrm{D})\). We also prove that the Langlands–Jacquet morphism from the Grothendieck group of finite length \(\ell \)-adic representations of \(\mathrm{GL}_n(\mathrm{F})\) to that of \(\mathrm{GL}_m(\mathrm{D})\) defined by Badulescu is compatible with reduction mod \(\ell \).
Similar content being viewed by others
References
Aubert, A.-M., Baum, P., Plymen, R., Solleveld, M.: Depth and the local Langlands correspondence (prépublication) (2013)
Badulescu, A.I.: Correspondance de Jacquet–Langlands en caractéristique non nulle. Ann. Sci. Écol. Norm. Sup. (4) 35, 695–747 (2002)
Badulescu, A.I.: Jacquet-Langlands et unitarisabilité. J. Inst. Math. Jussieu 6(3), 349–379 (2007)
Badulescu, A.I., Henniart, G., Lemaire, B., Sécherre, V.: Sur le dual unitaire de \({\rm GL}_r(D)\). Am. J. Math. 132(5), 1365–1396 (2010)
Broussous, P., Sécherre, V., Stevens, S.: Smooth representations of \({\rm GL}(m, D)\), V: endo-classes. Doc. Math. 17, 23–77 (2012)
Bushnell, C., Henniart, G.: Counting the discrete series for \({\rm GL}(n)\). Bull. Lond. Math. Soc. 39(1), 133–137 (2007)
Bushnell, C., Henniart, G.: The essentially tame Jacquet–Langlands correspondence for inner forms of \({\rm GL}(n)\). Pure Appl. Math. Q. 7(3), 469–538 (2011)
Bushnell, C., Henniart, G.: Modular local Langlands correspondence for \({\rm GL}_n\). Int. Math. Res. Not. 15, 4124–4145 (2014)
Bushnell, C., Henniart, G., Lemaire, B.: Caractère et degré formel pour les formes intérieures de \({\rm GL}(n)\) sur un corps local de caractéristique non nulle. Manuscr. Math. 131(1–2), 11–24 (2010)
Dat, J.-F.: Théorie de Lubin–Tate non-abélienne \(\ell \)-entière. Duke Math. J. 161(6), 951–1010 (2012)
Dat, J.-F.: Un cas simple de correspondance de Jacquet–Langlands modulo \(\ell \). Proc. Lond. Math. Soc. 104, 690–727 (2012)
Deligne, P., Kazhdan, D., Vignéras, M.-F.: Représentations des algèbres centrales simples \(p\)-adiques, Representations of Reductive Groups Over a Local Field. Hermann, Paris (1984)
Dipper, R.: On the decomposition numbers of the finite general linear groups. II. Trans. Am. Math. Soc. 292(1), 123–133 (1985)
Dipper, R., James, G.: Identification of the irreducible modular representations of \({\rm GL}_n(q)\). J. Algebra 104(2), 266–288 (1986)
Green, J.A.: The characters of the finite general linear groups. J. Algebra 184(3), 839–851 (1996)
James, G.: The irreducible representations of the finite general linear groups. Proc. Lond. Math. Soc. (3) 52(2), 236–268 (1986)
Jacquet, H., Langlands, R.P.: Automorphic forms on \({\rm GL}(2)\), Lecture Notes in Mathematics, vol. 114. Springer, New York (1970)
Kret, A., Lapid, E.: Jacquet modules of ladder representations. C. R. Math. Acad. Sci. Paris 350(21–22), 937–940 (2012)
Mínguez, A., Sécherre, V.: Représentations banales de \({\rm GL}(m, D)\). Compos. Math. 149, 679–704 (2013)
Mínguez, A., Sécherre, V.: Représentations lisses modulo \(\ell \) de \({{\rm GL}}_m({\rm D})\). Duke Math. J. 163, 795–887 (2014)
Mínguez, A., Sécherre, V.: Types modulo \(\ell \) pour les formes intérieures de \({\rm GL}_{n}\) sur un corps local non archimédien. Avec un appendice par V. Sécherre et S. Stevens. Proc. Lond. Math. Soc. 109(4), 823–891 (2014)
Mínguez, A., Sécherre, V.: Représentations modulaires de \({\rm GL}_n(q)\) en caractéristique non naturelle. Trends Number Theory Contemp. Math. 649, 163–183 (2015)
Mínguez, A., Sécherre, V.: L’involution de Zelevinski modulo \(\ell \). Represent. Theory 19, 236–262 (2015)
Rogawski, J.: Representations of \({\rm GL}(n)\) and division algebras over a \(p\)-adic field. Duke Math. J. 50, 161–196 (1983)
Sécherre, V., Stevens, S.: Smooth representations of \({\rm GL}(m, D)\), VI: semisimple types. Int. Math. Res. Not. 13, 2994–3039 (2012)
Sécherre, V., Stevens, S.: Block decomposition of the category of \(\ell \)-modular smooth representations of \({\rm GL}_n({\rm F})\) and its inner forms. Ann. Sci. Écol. Norm. Sup. 49(3), 669–709 (2016)
Vignéras, M.-F.: Représentations \(l\)-modulaires d’un groupe réductif \(p\)-adique avec \(l\ne p\), Progress in Mathematics, vol. 137. Birkhäuser Boston Inc., Boston (1996)
Vignéras, M.-F.: Correspondance de Langlands semi-simple pour \({\rm GL}_n({\rm F})\) modulo \(\ell \ne p\). Invent. Math 144, 177–223 (2001)
Vignéras, M.-F.: On Highest Whittaker Models and Integral Structures, Contributions to Automorphic Forms, Geometry and Number Theory. John Hopkins Univ. Press, Shalikafest, pp. 773–801 (2002)
Zelevinsky, A.V.: Induced representations of reductive \({\mathfrak{p}}\)-adic groups. II. On irreducible representations of \({\rm GL}(n)\). Ann. Sci. Écol. Norm. Sup (4) 13(2), 165–210 (1980)
Author information
Authors and Affiliations
Corresponding author
Additional information
Le séjour de V. Sécherre à l’University of East Anglia (Norwich) en mai 2014, durant lequel une partie de ce travail a été effectuée, a été financé par l’EPSRC grant EP/H00534X/1. Il remercie chaleureusement Shaun Stevens pour son invitation et les discussions à propos de ce travail. A. Mínguez remercie João Pedro Dos Santos et Erez Lapid pour les discussions à propos de ce travail.
Rights and permissions
About this article
Cite this article
Mínguez, A., Sécherre, V. Correspondance de Jacquet–Langlands locale et congruences modulo \(\ell \) . Invent. math. 208, 553–631 (2017). https://doi.org/10.1007/s00222-016-0696-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00222-016-0696-y