Purity for families of Galois representations

Saha, Jyoti Prakash

Annales de l'Institut Fourier, Tome 67 (2017) no. 2, pp. 879-910.

Nous formulons une notion de pureté pour les familles $p$-adiques de représentations galoisiennes et pseudo-caractères du groupe de Weil d’un corps de nombres $\ell$-adiques pour $\ell \ne p$. Ceci est obtenu en montrant que tous les puissances de la monodromie de toute représentation galoisienne restent aussi grandes que possible après spécialisations pures. En utilisant la pureté pour les familles, nous améliorons une partie de la correspondance de Langlands locale pour ${\mathrm{GL}}_n$ en familles formulée par Emerton et Helm. De plus, en utilisant les exemples de familles de Hida et variétés de Hecke, nous illustrons le rôle de pureté pour les familles dans l’étude de la variation des facteurs d’Euler locaux, types automorphes locaux le long des composantes irréductibles, les points d’intersection des composantes irréductibles de familles de représentations galoisiennes automorphes.

We formulate a notion of purity for $p$-adic big Galois representations and pseudorepresentations of Weil groups of $\ell$-adic number fields for $\ell \ne p$. This is obtained by showing that all powers of the monodromy of any big Galois representation stay “as large as possible” under pure specializations. Using purity for families, we improve a part of the local Langlands correspondence for ${\mathrm{GL}}_n$ in families formulated by Emerton and Helm. The role of purity for families in the study of variation of local Euler factors, local automorphic types along irreducible components, intersection points of irreducible components of $p$-adic families of automorphic Galois representations is illustrated using the examples of Hida families and eigenvarieties.

Classification : 11F41, 11F55, 11F80
Keywords: $p$-adic families of automorphic forms, Pure representations, Local Langlands correspondence, Euler factors
Mot clés : Familles $p$-adiques de formes automorphes, Représentations pures, Correspondance de Langlands, Facteurs d’Euler

[1] Andreatta, Fabrizio; Iovita, Adrian; Stevens, Glenn Overconvergent Eichler–Shimura isomorphisms, J. Inst. Math. Jussieu, Volume 14 (2015) no. 2, pp. 221-274 | DOI

[2] Bellaïche, Joël; Chenevier, Gaëtan Formes non tempérées pour $\mathrm{U}\left(3\right)$ et conjectures de Bloch-Kato, Ann. Sci. Éc. Norm. Supér., Volume 37 (2004) no. 4, pp. 611-662 | DOI

[3] Bellaïche, Joël; Chenevier, Gaëtan Families of Galois representations and Selmer groups, Astérisque (2009) no. 324, xii+314 pages

[4] Blasius, Don Hilbert modular forms and the Ramanujan conjecture, Noncommutative geometry and number theory (Aspects Math., E37), Vieweg, Wiesbaden, 2006, pp. 35-56 | DOI

[5] Blasius, Don; Rogawski, Jonathan D. Tate classes and arithmetic quotients of the two-ball, The zeta functions of Picard modular surfaces, Univ. Montréal, Montreal, QC, 1992, pp. 421-444

[6] Breuil, Christophe; Schneider, Peter First steps towards $p$-adic Langlands functoriality, J. Reine Angew. Math., Volume 610 (2007), pp. 149-180 | DOI

[7] Bushnell, Colin J.; Henniart, Guy The local Langlands conjecture for $\mathrm{GL}\left(2\right)$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 335, Springer-Verlag, Berlin, 2006, xii+347 pages | DOI

[8] Caraiani, Ana Local-global compatibility and the action of monodromy on nearby cycles, Duke Math. J., Volume 161 (2012) no. 12, pp. 2311-2413 | DOI

[9] Carayol, Henri Sur les représentations $l$-adiques associées aux formes modulaires de Hilbert, Ann. Sci. Éc. Norm. Supér., Volume 19 (1986) no. 3, pp. 409-468 | DOI

[10] Chenevier, Gaëtan Familles $p$-adiques de formes automorphes pour ${\mathrm{GL}}_n$, J. Reine Angew. Math., Volume 570 (2004), pp. 143-217 | DOI

[11] Chenevier, Gaëtan On the infinite fern of Galois representations of unitary type, Ann. Sci. Éc. Norm. Supér., Volume 44 (2011) no. 6, pp. 963-1019 | DOI

[12] Chenevier, Gaëtan; Harris, Michael Construction of automorphic Galois representations, II, Camb. J. Math., Volume 1 (2013) no. 1, pp. 53-73 | DOI

[13] Clozel, Laurent Motifs et formes automorphes: applications du principe de fonctorialité, Automorphic forms, Shimura varieties, and $L$-functions, Vol. I (Ann Arbor, MI, 1988) (Perspect. Math.), Volume 10, Academic Press, Boston, MA, 1990, pp. 77-159

[14] Clozel, Laurent Purity reigns supreme, Int. Math. Res. Not. IMRN (2013) no. 2, pp. 328-346

[15] Coleman, Robert; Mazur, Barry The eigencurve, Galois representations in arithmetic algebraic geometry (Durham, 1996) (London Math. Soc. Lecture Note Ser.), Volume 254, Cambridge University Press, 1998, pp. 1-113 | DOI

[16] Conrad, Brian Irreducible components of rigid spaces, Ann. Inst. Fourier (Grenoble), Volume 49 (1999) no. 2, pp. 473-541 | DOI

[17] Deligne, Pierre Formes modulaires et représentations $l$-adiques, Séminaire Bourbaki. Vol. 1968/69: Exposés 347–363 (Lecture Notes in Math.), Volume 175, Springer-Verlag, 1971, pp. Exp. No. 355, 139-172

[18] Deligne, Pierre Formes modulaires et représentations de $\mathrm{GL}\left(2\right)$, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer-Verlag, Berlin, 1973, p. 55-105. Lecture Notes in Math., Vol. 349

[19] Deligne, Pierre Les constantes des équations fonctionnelles des fonctions $L$, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer-Verlag, Berlin, 1973, p. 501-597. Lecture Notes in Math., Vol. 349

[20] Deligne, Pierre La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. (1980) no. 52, pp. 137-252 | DOI

[21] Eichler, Martin Quaternäre quadratische Formen und die Riemannsche Vermutung für die Kongruenzzetafunktion, Arch. Math., Volume 5 (1954), pp. 355-366 | DOI

[22] Emerton, Matthew Local-global compatibility in the $p$-adic Langlands programme for ${\mathrm{GL}}_{2/\mathbb{Q}}$ (2011) (preprint available at http://www.math.uchicago.edu/~emerton/pdffiles/lg.pdf)

[23] Emerton, Matthew; Helm, David The local Langlands correspondence for ${\mathrm{GL}}_n$ in families, Ann. Sci. Éc. Norm. Supér., Volume 47 (2014) no. 4, pp. 655-722 | DOI

[24] Emerton, Matthew; Pollack, Robert; Weston, Tom Variation of Iwasawa invariants in Hida families, Invent. Math., Volume 163 (2006) no. 3, pp. 523-580 | DOI

[25] Fouquet, Olivier Dihedral Iwasawa theory of nearly ordinary quaternionic automorphic forms, Compos. Math., Volume 149 (2013) no. 3, pp. 356-416 | DOI

[26] Fouquet, Olivier; Ochiai, Tadashi Control theorems for Selmer groups of nearly ordinary deformations, J. Reine Angew. Math., Volume 666 (2012), pp. 163-187 | DOI

[27] Gelbart, Stephen S. Automorphic forms on adèle groups, Princeton University Press, Princeton, N.J., 1975, x+267 pages (Annals of Mathematics Studies, No. 83)

[28] Geraghty, David James Modularity lifting theorems for ordinary Galois representations, ProQuest LLC, Ann Arbor, MI, 2010, 131 pages search.proquest.com/docview/612773827 Thesis (Ph.D.)–Harvard University

[29] Grothendieck, Alexander Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Inst. Hautes Études Sci. Publ. Math. (1967) no. 32, 361 pages

[30] Harris, Michael; Taylor, Richard The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, 151, Princeton University Press, Princeton, NJ, 2001, viii+276 pages (With an appendix by Vladimir G. Berkovich)

[31] Hida, Haruzo Galois representations into ${\mathrm{GL}}_2\left({\mathbf{Z}}_p\left[\left[X\right]\right]\right)$ attached to ordinary cusp forms, Invent. Math., Volume 85 (1986) no. 3, pp. 545-613 | DOI

[32] Hida, Haruzo Iwasawa modules attached to congruences of cusp forms, Ann. Sci. Éc. Norm. Supér., Volume 19 (1986) no. 2, pp. 231-273 | DOI

[33] Hida, Haruzo On $p$-adic Hecke algebras for ${\mathrm{GL}}_2$, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI (1987), pp. 434-443

[34] Hida, Haruzo Control theorems of $p$-nearly ordinary cohomology groups for $\mathrm{SL}\left(n\right)$, Bull. Soc. Math. France, Volume 123 (1995) no. 3, pp. 425-475 | DOI

[35] Illusie, Luc Autour du théorème de monodromie locale, Astérisque (1994) no. 223, pp. 9-57 Périodes $p$-adiques (Bures-sur-Yvette, 1988)

[36] Labesse, Jean-Pierre Changement de base CM et séries discrètes, On the stabilization of the trace formula (Stab. Trace Formula Shimura Var. Arith. Appl.), Volume 1, Int. Press, Somerville, MA, 2011, pp. 429-470

[37] Matsumura, Hideyuki Commutative algebra, Mathematics Lecture Note Series, 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980, xv+313 pages

[38] Matsumura, Hideyuki Commutative ring theory, Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, Cambridge, 1989, xiv+320 pages (Translated from the Japanese by M. Reid)

[39] Mazur, Barry Deforming Galois representations, Galois groups over $\mathbf{Q}$ (Berkeley, CA, 1987) (Math. Sci. Res. Inst. Publ.), Volume 16, Springer-Verlag, New York, 1989, pp. 385-437

[40] Nekovář, Jan Selmer complexes, Astérisque (2006) no. 310, viii+559 pages

[41] Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay Cohomology of number fields, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 323, Springer-Verlag, Berlin, 2008, xvi+825 pages

[42] Nyssen, Louise Pseudo-représentations, Math. Ann., Volume 306 (1996) no. 2, pp. 257-283 | DOI

[43] Ochiai, Tadashi On the two-variable Iwasawa main conjecture, Compos. Math., Volume 142 (2006) no. 5, pp. 1157-1200 | DOI

[44] Patrikis, Stefan; Taylor, Richard Automorphy and irreducibility of some $l$-adic representations, Compos. Math., Volume 151 (2015) no. 2, pp. 207-229 | DOI

[45] Paulin, Alexander G. M. Local to global compatibility on the eigencurve, Proc. Lond. Math. Soc., Volume 103 (2011) no. 3, pp. 405-440 | DOI

[46] Ribet, Kenneth A. Galois representations attached to eigenforms with Nebentypus, Modular functions of one variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), Springer-Verlag, Berlin, 1977, p. 17-51. Lecture Notes in Math., Vol. 601

[47] Scholze, Peter Perfectoid spaces, Publ. Math. Inst. Hautes Études Sci., Volume 116 (2012), pp. 245-313 | DOI

[48] Serre, Jean-Pierre Abelian $l$-adic representations and elliptic curves, Research Notes in Mathematics, 7, A K Peters Ltd., Wellesley, MA, 1998, 199 pages (With the collaboration of Willem Kuyk and John Labute, Revised reprint of the 1968 original)

[49] Serre, Jean-Pierre; Tate, John Good reduction of abelian varieties, Ann. Math., Volume 88 (1968), pp. 492-517 | DOI

[50] Shimura, Goro Correspondances modulaires et les fonctions $\zeta$ de courbes algébriques, J. Math. Soc. Japan, Volume 10 (1958), pp. 1-28 | DOI

[51] Shimura, Goro Collected papers. Vol. II. 1967–1977, Springer-Verlag, 2002, xiv+831 pages

[52] Shin, Sug Woo Galois representations arising from some compact Shimura varieties, Ann. Math., Volume 173 (2011) no. 3, pp. 1645-1741 | DOI

[53] Taylor, Richard Galois representations associated to Siegel modular forms of low weight, Duke Math. J., Volume 63 (1991) no. 2, pp. 281-332 | DOI

[54] Taylor, Richard; Yoshida, Teruyoshi Compatibility of local and global Langlands correspondences, J. Amer. Math. Soc., Volume 20 (2007) no. 2, pp. 467-493 | DOI

[55] Wiles, Andrew On ordinary $\lambda$-adic representations associated to modular forms, Invent. Math., Volume 94 (1988) no. 3, pp. 529-573 | DOI