Purity for families of Galois representations
Annales de l'Institut Fourier, Volume 67 (2017) no. 2, p. 879-910
We formulate a notion of purity for p-adic big Galois representations and pseudorepresentations of Weil groups of -adic number fields for 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 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.
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 -adiques pour 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 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.
Received : 2014-11-17
Revised : 2016-04-11
Accepted : 2016-06-21
Published online : 2017-05-31
DOI : https://doi.org/10.5802/aif.3099
Classification:  11F41,  11F55,  11F80
Keywords: p-adic families of automorphic forms, Pure representations, Local Langlands correspondence, Euler factors
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     author = {Saha, Jyoti Prakash},
     title = {Purity for families of Galois representations},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {2},
     year = {2017},
     pages = {879-910},
     doi = {10.5802/aif.3099},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2017__67_2_879_0}
}
Purity for families of Galois representations. Annales de l'Institut Fourier, Volume 67 (2017) no. 2, pp. 879-910. doi : 10.5802/aif.3099. https://aif.centre-mersenne.org/item/AIF_2017__67_2_879_0/

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

[2] Bellaïche, Joël; Chenevier, Gaëtan Formes non tempérées pour U(3) et conjectures de Bloch-Kato, Ann. Sci. Éc. Norm. Supér., Tome 37 (2004) no. 4, pp. 611-662 | Article

[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, Vieweg, Wiesbaden (Aspects Math., E37) (2006), pp. 35-56 | Article

[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., Tome 610 (2007), pp. 149-180 | Article

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

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

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

[10] Chenevier, Gaëtan Familles p-adiques de formes automorphes pour GL n , J. Reine Angew. Math., Tome 570 (2004), pp. 143-217 | Article

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

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

[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), Academic Press, Boston, MA (Perspect. Math.) Tome 10 (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), Cambridge University Press (London Math. Soc. Lecture Note Ser.) Tome 254 (1998), pp. 1-113 | Article

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

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

[18] Deligne, Pierre Formes modulaires et représentations de GL (2), 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 | Article

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

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

[23] Emerton, Matthew; Helm, David The local Langlands correspondence for GL n in families, Ann. Sci. Éc. Norm. Supér., Tome 47 (2014) no. 4, pp. 655-722 | Article

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

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

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

[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, Princeton University Press, Princeton, NJ, Annals of Mathematics Studies, Tome 151 (2001), viii+276 pages (With an appendix by Vladimir G. Berkovich)

[31] Hida, Haruzo Galois representations into GL 2 (Z p [[X]]) attached to ordinary cusp forms, Invent. Math., Tome 85 (1986) no. 3, pp. 545-613 | Article

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

[33] Hida, Haruzo On p-adic Hecke algebras for 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 SL (n), Bull. Soc. Math. France, Tome 123 (1995) no. 3, pp. 425-475 | Article

[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, Int. Press, Somerville, MA (Stab. Trace Formula Shimura Var. Arith. Appl.) Tome 1 (2011), pp. 429-470

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

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

[39] Mazur, Barry Deforming Galois representations, Galois groups over Q (Berkeley, CA, 1987), Springer-Verlag, New York (Math. Sci. Res. Inst. Publ.) Tome 16 (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, Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 323 (2008), xvi+825 pages

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

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

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

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

[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., Tome 116 (2012), pp. 245-313 | Article

[48] Serre, Jean-Pierre Abelian l-adic representations and elliptic curves, A K Peters Ltd., Wellesley, MA, Research Notes in Mathematics, Tome 7 (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., Tome 88 (1968), pp. 492-517 | Article

[50] Shimura, Goro Correspondances modulaires et les fonctions ζ de courbes algébriques, J. Math. Soc. Japan, Tome 10 (1958), pp. 1-28 | Article

[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., Tome 173 (2011) no. 3, pp. 1645-1741 | Article

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

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

[55] Wiles, Andrew On ordinary λ-adic representations associated to modular forms, Invent. Math., Tome 94 (1988) no. 3, pp. 529-573 | Article