Skip to Main Content

Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Cohomology of arithmetic families of $(\varphi , \Gamma )$-modules
HTML articles powered by AMS MathViewer

by Kiran S. Kedlaya, Jonathan Pottharst and Liang Xiao PDF
J. Amer. Math. Soc. 27 (2014), 1043-1115

Abstract:

We prove the finiteness and compatibility with base change of the $(\varphi , \Gamma )$-cohomology and the Iwasawa cohomology of arithmetic families of $(\varphi , \Gamma )$-modules. Using this finiteness theorem, we show that a family of Galois representations that is densely pointwise refined in the sense of Mazur is actually trianguline as a family over a large subspace. In the case of the Coleman-Mazur eigencurve, we determine the behavior at all points.
References
  • A. Grothendieck, Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Inst. Hautes Études Sci. Publ. Math. 11 (1961), 167. MR 217085
  • Joël Bellaïche, Ranks of Selmer groups in an analytic family, Trans. Amer. Math. Soc. 364 (2012), no. 9, 4735–4761. MR 2922608, DOI 10.1090/S0002-9947-2012-05504-8
  • Joël Bellaïche and Gaëtan Chenevier, Families of Galois representations and Selmer groups, Astérisque 324 (2009), xii+314 (English, with English and French summaries). MR 2656025
  • R. Bellovin, $p$-adic Hodge theory in rigid analytic families, preprint, arXiv:1306.5685v1.
  • Denis Benois, A generalization of Greenberg’s $\scr L$-invariant, Amer. J. Math. 133 (2011), no. 6, 1573–1632. MR 2863371, DOI 10.1353/ajm.2011.0043
  • Laurent Berger, An introduction to the theory of $p$-adic representations, Geometric aspects of Dwork theory. Vol. I, II, Walter de Gruyter, Berlin, 2004, pp. 255–292 (English, with English and French summaries). MR 2023292
  • Laurent Berger, Équations différentielles $p$-adiques et $(\phi ,N)$-modules filtrés, Astérisque 319 (2008), 13–38 (French, with English and French summaries). Représentations $p$-adiques de groupes $p$-adiques. I. Représentations galoisiennes et $(\phi ,\Gamma )$-modules. MR 2493215
  • Laurent Berger, Christophe Breuil, and Pierre Colmez (eds.), Représentations $p$-adiques de groupes $p$-adiques. I. Représentations galoisiennes et $(\phi ,\Gamma )$-modules, Société Mathématique de France, Paris, 2008 (French). Astérisque No. 319 (2008) (2008). MR 2482309
  • Laurent Berger, Christophe Breuil, and Pierre Colmez (eds.), Représentations $p$-adiques de groupes $p$-adiques I: Répresentations de $\mathrm {GL}_2(\mathbf {Q}_p)$ et $(\varphi ,\Gamma )$-modules, Astérisque 330 (2010).
  • Laurent Berger and Pierre Colmez, Familles de représentations de de Rham et monodromie $p$-adique, Astérisque 319 (2008), 303–337 (French, with English and French summaries). Représentations $p$-adiques de groupes $p$-adiques. I. Représentations galoisiennes et $(\phi ,\Gamma )$-modules. MR 2493221
  • S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean analysis, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 261, Springer-Verlag, Berlin, 1984. A systematic approach to rigid analytic geometry. MR 746961, DOI 10.1007/978-3-642-52229-1
  • N. Bourbaki, Espaces Vectoriels Topologiques, reprint of the 1981 original, Springer, Berlin, 2007.
  • Christophe Breuil and Matthew Emerton, Représentations $p$-adiques ordinaires de $\textrm {GL}_2(\mathbf Q_p)$ et compatibilité local-global, Astérisque 331 (2010), 255–315 (French, with English and French summaries). MR 2667890
  • Kevin Buzzard, Eigenvarieties, $L$-functions and Galois representations, London Math. Soc. Lecture Note Ser., vol. 320, Cambridge Univ. Press, Cambridge, 2007, pp. 59–120. MR 2392353, DOI 10.1017/CBO9780511721267.004
  • Gaëtan Chenevier, Sur la densité des représentations cristallines de $\text {Gal}(\overline {\Bbb Q}_p/\Bbb Q_p)$, Math. Ann. 355 (2013), no. 4, 1469–1525 (French, with English summary). MR 3037022, DOI 10.1007/s00208-012-0815-z
  • Frédéric Cherbonnier and Pierre Colmez, Théorie d’Iwasawa des représentations $p$-adiques d’un corps local, J. Amer. Math. Soc. 12 (1999), no. 1, 241–268 (French). MR 1626273, DOI 10.1090/S0894-0347-99-00281-7
  • Robert F. Coleman, Classical and overconvergent modular forms, Invent. Math. 124 (1996), no. 1-3, 215–241. MR 1369416, DOI 10.1007/s002220050051
  • Pierre Colmez, Théorie d’Iwasawa des représentations de de Rham d’un corps local, Ann. of Math. (2) 148 (1998), no. 2, 485–571 (French). MR 1668555, DOI 10.2307/121003
  • Pierre Colmez, Représentations triangulines de dimension 2, Astérisque 319 (2008), 213–258 (French, with English and French summaries). Représentations $p$-adiques de groupes $p$-adiques. I. Représentations galoisiennes et $(\phi ,\Gamma )$-modules. MR 2493219
  • Pierre Colmez, Représentations de $\textrm {GL}_2(\mathbf Q_p)$ et $(\phi ,\Gamma )$-modules, Astérisque 330 (2010), 281–509 (French, with English and French summaries). MR 2642409
  • Pierre Colmez, $(\phi ,\Gamma )$-modules et représentations du mirabolique de $\textrm {GL}_2(\mathbf Q_p)$, Astérisque 330 (2010), 61–153 (French, with English and French summaries). MR 2642405
  • Brian Conrad, Irreducible components of rigid spaces, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 2, 473–541 (English, with English and French summaries). MR 1697371, DOI 10.5802/aif.1681
  • Richard Crew, Finiteness theorems for the cohomology of an overconvergent isocrystal on a curve, Ann. Sci. École Norm. Sup. (4) 31 (1998), no. 6, 717–763 (English, with English and French summaries). MR 1664230, DOI 10.1016/S0012-9593(99)80001-9
  • David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
  • Jean-Marc Fontaine, Représentations $p$-adiques des corps locaux. I, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 249–309 (French). MR 1106901
  • J.-M. Fontaine and Y. Ouyang, Theory of $p$-adic Galois representations, book to appear, downloaded from http://staff.ustc.edu.cn/~yiouyang/galoisrep.pdf on 6 August 2013.
  • Ralph Greenberg, Iwasawa theory and $p$-adic deformations of motives, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 193–223. MR 1265554
  • Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
  • E. Hellmann, Families of trianguline representations and finite slope spaces, preprint, arXiv:1202.4408v1.
  • Laurent Herr, Sur la cohomologie galoisienne des corps $p$-adiques, Bull. Soc. Math. France 126 (1998), no. 4, 563–600 (French, with English and French summaries). MR 1693457, DOI 10.24033/bsmf.2337
  • Laurent Herr, Une approche nouvelle de la dualité locale de Tate, Math. Ann. 320 (2001), no. 2, 307–337 (French). MR 1839766, DOI 10.1007/PL00004476
  • Kiran S. Kedlaya, Finiteness of rigid cohomology with coefficients, Duke Math. J. 134 (2006), no. 1, 15–97. MR 2239343, DOI 10.1215/S0012-7094-06-13412-9
  • Kiran S. Kedlaya, Slope filtrations for relative Frobenius, Astérisque 319 (2008), 259–301 (English, with English and French summaries). Représentations $p$-adiques de groupes $p$-adiques. I. Représentations galoisiennes et $(\phi ,\Gamma )$-modules. MR 2493220
  • Kiran Kedlaya and Ruochuan Liu, On families of $\phi$, $\Gamma$-modules, Algebra Number Theory 4 (2010), no. 7, 943–967. MR 2776879, DOI 10.2140/ant.2010.4.943
  • Mark Kisin, Overconvergent modular forms and the Fontaine-Mazur conjecture, Invent. Math. 153 (2003), no. 2, 373–454. MR 1992017, DOI 10.1007/s00222-003-0293-8
  • Ruochuan Liu, Cohomology and duality for $(\phi ,\Gamma )$-modules over the Robba ring, Int. Math. Res. Not. IMRN 3 (2008), Art. ID rnm150, 32. MR 2416996, DOI 10.1093/imrn/rnm150
  • R. Liu, Triangulation of refined families, preprint, arXiv:1202.2188v3.
  • Hideyuki Matsumura, Commutative ring theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid. MR 1011461
  • Kentaro Nakamura, Classification of two-dimensional split trianguline representations of $p$-adic fields, Compos. Math. 145 (2009), no. 4, 865–914. MR 2521248, DOI 10.1112/S0010437X09004059
  • K. Nakamura, Deformations of trianguline $\mathbb {B}$-pairs and Zariski density of two dimensional crystalline representations, preprint. arXiv:1006.4891v5.
  • Jan Nekovář, Selmer complexes, Astérisque 310 (2006), viii+559 (English, with English and French summaries). MR 2333680
  • Jonathan Pottharst, Analytic families of finite-slope Selmer groups, Algebra Number Theory 7 (2013), no. 7, 1571–1612. MR 3117501, DOI 10.2140/ant.2013.7.1571
  • J. Pottharst, Cyclotomic Iwasawa theory of motives, preprint.
  • M. Raynaud, Flat modules in algebraic geometry, Compositio Math. 24 (1972), 11–31. MR 302645
  • Peter Schneider, Nonarchimedean functional analysis, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002. MR 1869547, DOI 10.1007/978-3-662-04728-6
  • Peter Schneider and Jeremy Teitelbaum, Algebras of $p$-adic distributions and admissible representations, Invent. Math. 153 (2003), no. 1, 145–196. MR 1990669, DOI 10.1007/s00222-002-0284-1
Similar Articles
Additional Information
  • Kiran S. Kedlaya
  • Affiliation: Department of Mathematics, University of California, San Diego, La Jolla, California 92093
  • MR Author ID: 349028
  • ORCID: 0000-0001-8700-8758
  • Email: kedlaya@ucsd.edu
  • Jonathan Pottharst
  • Affiliation: 5 Redwood Street, Boston, Massachusetts 02122
  • MR Author ID: 894842
  • Email: jay@vbrt.org
  • Liang Xiao
  • Affiliation: Department of Mathematics, University of California, Irvine, Rowland Hall 340, Irvine, California 92697
  • MR Author ID: 888789
  • Email: lxiao@math.uci.edu
  • Received by editor(s): June 26, 2012
  • Received by editor(s) in revised form: August 14, 2013
  • Published electronically: April 3, 2014
  • © Copyright 2014 by Kiran S. Kedlaya, Jonathan Pottharst, and Liang Xiao
  • Journal: J. Amer. Math. Soc. 27 (2014), 1043-1115
  • MSC (2010): Primary 11F33, 11R23, 11S25, 11S31
  • DOI: https://doi.org/10.1090/S0894-0347-2014-00794-3
  • MathSciNet review: 3821175