Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 

 

Hasse-Arf filtrations in $ p$-adic analytic geometry


Author: Lorenzo Ramero
Journal: J. Algebraic Geom. 21 (2012), 97-182
DOI: https://doi.org/10.1090/S1056-3911-2011-00568-7
Published electronically: May 9, 2011
MathSciNet review: 2846681
Full-text PDF

Abstract | References | Additional Information

Abstract: We develop a theory of local Fourier transforms for abelian sheaves on the étale site of a $ p$-adic punctured disc, and we prove a principle of stationary phase linking these local Fourier transforms to the global Fourier transform that was introduced in one of our earlier works. We use this theory to study the local monodromy of abelian sheaves on the étale site of a $ p$-adic punctured disc, and in particular we exhibit a natural slope decomposition for (germs of) such sheaves, with properties that are wholly analogous to those of the slope decomposition for representation of the Galois group of a local field. We conclude with an application to the study of the so-called cohomological epsilon factor of a locally constant abelian sheaf on the étale site of an affine curve defined over a $ p$-adic field: namely, we exhibit a decomposition of such factor as a tensor product of ``local epsilon factors'' that depend only on the local monodromies of the abelian sheaf.


References [Enhancements On Off] (What's this?)

  • [1] Yves André, Filtrations de type Hasse-Arf et monodromie 𝑝-adique, Invent. Math. 148 (2002), no. 2, 285–317 (French). MR 1906151, https://doi.org/10.1007/s002220100207
  • [2] Yves André, Period mappings and differential equations. From ℂ to ℂ_{𝕡}, MSJ Memoirs, vol. 12, Mathematical Society of Japan, Tokyo, 2003. Tôhoku-Hokkaidô lectures in arithmetic geometry; With appendices by F. Kato and N. Tsuzuki. MR 1978691
  • [3] M.Artin, Grothendieck topologies. Lecture Notes, Harvard Univ. (1962).
  • [4] M.Artin et al., Théorie des topos et cohomologie étale des schémas - tome 1. Springer Lect. Notes Math. 269 (1972).
  • [5] Théorie des topos et cohomologie étale des schémas. Tome 3, Lecture Notes in Mathematics, Vol. 305, Springer-Verlag, Berlin-New York, 1973 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4); Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de P. Deligne et B. Saint-Donat. MR 0354654
  • [6] A. Beilinson, Topological ℰ-factors, Pure Appl. Math. Q. 3 (2007), no. 1, Special Issue: In honor of Robert D. MacPherson., 357–391. MR 2330165, https://doi.org/10.4310/PAMQ.2007.v3.n1.a13
  • [7] Alexander Beilinson, ℰ-factors for the period determinants of curves, Motives and algebraic cycles, Fields Inst. Commun., vol. 56, Amer. Math. Soc., Providence, RI, 2009, pp. 15–82. MR 2562452
  • [8] Alexander Beilinson, Spencer Bloch, and Hélène Esnault, 𝜀-factors for Gauss-Manin determinants, Mosc. Math. J. 2 (2002), no. 3, 477–532. Dedicated to Yuri I. Manin on the occasion of his 65th birthday. MR 1988970
  • [9] Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, 1990. MR 1070709
  • [10] Francis Borceux, Handbook of categorical algebra. 1, Encyclopedia of Mathematics and its Applications, vol. 50, Cambridge University Press, Cambridge, 1994. Basic category theory. MR 1291599
    Francis Borceux, Handbook of categorical algebra. 2, Encyclopedia of Mathematics and its Applications, vol. 51, Cambridge University Press, Cambridge, 1994. Categories and structures. MR 1313497
    Francis Borceux, Handbook of categorical algebra. 3, Encyclopedia of Mathematics and its Applications, vol. 52, Cambridge University Press, Cambridge, 1994. Categories of sheaves. MR 1315049
  • [11] George Lusztig, Introduction to character sheaves, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986) Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 165–179. MR 933358
  • [12] P.Deligne, Séminaire à l'IHES sur les constantes des équations fonctionelles des fonctions $ L$. Typeset notes by L.Illusie (1980).
  • [13] Marco A. Garuti, Prolongement de revêtements galoisiens en géométrie rigide, Compositio Math. 104 (1996), no. 3, 305–331 (French, with English summary). MR 1424559
  • [14] Revêtements étales et groupe fondamental (SGA 1), Documents Mathématiques (Paris) [Mathematical Documents (Paris)], vol. 3, Société Mathématique de France, Paris, 2003 (French). Séminaire de géométrie algébrique du Bois Marie 1960–61. [Algebraic Geometry Seminar of Bois Marie 1960-61]; Directed by A. Grothendieck; With two papers by M. Raynaud; Updated and annotated reprint of the 1971 original [Lecture Notes in Math., 224, Springer, Berlin; MR0354651 (50 #7129)]. MR 2017446
  • [15] R. Huber, Continuous valuations, Math. Z. 212 (1993), no. 3, 455–477. MR 1207303, https://doi.org/10.1007/BF02571668
  • [16] R. Huber, A generalization of formal schemes and rigid analytic varieties, Math. Z. 217 (1994), no. 4, 513–551. MR 1306024, https://doi.org/10.1007/BF02571959
  • [17] Roland Huber, Étale cohomology of rigid analytic varieties and adic spaces, Aspects of Mathematics, E30, Friedr. Vieweg & Sohn, Braunschweig, 1996. MR 1734903
  • [18] R. Huber, Swan representations associated with rigid analytic curves, J. Reine Angew. Math. 537 (2001), 165–234. MR 1856262, https://doi.org/10.1515/crll.2001.063
  • [19] A. J. de Jong, Étale fundamental groups of non-Archimedean analytic spaces, Compositio Math. 97 (1995), no. 1-2, 89–118. Special issue in honour of Frans Oort. MR 1355119
  • [20] Nicholas M. Katz, Local-to-global extensions of representations of fundamental groups, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 4, 69–106 (English, with French summary). MR 867916
  • [21] Nicholas M. Katz, Gauss sums, Kloosterman sums, and monodromy groups, Annals of Mathematics Studies, vol. 116, Princeton University Press, Princeton, NJ, 1988. MR 955052
  • [22] G. Laumon, Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de Weil, Inst. Hautes Études Sci. Publ. Math. 65 (1987), 131–210 (French). MR 908218
  • [23] James S. Milne, Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
  • [24] Lorenzo Ramero, On a class of étale analytic sheaves, J. Algebraic Geom. 7 (1998), no. 3, 405–504. MR 1618148
  • [25] Lorenzo Ramero, Local monodromy in non-Archimedean analytic geometry, Publ. Math. Inst. Hautes Études Sci. 102 (2005), 167–280. MR 2217053, https://doi.org/10.1007/s10240-005-0036-z
  • [26] Neantro Saavedra Rivano, Catégories Tannakiennes, Lecture Notes in Mathematics, Vol. 265, Springer-Verlag, Berlin-New York, 1972 (French). MR 0338002
  • [27] J.P.Serre, Représentations linéaires des groupes finis - Cinquiéme edition. Hermann (1998).


Additional Information

Lorenzo Ramero
Affiliation: Université Lille I, Laboratoire Paul Painléve, UFR de Mathématiques, 59655, Villeneuve d’Ascq Cédex, France

DOI: https://doi.org/10.1090/S1056-3911-2011-00568-7
Received by editor(s): April 22, 2009
Received by editor(s) in revised form: March 22, 2010, and April 5, 2010
Published electronically: May 9, 2011

Journal of Algebraic Geometry
The Journal of Algebraic Geometry
is sponsored by the Department of Mathematical Sciences
of Tsinghua University
and is distributed by the American Mathematical Society
for University Press, Inc.
Online ISSN 1534-7486; Print ISSN 1056-3911
© 2017 University Press, Inc.
Comments: jag-query@ams.org
AMS Website