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
- Yves André, Filtrations de type Hasse-Arf et monodromie $p$-adique, Invent. Math. 148 (2002), no. 2, 285–317 (French). MR 1906151, DOI https://doi.org/10.1007/s002220100207
- Yves André, Period mappings and differential equations. From $\Bbb C$ to $\Bbb C_p$, 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
- M.Artin, Grothendieck topologies. Lecture Notes, Harvard Univ. (1962).
- M.Artin et al., Théorie des topos et cohomologie étale des schémas – tome 1. Springer Lect. Notes Math. 269 (1972).
- 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
- A. Beilinson, Topological $\scr E$-factors, Pure Appl. Math. Q. 3 (2007), no. 1, Special Issue: In honor of Robert D. MacPherson., 357–391. MR 2330165, DOI https://doi.org/10.4310/PAMQ.2007.v3.n1.a13
- Alexander Beilinson, $\scr E$-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
- Alexander Beilinson, Spencer Bloch, and Hélène Esnault, $\epsilon $-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, DOI https://doi.org/10.17323/1609-4514-2002-2-3-477-532
- 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
- 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
- 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
- P.Deligne, Séminaire à l’IHES sur les constantes des équations fonctionelles des fonctions $L$. Typeset notes by L.Illusie (1980).
- Marco A. Garuti, Prolongement de revêtements galoisiens en géométrie rigide, Compositio Math. 104 (1996), no. 3, 305–331 (French, with English summary). MR 1424559
- 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
- R. Huber, Continuous valuations, Math. Z. 212 (1993), no. 3, 455–477. MR 1207303, DOI https://doi.org/10.1007/BF02571668
- R. Huber, A generalization of formal schemes and rigid analytic varieties, Math. Z. 217 (1994), no. 4, 513–551. MR 1306024, DOI https://doi.org/10.1007/BF02571959
- Roland Huber, Étale cohomology of rigid analytic varieties and adic spaces, Aspects of Mathematics, E30, Friedr. Vieweg & Sohn, Braunschweig, 1996. MR 1734903
- R. Huber, Swan representations associated with rigid analytic curves, J. Reine Angew. Math. 537 (2001), 165–234. MR 1856262, DOI https://doi.org/10.1515/crll.2001.063
- 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
- 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
- Nicholas M. Katz, Gauss sums, Kloosterman sums, and monodromy groups, Annals of Mathematics Studies, vol. 116, Princeton University Press, Princeton, NJ, 1988. MR 955052
- 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
- James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
- Lorenzo Ramero, On a class of étale analytic sheaves, J. Algebraic Geom. 7 (1998), no. 3, 405–504. MR 1618148
- Lorenzo Ramero, Local monodromy in non-Archimedean analytic geometry, Publ. Math. Inst. Hautes Études Sci. 102 (2005), 167–280. MR 2217053, DOI https://doi.org/10.1007/s10240-005-0036-z
- Neantro Saavedra Rivano, Catégories Tannakiennes, Lecture Notes in Mathematics, Vol. 265, Springer-Verlag, Berlin-New York, 1972 (French). MR 0338002
- J.P.Serre, Représentations linéaires des groupes finis – Cinquiéme edition. Hermann (1998).
References
- Y.André, Filtrations de type Hasse-Arf et monodromie $p$-adique. Invent. Math. 148 (2002), pp. 285–317. MR 1906151 (2003k:12011)
- Y.André, Period mappings and differential equations. From $\mathbb {C}$ to $\mathbb {C}_p$. MSJ Memoirs 12 (2003). MR 1978691 (2004g:14014)
- M.Artin, Grothendieck topologies. Lecture Notes, Harvard Univ. (1962).
- M.Artin et al., Théorie des topos et cohomologie étale des schémas – tome 1. Springer Lect. Notes Math. 269 (1972).
- M.Artin et al., Théorie des topos et cohomologie étale des schémas – tome 3. Springer Lect. Notes Math. 305 (1973). MR 0354654 (50:7132)
- A.Beilinson, Topological $\varepsilon$-factors. Pure Appl. Math. Q. 3 (2007), pp. 357–391. MR 2330165 (2008m:32051)
- A.Beilinson, $\varepsilon$-factors for the period determinants of curves. Preprint \ttfamily arXiv:0903.2674 (2009). MR 2562452
- A.Beilinson, S.Bloch, H.Esnault, $\varepsilon$-factors for Gauss-Manin determinants. Moscow Math. J. 23 (2002), pp. 477–532. MR 1988970 (2004m:14011)
- V.Berkovich, Spectral theory and analytic geometry over non-Archimedean fields. AMS Math. Surveys and Monographs 33 (1990). MR 1070709 (91k:32038)
- F.Borceux, Handbook of Categorical Algebra I. Basic Category Theory. Cambridge Univ. Press Encycl. of Math. and Its Appl. 50 (1994). MR 1291599 (96g:18001a)
- M.Boyarchenko, V.Drinfeld, A motivated introduction to character sheaves and the orbit method for unipotent groups in positive characteristic. Preprint \ttfamily arXiv:math.RT/0609769. MR 933358 (89e:20079)
- P.Deligne, Séminaire à l’IHES sur les constantes des équations fonctionelles des fonctions $L$. Typeset notes by L.Illusie (1980).
- M.Garuti, Prolongement de revétements galoisiens en géométrie rigide. Comp. Math. 104 (1996), pp. 305–331. MR 1424559 (98m:14023)
- A.Grothendieck et al., Revêtements Étales et Groupe Fondamental. Documents Math. Soc. Math. France 3 (2003). MR 2017446 (2004g:14017)
- R.Huber, Continuous valuations. Math. Zeit. 212 (1993), pp. 445–477. MR 1207303 (94e:13041)
- R.Huber, A generalization of formal schemes and rigid analytic geometry. Math.Zeit. 217 (1994), pp. 513–551. MR 1306024 (95k:14001)
- R.Huber, Étale cohomology of rigid analytic varieties and adic spaces. Vieweg Aspects of Math. 30 (1996). MR 1734903 (2001c:14046)
- R.Huber, Swan representations associated with rigid analytic curves. J. reine angew. Math. 537 (2001), pp. 165–234. MR 1856262 (2002h:14036)
- J. de Jong, Étale fundamental groups of non-Archimedean analytic spaces. Comp. Math. 97 (1995), pp. 89–118. MR 1355119 (97c:32047)
- N.M.Katz Local-to-global extensions of representations of fundamental groups. Ann. Inst. Fourier 36 (1986), pp. 69–106. MR 867916 (88a:14032)
- N.M.Katz, Gauss Sums, Kloosterman Sums, and Monodromy Groups. Princeton U. Ann. of Math. Studies 116 (1988). MR 955052 (91a:11028)
- G.Laumon, Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de Weil. Publ. Math. IHES 65 (1987), pp. 131–210. MR 908218 (88g:14019)
- J.S.Milne, Étale cohomology. Princeton Math. Series 33 (1980). MR 559531 (81j:14002)
- L.Ramero, On a class of étale analytic sheaves. J. of Alg.Geom. 7 (1998), pp. 405–504. MR 1618148 (2000b:14026)
- L.Ramero, Local monodromy in non-Archimedean analytic geometry. Publ. Math. IHES 102 (2005), pp. 167–280. MR 2217053 (2007e:14036)
- N. Saavedra Rivano, Catégories tannakiennes. Springer Lect. Notes Math 265 (1972). MR 0338002 (49:2769)
- 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
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