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Local Fourier transform and epsilon factors

Part of: (Co)homology theory Algebraic number theory: local and $p$-adic fields

Published online by Cambridge University Press:  05 August 2010

Ahmed Abbes  and
Takeshi Saito
Ahmed Abbes
Affiliation:
CNRS UMR 6625, IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes cedex, France (email: [email protected])
Takeshi Saito
Affiliation:
Department of Mathematical Sciences, University of Tokyo, Tokyo 153-8914, Japan (email: [email protected])
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Abstract

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Laumon introduced the local Fourier transform for ℓ-adic Galois representations of local fields, of equal characteristic p different from ℓ, as a powerful tool for studying the Fourier–Deligne transform of ℓ-adic sheaves over the affine line. In this article, we compute explicitly the local Fourier transform of monomial representations satisfying a certain ramification condition, and deduce Laumon’s formula relating the ε-factor to the determinant of the local Fourier transform under the same condition.

Keywords

local Fourier transformε-factorsramificationvanishing cyclesArtin–Schreier–Witt theory

MSC classification

Secondary: 14F20: Étale and other Grothendieck topologies and (co)homologies 11S15: Ramification and extension theory
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 6 , November 2010 , pp. 1507 - 1551
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Abbes, A. and Saito, T., Ramification of local fields with imperfect residue fields, Amer. J. Math. 124 (2002), 879–920.CrossRefGoogle Scholar
[2]Abbes, A. and Saito, T., Ramification of local fields with imperfect residue fields II, Documenta Math. Extra Volume K. Kato (2003), 3–70.Google Scholar
[3]Abbes, A. and Saito, T., Analyse micro-locale ℓ-adique en caractéristique p>0. Le cas d’un trait, Publ. Res. Inst. Math. Sci., Kyoto Univ. 45 (2009), 25–74.CrossRefGoogle Scholar
[4]Beilinson, A., Bernstein, J. and Deligne, P., Faisceaux pervers, Analysis and topology on singular spaces I, Astérisque 100 (1982), 5–171.Google Scholar
[5]Brylinski, J. L., Théorie du corps de classes de Kato et revêtements abéliens de surfaces, Ann. Inst. Fourier 33 (1983), 23–38.CrossRefGoogle Scholar
[6]Bourbaki, N., Algèbre commutative, chapitres 1–9 (Hermann, Paris, 1985).Google Scholar
[7]Deligne, P., Les constantes locales des équations fonctionnelles des fonctions L, in Modular functions of one variable II, Lecture Notes in Mathematics, vol. 349 (Springer, Berlin, 1973), 501–597.CrossRefGoogle Scholar
[8]Deligne, P., Les constantes locales de l’équation fonctionnelle de la fonction L d’Artin d’une représentation orthogonale, Invent. Math. 35 (1976), 299–316.CrossRefGoogle Scholar
[9]Deligne, P., Cohomologie étale, SGA 4 1/2, Lecture Notes in Mathematics, vol. 569 (Springer, Berlin, 1977).Google Scholar
[10]Deligne, P., La conjecture de Weil. II, Publ. Math. Inst. Hautes Études Sci. 52 (1980), 137–252.CrossRefGoogle Scholar
[11]Deligne, P. and Katz, N., Groupes de Monodromie en géométrie algébrique, SGA 7 II, Lecture Notes in Mathematics, vol. 340 (Springer, Berlin, 1973).CrossRefGoogle Scholar
[12]Dieudonné, J., Calcul infinitésimal (Hermann, Paris, 1968).Google Scholar
[13]Epp, H., Eliminating wild ramification, Invent. Math. 19 (1973), 235–249.CrossRefGoogle Scholar
[14]Fontaine, J.-M., Représentations p-adiques des corps locaux. I, in Grothendieck Festschrift, Progress in Mathematics, vol. 87 (Birkhäuser, Boston, MA, 1990), 249–309.Google Scholar
[15]Fu, L., Calculation of ℓ-adic local Fourier transformations. Preprint (2007),http://arxiv.org/abs/math/0702436.Google Scholar
[16]Henniart, G., Galois ε-factors modulo root of unity, Invent. Math. 78 (1984), 117–126.CrossRefGoogle Scholar
[17]Illusie, L., Autour du théorème de monodromie locale, Périodes p-adiques, Astérisque 223 (1994), 9–57.Google Scholar
[18]Kato, K., Vanishing cycles, ramification of valuations, and class field theory, Duke Math. J. 55 (1987), 629–659.CrossRefGoogle Scholar
[19]Kato, K., Swan conductors with differential values, in Galois representations and arithmetic algebraic geometry, Advanced Studies in Pure Mathematics, vol. 12 (North-Holland, Amsterdam, 1987), 315–342.CrossRefGoogle Scholar
[20]Kato, K., Swan conductors for characters of degree one in the imperfect residue field case, in Algebraic K-theory and algebraic number theory, Contemporary Mathematics, vol. 83 (American Mathematical Society, 1989), 101–131.CrossRefGoogle Scholar
[21]Katz, N., Gauss sums, Kloosterman sums, and monodromy groups, Annals of Mathematics Studies, vol. 116 (Princeton University Press, Princeton, NJ, 1988).CrossRefGoogle Scholar
[22]Katz, N., Travaux de Laumon, in Séminaire Bourbaki, Exp. No 691 (1988), Astérisque, 161–162 (Société Mathématique de France, Paris, 1989), 105–132.Google Scholar
[23]Laumon, G., Semi-continuité du conducteur de Swan (d’après P. Deligne), in The Euler–Poincaré characteristic, Astérisque, 82–83 (Société Mathématique de France, Paris, 1981), 173–219.Google Scholar
[24]Laumon, G., Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de Weil, Publ. Math. Inst. Hautes Études Sci. 65 (1987), 131–210.CrossRefGoogle Scholar
[25]Sabbah, C., An explicit stationary phase formula for the local formal Fourier–Laplace transform, in Singularities, Vol. 1, Contemporary Mathematics (American Mathematical Society, Providence, RI, 2008), 300–330.Google Scholar
[26]Saito, T., Trace formula for vanishing cycles of curves, Math. Ann. 276 (1987), 311–315.CrossRefGoogle Scholar
[27]Serre, J. P., Corps locaux, third edition (Hermann, Paris, 1968).Google Scholar
[28]Serre, J. P., Conducteur d’Artin des caractères réels, Invent. Math. 14 (1971), 173–183.CrossRefGoogle Scholar
[29]Serre, J. P., L’invariant de Witt de la forme Tr(x 2), Comment. Math. Helv. 59 (1984), 651–676.CrossRefGoogle Scholar
[30]Grothendieck, A. and Dieudonné, J. A., Éléments de Géométrie Algébrique, IV Étude locale des schémas et des morphismes de schémas, Publ. Math. Inst. Hautes Études Sci. 20 (1964), 24 (1965), 28 (1966), 32 (1967).CrossRefGoogle Scholar
[31]Grothendieck, A., Cohomologie locale des faisceux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), Lecture Notes in Mathematics, Advanced Studies in Pure Mathematics (North-Holland, Amsterdam, 1968).Google Scholar
[32]Artin, M., Grothendieck, A. and Verdiev, J. L., Théorie des topos et cohomologie étale des schémas (SGA 6), Lecture Notes in Mathematics, Tome 1: 269 (1972); Tome 2: 270 (1972); Tome 3: 305 (Springer, Berlin–New York, 1973).Google Scholar