[Caractéristiques d’Euler équivariantes et faisceau résolvant.]
Nous obtenons pour certains revêtements modérés de surfaces arithmétiques des expressions des caractéristiques d’Euler équivariantes du faisceau canonique et de sa racine carrée qui font apparaître une forme quadratique décrite en terme de nombres d’intersection. Ces formules se prêtent au calcul. Elles nous permettent notamment de donner des exemples où ces caractéristiques ainsi que celle du faisceau structural sont deux à deux distinctes et non triviales. Nos résultats s’obtiennent par l’utilisation du théorème de Riemann-Roch local et par un calcul de résolvantes.
For certain tame abelian covers of arithmetic surfaces we obtain formulas, involving a quadratic form derived from intersection numbers, for the equivariant Euler characteristics of both the canonical sheaf and also its square root. These formulas allow us to carry out explicit calculations; in particular, we are able to exhibit examples where these two Euler characteristics and that of the structure sheaf are all different and non-trivial. Our results are obtained by using resolvent techniques together with the local Riemann-Roch Theorem.
Keywords: Euler characteristic, resolvent, intersection numbers.
Mot clés : caratéristique d’Euler, résolvante, nombre d’intersection.
@article{AIF_2012__62_6_2315_0,
author = {Cassou-Nogu\`es, Ph. and Taylor, M.J.},
title = {Equivariant {Euler} characteristics and sheaf resolvents},
journal = {Annales de l'Institut Fourier},
pages = {2315--2345},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {62},
number = {6},
year = {2012},
doi = {10.5802/aif.2750},
mrnumber = {3060759},
language = {en},
url = {http://www.numdam.org/articles/10.5802/aif.2750/}
}
TY - JOUR AU - Cassou-Noguès, Ph. AU - Taylor, M.J. TI - Equivariant Euler characteristics and sheaf resolvents JO - Annales de l'Institut Fourier PY - 2012 SP - 2315 EP - 2345 VL - 62 IS - 6 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2750/ DO - 10.5802/aif.2750 LA - en ID - AIF_2012__62_6_2315_0 ER -
%0 Journal Article %A Cassou-Noguès, Ph. %A Taylor, M.J. %T Equivariant Euler characteristics and sheaf resolvents %J Annales de l'Institut Fourier %D 2012 %P 2315-2345 %V 62 %N 6 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2750/ %R 10.5802/aif.2750 %G en %F AIF_2012__62_6_2315_0
Cassou-Noguès, Ph.; Taylor, M.J. Equivariant Euler characteristics and sheaf resolvents. Annales de l'Institut Fourier, Tome 62 (2012) no. 6, pp. 2315-2345. doi : 10.5802/aif.2750. http://www.numdam.org/articles/10.5802/aif.2750/
[1] Galois module structure of de Rham cohomology, J. de Théorie des Nombres de Bordeaux, Volume 4 (1991), pp. 1-18 | Numdam | MR | Zbl
[2] Galois structure of the de Rham cohomology of tame covers of schemes, Ann. of Math., Volume 139 (1994), pp. 443-490 | MR | Zbl
[3] Equivariant Euler-Poincaré characteristics and tameness, Astérisque, Volume 209 (1992), pp. 179-194 | MR | Zbl
[4] Tame actions of group schemes: integrals amd slices, Duke Math. J., Volume 82 (1996), pp. 269-308 | MR | Zbl
[5] Riemann-Roch type theorems for arithmetic schemes with a finite group action, J. Reine Angew. Math., Volume 489 (1997), pp. 151-187 | MR | Zbl
[6] -constants and the Galois structure of de Rham cohomology, Ann. of Math., Volume 146 (1997), pp. 411-473 | MR | Zbl
[7] Cubic structures, equivariant Euler characteristics and lattices of modular forms, Ann. of Math., Volume 170 (2009), pp. 561-608 | MR
[8] Hermitian modules in Galois extensions of number fields and Adams operations, Ann. of Math., Volume 135 (1992) no. 2, pp. 271-296 | MR | Zbl
[9] Arithmetic and Galois module structure for tame extensions, J. Crelle, Volume 286/287 (1976), pp. 380-440 | MR | Zbl
[10] Galois module structure of algebraic integers, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 1, Springer-Verlag, 1983 | MR | Zbl
[11] Intersection theory, 2nd edition, Ergebnisse der Mathematik und ihrer Grenzgebiete 3, Folge, Springer-Verlag, 1998 | MR | Zbl
[12] Residues and Duality, Lect. Notes in Math., 20, Springer-Verlag, 1966 | MR
[13] Algebraic Geometry, Grad. Texts Math., 52, Springer-Verlag, 1977 | MR | Zbl
[14] Introduction to Arakelov theory, Springer-Verlag, 1988 | MR
[15] Galois modules and the Theorem of the Cube, Invent. Math., Volume 133 (1998), pp. 193-225 | MR | Zbl
[16] Galois module structure and the -filtration, Compositio Mathematica, Volume 121 (2000), pp. 79-104 | MR | Zbl
[17] Revêtements à ramification impaire et thêta-caractéristiques, C. R. Acad. Sci. Paris, Volume 311, Série 1 (1990), pp. 547-552 | MR | Zbl
[18] On the self duality of a ring of integers as a Galois module, Invent. Math., Volume 46 (1978), pp. 173-177 | MR | Zbl
[19] On Fröhlich’s conjecture for rings of integers of tame extensions, Invent. Math., Volume 63 (1981), pp. 41-79 | MR | Zbl
[20] Class groups of group rings, London Math. Soc. Lect. Notes, C. U. P., 1983 no. 91 | Zbl
[21] Introduction to cyclotomic fields, Graduate Texts in Mathematics, Springer Verlag, 1980 | MR | Zbl
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