Il est conjecturé que si est un corps algébriquement clos de caractéristique , alors tout -revêtement ramifié de courbes projectives lisses sur pour lequel l’obstruction « KGB » s’annule et tel qu’un -sous-groupe de Sylow de est cyclique peut être relevé en caractéristique . Obus a démontré que cette conjecture est vraie si l’on suppose l’existence de certaines formes différentielles méromorphes sur dont les propriétés sont détérminées par la filtration de ramification du revêtement. Nous présentons ici un algorithme plus efficace pour calculer ces formes. En conséquence, nous pouvons prouver que tous les -revêtements et tous les -revêtements se relèvent en caractéristique zéro.
It is conjectured that if is an algebraically closed field of characteristic , then any branched -cover of smooth projective -curves where the “KGB” obstruction vanishes and where a -Sylow subgroup of is cyclic lifts to characteristic . Obus has shown that this conjecture holds given the existence of certain meromorphic differential forms on with behavior determined by the ramification data of the cover. We give a more efficient procedure to compute these forms than was previously known. As a consequence, we show that all -covers and -covers lift to characteristic zero.
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Mots clés : local lifting problem, local Oort group, differential data, Vandermonde determinant
@article{JTNB_2022__34_1_251_0, author = {Dang, Huy and Das, Soumyadip and Karagiannis, Kostas and Obus, Andrew and Thatte, Vaidehee}, title = {Local {Oort} groups and the isolated differential data criterion}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {251--269}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {34}, number = {1}, year = {2022}, doi = {10.5802/jtnb.1200}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jtnb.1200/} }
TY - JOUR AU - Dang, Huy AU - Das, Soumyadip AU - Karagiannis, Kostas AU - Obus, Andrew AU - Thatte, Vaidehee TI - Local Oort groups and the isolated differential data criterion JO - Journal de théorie des nombres de Bordeaux PY - 2022 SP - 251 EP - 269 VL - 34 IS - 1 PB - Société Arithmétique de Bordeaux UR - http://archive.numdam.org/articles/10.5802/jtnb.1200/ DO - 10.5802/jtnb.1200 LA - en ID - JTNB_2022__34_1_251_0 ER -
%0 Journal Article %A Dang, Huy %A Das, Soumyadip %A Karagiannis, Kostas %A Obus, Andrew %A Thatte, Vaidehee %T Local Oort groups and the isolated differential data criterion %J Journal de théorie des nombres de Bordeaux %D 2022 %P 251-269 %V 34 %N 1 %I Société Arithmétique de Bordeaux %U http://archive.numdam.org/articles/10.5802/jtnb.1200/ %R 10.5802/jtnb.1200 %G en %F JTNB_2022__34_1_251_0
Dang, Huy; Das, Soumyadip; Karagiannis, Kostas; Obus, Andrew; Thatte, Vaidehee. Local Oort groups and the isolated differential data criterion. Journal de théorie des nombres de Bordeaux, Tome 34 (2022) no. 1, pp. 251-269. doi : 10.5802/jtnb.1200. http://archive.numdam.org/articles/10.5802/jtnb.1200/
[1] Obstructions locales au relèvement de revêtements galoisiens de courbes lisses, C. R. Math. Acad. Sci. Paris, Volume 326 (1998) no. 1, pp. 55-58 | DOI | MR | Zbl
[2] Déformations formelles des revêtements sauvagement ramifiés de courbes algébriques, Invent. Math., Volume 141 (2000) no. 1, pp. 195-238 | DOI | MR | Zbl
[3] The local lifting problem for dihedral groups, Duke Math. J., Volume 134 (2006) no. 3, pp. 421-452 | DOI | MR | Zbl
[4] Deformation data, Belyi maps, and the local lifting problem, Trans. Am. Math. Soc., Volume 361 (2009) no. 12, pp. 6645-6659 | DOI | MR | Zbl
[5] Artin characters, Hurwitz trees and the lifting problem, Math. Ann., Volume 345 (2009) no. 3, pp. 711-730 | DOI | MR | Zbl
[6] Une nouvelle opération sur les formes différentielles, C. R. Acad. Sci. Paris, Volume 244 (1957), pp. 426-428 | MR | Zbl
[7] The local lifting problem for actions of finite groups on curves, Ann. Sci. Éc. Norm. Supér., Volume 44 (2011) no. 4, pp. 537-605 | DOI | Numdam | MR | Zbl
[8] Abstract algebra, John Wiley & Sons, 2004, xii+932 pages | MR
[9] Prolongement de revêtements galoisiens en géométrie rigide, Compos. Math., Volume 104 (1996) no. 3, pp. 305-331 | Numdam | MR | Zbl
[10] Liftings of Galois covers of smooth curves, Compos. Math., Volume 113 (1998) no. 3, pp. 237-272 | DOI | MR | Zbl
[11] Séminaire de géométrie algébrique du Bois Marie 1960–61. Revêtements étales et groupe fondamental (SGA 1) (Grothendieck, Alexander, ed.), Documents Mathématiques, 3, Société Mathématique de France, 2003, xviii+327 pages | MR
[12] Generalized Vandermonde determinants, Trans. Am. Math. Soc., Volume 31 (1929) no. 3, pp. 464-476 | DOI | MR | Zbl
[13] Elliptic functions, Graduate Texts in Mathematics, 112, Springer, 1987, xii+326 pages (With an appendix by J. Tate) | DOI | MR
[14] The (local) lifting problem for curves, Galois-Teichmüller theory and arithmetic geometry (Advanced Studies in Pure Mathematics), Volume 63, Mathematical Society of Japan, 2012, pp. 359-412 | DOI | MR | Zbl
[15] The local lifting problem for , Algebra Number Theory, Volume 10 (2016) no. 8, pp. 1683-1693 | DOI | MR | Zbl
[16] A generalization of the Oort conjecture, Comment. Math. Helv., Volume 92 (2017) no. 3, pp. 551-620 | DOI | MR | Zbl
[17] Lifting of curves with automorphisms, Open Problems in Arithmetic Algebraic Geometry (Advanced Lectures in Mathematics (ALM)), Volume 46, International Press, 2019, pp. 9-59 | MR | Zbl
[18] Wild tame-by-cyclic extensions, J. Pure Appl. Algebra, Volume 214 (2010) no. 5, pp. 565-573 | DOI | MR | Zbl
[19] Cyclic extensions and the local lifting problem, Ann. Math., Volume 180 (2014) no. 1, pp. 233-284 | DOI | MR | Zbl
[20] Relèvement en caractéristique zéro d’actions de groupes abéliens de type , Ph. D. Thesis, Université Bordeaux I (France) (2002) (available at http://www.math.u-bordeaux1.fr/~mmatigno/Pagot-These.pdf)
[21] The Oort conjecture on lifting covers of curves, Ann. Math., Volume 180 (2014) no. 1, pp. 285-322 | DOI | MR | Zbl
[22] Wildly ramified covers with large genus, J. Number Theory, Volume 119 (2006) no. 2, pp. 194-209 | DOI | MR | Zbl
[23] Corps locaux, Publications de l’Institut de Mathématique de l’Université de Nancago, 8, Hermann, 1968, 245 pages
[24] The local lifting problem for , Isr. J. Math., Volume 228 (2018) no. 2, pp. 587-626 | DOI | MR | Zbl
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