Local Oort groups and the isolated differential data criterion

Huy Dang; Soumyadip Das; Kostas Karagiannis; Andrew Obus; Vaidehee Thatte

doi:10.5802/jtnb.1200

Huy Dang ¹; Soumyadip Das ²; Kostas Karagiannis ³; Andrew Obus ⁴; Vaidehee Thatte ⁵

¹ Institute of Mathematics Vietnam Academy of Science and Technology 18 Hoang Quoc Viet Road Cau Giay District Hanoi 10307, Vietnam
² Tata Institute of Fundamental Research Homi Bhaba Road, Colaba Mumbai 400005, India
³ Aristotle University of Thessaloniki Department of Mathematics School of Sciences 54124 Thessaloniki, Greece
⁴ Baruch College 1 Bernard Baruch Way New York, NY 10010, USA
⁵ Binghamton University Binghamton New York 13902-6000, USA

Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 1, pp. 251-269.

Abstract
Résumé

It is conjectured that if $k$ is an algebraically closed field of characteristic $p > 0$ , then any branched $G$ -cover of smooth projective $k$ -curves where the “KGB” obstruction vanishes and where a $p$ -Sylow subgroup of $G$ is cyclic lifts to characteristic $0$ . Obus has shown that this conjecture holds given the existence of certain meromorphic differential forms on $ℙ_{k}^{1}$ 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 $D_{25}$ -covers and $D_{27}$ -covers lift to characteristic zero.

Il est conjecturé que si $k$ est un corps algébriquement clos de caractéristique $p > 0$ , alors tout $G$ -revêtement ramifié de courbes projectives lisses sur $k$ pour lequel l’obstruction « KGB » s’annule et tel qu’un $p$ -sous-groupe de Sylow de $G$ est cyclique peut être relevé en caractéristique $0$ . Obus a démontré que cette conjecture est vraie si l’on suppose l’existence de certaines formes différentielles méromorphes sur $ℙ_{k}^{1}$ 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 $D_{25}$ -revêtements et tous les $D_{27}$ -revêtements se relèvent en caractéristique zéro.

Received: 2020-09-22
Accepted: 2021-06-05
Published online: 2022-07-07

DOI: 10.5802/jtnb.1200

Classification: 14H37, 12F10, 13B05
Mots-clés : local lifting problem, local Oort group, differential data, Vandermonde determinant

Author's affiliations:

Huy Dang ¹; Soumyadip Das ²; Kostas Karagiannis ³; Andrew Obus ⁴; Vaidehee Thatte ⁵

¹ Institute of Mathematics Vietnam Academy of Science and Technology 18 Hoang Quoc Viet Road Cau Giay District Hanoi 10307, Vietnam
² Tata Institute of Fundamental Research Homi Bhaba Road, Colaba Mumbai 400005, India
³ Aristotle University of Thessaloniki Department of Mathematics School of Sciences 54124 Thessaloniki, Greece
⁴ Baruch College 1 Bernard Baruch Way New York, NY 10010, USA
⁵ Binghamton University Binghamton New York 13902-6000, USA

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Huy Dang and Soumyadip Das and Kostas Karagiannis and Andrew Obus and Vaidehee Thatte},
     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},
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     doi = {10.5802/jtnb.1200},
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Huy Dang; Soumyadip Das; Kostas Karagiannis; Andrew Obus; Vaidehee Thatte. Local Oort groups and the isolated differential data criterion. Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 1, pp. 251-269. doi : 10.5802/jtnb.1200. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1200/

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