Local Oort groups and the isolated differential data criterion

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

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

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 ${\mathbb{P}}_k^1$ 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.

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 ${\mathbb{P}}_k^1$ 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.

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

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