K3 surfaces with maximal finite automorphism groups containing M 20
[Surfaces K3 avec un groupe fini d’automorphismes maximal contenant M 20 ]
Annales de l'Institut Fourier, Online first, 20 p.

Mukai a montré que l’ordre maximal d’un groupe fini agissant fidèlement et symplectiquement sur une surface K3 est 960 et que, si un tel groupe a pour ordre 960, alors il est isomorphe au groupe de Mathieu M 20 . Kondo a ensuite montré que l’ordre maximal d’un groupe fini agissant fidèlement sur une K3 surface est 3840 et qu’un tel groupe contient M 20 comme sous-groupe d’indice 4. Kondo a aussi montré qu’il existe une unique surface K3 sur laquelle ce groupe agit fidèlement : c’est la surface de Kummer Km(E i ×E i ). Dans cet article, nous décrivons deux autres surfaces K3 admettant un groupe fini d’automorphismes d’ordre 1920, ces deux groupes et ces deux surfaces K3 étant uniques. Ce résultat a été obtenu indépendamment par S. Brandhorst and K. Hashimoto dans un article à venir, dont le but est de classifier les groupes finis agissant fidèlement sur des K3 surfaces et dont la partie symplectique est maximale.

It was shown by Mukai that the maximum order of a finite group acting faithfully and symplectically on a K3 surface is 960 and that if such a group has order 960, then it is isomorphic to the Mathieu group M 20 . Then Kondo showed that the maximum order of a finite group acting faithfully on a K3 surface is 3840 and this group contains M 20 with index four. Kondo also showed that there is a unique K3 surface on which this group acts faithfully, which is the Kummer surface Km(E i ×E i ). In this paper we describe two more K3 surfaces admitting a big finite automorphism group of order 1920, both groups contains M 20 as a subgroup of index 2. We show moreover that these two groups and the two K3 surfaces are unique. This result was shown independently by S. Brandhorst and K. Hashimoto in a forthcoming paper, with the aim of classifying all the finite groups acting faithfully on K3 surfaces with maximal symplectic part.

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DOI : https://doi.org/10.5802/aif.3411
Classification : 14J28,  14J40,  14J10
Mots clés : Surfaces K3, automorphismes
@unpublished{AIF_0__0_0_A7_0,
     author = {Bonnaf\'e, C\'edric and Sarti, Alessandra},
     title = {K3 surfaces with maximal finite automorphism groups containing $M_{20}$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2021},
     doi = {10.5802/aif.3411},
     language = {en},
     note = {Online first},
}
Bonnafé, Cédric; Sarti, Alessandra. K3 surfaces with maximal finite automorphism groups containing $M_{20}$. Annales de l'Institut Fourier, Online first, 20 p.

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