Generalized Shioda–Inose structures of order 3

Alice Garbagnati; Yulieth Prieto-Montañez

doi:10.1515/advgeom-2024-0005

Article

Generalized Shioda–Inose structures of order 3

Alice Garbagnati and Yulieth Prieto-Montañez

Published/Copyright: April 26, 2024

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From the journal Advances in Geometry Volume 24 Issue 2

Abstract

A Shioda–Inose structure is a geometric construction which associates to an Abelian surface a projective K3 surface in such a way that their transcendental lattices are isometric. This geometric construction was described by Morrison by considering special symplectic involutions on the K3 surfaces. After Morrison several authors provided explicit examples. The aim of this paper is to generalize Morrison’s results and some of the known examples to an analogous geometric construction involving not involutions, but order 3 automorphisms. Therefore, we define generalized Shioda–Inose structures of order 3, we identify the K3 surfaces and the Abelian surfaces which appear in these structures and we provide explicit examples.

Keywords: K3 surface; symplectic automorphism; Abelian surface; Shioda–Inose structure; elliptic K3 surface

MSC 2010: 14J28; 14J50

Acknowledgements

The authors thank Bert van Geemen for many suggestions on the preliminary version of the paper and Benedetta Piroddi for useful remarks on the embeddings of some lattices. We also thank the anonymous referee.

Communicated by: I. Coskun

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Received: 2023-03-19

Revised: 2023-10-31

Published Online: 2024-04-26

Published in Print: 2024-04-25

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https://doi.org/10.1515/advgeom-2024-0005

Keywords for this article

K3 surface; symplectic automorphism; Abelian surface; Shioda–Inose structure; elliptic K3 surface