Generalized Shioda--Inose structures of order 3

TL;DR

This paper generalizes the classical Shioda--Inose construction from involutions to order 3 automorphisms, linking Abelian surfaces and K3 surfaces through explicit geometric structures.

Contribution

It introduces generalized Shioda--Inose structures of order 3, extending Morrison's work to new automorphisms and providing explicit examples and classifications.

Findings

Defined generalized structures of order 3

Identified associated K3 and Abelian surfaces

Provided explicit examples of these structures

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 the 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.