Isometries of ideal lattices and hyperk\"ahler manifolds

TL;DR

This paper demonstrates the existence of a special hyperk"ahler manifold with a non-symplectic automorphism of maximal prime order, using ideal lattice theory in cyclotomic fields.

Contribution

It introduces a novel application of ideal lattice theory to construct hyperk"ahler manifolds with specific automorphism properties.

Findings

Existence of a hyperk"ahler manifold with a non-symplectic automorphism of order 23.

Use of ideal lattices in cyclotomic fields to prove automorphism existence.

Maximal prime order automorphism in this deformation family.

Abstract

We prove that there exists a holomorphic symplectic manifold deformation equivalent to the Hilbert scheme of two points on a K3 surface that admits a non-symplectic automorphism of order 23, that is the maximal possible prime order in this deformation family. The proof uses the theory of ideal lattices in cyclomotic fields.