Fourier-Mukai transformations on K3 surfaces with $\rho =1$ and Atkin-Lehner involutions

TL;DR

The paper establishes a connection between Fourier-Mukai transformations on certain K3 surfaces and Atkin-Lehner involutions, confirming a previously expected relationship.

Contribution

It demonstrates a surjective relationship between Fourier-Mukai transformations on K3 surfaces with Picard number one and Atkin-Lehner involutions, confirming a conjecture.

Findings

Surjection from Fourier-Mukai transformations to Atkin-Lehner involutions

Confirmation of the expected relationship in prior conjecture

Focus on K3 surfaces with Picard number one

Abstract

We show that there is a surjection from the Fourier-Mukai transformations on projective K3 surfaces with the Picard number $\rho (X)=1$ to so called to the group of Atkin-Lehner involutions. This was expected in Hosono-Lian-Oguiso-Yau's paper.