Polarized K3 surfaces with an automorphism of order 3 and low Picard number

TL;DR

This paper investigates the minimal Picard number for polarized K3 surfaces with an automorphism of order 3, providing bounds, explicit examples, and distinctions based on automorphism action on the Picard lattice.

Contribution

It determines the minimal Picard numbers for such K3 surfaces with automorphisms of order 3, including cases with automorphisms acting trivially on the Picard lattice, and constructs explicit examples.

Findings

h_{3,2} is either 4 or 6; h_{3,2d}=2 for d>1

h^*_{3,2d} equals 2 for d>1 and 6 for d=1

Explicit examples of K3 surfaces over Q are provided

Abstract

In this paper, for each $d>0$, we study the minimum integer $h_{3,2d}\in \mathbb{N}$ for which there exists a complex polarized K3 surface $(X,H)$ of degree $H^2=2d$ and Picard number $\rho (X):=\textrm{rank } \textrm{Pic } X = h_{3,2d}$ admitting an automorphism of order $3$. We show that $h_{3,2}\in\{ 4,6\}$ and $h_{3,2d}=2$ for $d>1$. Analogously, we study the minimum integer $h^*_{3,2d}\in \mathbb{N}$ for which there exists a complex polarized K3 surface $(X,H)$ as above plus the extra condition that the automorphism acts as the identity on the Picard lattice of $X$. We show that $h^*_{3,2d}$ is equal to $2$ if $d>1$ and equal to $6$ if $d=1$. We provide explicit examples of K3 surfaces defined over $\mathbb{Q}$ realizing these bounds.