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

Abstract

In this paper, for each 𝑑 > 0, we study the minimum integer ℎ_3,2𝑑 ∈ ℕ for which there exists a complex polarized K3 surface (𝑋, 𝐻) of degree 𝐻² = 2𝑑 and Picard number 𝜌(𝑋) := rank Pic 𝑋 = ℎ_3,2𝑑 admitting an automorphism of order 3. We show that ℎ_3,2 ∈ {4, 6} and ℎ_3,2𝑑 = 2 for 𝑑 > 1. Analogously, we study the minimum integer ℎ^*_3,2𝑑 ∈ 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 ℚ realizing these bounds.