K3 surfaces with a symplectic automorphism of order 11

Abstract

We classify possible finite groups of symplectic automorphisms of K3 surfaces of order divisible by 11. The characteristic of the ground field must be equal to 11. The complete list of such groups consists of five groups: the cyclic group C_{11} of order 11, C_{11} ⋊ C_5 , \mathrm{PSL2}(\mathbb F_{11}) and the Mathieu groups M_{11} , M_{22} . We also show that a surface X admitting an automorphism g of order 11 admits a g -invariant elliptic fibration with the Jacobian fibration isomorphic to one of explicitly given elliptic K3 surfaces.