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 C11 of order 11, C11 ⋊ C5 , PSL2(F11) 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.
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Igor V. Dolgachev, JongHae Keum, K3 surfaces with a symplectic automorphism of order 11. J. Eur. Math. Soc. 11 (2009), no. 4, pp. 799–818