The automorphism group of the singular \(K3\) surface of discriminant \(7\) (Q2866580)

The aim of the paper is to show a system of generators of the automorphism group of a \(K3\) surface which is singular (i.e. of Picard number \(20\)) and has discriminant \(7\). The strategy to do this is as follows. First the author describes the Naruki \(K3\) surface (see \textit{I. Naruki}'s paper [``On a K3 surface which is a ball quotient'', MPIM preprint series, No. 1985-52 (1985)]) and he shows that the singular \(K3\) surface of discriminant 7 \(X\) can be identified to the Naruki \(K3\) surface. This gives information on the geometry of the surface.NEWLINENEWLINEThen the author shows that the automorphism group of \(X\) is isomorphic to the automorphism group of the ample cone \(D(S_X)\) of \(X\) and in order to study this group, he embeds the Néron-Severi lattice \(S_X\) of \(X\) in the even unimodular lattice \(II_{1,25}\) of signature \((1,25)\) whose description can be found in the paper. This embedding is done in such a way that \(S_X\) is the orthogonal complement of a \(A_6\) lattice. Then thanks to the notion of fundamental domain and some other tools, the paper shows some elliptic fibrations on the surface \(X\). When taking into account the inversion involutions obtained by these elliptic fibrations, they give a set of generators for the automorphism group of \(X\) together with \(\mathrm{PGL}_2(7)\).