On correspondences between K3 surfaces

Using arithmetic of integral quadratic forms and results of Mukai, it is proved among other things that an endomorphism over $\mathbf Q$ of the cohomology lattice of a $K3$ surface over $\mathbf C$ preserving the Hodge structure and the intersection form is induced by an algebraic cycle (as was conjectured in [2]) provided that the Picard lattice $S_X$ of the surface $X$ represents zero (in particular, this is so if $\operatorname{rg}S_X\geqslant5$). Previously this result was obtained by Mukai under the assumption that $\operatorname{rg}S_X\geqslant11$.
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