Hypergeometric groups and dynamics on \(K3\) surfaces (Q2126091)

The paper under review explores interactions between hypergeometric groups and dynamics on \(K3\) surfaces, especially, showing that a certain class of hypergeometic groups and the related hypergeometric lattices yield \(K3\) surface autormorphisms (e.g., with Siegel disks) of positive entropy. A hypergeometric group is a matrix group modelled after monodromy group of a generalized hypergeometic differential equation. Let \(H=<A,B> \subset GL(n,{\mathbb(C)})\) be a hypergeometric group generated by two invertible matrices \(A\) and \(B\) such that \(rank(A-B)=1\) (or equivalently, \(rank(I-C)=1\) for \(C=A^{-1}B\)). There is a unique unimodular even lattice \(L\) of rank \(n\) equipped with the \(H\)-invariant form, called the hypergeometic lattice. Let \(X\) be a \(K3\) surface with the \(K3\) lattice \(L=H^2(X,\mathbb{Z})\) (an even unimodular lattice of rank \(22\) and signature \((3,19)\)). Any \(K3\) surface automorphism \(f: X\to X\) induces a lattice automorphism \(f^*:L\to L\) preserving the \(K3\) structure. An automorphism \(f: X\to X\) gives rise to the various invariants: the entropy \(h(f)=\mbox{log}\,\lambda(f)\) where \(\lambda(f)\) is the spectral radius of \(f^*|H^2(X)\), the special eigenvalue \(\delta(f)\in S^1\) defined by \(f^*\eta=\delta(f)\eta\) for any nowhere vanishing holomorphic \(2\)-form \(\eta\) on \(X\), and the special trace \(\tau(f)=\delta(f)+\delta(f)^{-1}\). Note that \(\lambda(f)=1\) or a Salem number \(\lambda>1\), while \(\delta(f)\) is either a root of \(1\) or a conjudate of the Salem number \(\lambda\). A question: In the case of \(n=22\), when does a hypergeometic lattice \(L\) or its negative \(L(-1)\) become a \(K3\) lattice with a Hodge structure such that the matrix \(A\) or \(B\) is a Hodge isometry (of elliptic, parabolic or hyperbolic type)? The paper under review presents examples answering this question, illustrating the existence of non-projective \(K3\) surface automorphisms of the smallest possible entropy \(\mbox{log}\,\lambda_L\) where \(\lambda_L\) is a Lehmer's nubmer which admits a \(3\)-periodic cycle of Siegel disks.