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Automorphisms of projective K3 surfaces with minimum entropy

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Notes

  1. The lattice \(\mathrm{Pic}(X)\) can be degenerate when \(X\) is not projective, e.g. one can have \(\mathrm{Pic}(X) = {\mathbb {Z}}E\) with \(E^2=0\).

  2. If we had chosen \(v|23\) of degree \(4\), then \(f_1\) would have period \(n=25440\) on \(G(L_1) \cong {\mathbb {F}}_{23}^8\), which would give \(D(n) = 74 > 12\). Any candidate for \(f|L_1^\perp \) would then be too large to fit inside the cohomology of a K3 surface.

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Acknowledgments

I would like to thank E. Bedford and B. Gross for useful conversations related to this work.

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Authors and Affiliations

  1. Mathematics Department, Harvard University, 1 Oxford St, Cambridge, MA, 02138-2901, USA

    Curtis T. McMullen

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Research supported by the NSF, MPI and the Humboldt Foundation.

Appendix: Small Salem numbers

Appendix: Small Salem numbers

See Table 1

Table 1 Small Salem numbers
Full size table

This table gives the value of the smallest Salem number \(\lambda _d\) of degree \(d\), its minimal polynomial \(S(x)\), and the determinant \(\Delta = |S(1)S(-1)|\) of its principal lattice, for all even \(d \le 20\). The entries are listed in increasing order as real numbers.

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McMullen, C.T. Automorphisms of projective K3 surfaces with minimum entropy. Invent. math. 203, 179–215 (2016). https://doi.org/10.1007/s00222-015-0590-z

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