Using elliptic structures, we show that any supersingular K3 surface of Artin invariant 1 in characteristic p = 5, 7, 13 has an automorphism the entropy of which is the natural logarithm of a Salem number of degree 22.

In this note, we consider crepant resolutions of the quotient varieties of smooth quintic threefolds by Gorenstein group actions. We compute their Hodge numbers via McKay correspondence. In this way, we find some new pairs (h 1,1 , h 2,1 ) of Hodge numbers of Calabi-Yau threefolds.

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We study automorphism groups of smooth quintic threefolds. Especially, we describe all the maximal ones with explicit examples of target quintic threefolds. There are exactly 22 such groups.