Title: Prime Fano threefolds and Leech-like lattices

Abstract: I computed the automorphism groups of certain two models of supersingular K3 surfaces with Artin invariant one in Beijing 2019.  One is over the binary and the other over ternary field.  Recently I found two more models over the binary and quaternary fields.  I will report their relation with prime Fano threefolds classified into 10 deformation types of degree (-K)^3=2g-2 with g=2,…,10 and 12, by Fano-Iskovskih.  Our four models are related with g=7, 8, 9, 10 via three Leech-like lattices; Leech, Barnes-Wall and Coxeter-Todd.  Another Leech-like lattice is related with g=12 via the regular 4-polytope {3,3,5}.