The elliptic modular surface of level 4 and its reduction modulo 3

Abstract

The elliptic modular surface of level 4 is a complex K3 surface with Picard number 20. This surface has a model over a number field such that its reduction modulo 3 yields a surface isomorphic to the Fermat quartic surface in characteristic 3, which is supersingular. The specialization induces an embedding of the Neron–Severi lattices. Using this embedding, we determine the automorphism group of this K3 surface over a discrete valuation ring of mixed characteristic whose residue field is of characteristic 3. The elliptic modular surface of level 4 has a fixed-point-free involution that gives rise to the Enriques surface of type IV in Nikulin–Kondo–Martin’s classification of Enriques surfaces with finite automorphism group. We investigate the specialization of this involution to characteristic 3.