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arXiv · 1806.05787

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

Ichiro Shimada
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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 N\'eron-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.

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Ichiro Shimada. 2018-06-15. The elliptic modular surface of level 4 and its reduction modulo 3. https://arxiv.org/abs/1806.05787

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