There is no Enriques surface over the integers

• 成果类型: Article
• 署名作者: Schroeer, Stefan
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2023.197.1.1
• 发表日期: 2023
• 页码: 1-63
• 关键词: abelian-varieties; number-fields; COHOMOLOGY; FINITENESS; CURVES; fibers

摘要

We show that there is no family of Enriques surfaces over the ring of integers. This extends non-existence results of Minkowski for families of finite e ' tale schemes, of Tate and Ogg for families of elliptic curves, and of Fontaine and Abrashkin for families of abelian varieties and more general smooth proper schemes with certain restrictions on Hodge numbers. Our main idea is to study the local system of numerical classes of invertible sheaves. Among other things, our result also hinges on counting rational points, Lang's classification of rational elliptic surfaces in characteristic two, the theory of exceptional Enriques surfaces due to Ekedahl and ShepherdBarron, some recent results on the base of their versal deformation, Shioda's theory of Mordell-Weil lattices, and an extensive combinatorial study for the pairwise interaction of genus-one fibrations.