论文标题
整数上没有Enriques表面
There is no Enriques surface over the integers
论文作者
论文摘要
我们表明，整数上没有一个富度的表面。这扩展了Minkowski的不存在结果，用于有限的典型计划，TATE和OGG的家族的椭圆曲线家族，以及Fontaine和Abrashkin的Abelian品种家庭以及更一般的平滑顺畅方案，以及对Hodge数字的某些限制。我们的主要思想是研究可逆滑轮的数值类别的本地系统。 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 Shepherd-Barron, 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.
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 étale 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 Shepherd-Barron, 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.
