论文标题
何时由$ \ boldsymbol {p} $ - 可划分的群体确定？
When is a polarised abelian variety determined by its $\boldsymbol{p}$-divisible group?
论文作者
论文摘要
我们研究Siegel模块化品种$ \ MATHCAL {a} _g \ otimes \ Overline {\ Mathbb {f}} _ p $ of属$ g $及其suppersingular locus $ \ mathcal $ \ mathcal {s} _g $。作为我们的主要结果，我们精确地确定了$ \ nathcal {s} _g $是不可记论的，我们列出了$ \ \ \ m nathcal {a} _g \ otimes \ otimies \ edline {\ mathbb {f}} _ p $的所有$ x $ in Mathcal {a} _g \ otimes \ otimes \ otline {\ mathbb {f} _ p $，其中相应的中央叶$ \ nath $ cons $ cons $ cons $ cons $ cons $ consection in sho possions对于一个两极分化的阿贝尔品种，它由其相关的两极分化$ p $可分别的群体唯一决定。第一个问题转化为四季度Hermitian Lattices的第一类问题。第二个问题也转化为第一类问题，其解决方案涉及质量公式，自动形态组，以及对属属$ g = 4 $的Ekedahl-oort阶层的仔细分析。
We study the Siegel modular variety $\mathcal{A}_g \otimes \overline{\mathbb{F}}_p$ of genus $g$ and its supersingular locus $\mathcal{S}_g$. As our main result we determine precisely when $\mathcal{S}_g$ is irreducible, and we list all $x$ in $\mathcal{A}_g \otimes \overline{\mathbb{F}}_p$ for which the corresponding central leaf $\mathcal{C}(x)$ consists of one point, that is, for which $x$ corresponds to a polarised abelian variety which is uniquely determined by its associated polarised $p$-divisible group. The first problem translates to a class number one problem for quaternion Hermitian lattices. The second problem also translates to a class number one problem, whose solution involves mass formulae, automorphism groups, and a careful analysis of Ekedahl-Oort strata in genus $g=4$.
