We describe the structure of the supersingular locus of a Shimura variety for a quaternionic unitary similitude group of degree 2 over a ramified odd prime p if the level at p is given by a special maximal compact open subgroup. More precisely, we show that such a locus is purely 2-dimensional, and every irreducible component is birational to the Fermat surface. Furthermore, we have an estimation of the numbers of connected and irreducible components. To prove these assertions, we completely determine the structure of the underlying reduced scheme of the Rapoport–Zink space for the quaternionic unitary similitude group of degree 2, with a special parahoric level. We prove that such a scheme is purely 2-dimensional, and every irreducible component is isomorphic to the Fermat surface. We also determine its connected components, irreducible components and their intersection behaviors by means of the Bruhat–Tits building of $${{\,\mathrm{PGSp}\,}}_4({\mathbb {Q}}_p)$$ PGSp 4 ( Q p ) . In addition, we compute the intersection multiplicity of the GGP cycles associated to an embedding of the considering Rapoport–Zink space into the Rapoport–Zink space for the unramified $${{\,\mathrm{GU}\,}}_{2,2}$$ GU 2 , 2 with hyperspecial level for the minuscule case.

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关于 2 次四元幺正群的 Shimura 变体的超奇异位点

如果 p 处的水平由一个特殊的极大紧致开子群给出，我们描述了一个 Shimura 变体的超奇异轨迹结构，用于在一个分支奇素数 p 上的 2 次四元数酉相似群。更准确地说，我们证明了这样的轨迹是纯二维的，并且每个不可约分量对于费马表面都是双有理的。此外，我们估计了连接和不可约组件的数量。为了证明这些断言，我们完全确定了 2 次四元数幺正群的 Rapoport-Zink 空间的基本简化方案的结构，具有特殊的超平行水平。我们证明了这样的方案是纯二维的，并且每个不可约分量都与费马面同构。我们还确定了它的连接组件，通过 $${{\,\mathrm{PGSp}\,}}_4({\mathbb {Q}}_p)$$ PGSp 4 ( Q p ) 的 Bruhat-Tits 构建的不可约分量及其相交行为。此外，我们计算了与考虑的 Rapoport-Zink 空间嵌入到未分枝的 $${{\,\mathrm{GU}\,}}_{2 的 Rapoport-Zink 空间相关的 GGP 循环的交集多重性,2}$$ GU 2 , 2 具有极小情况的超特殊级别。