superspecial abelian varieties 的 isogeny 图和 Brandt 矩阵
Isogeny graphs of superspecial abelian varieties and Brandt matrices
AI总结：
本文研究了 superspecial abelian varieties 的 isogeny 图和 Brandt 矩阵，证明了三种 isogeny 图的连通性，并探讨了它们的性质和 Ramanujan 性质。
摘要：
固定素数 p 和 ℓ，其中 ℓ≠p。如果 (A,λ) 是一个 g 维的 principally polarized abelian variety，则一个 (ℓ)^g-isogeny 的核是一个 maximal isotropic subgroup of the ℓ-torsion of A；图像有一个 natural principal polarization。我们定义了三种与这样的 (ℓ)^g-isogenies 相关的 isogeny 图——big isogeny graph Gr_g(ℓ,p)，little isogeny graph gr_g(ℓ,p)，以及 enhanced isogeny graph ˜gr_g(ℓ,p)。我们证明了这三个 isogeny 图都是 connected。证明的一个成分是 quaternionic unitary group 的 strong approximation，这之前已经被应用于 characteristic p 的 abelian varieties 的 moduli 由 Chai, Ekedahl/Oort, 和 Chai/Oort 应用。这三个 isogeny 图的 adjacency matrices 用 Hashimoto, Ibukiyama, Ihara, 和 Shimizu 定义的 Brandt matrices 表示。我们研究了这些 Brandt matrices 的一些基本性质，并使用 Brandt graphs 的概念重新阐述理论。我们证明了 isogeny 图 Gr_g(ℓ,p) 和 gr_g(ℓ,p) 实际上是我们的 Brandt graphs。我们给出了 gr_g(ℓ,p) 和 ˜gr_g(ℓ,p) 的 ℓ-adic uniformization。对于 supersingular elliptic curves 的 (ℓ+1)-regular isogeny 图 Gr_1(ℓ,p) 是 well known to be Ramanujan。我们计算了 g>1, ℓ 和 p 的范围内 Brandt matrices。这些计算给出了四个例子，其中 g>1 的 regular 图 Gr_g(ℓ,p) 有两个顶点并且是 Ramanujan，而所有其他计算的 g>1 和两个或更多顶点的例子都不是 Ramanujan。特别是，对于 g>1，(ℓ)^g-isogeny 图不一般是 Ramanujan。
英文摘要：
Fix primes $p$ and $\ell$ with $\ell\neq p$. If $(A,λ)$ is a $g$-dimensional principally polarized abelian variety, an $(\ell)^g$-isogeny of $(A,λ)$ has kernel a maximal isotropic subgroup of the $\ell$-torsion of $A$; the image has a natural principal polarization. We define three isogeny graphs associated to such $(\ell)^g$-isogenies -- the big isogeny graph $\mathit{Gr}_{\!g}(\ell,p)$, the little isogeny graph $\mathit{gr}_{\!g}(\ell,p)$, and the enhanced isogeny graph $\widetilde{\mathit{gr}}_{\!g}(\ell, p)$. We prove that all three isogeny graphs are connected. One ingredient of the proof is strong approximation for the quaternionic unitary group, which has previously been applied to moduli of abelian varieties in charateristic $p$ by Chai, Ekedahl/Oort, and Chai/Oort. The adjacency matrices of the three isogeny graphs are given in terms of the Brandt matrices defined by Hashimoto, Ibukiyama, Ihara, and Shimizu. We study some basic properties of these Brandt matrices and recast the theory using the notion of Brandt graphs. We show that the isogeny graphs $\mathit{Gr}_{\!g}(\ell, p)$ and $\mathit{gr}_{\!g}(\ell, p)$ are in fact our Brandt graphs. We give the $\ell$-adic uniformization of $\mathit{gr}_{\!g}(\ell,p)$ and $\widetilde{\mathit{gr}}_{\!g}(\ell,p)$. The $(\ell+1)$-regular isogeny graph $\mathit{Gr}_1(\ell,p)$ for supersingular elliptic curves is well known to be Ramanujan. We calculate the Brandt matrices for a range of $g>1$, $\ell$, and $p$. These calculations give four examples with $g>1$ where the regular graph $\mathit{Gr}_{\!g}(\ell,p)$ has two vertices and is Ramanujan, and all other examples we computed with $g>1$ and two or more vertices were not Ramanujan. In particular, the $(\ell)^g$-isogeny graph is not in general Ramanujan for $g>1$.
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