论文标题
超级特色阿伯利亚品种和勃兰特矩阵的等不良图
Isogeny graphs of superspecial abelian varieties and Brandt matrices
论文作者
论文摘要
修复Primes $ P $和$ \ ell $，带有$ \ ell \ neq p $。如果$（a，λ）$是$ g $维主要是极化的阿贝利亚品种，则$（\ ell）^g $ - $（a，λ）$的发育率具有核的最大同位素亚组，$ \ ell $ torsion $ a $ a $ a $ a $ a $ a $;该图像具有自然的主偏振。我们定义了与此类$（\ ell）^g $ - 异基因相关的三个同基因图 - 大的同学图$ \ Mathit {gr} _ {\！g}（\！ $ \ widetilde {\ mathit {gr}} _ {\！g}（\ ell，p）$。我们证明所有三个均质图都连接了。证明的一种成分是季度单位群的强近似值，以前已将其用于柴，ekedahl/oort和chai/oort的Abelian品种的模量。这三个等级图的邻接矩阵是根据Hashimoto，Ibukiyama，Ihara和Shimizu定义的Brandt矩阵给出的。我们研究了这些Brandt矩阵的一些基本特性，并使用Brandt图的概念重塑了理论。我们表明，等级图$ \ mathit {gr} _ {\！g}（\ ell，p）$和$ \ mathit {gr} _ {\！g}（\ ell，p）$实际上是我们的品牌图。我们给出了$ \ mathit {gr} _ {\！g}（\ ell，p）$和$ \ widetilde {\ mathit {gr}} _ {\！g}（\ ell，p）$的$ \ ell $ -adic统一化{\！g}（\ ell，p）$。 $（\ ell+1）$ - 常规的iSGEOG图$ \ mathit {gr} _1（\ ell，p）$用于超明椭圆曲线是Ramanujan。我们计算了$ g> 1 $，$ \ ell $和$ p $的Brandt矩阵。这些计算给出了四个示例，其中$ g> 1 $，其中常规图$ \ mathit {gr} _ {\！g}（\ ell，p）$具有两个顶点，是Ramanujan，我们使用$ g> 1 $的所有其他示例，两个或多个或多个顶点不是Ramanujan。特别是，$（\ ell）^g $ - 发育图并不是$ g> 1 $的一般Ramanujan。
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 $g>1$.
