论文标题
超级亚巴亚品种的同性基因复合物
Isogeny complexes of superspecial abelian varieties
论文作者
论文摘要
我们考虑由极化不一定是原理的阿伯利亚品种的同基因形成的结构，特别是我们以前定义的$ [\ ell] $ - 极化。我们的主要兴趣是在超级特殊的阿贝尔品种中，在这些品种中，同源性与季节性遗传学形式有关。我们首先考虑降低图。我们表明，这些$ [\ ell] $ - 同学图形是一个广义的勃兰特图，并完全根据确定的四元组代数来构造它们。我们证明它们是连接的，并举例说明了所获得的常规图是有时的Ramanujan，有时不是Ramanujan。 $ [\ ell] $ - 两极分化的阿贝尔品种的同生可以在组成下关闭，结果是，这种ISEGEN自然形成了Eilenberg和Zilber在1950年引入的半动态复合物（后来也称为$δ$ -Complexes） - 多数族的高维类似物。 我们表明，这些同性恋复合物可以是通过赫尔米尔人形式而在确定的四元组代数上构建的，并且它们是通过Quaternionic统一组的作用来构建符号群体的商的商。使用四元组合这些同等基础图和复合物可容纳机器计算，我们包括许多示例，以详细检查$ [2] $ - 同类Abelian表面的$ [2] $同等激素复合物，特征性$ 7 $。
We consider the structures formed by isogenies of abelian varieties with polarizations that are not necessarily principal, specifically with the $[\ell]$-polarizations we have previously defined. Our primary interest is in superspecial abelian varieties, where the isogenies are related to quaternionic hermitian forms. We first consider isogeny graphs. We show that these $[\ell]$-isogeny graphs are a generalized Brandt graph and construct them entirely in terms of definite quaternion algebras. We prove that they are connected and give examples to show that the regular graphs obtained are sometimes Ramanujan and sometimes not. Isogenies of $[\ell]$-polarized abelian varieties can be closed under composition, with the consequence that such isogenies naturally form semi-simplicial complexes as introduced by Eilenberg and Zilber in 1950 (later also called $Δ$-complexes) -- the higher-dimensional analogues of multigraphs. We show that these isogeny complexes can be constructed from the arithmetic of hermitian forms over definite quaternion algebras and that they are quotients of the Bruhat-Tits building of the symplectic group by the action of a quaternionic unitary group. Working with quaternions these isogeny graphs and complexes are amenable to machine computation and we include many examples, concluding with a detailed examination of the $[2]$-isogeny complexes of superspecial abelian surfaces in characteristic $7$.
