论文标题
超级特色阿贝尔品种上的同学图：特征值和与Bruhat-tits建筑物的连接
Isogeny graphs on superspecial abelian varieties: Eigenvalues and Connection to Bruhat-Tits buildings
论文作者
论文摘要
我们研究每个固定整数$ g \ ge 2 $，对于所有Primes $ \ ell $和$ \ ell \ ell \ neq p $ $ P $。这意味着在椭圆曲线案例之外的同一基因谱系上，自然随机步行的快速混合特性，表明Charles-Goren-Lauter型加密哈希功能的潜在构建是Abelian品种的。当$ g \ ge 2 $时，我们会根据simbletectic群体的kazhdan常数为差距提供明确的下限，并在$ g = 2 $时讨论自动形态表示理论的最佳值。作为副产品，我们还表明，由约旦 - 扎伊特曼（Jordan-Zaytman）构建的有限规则定向图也具有相同的属性。
We study for each fixed integer $g \ge 2$, for all primes $\ell$ and $p$ with $\ell \neq p$, finite regular directed graphs associated with the set of equivalence classes of $\ell$-marked principally polarized superspecial abelian varieties of dimension $g$ in characteristic $p$, and show that the adjacency matrices have real eigenvalues with spectral gaps independent of $p$. This implies a rapid mixing property of natural random walks on the family of isogeny graphs beyond the elliptic curve case and suggests a potential construction of the Charles-Goren-Lauter type cryptographic hash functions for abelian varieties. We give explicit lower bounds for the gaps in terms of the Kazhdan constant for the symplectic group when $g \ge 2$, and discuss optimal values in view of the theory of automorphic representations when $g=2$. As a by-product, we also show that the finite regular directed graphs constructed by Jordan-Zaytman also has the same property.
