论文标题
拓扑结构不变和算术统计
Topological Iwasawa invariants and Arithmetic Statistics
论文作者
论文摘要
给定质数$ p $，我们研究了与$ \ mathbb {z} _p $ - $ 3 $ - sphere的股份相关的iWasawa不变性的拓扑类似物。我们证明了检测这些Iwasawa不变性的明确标准，并将其应用于由$ 2 $组成结组成的链接的研究。修复Prime $ p $，我们证明了$ P $ - 主要的Iwasawa不变性的统计结果，以$ 2 $ - 桥的链接为$ schubert normal形式。我们的主要结果完全是无条件的，表明$ 2 $桥的密度为$ $ $ $ invariant消失，而$λ$ -Invariant等于$ 1 $，为$（1- \ frac {1} {p}）$。我们还猜想，$ 2 $桥的密度为$ $ $ $ invariant Visanes是$ 1 $，这得到了计算证据的很大支持。我们的结果在拓扑环境中得到了证明，但具有算术意义，因为我们列出了算术统计和算术拓扑的新方向。
Given a prime number $p$, we study topological analogues of Iwasawa invariants associated to $\mathbb{Z}_p$-covers of the $3$-sphere that are branched along a link. We prove explicit criteria to detect these Iwasawa invariants, and apply them to the study of links consisting of $2$ component knots. Fixing the prime $p$, we prove statistical results for the average behaviour of $p$-primary Iwasawa invariants for $2$-bridge links that are in Schubert normal form. Our main result, which is entirely unconditional, shows that the density of $2$-bridge links for which the $μ$-invariant vanishes, and the $λ$-invariant is equal to $1$, is $(1-\frac{1}{p})$. We also conjecture that the density of $2$-bridge links for which the $μ$-invariant vanishes is $1$, and this is significantly backed by computational evidence. Our results are proven in a topological setting, yet have arithmetic significance, as we set out new directions in arithmetic statistics and arithmetic topology.
