论文标题
弗里德曼和华盛顿在随机$ p $ adadic矩阵上的结果的概括
Generalizations of results of Friedman and Washington on cokernels of random $p$-adic matrices
论文作者
论文摘要
令$ p $为prime，$ x $是$ \ mathbb {z} _ {p} $，$ p $ - ad-adic Integers的环。令$ p_ {1}（t），\ dots，p_ {l}（t）\ in \ mathbb {z} _ {p} [p} [t] $是$ 2 $的一元多项式，其图像的modulo $ p $在$ \ mathbb中是独特的，在$ \ mathbb in v}中是独特的，并且是不可修复的。对于每个$ j $，令$ g_ {j} $为$ \ mathbb {z} _ {p} [t]/（p_ {j}（t））$的有限模块。我们表明，随着$ n $进入无限，$ \ mathrm {cok}（p_ {j}（x））\ simeq g_ {j} $是独立的，并且可以用cohen-lenstra分布来描述每个概率的概率。我们还表明，对于任何固定的$ n $，$ \ mathrm {cok}（p_ {j}（x））\ simeq g_ {j $ j $的可能性是$ \ mathrm {cok}（cok}（p_ {j}（j}（j}（x}）， g_ {j}/pg_ {j} $对于每个$ j $，其中$ \ bar {x} $是$ n \ times n $均匀的随机矩阵，$ \ mathbb {f} _ {p} $。这些结果概括了弗里德曼和华盛顿的工作，并证明了猜想的新案例。
Let $p$ be prime and $X$ be a Haar-random $n \times n$ matrix over $\mathbb{Z}_{p}$, the ring of $p$-adic integers. Let $P_{1}(t), \dots, P_{l}(t) \in \mathbb{Z}_{p}[t]$ be monic polynomials of degree at most $2$ whose images modulo $p$ are distinct and irreducible in $\mathbb{F}_{p}[t]$. For each $j$, let $G_{j}$ be a finite module over $\mathbb{Z}_{p}[t]/(P_{j}(t))$. We show that as $n$ goes to infinity, the probabilities that $\mathrm{cok}(P_{j}(X)) \simeq G_{j}$ are independent, and each probability can be described in terms of a Cohen-Lenstra distribution. We also show that for any fixed $n$, the probability that $\mathrm{cok}(P_{j}(X)) \simeq G_{j}$ for each $j$ is a constant multiple of the probability that that $\mathrm{cok}(P_{j}(\bar{X})) \simeq G_{j}/pG_{j}$ for each $j$, where $\bar{X}$ is an $n \times n$ uniformly random matrix over $\mathbb{F}_{p}$. These results generalize work of Friedman and Washington and prove new cases of a conjecture of Cheong and Huang.
