有限局部环Z/p^kZ上辛几何中计数定理

SOME ANZAHL THEOREMS IN SYMPLECTIC GEOMETRY OVER FINITE LOCAL RING Z/P kZ

本文计算了Ｎ（ｍ，ｓ；２ｖ）与ｎ（ｍ，ｓ，ｔ，ｒ１，…，ｒｔ；２ｖ）．并以推论形式得到Ｓｐ２ｖ（Ｚ／ｐｋＺ）的阶．Ｎ（ｍ，ｓ；２ｖ）表示环Ｚ／ｐｋＺ上２ｖ维向量空间Ｖ２ｖ（Ｚ／ｐｋＺ）上的指数为ｓ的ｍ维子空间的个数；ｎ（ｍ，ｓ，ｔ，ｒ１，…，ｒ１，２ｖ）是秩为ｍ，不变因子为（ｒ，ｓ，ｔ，ｒ１。

Denote by N (m,s;2v) and n ( m,s,t,r 1,…,r t;2v) , respectively, the number of m dimensional subspaces in v 2v (Z/P kZ) with index s and the number of m by 2 v matrices its rank is m with invariants factors ( r,s,t,r 1,…,r t ).In the present paper,we compute N(m,s;2v )and n(m,s,t,r 1,…,r t;2v) , and also get the order of S p 2v (Z/P kZ ) by a corollary form.