A note on representation numbers of quadratic forms modulo prime powers

Let $f$ be an integral quadratic form in $k$ variables, $F$ the Gram matrix corresponding to a $\Z$-basis of $\Z^{k}$. For $r\in F^{-1}\Z^{k}$, a rational number $n$ with $f(r)\equiv n\bmod\Z$ and a positive integer $c$, set $N_{f}(n,r;c):=\sharp\{x\in\Z^{k}/c\Z^{k}: f(x+r)\equiv n\bmod c\}$. Siegel showed that for each prime $p$, there is a number $w$ depending on $r$ and $n$ such that $N_{f}(n,r;p^{\nu+1})=p^{k-1}N_{f}(n,r;p^{\nu})$ holds for every integer $\nu>w$ and gave a rough estimation on the upper bound for such $w$. In this short note, we give a more explicit estimation on this bound than Siegel's. KCI Citation Count: 0