Exact bounds for even vanishing of $K_{\ast }(\mathbb{Z}\mathrm{/}{p}^n)$

In this note, we prove that $K_{2i}(\mathbb{Z}\mathrm{/}{p}^n)\mathrm{\ne }0$ if and only if $p-1$ divides $i$ and $0\le i\le (p-1)p^{n-2}$, refining the even vanishing theorem of Antieau, Nikolaus and the first author in this case. As a corollary of our proof, we determine that the nilpotence order of $v_1$ in ${\pi }_{\ast }K(\mathbb{Z}\mathrm{/}{p}^n)\mathrm{/}p$ is equal to $\frac{p^n-1}{p-1}$. Our proof combines the recent crystallinity result for reduced syntomic cohomology of Hahn, Levy and the second author with the explicit complex computing the syntomic cohomology of ${\mathcal{O}}_K\mathrm{/}{\varpi }^n$ constructed by Antieau, Nikolaus and the first author.