The Local Four-Square Problem over \(\mathbb{Z}_{p^k}\)

The norm map $N:\mathcal{H}_{\mathbb{Z}_{p^k}}\to \mathbb{Z}_{p^k}$ is studied on the quaternion ring over $\mathbb{Z}_{p^k}$, where $p$ is an odd prime and $k\ge 1$ an integer. By means of the isomorphism $\mathcal{H}_{\mathbb{Z}_{p^k}}\cong M_2(\mathbb{Z}_{p^k})$, quaternions are investigated using matrix methods. It is shown that the fibre size $$a_{p^k}(m)=|\{q\in \mathcal{H}_{\mathbb{Z}_{p^k}}:N(q)=m\}|$$ depends only on the $p$-adic valuation $v_p(m)$ of $m$. Explicit formulas for the fibre sizes are derived for every $m\in\mathbb{Z}_{p^k}$: $$a_{p^k}(m)= \begin{cases} p^{3k-2}(p^2-1), & t=0,\\[6pt] p^{3k-2-t}(p+1)(p^{t+1}-1), & 0<t<k,\\[6pt] p^{2k-1}(p^{k+1}+p^k-1), & t=k, \end{cases}$$ where $t=v_p(m)$ (with the convention $v_p(0)=k$). The main result of the paper is a complete solution to the \emph{local four-square problem} over the ring $\mathbb{Z}_{p^k}$: the number $a_{p^k}(m)$ gives the exact number of representations of an arbitrary element $m\in \mathbb{Z}_{p^k}$ as a sum of four squares, $$x_1^2+x_2^2+x_3^2+x_4^2=m.$$ The proof is purely algebraic; it relies only on matrix theory and Smith normal form, thus avoiding the abstract machinery of number theory. This preprint has not undergone peer review (when applicable) or any post-submission improvements or corrections