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

Abstract

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