Integral quadratic forms over a ring of $p$-adic integers

Abstract

Jungin Lee in 2018 proved a necessary and sufficient condition that an integral quadratic form ${\sum }_{i=1}^m{a}_i{X}_i^2$ is universal over $M_2(\mathbb{Z})$. For a positive integer $n\ge 2$, Lee defined $f(n)$ to be the smallest positive integer $m$ such that for every pairwise coprime $a_1,a_2,\dots a_m\in \mathbb{Z}$, ${\sum }_{i=1}^m{a}_i{X}_i^2$ is universal over $M_n(\mathbb{Z})$. He gave bounds on $f(n)$ too. Koo and Lee further improved the bounds in 2025. This paper is organized as follows. Let ${\mathbb{Z}}_p$ be the ring of $p$-adic integers. In the first section we give necessary and sufficient condition for a quadratic form ${\sum }_{i=1}^m{a}_i{X}_i^2$ to be universal over $M_2({\mathbb{Z}}_p)$ for $p=2$ and for an odd prime $p$. Consequently we express matrices of over ${\mathbb{Z}}_p$ as sum of squares. For a positive integer $n\ge 3$, we define $f(n)$ to be the smallest positive integer $m$ such that for every $p$-adic integers $a_1,a_2,\dots a_m$ with at least three of them units, the diagonal quadratic form ${\sum }_{i=1}^m{a}_i{X}_i^2$ is universal over $M_n({\mathbb{Z}}_p)$. In the next section we find bounds on $f(n)$ for $n\ge 3$.