On a Waring's problem for Hermitian lattices

Abstract

Assume E is an imaginary quadratic field and $\mathcal{O}$ is its ring of integers. For each positive integer m, let $I_m$ be the free Hermitian lattice of rank m over $\mathcal{O}$ having an orthonormal basis. For each positive integer n, let ${\mathfrak{S}}_{\mathcal{O}}(n)$ be the set of all Hermitian lattices of rank n over $\mathcal{O}$ that can be represented by some $I_m$. Denote by $g_{\mathcal{O}}(n)$ the smallest positive integer g such that each Hermitian lattice in ${\mathfrak{S}}_{\mathcal{O}}(n)$ can be represented by $I_g$. In this paper, we shall provide an explicit upper bound for $g_{\mathcal{O}}(n)$ for all imaginary quadratic fields E and all positive integers n.

Introduction

A positive integer a is said to be represented by a quadratic form f, written as $a\to f$, if there are some integers $x_1,\dots ,x_m$ such that $f(x_1,\dots ,x_m)=a$. A typical question is to determine those positive integers that can be represented by the quadratic form $I_m:=x_1^2+\cdots +x_m^2$ for a positive integer m. In light of the celebrated work of Fermat, Euler, Legendre and Lagrange on sums of squares, it is well-known that every positive integer can be written as a sum of at most four squares, and the number four is optimal.

This result has been extended in several different directions. For example, Waring's Problem seeks the smallest positive integer $r(k)$ such that every positive integer is a sum of r powers of order k. There is also a higher dimensional generalization in terms of representations of positive definite integral quadratic forms, where a quadratic form $f(x_1,\dots ,x_m)$ is said to be represented by another quadratic form $\tilde{f}(y_1,\dots ,y_n)$ with $n\ge m$ if one finds an $m\times n$ integral matrix A such that $f(\mathbf{x})=\tilde{f}(\mathbf{x}A)$; f is said to be equivalent to $\tilde{f}$ if A is a unimodular matrix.

In general, a quadratic form can be represented by the quadratic form $I_m$ if and only if it is a sum of m squares of integral linear forms. The renowned Lagrange's Four-Square Theorem implies that all unary positive definite integral quadratic forms can be represented by $I_4$. In the 1930s, Mordell [16] and Ko [14] successively extended this result to quadratic forms in more variables, and they showed that all positive definite integral quadratic forms in n variables can be represented by $I_{n+3}$ if $2\le n\le 5$. All the preceding results can be derived directly from the observations that every positive definite integral quadratic form in n variables is represented by $I_{n+3}$ locally everywhere, and that there is exactly one equivalence class in the genus of $I_{n+3}$ when $1\le n\le 5$. Here, the genus of f, written as $\mathrm{gen}(f)$, is the set of all quadratic forms that are equivalent to f over the ring ${\mathbb{Z}}_p$ of p-adic integers at each prime number p and over the field $\mathbb{R}$ of real numbers.

This approach does not work when $n\ge 6$, since in this case there are more than one equivalence classes in the genus of $I_{n+3}$; even worse, Mordell [17] found that the quadratic form

$$\sum _{l=1}^6{x}_l^2+{\left(\sum _{l=1}^6{x}_l\right)}^2-2x_1{x}_2-2x_2{x}_6$$

corresponding to the root system $E_6$ cannot be represented by any sum of squares. This subtlety leads to the consideration of the set ${\mathfrak{S}}_{\mathbb{Z}}(n)$ consisting of all quadratic forms in n variables that can be represented by some sums of squares.

Define the g-invariant $g_{\mathbb{Z}}(n)$ to be the smallest positive integer g such that $I_g$ represents every quadratic form in ${\mathfrak{S}}_{\mathbb{Z}}(n)$. Icaza [8] first found the existence of $g_{\mathbb{Z}}(n)$ and also derived an explicit upper bound for it by computing the constant c in the Hisa-Kitaoka-Kneser Theorem [7, Theorem 1] with respect to $I_{n+3}$. Kim and Oh [11] later improved the upper bound for $g_{\mathbb{Z}}(n)$ to an exponential of n through the theory of neighbors of quadratic forms. Recently, in [1], with my coauthors, we further improved the upper bound for $g_{\mathbb{Z}}(n)$ to an exponential of $\sqrt{n}$ and also observed that our method can be used to study a similar problem of Hermitian forms over an imaginary quadratic field with class number 1. Unfortunately, we cannot apply our method to general imaginary quadratic fields due to some emerged complications if the class numbers of those fields are bigger than 1; for instance, when the ring of integers is not a principle ideal domain, then a positive definite integral Hermitian form may not attain its minimum at a primitive vector.

In this paper, we shall consider the Hermitian analog of this representation problem in general situations. Let $E:=\mathbb{Q}(\sqrt{-\ell })$ be an imaginary quadratic field with ℓ a square-free positive integer, and let $\mathcal{O}$ be its ring of integers. Because $\mathcal{O}$ is not necessarily a principle ideal domain in general, a Hermitian form corresponds to a Hermitian lattice but not vice visa; so, instead of considering only Hermitian forms, the more general notion of Hermitian lattices will be utilized. For each integer $m\ge 1$, denote by $I_m$ the free Hermitian lattice of rank m having an orthonormal basis. From the standard correspondence between Hermitian forms and free Hermitian lattices, $I_m$ corresponds to the Hermitian form $x_1\overline{x_1}+\cdots +x_m\overline{x_m}$. For each integer $n\ge 1$, let ${\mathfrak{S}}_{\mathcal{O}}(n)$ be the set consisting of all Hermitian lattices of rank n that can be represented by some $I_m$, and write

$$g_{\mathcal{O}}(n):=\min \{g\in \mathbb{Z}:L\to I_g\text{for all}L\in {\mathfrak{S}}_{\mathcal{O}}(n)\}.$$

The finiteness of $g_{\mathcal{O}}(n)$ was observed in [1, Appendix A].

In this paper, we derive an explicit upper bound for $g_{\mathcal{O}}(n)$ that only depends on the imaginary quadratic field E and the rank n, and our main result reads as follows.

Theorem 1.1

Let $E=\mathbb{Q}(\sqrt{-\ell })$ be an imaginary quadratic field for a square-free positive integer ℓ, and let $n\ge 1$ be an integer. Then, for $g_{\mathcal{O}}(n)$ defined in (1.1), we have

$$g_{\mathcal{O}}(n)\le 2\cdot 3^n\delta (n+1)+\frac{3(p_0-1){(p_0+5)}^n}{2^{n-1}}(\sum _{j=3}^{n+1}\lceil 7{\log }_{p_0}(\mathcal{C}j\ell )\rceil +G(\ell ))+n+6.$$

Here, $\delta (n):=\{\begin{array}{cc}n-3\hfill & \mathit{\text{if}}n\ge 3\mathit{\text{and}}2\mathit{\text{is ramified}}\hfill \\ 0\hfill & \mathit{\text{otherwise}}\hfill \end{array}$, $p_0$ is the smallest inert prime number in the extension $E/\mathbb{Q}$, $G(\ell ):=\{\begin{array}{cc}\ell B_{3,\chi }/36+64{\ell }^3\hfill & \mathit{\text{if}}2\mathit{\text{is ramified}}\hfill \\ \ell B_{3,\chi }/144+{\ell }^3\hfill & \mathit{\text{otherwise}}\hfill \end{array}$, χ is the Dirichlet character associated with E, $B_{3,\chi }$ is the third Bernoulli number twisted by χ, and $\mathcal{C}>0$ is an absolute constant.

In Section 2, we introduce the geometric language of Hermitian spaces and Hermitian lattices. In Section 3, we consider the local density of a positive integer represented by a quadratic lattice $\hat{K}$ (which is associated with a Hermitian lattice K in the genus of $I_m$) at each prime number p. In Section 4, we find a positive integer $c(m)$ such that an integer t is represented by every Hermitian lattice $K\in \mathrm{gen}(I_m)$ when $t\ge c(m)$. In Section 5, we briefly review the theory of neighbors of Hermitian lattices over E following Schiemann [20]. In Section 6, we prove our main Theorem 1.1. Notice that the main scheme of Kim and Oh [11] is utilized in proving Theorem 1.1, but there are a number of difficulties that need to be overcome in the current setting based on [1], [15].