Almost 2-universal diagonal quinary quadratic forms

TL;DR

This paper proves that three specific quinary diagonal quadratic forms are almost 2-universal, meaning they represent all but finitely many binary quadratic forms, advancing the classification of such forms.

Contribution

It confirms that the three remaining candidate forms are indeed almost 2-universal, completing their classification.

Findings

All three candidate forms are proven to be almost 2-universal.

The classification of almost 2-universal quinary diagonal quadratic forms is completed.

The results refine understanding of representation properties of quadratic forms.

Abstract

A (positive definite integral) quadratic form is called almost 2-universal if it represents all (positive definite integral) binary quadratic forms except those in only finitely many equivalence classes. Oh [7] determined all almost 2-universal quinary diagonal quadratic forms remaining three as candidates. In this article, we prove that those three candidates are indeed almost 2-universal.

Tables

Table 1. Table 1 . Almost 2-universal diagonal quinary ℤ ℤ {\mathbb{Z}} -lattices

$⟨1,1,1,1,1⟩,⟨1,1,1,1,2⟩,⟨1,1,1,1,3⟩,⟨1,1,1,2,2⟩,⟨1,1,1,2,3⟩$
$⟨1,1,1,1,5⟩,⟨1,1,1,2,4⟩,⟨1,1,1,2,5⟩,⟨1,1,1,2,7⟩,⟨1,1,2,2,3⟩,⟨1,1,2,2,5⟩$
$⟨1,1,1,3,7⟩,⟨1,1,2,3,5⟩,⟨1,1,2,3,8⟩$

Equations

$$L=\big{(}B(\mathbf{e}_{i},\mathbf{e}_{j})\big{)}.$$

$$L=\big{<}Q(\mathbf{e}_{1}),Q(\mathbf{e}_{2}),\cdots,Q(\mathbf{e}_{n})\big{>}.$$

$$[a,b,c]:=\left(\begin{array}{cc}a& b\\ b& c\end{array}\right).$$

$${\ell }_{s,t}^n:=\left(\begin{array}{cc}a-n{s}^2& b-nst\\ b-nst& c-n{t}^2\end{array}\right).$$

$$[2,1,3],[4,1,4],⟨1,6⟩,⟨4,6⟩,[2,1,7],[3,1,7],[4,2,7],$$

$$[6,3,7],[7,2,7],[7,3,9],[7,1,10],[10,5,10],[7,2,15],$$

$$⟨10,15⟩,⟨6,16⟩,[7,2,22],[7,1,26],[7,3,34],[10,5,47].$$

$${\ell }_{s,t}:=[a-7s^2,b-7st,c-7t^2]$$

$$[3,1,c]⟶M\text{if}c\equiv 0,1,4,5,6(\mathrm{mod}8),$$

$$⟨3,c⟩⟶M\text{if}c\equiv 1,2,3,5,6(\mathrm{mod}8).$$

$$[10,5,c]⟶M\perp ⟨7⟩\forall c\ne 3,10,47.$$

$$⟨1,6⟩,[2,1,3],⟨4,6⟩,[4,1,4],⟨6,16⟩,⟨10,15⟩,[10,5,10],[10,5,47],$$

$$[7,1,c_1](c_1=2,3,10,26),[7,2,c_2](c_2=4,7,15,22),[7,3,c_3](c_3=6,9,34).$$

$$\big{(}K\perp\langle 21\rangle\big{)}^{7}{\longrightarrow}L.$$

$$d{\ell }^{\prime }\equiv 1,2,4(\mathrm{mod}7).$$

$$[2,1,2],[5,2,5],[6,3,6].$$

$$[2,1,2],⟨2,6⟩,⟨5,6⟩,[5,2,5],[5,1,8],[6,3,6],[6,3,8],[6,3,14],$$

$$⟨6,26⟩,⟨7,7⟩,[8,4,8],⟨10,33⟩,⟨11,14⟩,⟨14,14⟩,[25,3,25].$$

Full text

1. Introduction

M.-H. Kim and his collaborators proved in [4] that there are exactly 11 quinary 2-universal quadratic forms. Hwang [3] proved that there are exactly 3 quinary diagonal quadratic forms that represents all binary quadratic forms except only one. Oh [7] proved that there exist only finitely many quinary quadratic forms that represent all but at most finitely many equivalence classes of binary quadratic forms. Such quadratic forms are called almost 2-universal quadratic forms. And he provided a list of almost 2-universal quinary diagonal quadratic forms, including 3 unconfirmed candidates. In this article, we show that those 3 candidates are indeed almost 2-universal.

2. Preliminaries and tools

We adopt lattice theoretic language. Let ${\mathbb{Q}}$ be the rational number field. For a prime (including $\infty$), let ${\mathbb{Q}}_{p}$ be the fields of $p$-adic completions of ${\mathbb{Q}}$, in particular ${\mathbb{Q}}_{\infty}=\mathbb{R}$, field of real numbers. For a finite prime $p$, ${\mathbb{Z}}_{p}$ denotes the $p$-adic integer ring. Let $R$ be the ring of integers ${\mathbb{Z}}$ or the ring of $p$-adic integers ${\mathbb{Z}}_{p}$. An $R$-lattice $L$ is a free $R$-module of finite rank equipped with a non-degenerate symmetric bilinear form $B:L\times L\rightarrow R$. The corresponding quadratic map is denoted by $Q$. For a $R$-lattice $L=R\mathbf{e}_{1}+R\mathbf{e}_{2}+\cdots+R\mathbf{e}_{n}$ with basis $\mathbf{e}_{1},\mathbf{e}_{2},\cdots,\mathbf{e}_{n}$, we write

[TABLE]

For $R$-sublattices $L_{1},L_{2}$ of $L$, we write $L=L_{1}\perp L_{2}$ when $L=L_{1}\oplus L_{2}$ and $B(\mathbf{v}_{1},\mathbf{v}_{2})=0$ for all $\mathbf{v}_{1}\in L_{1},\,\mathbf{v}_{2}\in L_{2}$. If $L$ admits an orthogonal basis $\{\mathbf{e}_{1},\mathbf{e}_{2},\cdots,\mathbf{e}_{n}\}$, we call $L$ diagonal and simply write

[TABLE]

We call $L$ non-diagonal otherwise. Define the discriminant $dL$ of $L$ to be the determinant of the matrix \big{(}B(\mathbf{e}_{i},\mathbf{e}_{j})\big{)}. Note that $dL$ is independent of the choice of a basis up to unit squares of $R$. We define scale $\mathfrak{s}L$ of $L$ to be the ideal of $R$ generated by $B(\mathbf{v},\mathbf{w})$ for all $\mathbf{v,w}\in L$, norm $\mathfrak{n}L$ of $L$ to be the ideal of $R$ generated by $Q(\mathbf{v})$ for all $\mathbf{v}\in L$. For $a\in R^{\times}$, we denote by $L^{a}$ the $R$-lattice obtained from scaling $L$ by $a$.

Let $\ell,L$ be $R$-lattices. We say $L$ represents $\ell$ if there is an injective linear map from $\ell$ into $L$ that preserves the bilinear form, and write $\ell{\longrightarrow}L$. Such a map will be called a representation. A representation is called isometry if it is surjective. We say two $R$-lattices $L,K$ are isometric if there is an isometry between them, and wrtie $L\cong K$.

For a ${\mathbb{Z}}$-lattice $L$ and a prime $p$, we define the ${\mathbb{Z}}_{p}$-lattice $L_{p}:={\mathbb{Z}}_{p}\otimes L$ and call it the localization of $L$ at $p$. The set of all ${\mathbb{Z}}$-lattices that are isometric to $L$ is called the class of $L$, denoted by $\text{cls}(L)$. The set of all ${\mathbb{Z}}$-lattices $K$ such that $L_{p}\cong K_{p}$ for all prime spots $p$ (including $\infty$) is called genus of $L$, denoted by $\text{gen}(L)$. The number of non-isometric classes in $\text{gen}(L)$ is called the class number of $L$, denoted by $h(L)$. The following two properties are well-known. (see [8, 103, 102:5])

• (1)

$h(L)$ is finite for any ${\mathbb{Z}}$-lattice $L$.

• (2)

For two lattices $\ell,L$, if $\ell_{p}{\longrightarrow}L_{p}$ for all prime $p$ (including $\infty$), then $\ell{\longrightarrow}K$ for some lattice $K\in\text{gen}(L)$.

For a ${\mathbb{Z}}$-lattice $L$, we say that $L$ is positive definite or simply positive if $Q(\mathbf{v})>0$ for any $\mathbf{v}\in L,\,\mathbf{v\neq 0}$. Let $L$ be a positive ${\mathbb{Z}}$-lattice. $L$ is called n-universal if $L$ represents all $n$-ary positive ${\mathbb{Z}}$-lattices. And $L$ is called almost n-universal if $L$ represents all $n$-ary positive ${\mathbb{Z}}$-lattices except those in only finitely many equivalence classes. For a fixed prime $p$, $L$ is called n-universal over ${\mathbb{Z}}_{p}$ if its localization $L_{p}$ represents all $n$-ary ${\mathbb{Z}}_{p}$-lattices. And $L$ is called locally n-universal if it is $n$-universal over ${\mathbb{Z}}_{p}$ for all primes $p$.

We will denote for convenience

[TABLE]

Any unexplained notations and terminologies can be found in [5] or [8].

Now we provide a technique for representations of binary ${\mathbb{Z}}$-lattices by certain quinary ${\mathbb{Z}}$-lattices, which is based on the proof of the main theorems in [3] and [4]. Let $\ell=[a,b,c]$ be a binary ${\mathbb{Z}}$-lattice. For any integers $n,s,t$, we define

[TABLE]

Let $M$ be a ${\mathbb{Z}}$-lattice. It can be verified that $\ell{\longrightarrow}M\perp\langle n\rangle$ if and only if there exist integers $s,t$ such that $\ell_{s,t}^{n}{\longrightarrow}M$. If the class number of the lattice $M$ is one, we can classify all binary lattices which are represented by $M$ using the local representation theory. In this thesis, we only consider the case that $M$ is quaternary. Let $p$ be a prime such that $p\nmid 2dM$. Then $M_{p}\simeq\langle 1,1,1,dM\rangle$. If $dM$ is a square in ${\mathbb{Z}}_{p}$, then $M$ is 2-universal over ${\mathbb{Z}}_{p}$. If $dM$ is not a square, then $\ell_{p}{\longrightarrow}M_{p}$ if and only if $\ell_{p}$ is not isometric to any sublattices of the ${\mathbb{Z}}_{p}$-lattice $\langle p,-p\varDelta\rangle$, where $\varDelta$ is a non-square unit in ${\mathbb{Z}}_{p}$. In particular, $\mathfrak{s}\ell\not\subseteq p{\mathbb{Z}}$ implies that $\ell_{p}{\longrightarrow}M_{p}$.

Let $\mathfrak{P}$ be the set of primes $p$ such that \displaystyle\bigg{(}\frac{dM}{p}\bigg{)}=-1. We will choose $s,t$ such that $\gcd(a-ns^{2},b-nst)$ has no prime factors in $\mathfrak{P}$. Then the scale of $\ell_{s,t}=[a-ns^{2},b-nst,c-nt^{2}]$ is not contained in $p{\mathbb{Z}}$ for any prime $p\in\mathfrak{P}$. Thus we may only consider the ${\mathbb{Z}}_{p}$-structure for primes $p\mid 2dM$.

Consider the case that $n$ has a prime factor $q\in\mathfrak{P}$, which is a hard case. If $\mathfrak{s}\ell\subseteq q{\mathbb{Z}}$, then $\mathfrak{s}\ell_{s,t}\subseteq q{\mathbb{Z}}$ for all $s,t$. For this reason, we have to prove this case separately.

Suppose that $\ell_{s,t}^{n}{\longrightarrow}M$ over ${\mathbb{Z}}_{p}$ for all primes $p$. If $\ell_{s,t}$ is positive, then we conclude that $\ell_{s,t}^{n}{\longrightarrow}M$, and that $\ell{\longrightarrow}M\perp\langle n\rangle$. Following lemma says that $\ell_{s,t}^{n}$ is positive for sufficiently large $a$.

Lemma 2.1**.**

Let $\ell=[a,b,c]$ be a Minkowski reduced binary ${\mathbb{Z}}$-lattice, that is, $2|b|\leq a\leq c$. If $a>\displaystyle\frac{4}{3}n(s^{2}+|st|+t^{2})$, then $\ell_{s,t}^{n}$ is positive.

Proof.

$d\ell_{s,t}^{n}=ac-b^{2}-ns^{2}c+2nstb-nt^{2}a=\displaystyle\frac{1}{4}ac-b^{2}+\frac{3}{4}ac-n(s^{2}c-2stb+t^{2}a)$

\geq 0+\displaystyle\frac{3}{4}ac-n(s^{2}c+|st|c+t^{2}c)=\frac{3}{4}c\bigg{(}a-\frac{4}{3}n(s^{2}+|st|+t^{2})\bigg{)}>0. ∎

3. Main results

Theorem 3.1**.**

There are exactly 14 almost 2-universal quinary diagonal ${\mathbb{Z}}$-lattices. Those lattices are:

By [7], there are eleven almost 2-universal quinary diagonal ${\mathbb{Z}}$-lattices and there can be at most three more. Among the quinary diagonal ${\mathbb{Z}}$-lattices listed in Table 1, five lattices in the first box are, in fact 2-universal, and six lattices in the second box are almost 2-universal (see [4], [3] and [7]). Three lattices in the third box are candidates for almost 2-universal quinary diagonal ${\mathbb{Z}}$-lattices provided in [7]. Now we prove that those three lattices are indeed almost 2-universal.

Theorem 3.2**.**

The quinary ${\mathbb{Z}}$-lattice $L=\langle 1,1,1,3,7\rangle$ represents all binary lattices except following 19 binary lattices:

[TABLE]

[TABLE]

[TABLE]

Proof.

Consider the quaternary sublattice $M=\langle 1,1,1,3\rangle$ of $L$, which has class number one. Let $\ell:=[a,b,c]$ a binary lattice such that $0\leq 2b\leq a\leq c$. And we define the binary lattice

[TABLE]

for each integers $s,t$. Note that $\ell{\longrightarrow}M\perp\langle 7\rangle$ if and only if $\ell_{s,t}{\longrightarrow}M$ for some integers $s,t$.

(Step 1) First, for the case that $a<30$, we verify that $\ell=[a,b,c]{\longrightarrow}M\perp\langle 7\rangle$. As a sample, we only consider the case that $a=10,\,b=5$. Other cases can be verified similarly. We use the fact that

[TABLE]

[TABLE]

For a binary lattice $\ell=[10,5,c]$, we can check the followings:

• •

If $c\not\equiv 3,6,7\pmod{8}$, then $\ell_{1,2}\simeq\langle 3,c-55\rangle{\longrightarrow}M$. ($c>55$)

• •

If $c\equiv 6\pmod{8}$, then $\ell_{1,1}\simeq[3,1,c-8]{\longrightarrow}M$. ($c>8$)

• •

If $c\equiv 7,11\pmod{16}$, then $\ell_{1,1}\subseteq[3,1,\frac{1}{4}(c-7)]{\longrightarrow}M$. ($c>7$)

• •

If $c\equiv 15\pmod{16}$, then $\ell_{1,2}\subseteq\langle 3,\frac{1}{4}(c-55)\rangle{\longrightarrow}M$. ($c>55$)

• •

If $c\equiv 3\pmod{16}$, then $\ell_{1,3}\subseteq\ [3,1,\frac{1}{4}(c-147)]{\longrightarrow}M$. ($c>147$)

For a small $c$ such that $\ell_{1,t}$ is not positive, we can also check it by a direct calculation. In this case, it can be verified that three binary lattices $[10,5,3]\simeq[2,1,3],[10,5,10],[10,5,47]$ are not represented by $M\perp\langle 7\rangle$. In short,

[TABLE]

For $a<30$, we can verify $\ell{\longrightarrow}L$ except followings:

[TABLE]

[TABLE]

By a direct calculation, we can verify that sublattices of above exceptions with index a power of 2 are represented by $L$. Hereafter, we only consider ${\mathbb{Z}}_{2}$-primtive binary lattices $\ell$.

(Step 2) For a binary lattice $\ell=[a,b,c]$, suppose that $a,c\geq 30$ and $0\leq 2b\leq a\leq c$. And suppose that $\mathfrak{s}\ell\not\subseteq 7{\mathbb{Z}}$.

By checking the local structure of $\ell_{s,t}$ and $M$ over ${\mathbb{Z}}_{2},{\mathbb{Z}}_{3}$, we obtain the following properties.

• (1.1)

If $(a,b,c)\equiv(1,0,1)\pmod{2}$ and $(s,t)\equiv(1,1)\pmod{2}$, then $\ell_{s,t}{\longrightarrow}M$ over ${\mathbb{Z}}_{2}$.

• (1.2)

If $(a,b,c)\equiv(0,1,0)\pmod{2}$ and $(s,t)\equiv(0,0)\pmod{2}$, then $\ell_{s,t}{\longrightarrow}M$ over ${\mathbb{Z}}_{2}$.

• (1.3)

If $(a,b,c)\equiv(1,1,0)\pmod{2}$ and $(s,t)\equiv(1,0)\pmod{2}$, then $\ell_{s,t}{\longrightarrow}M$ over ${\mathbb{Z}}_{2}$.

• (1.4)

If $(a,b,c)\equiv(0,1,1)\pmod{2}$ and $(s,t)\equiv(0,1)\pmod{2}$, then $\ell_{s,t}{\longrightarrow}M$ over ${\mathbb{Z}}_{2}$.

• (1.5)

If $a\equiv 5\pmod{8}$ or $c\equiv 5\pmod{8}$, then $\ell_{s,t}{\longrightarrow}M$ over ${\mathbb{Z}}_{2}$ for any $s,t$.

• (1.6)

If $a\equiv 1\pmod{8},\,2\mid s$ or $c\equiv 1\pmod{8},\,2\mid t$, then $\ell_{s,t}{\longrightarrow}M$ over ${\mathbb{Z}}_{2}$.

• (1.7)

If $4\mid a,\,2\nmid s$ or $4\mid c,\,2\nmid t$, then $\ell_{s,t}{\longrightarrow}M$ over ${\mathbb{Z}}_{2}$.

• (1.8)

If $a\equiv 3\pmod{4}$ or $c\equiv 3\pmod{4}$, $2\mid b$ and $2\nmid st$, then $\ell_{s,t}{\longrightarrow}M$ over ${\mathbb{Z}}_{2}$.

• (2.1)

If $3\mid ac$ and $3\nmid st$, then $\ell_{s,t}{\longrightarrow}M$ over ${\mathbb{Z}}_{3}$.

• (2.2)

If $(a,b,c)\equiv(1,0,1),(1,0,2),(2,0,1)\pmod{3}$ and $3\nmid st$, then $\ell_{s,t}{\longrightarrow}M$ over ${\mathbb{Z}}_{3}$.

• (2.3)

If $(a,b,c)\equiv(1,2,1),(1,2,2),(2,1,2),(2,2,1)\pmod{3}$ and $st\equiv 1\pmod{3}$, then $\ell_{s,t}{\longrightarrow}M$ over ${\mathbb{Z}}_{3}$.

• (2.4)

If $(a,b,c)\equiv(1,1,1),(1,1,2),(2,1,1),(2,2,2)\pmod{3}$ and $st\equiv 2\pmod{3}$, then $\ell_{s,t}{\longrightarrow}M$ over ${\mathbb{Z}}_{3}$.

• (2.5)

If $(a,b,c)\equiv(2,0,2)\pmod{3}$ and $st\equiv 0\pmod{3}$, then $\ell_{s,t}{\longrightarrow}M$ over ${\mathbb{Z}}_{3}$.

Under the assumption that the binary lattice $\ell$ is ${\mathbb{Z}}_{2}$-primitive, above conditions cover all cases. For example, the case that $a\equiv c\equiv 3\pmod{4}$ and $2\nmid b$ is not contained in above. However in this case $\ell$ is not ${\mathbb{Z}}_{2}$-primitive. ${\mathbb{Z}}_{3}$-primitivity of $\ell$ is not necessary. Regardless of ${\mathbb{Z}}_{3}$-primitivity of $\ell$, all cases are contained in above.

For all cases, we may choose $s=1,2$ and $t\equiv i\pmod{6}$ for some $i$. Each case can be proved similarly. We only consider the case that $\ell=[a,b,c]$ satisfies the conditions given in both (1.4) and (2.3). In this case, $\ell_{s,t}{\longrightarrow}M$ over ${\mathbb{Z}}_{2},{\mathbb{Z}}_{3}$ if $s=2$ and $t\equiv 1\pmod{6}$.

Let $\mathfrak{P}=\{5,7,17,19,29,31,...\}$ be the set of primes $p$ such that \displaystyle\bigg{(}\frac{3}{p}\bigg{)}=-1. From the assumption that $\mathfrak{s}\ell\not\subseteq 7{\mathbb{Z}}$, we get $\mathfrak{s}\ell_{s,t}\not\subseteq 7{\mathbb{Z}}$ for all $s,t$, and hence $\ell_{s,t}{\longrightarrow}M$ over ${\mathbb{Z}}_{7}$. Let $p_{1},p_{2},...,p_{k}$ be the primes in $\mathfrak{P}-\{7\}$ dividing $a-28$. We want choose a suitable $t$ such that $b-14t$ is relatively prime to $p_{1}p_{2}...p_{k}$. Then we get $\ell_{s,t}{\longrightarrow}M$ over ${\mathbb{Z}}_{p}$ for all $p\neq 2,3$.

If $k=0$, then $\ell_{2,1}{\longrightarrow}M$. By lemma 2.1, $\ell_{2,1}$ is positive if $a\geq 66$. In the case that $30\leq a\leq 65$, one can show that $\ell_{2,1}$ is also positive for sufficiently large $c$. In the case that $a=34$, for example, $\ell_{2,1}$ is positive whenever $c>39$. The remaining cases are finitely many and we can check it by a direct calculation.

If $1\leq k\leq 4$, then $\ell_{2,t}{\longrightarrow}M$ for some t\in\{6m+1\mid\displaystyle-\Big{[}\frac{k+1}{2}\Big{]}\leq m\leq\Big{[}\frac{k}{2}\Big{]}\}. Note that $a\geq 28+p_{1}\cdots p_{k}$. In the case that $k=3,4,5$, all $\ell_{2,t}$ are positive by lemma 2.1. In case that $k=1,2$, however, the positiveness of $\ell_{2,t}$ is not guaranteed. There are only finitely many cases such that $\ell_{2,t}$ is not positive. When $\ell_{2,t}$ is not positive, we can check that $\ell{\longrightarrow}L$ by a direct calculation.

If $k=5$, then $\ell_{2,t}{\longrightarrow}M$ for some $t\in\{-17,-11,...13,19\}$. Since $a\geq 28+5\cdot 17\cdot 19\cdot 29\cdot 31$, $\ell_{2,19}$ is positive.

If $k\geq 6$, then $\ell_{2,t}{\longrightarrow}M$ for some $t\in\{-3k2^{k}+1,...-5,1,...,3k2^{k}-5\}$. Since $a\geq 28+5\cdot 17\cdot 19\cdot 29\cdot 31\cdot 41\cdot 43^{k-6}$, all $\ell_{2,t}$ are positive by lemma 3 of [4].

(Step 3) We show that $\ell{\longrightarrow}L$ when \mathfrak{s}\ell\subseteq 7\, that is, $\ell$ is the form of $[7a,7b,7c]$. Let $\ell^{\prime}=[a,b,c]$, then $\ell=\ell^{\prime 7}$. Consider the quaternary lattice $K=\langle 1\rangle\perp\begin{pmatrix}2&1&0\\ 1&2&1\\ 0&1&3\end{pmatrix}$. Note that $K$ has class number one and

[TABLE]

If $\ell^{\prime}{\longrightarrow}K\perp\langle 21\rangle$, then $\ell^{\prime 7}=\ell{\longrightarrow}L$. We only consider the case that $\ell^{\prime 7}$ is ${\mathbb{Z}}_{7}$-primitive. Thus we may assume that $\ell^{\prime}\simeq\langle 1,-\varDelta\rangle$ over ${\mathbb{Z}}_{7}$, where $\varDelta$ is a nonsquare unit in ${\mathbb{Z}}_{7}$. This is equivalent to

[TABLE]

Define $\ell^{\prime}_{s,t}=[a-21s^{2},b-21st,c-21t^{2}]$. From the fact that $7\nmid d\ell^{\prime}$ we get $7\nmid d\ell^{\prime}_{s,t}$ and $\ell^{\prime}_{s,t}{\longrightarrow}K$ over ${\mathbb{Z}}_{7}$ for any $s,t$. The followings are the sufficient conditions such that $\ell^{\prime}_{s,t}{\longrightarrow}K$ over ${\mathbb{Z}}_{2}$ assuming that $\ell^{\prime}$ is ${\mathbb{Z}}_{2}$-primitive.

• (1)

$(a,b,c)\equiv(1,0,1)\pmod{2}$ and $(s,t)\equiv(1,1)\pmod{2}$.

• (2)

$(a,b,c)\equiv(0,0,1),(0,1,0),(1,0,0),(1,1,1)\pmod{2}$ and $(s,t)\equiv(0,0)\pmod{2}$.

• (3)

$(a,b,c)\equiv(1,1,0)\pmod{2}$ and $(s,t)\equiv(1,0)\pmod{2}$.

• (4)

$(a,b,c)\equiv(0,1,1)\pmod{2}$ and $(s,t)\equiv(0,1)\pmod{2}$.

Using the same method as step 1,2, we get $\ell^{\prime}{\longrightarrow}K\perp\langle 21\rangle$ with finitely many exceptions. For example, $\langle 5,61\rangle,\,[9,2,13]{\longarrownot\longrightarrow}K\perp\langle 21\rangle$. By brute force computation, one can show that the binary lattices obtained by scaling these excepstions are indeed represented by $L$. Therefore we conclude that $\ell{\longrightarrow}L$ for all binary lattices whose scale is contained in $7{\mathbb{Z}}$. ∎

For the other two lattices, the proofs are quite similar to the above. We only provide following data for the proof of almost 2-universality of $L$:

• (1)

Quternary sublattice $M$ which has class number one

• (2)

The integer $n$ satisfying $M\perp\langle n\rangle{\longrightarrow}L$

• (3)

Conditions such that $\ell_{s,t}^{n}{\longrightarrow}M$ over ${\mathbb{Z}}_{p}$ where $p\mid 2dM$

• (4)

Data for the case that $\mathfrak{s}\ell\subseteq q{\mathbb{Z}}$ where $q\mid n$ and \displaystyle\bigg{(}\frac{dM}{q}\bigg{)}=-1

Theorem 3.3**.**

(1)* The quinary ${\mathbb{Z}}$-lattice $\langle 1,1,2,3,5\rangle$ represents all binary lattices except*

[TABLE]

(2)* The quinary ${\mathbb{Z}}$-lattice $\langle 1,1,2,3,8\rangle$ represents all binary lattices except the following 15 binary lattices:*

[TABLE]

[TABLE]

Proof.

Set $M=\langle 1,1,2,3\rangle,\,n=5,8$

Conditions such that $\ell_{s,t}^{n}{\longrightarrow}M$ over ${\mathbb{Z}}_{3}$ where $n=5,8$: (${\mathbb{Z}}_{3}$-primitivity of $\ell$ is not necessary)

• •

$3\mid ac$ and $3\nmid st$

• •

$(a,b,c)\equiv(1,0,2),(2,0,1),(2,0,2)\pmod{3}$ and $3\nmid st$

• •

$(a,b,c)\equiv(1,1,2),(1,2,1),(2,1,1),(2,1,2)\pmod{3}$ and $st\equiv 1\pmod{3}$

• •

$(a,b,c)\equiv(1,1,1),(1,2,2),(2,2,1),(2,2,2)\pmod{3}$ and $st\equiv 2\pmod{3}$

• •

$(a,b,c)\equiv(1,0,1)\pmod{3}$ and $st\equiv 0\pmod{3}$

Conditions such that $\ell_{s,t}^{5}{\longrightarrow}M$ over ${\mathbb{Z}}_{2}$: ($\ell$ is ${\mathbb{Z}}_{2}$-primitive)

• •

$(a,b,c)\equiv(0,1,0)\pmod{2}$ and $\forall s,t$

• •

$(a,b,c)\equiv(0,0,1),(1,0,0),(1,0,1)\pmod{2}$ and $(s,t)\equiv(1,1)\pmod{2}$

• •

$(a,b,c)\equiv(0,1,1),(1,1,1)\pmod{2}$ and $(s,t)\equiv(0,1)\pmod{2}$

• •

$(a,b,c)\equiv(1,1,0),(1,1,1)\pmod{2}$ and $(s,t)\equiv(1,0)\pmod{2}$

Conditions such that $\ell_{s,t}^{8}{\longrightarrow}M$ over ${\mathbb{Z}}_{2}$: ($\ell$ is ${\mathbb{Z}}_{2}$-primitive)

• •

$(a,b,c)\equiv(0,1,0),(0,1,1),(1,1,0),(1,0,1)\pmod{2}$ and $\forall s,t$

• •

$a\equiv 5,7\pmod{8}$ or $c\equiv 5,7\pmod{8}$ and $\forall s,t$

• •

$a\equiv 10,14\pmod{16},\,2\nmid s$ or $c\equiv 10,14\pmod{16},\,2\nmid t$

• •

$(a,c)\equiv(1,3),(9,11)\pmod{16},\,b\equiv\pm 1\pmod{8}$ and $(s,t)\equiv(1,1)\pmod{2}$

• •

$(a,c)\equiv(1,11),(3,9)\pmod{16},\,b\equiv\pm 3\pmod{8}$ and $(s,t)\equiv(1,1)\pmod{2}$

• •

$(a,c)\equiv(1,3),(9,11)\pmod{16},\,b\equiv\pm 3\pmod{8}$ and $(s,t)\equiv(0,1),(1,0)\pmod{2}$

• •

$(a,c)\equiv(1,11),(3,9)\pmod{16},\,b\equiv\pm 1\pmod{8}$ and $(s,t)\equiv(0,1),(1,0)\pmod{2}$

• •

$(a,c)\equiv(1,6),(9,6),(3,2),(11,2)\pmod{16},\,b\equiv 2\pmod{4}$ and $(s,t)\equiv(1,0)\pmod{2}$

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$(a,c)\equiv(1,2),(9,2),(3,6),(11,6)\pmod{16},\,b\equiv 0\pmod{4}$ and $(s,t)\equiv(1,0)\pmod{2}$

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$(a,c)\equiv(1,6),(9,6),(3,2),(11,2)\pmod{16},\,b\equiv 0\pmod{4}$ and $(s,t)\equiv(1,1)\pmod{2}$

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$(a,c)\equiv(1,2),(9,2),(3,6),(11,6)\pmod{16},\,b\equiv 2\pmod{4}$ and $(s,t)\equiv(1,1)\pmod{2}$

Since there are no prime factors $q$ such that $q\mid n$ and \displaystyle\bigg{(}\frac{dM}{q}\bigg{)}=-1, in this case, the process such as step 3 in the proof of Theorem 3.2 is not necessary. ∎