# Exceptional sets for spinor regular ternary quadratic forms

A .G. Earnest  Department of Mathematics, Southern Illinois University,
Carbondale, IL, 62901, U.S.A<EMAIL_ADDRESS>
###### Abstract.

The goal of this note is to provide an analysis of the positive integers that
are represented everywhere locally, but not globally, by each of the 29 spinor
regular ternary quadratic forms that are not regular.

###### Key words and phrases:

Spinor exceptional integers, spinor regular ternary quadratic forms

###### 2010 Mathematics Subject Classification:

Primary 11E12; Secondary 11E08 11E20 11E25

##  1\. Introduction

It is well-known that there is no local-global principle for the
representation of integers by integral quadratic forms. Consequently, it is of
interest to identify the exceptional set of integers that are represented
everywhere locally, but not globally, by such a form. For ternary forms this
set can be either finite or infinite; when it is empty, Dickson called the
form regular [5]. That is, the regular forms are those that represent all
integers represented by their genus. In terminology later introduced by
Benham, Hsia, Hung and the author [2], a ternary form is said to be spinor
regular if it represents all the integers represented by its spinor genus.

An integer that is represented by a genus of ternary integral quadratic forms
but not by every spinor genus in that genus is referred to as a spinor
exceptional integer for the genus. In this note, the general theory of spinor
exceptional integers will be used to determine the exceptional sets for each
of the spinor regular positive definite ternary primitive integral quadratic
forms for which the exceptional set is nonempty. In light of a result of the
author and Haensch [7], it is known that there are exactly 29 inequivalent
forms with this property. For the 27 of these forms that are alone in their
spinor genus, explicit formulas for the numbers of representations of all
positive integers by each such form were recently obtained by Aygin, Doyle,
Münkel, Pehlivan and Williams [1].

For each of the 29 forms, the exceptional set will be seen to consist of the
integers lying in one or more squareclasses whose prime divisors satisfy
certain congruence conditions. This phenomenon was observed by Jones and Pall
[11] in their study of the forms in the genera of regular diagonal ternary
forms. For example, they observed that the form

$4x^{2}+9y^{2}+9z^{2}+4xy+4xz+2yz$

represents all those integers represented by its genus except for the odd
integers $m^{2}$ for which every prime factor $p$ of $m$ satisfies $p\equiv
1\,(\text{mod}\,4)$. In all, Jones and Pall listed seven forms with similar
properties. A full explanation for this type of behavior later emerged through
spinor genus theory. Schulze-Pillot [14] proved that each of these seven forms
fails to represent precisely those integers that are spinor exceptional
integers for their genus. The present paper can be viewed as an extension of
the work of Schulze-Pillot applying the theory of spinor exceptional integers
to complete the determination of the exceptional sets for each of the
remaining spinor regular ternaries that are not regular.

We will follow the notations describing the spinor regular ternary forms in
[1]. In that paper, the forms are labelled as A1–A13, B1–B12 and C1–C4, where
the forms are grouped according to the prime factors of their discriminant. In
this paper, the main results for the three groups appear in Propositions 4.1,
5.1 and 6.1, respectively.

The remainder of the paper is organized as follows. Section 2 contains
notations and conventions that will be used throughout the paper. Some
pertinent facts from the general theory of spinor exceptional integers will be
reviewed in Section 3. The statements given there are special cases of general
results from [14], stated here only in the generality needed to analyze the
genera of the spinor regular ternary forms. On the topic of spinor exceptional
integers, the interested reader may also wish to consult the excellent survey
[15]. Results for the forms A1–A13, B1–B12 and C1–C4 are given in Sections 4,
5 and 6, respectively. In the final section of the paper, the spinor regular
forms that are not alone in their spinor genus, namely B4 and B11, are
analyzed in more detail. Since the results on the positive integers not
represented by these forms are obtained only for even integers in [1], we give
the full statements here as Propositions 7.4 and 7.5.

##  2\. Ternary quadratic forms and lattices

A ternary quadratic form will be described by the sextuple of integers
appearing as its coefficients, where $(a,b,c,d,e,f)$ denotes the quadratic
form

$F=ax^{2}+by^{2}+cz^{2}+dyz+exz+fxy.$

This form is referred to as classic if $d$, $e$ and $f$ are even, and non-
classic otherwise. The discriminant $\Delta$ of $F$ is defined to be
$\frac{1}{2}\textrm{det}M_{F}$, where $M_{F}$ is the matrix of second partial
derivatives of the form.

For referencing the literature on spinor exceptional integers, it will be
convenient to freely switch between the language of quadratic forms and
lattices. For quadratic lattices, we will follow the terminology and notation
of O’Meara’s book [10]. To the ternary form $F$ given above, we associate the
ternary lattice $L$ having Gram matrix $\frac{1}{2}M_{F}$ with respect to some
basis. If $dL$ denotes the discriminant of this lattice, in the sense of [10],
then $\Delta=4dL$. Let $\text{gen}\mspace{1.0mu}L$ and
$\text{spn}\mspace{1.0mu}L$ denote the genus and spinor genus of $L$,
respectively. For an integer $n$, we will use the notation $n\rightarrow N$ to
indicate that $n$ is represented by $N$, where $N$ can be the lattice $L$, its
$p$-adic completion $L_{p}$ with respect to some prime $p$, or its genus
$\text{gen}\mspace{1.0mu}L$ or spinor genus $\text{spn}\mspace{1.0mu}L$. Note
that $n\rightarrow L$ if and only if $n$ is represented by the original form
$F$, and $n\rightarrow\text{gen}\mspace{1.0mu}L$ if and only if $n\rightarrow
L_{p}$ for all primes $p$.

##  3\. Determination of spinor exceptional integers

Throughout this section, $L$ will denote an arbitrary positive definite
integral ternary quadratic $\mathbb{Z}$-lattice, $\Delta$ will be the positive
integer $4dL$, and $n$ will be a positive integer represented by
$\text{gen}\mspace{1.0mu}L$.

###  3.1. General criteria for spinor exceptional integers

In light of the results of [14], the spinor exceptional integers for a genus
are determined by essentially local information. For a prime $p$, we denote
the $p$-adic numbers and $p$-adic integers by $\mathbb{Q}_{p}$ and
$\mathbb{Z}_{p}$, respectively, and the group of units of $\mathbb{Z}_{p}$ by
$\mathbb{Z}_{p}^{\times}$. The order of an element $\lambda$ of
$\dot{\mathbb{Q}}_{p}$ will be denoted by $\operatorname{ord}_{p}(\lambda)$;
so $\lambda=p^{\operatorname{ord}_{p}(\lambda)}\lambda_{0}$, with
$\lambda_{0}\in\mathbb{Z}_{p}^{\times}$.

Let $\theta(O^{+}(L_{p}))$ denote the group of spinor norms of rotations on
$L_{p}$, $\theta(L_{p},n)$ the relative spinor norm group defined in [14,
Definition 1, p. 531], and $N_{p}(-n\Delta)$ the group of local norms at $p$
of $\mathbb{Q}(\sqrt{-\text{sqf}(n\Delta)})$, where $\text{sqf}(\gamma)$
denotes the squarefree part of the positive integer $\gamma$. Concretely,

$N_{p}(-n\Delta)=\\{\gamma\in\dot{\mathbb{Q}}_{p}:(\gamma,-n\Delta)_{p}=+1\\},$

where $(\cdot,\cdot)_{p}$ denotes the Hilbert symbol at $p$. Note that
$N_{p}(-n\Delta)$ is unchanged when the argument is changed by a square
factor; in particular, $N_{p}(-n\Delta)=N_{p}(-\text{sqf}(n\Delta))$.

######  Lemma 3.1.

If $-n\Delta\in\dot{\mathbb{Q}}_{p}^{2}$, then
$\theta(L_{p},n)=N_{p}(-n\Delta)$.

###### Proof.

It follows from the definition of $\theta(L_{p},n)$ that

$N_{p}(-n\Delta)\subseteq\theta(L_{p},n)\subseteq\dot{\mathbb{Q}}_{p}.$

The result is immediate from this. ∎

######  Lemma 3.2.

If $\mathbb{Z}_{p}^{\times}\dot{\mathbb{Q}}_{p}^{2}\subseteq N_{p}(-n\Delta)$,
then $\operatorname{ord}_{p}(n\Delta)$ is even. Moreover, the reverse
implication is true when $p$ is odd.

###### Proof.

Suppose first that $\operatorname{ord}_{p}(-n\Delta)$ is odd. Then
$(\varepsilon_{p},-n\Delta)_{p}=(\varepsilon_{p},p)_{p}=-1$ by [10, 63:11a],
where $\varepsilon_{p}$ denotes a unit of $\mathbb{Z}_{p}$ of quadratic defect
$4\mathbb{Z}_{p}$. This establishes the first implication. If $p$ is odd and
$\operatorname{ord}_{p}(n\Delta)$ is even, then
$-n\Delta\in\mathbb{Z}_{p}^{\times}\dot{\mathbb{Q}}_{p}^{2}$. Hence
$(\gamma,-n\Delta)_{p}=+1$ for any
$\gamma\in\mathbb{Z}_{p}^{\times}\dot{\mathbb{Q}}_{p}^{2}$ by [10, 63:12]. ∎

The following theorem of Schulze-Pillot [14, Satz 2] gives complete criteria
for $n$ to be a spinor exceptional integer for $\text{gen}\mspace{1.0mu}L$.

######  Theorem 3.3.

The integer $n$ is a spinor exceptional integer for
$\text{gen}\mspace{1.0mu}L$ if and only if

$\theta(O^{+}(L_{p}))\subseteq
N_{p}(-n\Delta)\quad\textrm{and}\quad\theta(L_{p},n)=N_{p}(-n\Delta)$

for all primes $p$.

The local conditions appearing in the statement of this theorem will be
discussed in more detail in the following two subsections.

###  3.2. Primes not dividing $2\Delta$

Throughout this subsection, $p$ will denote a prime not dividing $2\Delta$.
Consequently, $p$ is odd, $\operatorname{ord}_{p}(\Delta)=0$ and $L_{p}$ is
unimodular. So
$\theta(O^{+}(L_{p}))=\mathbb{Z}_{p}^{\times}\dot{\mathbb{Q}}_{p}^{2}$ by [10,
92:5].

######  Lemma 3.4.

$\theta(O^{+}(L_{p}))\subseteq
N_{p}(-n\Delta)\iff\operatorname{ord}_{p}(n)\,\textrm{is even}$.

###### Proof.

This follows from Lemma 3.2 since
$\operatorname{ord}_{p}(n\Delta)=\operatorname{ord}_{p}(n)$. ∎

######  Lemma 3.5.

Assume that $\operatorname{ord}_{p}(n)$ is even.

  * (i)

If $\operatorname{ord}_{p}(n)=0$, then $\theta(L_{p},n)=N_{p}(-n\Delta)$.

  * (ii)

If $\operatorname{ord}_{p}(n)>0$, then

$\theta(L_{p},n)=N_{p}(-n\Delta)\iff-n\Delta\in\dot{\mathbb{Q}}_{p}^{2}.$

###### Proof.

If $-n\Delta\in\dot{\mathbb{Q}}_{p}^{2}$, then
$\theta(L_{p},n)=N_{p}(-n\Delta)$ by Lemma 3.1. So suppose that
$-n\Delta\notin\dot{\mathbb{Q}}_{p}^{2}$. Here $\operatorname{ord}_{p}(n)$
even implies that $\theta(O^{+}(L_{p}))\subseteq N_{p}(-n\Delta)$ by Lemma
3.4. Since $p\nmid\Delta$ and $\operatorname{ord}_{p}(n)$ is even, $p$ is
unramified in $\mathbb{Q}(\sqrt{-\text{sqf}(n\Delta)})$. So [14, Satz 3(a)]
applies with $r=s=0$. Hence, $\theta(O^{+}(L_{p}))\subseteq N_{p}(-n\Delta)$
holds if and only if $\operatorname{ord}_{p}(n)=0$. ∎

For a positive integer $t$, let $M_{t}$ denote the multiplicative semigroup
generated by $1$ and the set of all primes $p$ such that
$-t\in\dot{\mathbb{Q}}_{p}^{2}$. For any $s\in\dot{\mathbb{Q}}$, note that
$M_{t}=M_{s^{2}t}$; in particular, $M_{t}=M_{\text{sqf}(t)}$. Also,
$M_{t}^{2}$ will denote $\\{w^{2}:w\in M_{t}\\}$.

######  Corollary 3.6.

Assume that $\operatorname{ord}_{p}(n)$ is even for all $p\nmid 2\Delta$.
Write $n=kw^{2}$, where all prime divisors of $k$ divide $2\Delta$ and
$\textrm{g.c.d.}(w,2\Delta)=1$. Then

$\theta(L_{p},n)=N_{p}(-n\Delta)\,\,\textrm{for all}\,\,p\nmid 2\Delta\iff
w\in M_{n\Delta}.$

###  3.3. Primes dividing $2\Delta$

Throughout this subsection, $p$ will denote a prime divisor of $2\Delta$.

######  Lemma 3.7.

Let $p$ be an odd prime such that $-n\Delta\notin\dot{\mathbb{Q}}_{p}^{2}$ and
$\theta(O^{+}(L_{p}))\subseteq N_{p}(-n\Delta)$.

  * (i)

If $L_{p}\cong\langle 1,p,p^{2}\rangle$ or $L_{p}\cong\langle
1,p^{2},p^{3}\rangle$, then

$\theta(L_{p},n)=N_{p}(-n\Delta)\iff\operatorname{ord}_{p}(n)\leq 1.$

  * (ii)

If $L_{p}\cong\langle 1,p,p^{3}\rangle$, then

$\theta(L_{p},n)=N_{p}(-n\Delta)\iff\operatorname{ord}_{p}(n)\leq 2.$

###### Proof.

By the assumptions, we are in the situation of Satz 3(b) of [14]. For
$L_{p}\cong\langle 1,p,p^{2}\rangle$ or $L_{p}\cong\langle 1,p,p^{3}\rangle$,
the stated result follows from subcase (ii) of Satz 3(b) with
$r=1,s=2\,\textrm{or}\,3$. For $L_{p}\cong\langle 1,p^{2},p^{3}\rangle$, it
follows from subcase (i) with $r=2,s=3$. ∎

For the prime $2$, there are numerous possibilities for the nature of a Jordan
splitting of $L_{2}$. So in this case, we will state only the general form of
the needed result, and provide specific references to the relevant subcases of
the original result in [14, Satz 4] as they arise.

######  Lemma 3.8.

Assume that $-n\Delta\notin\dot{\mathbb{Q}}_{2}^{2}$ and
$\theta(O^{+}(L_{2}))\subseteq N_{2}(-n\Delta)$. Then there exists a
nonnegative integer $\lambda$ such that

$\theta(L_{2},n)=N_{2}(-n\Delta)\iff\operatorname{ord}_{2}(n)\leq\lambda.$

######  Corollary 3.9.

Let $p$ be a prime dividing $2\Delta$ and assume that $\lambda\geq 1$ if
$p=2$. If $\theta(O^{+}(L_{p}))\subseteq N_{p}(-n\Delta)$ and
$\operatorname{ord}_{p}(n)=0$, then $\theta(L_{p},n)=N_{p}(-n\Delta)$.

###### Proof.

For odd $p$, this follows either from Lemma 3.1 or from Satz 3(b) of [14]. For
$p=2$, it follows either from Lemma 3.1 or Lemma 3.8. ∎

##  4\. Discriminants divisible only by 2

The spinor regular ternaries with discriminants divisible only by 2 are
labelled as A1–A13 in [1]. Table 1 contains a list of representatives for all
of the classes of forms in the genus of each of these forms, along with their
separation into spinor genara and their discriminant. These representatives
and their separation into spinor genera were obtained by the use of Magma [4].
In each case, the spinor regular form is alone in its spinor genus, which is
labelled Spinor Genus I, and the remaining classes constitute a second spinor
genus labelled Spinor Genus II. These spinor genera will subsequently be
referred to simply as SGI and SGII.

# | Spinor Genus I | Spinor Genus II | $\Delta$
---|---|---|---
A1 | (2,2,5,2,2,0) | (1,1,16,0,0,0) | $2^{6}$
A2 | (1,4,9,4,0,0) | (1,1,32,0,0,0) | $2^{7}$
|  | (2,2,9,-2,2,0) |
A3 | (2,5,8,4,0,2) | (1,1,64,0,0,0) | $2^{8}$
|  | (2,2,17,2,-2,0) |
|  | (1,4,17,-4,0,0) |
A4 | (4,4,5,0,4,0) | (1,4,16,0,0,0) | $2^{8}$
A5 | (4,9,9,2,4,4) | (1,16,16,0,0,0) | $2^{10}$
A6 | (4,5,13,2,0,0) | (1,16,20,-16,0,0) | $2^{10}$
|  | (4,5,17,2,-4,-4) |
A7 | (5,8,8,0,4,4) | (1,4,64,0,0,0) | $2^{10}$
|  | (4,4,17,0,4,0) |
A8 | (4,8,17,0,4,0) | (1,8,64,0,0,0) | $2^{11}$
A9 | (9,9,16,8,8,2) | (1,16,64,0,0,0) | $2^{12}$
|  | (4,16,17,0,-4,0) |
A10 | (4,9,32,0,0,4) | (1,32,32,0,0,0) | $2^{12}$
|  | (4,17,17,2,4,4) |
A11 | (5,13,16,0,0,2) | (4,16,21,16,4,0) | $2^{12}$
|  | (4,5,64,0,0,-4) |
A12 | (9,17,32,-8,8,6) | (1,16,256,0,0,0) | $2^{14}$
|  | (16,16,17,-8,0,0) |
|  | (4,16,65,0,4,0) |
A13 | (9,16,36,16,4,8) | (1,64,64,0,0,0) | $2^{14}$
|  | (4,33,33,2,4,4) |
|  | (4,17,64,0,0,-4) |
Table 1. Genera containing spinor regular ternaries with 2-power discriminant

For each of the forms A1–A13, the data needed to determine the candidates for
spinor exceptional integers for their genus is given in Table 2. The second
column of the table gives the splitting of $L_{2}$ for the ternary lattice $L$
corresponding to the form. These splittings can be obtained by routine
calculation using one of the representative forms in the genus. In all cases,
the $2$-adic splitting is of the type

$\langle b_{1},b_{2}2^{r},b_{3}2^{s}\rangle,$

where $b_{i}\in\mathbb{Z}_{2}^{\times}$ and $r\leq s$ are nonnegative
integers. The third column contains the local spinor norm group
$\theta(O^{+}(L_{2}))$ for each lattice $L$. For the lattices for which the
splitting of $L_{2}$ has $0<r<s$ (that is, A4, A6, A7, A8, A9, A11 and A12),
the corresponding spinor norm groups $\theta(O^{+}(L_{2}))$ can be determined
by Propositions 1.4, 1.6, 1.7 and 1.8 and Theorem 2.7 of [8]. For the
remainder of the cases, $L_{2}$ is split by a multiple of $\langle
1,1\rangle$. Here $\theta(O^{+}(\langle
1,1\rangle)=\\{\gamma\in\dot{Q_{2}}:(\gamma,-1)_{2}=+1\\}=\\{1,2,5,10\\}\dot{\mathbb{Q}}_{2}^{2}$
by Proposition B of [9] 111Throughout the paper,
$\\{a_{1},\ldots,a_{k}\\}\dot{F}^{2}$ will be used to denote
$a_{1}\dot{F}^{2}\cup\cdots\cup a_{k}\dot{F}^{2}$.. Then
$\theta(O^{+}(L_{2}))=\\{1,2,5,10\\}\dot{\mathbb{Q}}_{2}^{2}$ follows from
Theorem 3.14(iv) of [8]. The value of $\lambda$ in Lemma 3.8 can be determined
by one of the subcases of [14, Satz 4]. For each form, the specific subcase
that applies is identified in the fourth column of the table, and the value of
$\lambda$ appears in the last column.

# | $L_{2}$ | $\theta(O^{+}(L_{2}))$ | subcase | $\lambda$
---|---|---|---|---
A1 | $\langle 1,1,2^{4}\rangle$ | $\\{1,2,5,10\\}\dot{\mathbb{Q}}_{2}^{2}$ | (b)(iii) | 1
A2 | $\langle 1,1,2^{5}\rangle$ | $\\{1,2,5,10\\}\dot{\mathbb{Q}}_{2}^{2}$ | (b)(iii) | 2
A3 | $\langle 1,1,2^{6}\rangle$ | $\\{1,2,5,10\\}\dot{\mathbb{Q}}_{2}^{2}$ | (b)(iii) | 3
A4 | $\langle 1,2^{2},2^{4}\rangle$ | $\\{1,5\\}\dot{\mathbb{Q}}_{2}^{2}$ | (b)(iii) | 1
A5 | $\langle 1,2^{4},2^{4}\rangle$ | $\\{1,2,5,10\\}\dot{\mathbb{Q}}_{2}^{2}$ | (b)(i) | 1
A6 | $\langle 5,2^{2},5\cdot 2^{6}\rangle$ | $\\{1,5\\}\dot{\mathbb{Q}}_{2}^{2}$ | (b)(ii) | 1
A7 | $\langle 1,2^{2},2^{6}\rangle$ | $\\{1,5\\}\dot{\mathbb{Q}}_{2}^{2}$ | (b)(iii) | 3
A8 | $\langle 1,2^{3},2^{6}\rangle$ | $\\{1,2,3,6\\}\dot{\mathbb{Q}}_{2}^{2}$ | (c)(iii) | 1
A9 | $\langle 1,2^{4},2^{6}\rangle$ | $\\{1,5\\}\dot{\mathbb{Q}}_{2}^{2}$ | (b)(iii) | 3
A10 | $\langle 1,2^{5},2^{5}\rangle$ | $\\{1,2,5,10\\}\dot{\mathbb{Q}}_{2}^{2}$ | (b)(iv) | 1
A11 | $\langle 5,2^{4},5\cdot 2^{6}\rangle$ | $\\{1,5\\}\dot{\mathbb{Q}}_{2}^{2}$ | (b)(ii) | 3
A12 | $\langle 1,2^{4},2^{8}\rangle$ | $\\{1,5\\}\dot{\mathbb{Q}}_{2}^{2}$ | (b)(iii) | 5
A13 | $\langle 1,2^{6},2^{6}\rangle$ | $\\{1,2,5,10\\}\dot{\mathbb{Q}}_{2}^{2}$ | (b)(i) | 3
Table 2. Data for A1–A13

######  Proposition 4.1.

Let $f$ be one of the forms A1–A13, and let $n$ be a positive integer. Then
$n$ is represented everywhere locally, but not globally, by $f$ if and only if
$n$ lies in:

  * (i)

$M_{1}^{2}$ for A1, A4, A5, A6, A10;

  * (ii)

$2M_{1}^{2}$ for A2;

  * (iii)

$M_{1}^{2},4M_{1}^{2}$ for A3, A7, A9, A13;

  * (iv)

$M_{2}^{2}$ for A8;

  * (v)

$4M_{1}^{2}$ for A11;

  * (vi)

$M_{1}^{2},4M_{1}^{2},16M_{1}^{2}$ for A12.

###### Proof.

First consider the lattice $L$ corresponding to the form A1. In this case, we
seek to show that

$n\not\rightarrow L\iff n\in M_{1}^{2}.$

We first prove the forward implication. From $n\not\rightarrow L$ it follows
that $n\not\rightarrow\text{spn}\mspace{1.0mu}L$, since $L$ is spinor regular.
So $n$ is a spinor exceptional integer for $\text{gen}\mspace{1.0mu}L$. So by
Theorem 3.3 and Lemma 3.4 we see that $\operatorname{ord}_{p}(n)$ is even for
all odd $p$; thus,

$n=2^{t}w^{2},$

with $w$ odd. As $\Delta=2^{6}$, we then have $\text{sqf}(n\Delta)=1$ or $2$,
depending upon whether $t$ is even or odd, respectively. Since
$\theta(O^{+}(L_{2}))\subseteq N_{2}(-n\Delta)$ and
$5\in\theta(O^{+}(L_{2}))\smallsetminus N_{2}(-2)$, it must be that $t$ is
even and $M_{n\Delta}=M_{1}$. By Lemma 3.5 it follows that $w\in M_{1}$. Since
$-1\notin\dot{\mathbb{Q}}_{2}^{2}$, it follows from Lemma 3.8 that

$t=\operatorname{ord}_{2}(n)\leq\lambda=1.$

Since $t$ is even, this implies that $t=0$. Hence, $n\in M_{1}^{2}$ as
claimed.

For the reverse implication, assume that $n\in M_{1}^{2}$. So $n=w^{2}$, $w\in
M_{1}$ and $\text{sqf}(n\Delta)=1$. Note that $w$ is odd since $2\notin
M_{1}$, as $-1\notin\dot{\mathbb{Q}}_{2}^{2}$. For $p$ odd,
$\operatorname{ord}_{p}(n)$ even implies that $\theta(O^{+}(L_{p}))\subseteq
N_{p}(-n\Delta)$, by Lemma 3.4, and $\theta(L_{p},n)=N_{p}(-n\Delta)$ by Lemma
3.6. Also $\theta(O^{+}(L_{2}))\subseteq N_{2}(-1)$. Since
$\operatorname{ord}_{2}(n)=0$, it follows from Lemma 3.8 that
$\theta(L_{2},n)=N_{2}(-n\Delta)$. By Theorem 3.3, $n$ is a spinor exceptional
integer for $\text{gen}\mspace{1.0mu}L$. The form $\langle 1,1,16\rangle$ lies
in SGII for the genus of A1. So $1\rightarrow SGII$ and hence $n\rightarrow
SGII$ since $n$ is a square. But then it must be that $n\not\rightarrow SGI$
and so $n\not\rightarrow L$. This completes the proof for A1.

The proofs for the forms A4, A5, A6 and A10 are identical. For A3, A7, A9, A11
and A13, the proof remains the same except that $\lambda=3$ for these cases.
Thus the conditions for $\theta(L_{2},n)=N_{2}(-n\Delta)$ in Lemma 3.8 hold
also with $t=2$, and integers of the type $4M_{1}^{2}$ are also spinor
exceptional integers for these genera. Note that for A11,
$1\not\rightarrow\text{gen}\mspace{1.0mu}L$, so the integers in $M_{1}^{2}$
need not be listed among the spinor exceptional integers. In this one
instance, $4\rightarrow SGII$, and so all integers of the type $4M_{1}^{2}$
are represented by SGII, and hence not by $L$. For the form A12, the proof is
again unchanged except that in this case $\lambda=5$; thus the conditions for
$\theta(L_{2},n)=N_{2}(-n\Delta)$ in Lemma 3.8 hold also with $t=4$, and the
integers of the type $M_{1}^{2},4M_{1}^{2}$ and $16M_{4}^{2}$ are spinor
exceptional integers for the genus.

This leaves two remaining cases, A2 and A8. The only difference for the proof
of A2 is that $\operatorname{ord}_{2}(\Delta)$ is odd. Since the other aspects
of the proof remain the same, including that $\text{sqf}(n\Delta)=1$, we see
in this case that $n=2^{t}w^{2}$, where $t$ and $w$ are odd integers. As
$\lambda=1$, it must be that $t=1$ and so $n=2w^{2}$. Here $2\rightarrow
SGII$, so we conclude that no elements of the type $2M_{1}^{2}$ are
represented by $L$.

Finally, in the case of A8, we have
$\theta(O^{+}(L_{2}))=\\{1,2,3,6\\}\dot{\mathbb{Q}}_{2}^{2}$. So
$\theta(O^{+}(L_{p}))\subseteq N_{p}(-n\Delta)$ holds for all $p$ if and only
if $\text{sqf}(n\Delta)=2$. Hence $\text{sqf}(n)=1$ for any spinor exceptional
integer $n$ for this genus, since and $\operatorname{ord}_{2}(\Delta)=11$ is
odd. As $\lambda=1$, the spinor exceptional integers for the genus must be of
the form $w^{2}$, for some odd integer $w$. In this case, the prime divisors
$p$ of $w$ must satisfy the condition that $-2\in\dot{\mathbb{Q}}_{p}^{2}$.
Hence, $w\in M_{2}$, as asserted. ∎

######  Remark.

The set $M_{1}$ appearing in the statement of the proposition is generated by
1 and the primes congruent to 1 modulo 4, and the set $M_{2}$ is generated by
1 and the primes congruent to 1 or 3 modulo 8.

##  5\. Discriminants divisible by 2 and 3

The spinor regular ternaries with discriminants divisible by both 2 and 3 are
labelled as B1–B12 in [1]. Table 3 contains a list of representatives for all
of the classes of forms in the genus of each of these forms, along with their
separation into spinor genara and their discriminant. In each case, as for the
previous grouping, the genus splits into two spinor genera, which are labelled
as before. The spinor regular forms B4 and B11 are the first forms listed in
SGI for their genus.

# | Spinor Genus I | Spinor Genus II | $\Delta$
---|---|---|---
B1 | (3,3,4,0,0,3) | (1,1,36,0,0,1) | $2^{2}3^{3}$
B2 | (3,4,4,4,3,3) | (1,3,10,-3,1,0) | $2^{2}3^{3}$
B3 | (1,7,12,0,0,1) | (3,3,13,-3,3,-3) | $2^{2}3^{4}$
|  | (1,1,108,0,0,1) |
B4 | (3,7,7,5,3,3) | (1,1,144,0,0,1) | $2^{4}3^{3}$
| (3,3,16,0,0,-3) | (1,3,37,3,1,0) |
B5 | (4,4,9,0,0,4) | (1,12,12,12,0,0) | $2^{4}3^{3}$
B6 | (3,4,9,0,0,0) | (1,3,36,0,0,0) | $2^{4}3^{3}$
B7 | (4,9,12,0,0,0) | (1,12,36,0,0,0) | $2^{6}3^{3}$
B8 | (4,9,28,0,4,0) | (1,36,36,-36,0,0) | $2^{4}3^{5}$
|  | (9,13,13,-10,-6,-6) |
B9 | (9,16,16,16,0,0) | (1,48,48,-48,0,0) | $2^{8}3^{3}$
B10 | (13,13,16,-8,8,10) | (4,13,37,-2,4,-4) | $2^{8}3^{3}$
B11 | (9,16,48,0,0,0) | (1,48,144,0,0,0) | $2^{10}3^{3}$
| (16,25,25,-14,16,-16) | (4,49,49,-46,4,4) |
B12 | (9,16,112,16,0,0) | (1,144,144,144,0,0) | $2^{8}3^{5}$
|  | (9,49,49,-46,6,6) |
Table 3. Genera containing spinor regular ternaries with discriminant
divisible by 2 and 3

The data necessary to determine the spinor exceptional integers for these
genera is summarized in Table 4. We first record the local splittings at the
primes 2 and 3 for the corresponding lattices. The splittings for $L_{3}$ are
given in the second column of Table 4. In every case, it follows from [13,
Satz 3] that

$\theta(O^{+}(L_{3}))=\\{1,3\\}\dot{\mathbb{Q}}_{3}^{2}.$

The $2$-adic splittings appear in the third column. Some additional
explanation is in order here. This grouping contains both classic and non-
classic forms. For the non-classic forms B1–B4, the splitting given in the
table is for the lattice $L_{2}^{\prime}$ obtained by scaling $L_{2}$ by $2$.
For the remaining forms B5–B12, which are classic, the splitting of
$L_{2}^{\prime}=L_{2}$ is given. Since the spinor norm group
$\theta(O^{+}(L_{2}))$ is unaffected by scaling, this makes it possible to
directly apply the results of [8] to determine $\theta(O^{+}(L_{2}))$ from
$L_{2}^{\prime}$ in all cases. We use the notations $\mathbb{H}$ and
$\mathbb{A}$ to denote the binary $\mathbb{Z}_{2}$-lattices with Gram matrices
$\left(\begin{smallmatrix}0&1\\\ 1&0\end{smallmatrix}\right)$ and
$\left(\begin{smallmatrix}2&1\\\ 1&2\end{smallmatrix}\right)$, respectively.
If $M$ is one of these matrices and $\alpha\in\mathbb{Z}$, $\alpha M$ will
denote the matrix resulting from multiplying each entry of $M$ by $\alpha$. In
all cases occurring in Table 4, $L_{2}^{\prime}$ has a binary Jordan component
either of odd order or of even order (see [8, Definition 3.1, p. 79] for an
explanation of this terminology), and the complementary component has the same
order. It follows from Theorem 3.14(i) of [8] that in all cases

$\theta(O^{+}(L_{2}))=\mathbb{Z}_{2}^{\times}\dot{\mathbb{Q}}_{2}^{2}.$

The value of $\lambda$ in Lemma 3.8 can be determined by one of the subcases
of [14, Satz 4(a)]. For each form, the specific subcase that applies is
identified in the fourth column of the table, and the value of $\lambda$
appears in the last column.

# | $L_{3}$ | $L^{\prime}_{2}$ | subcase | $\lambda$
---|---|---|---|---
B1 | $\langle 1,3,3^{2}\rangle$ | $\mathbb{A}\perp\langle 2^{3}\rangle$ | (ii)($\beta$) | 1
B2 | $\langle 1,3,3^{2}\rangle$ | $\mathbb{H}\perp\langle 5\cdot 2^{3}\rangle$ | (ii)($\alpha$) | 1
B3 | $\langle 1,3,3^{3}\rangle$ | $\mathbb{A}\perp\langle 3\cdot 2^{3}\rangle$ | (ii)($\beta$) | 1
B4 | $\langle 1,3,3^{2}\rangle$ | $\mathbb{A}\perp\langle 2^{5}\rangle$ | (ii)($\beta$) | 3
B5 | $\langle 1,3,3^{2}\rangle$ | $\langle 1\rangle\perp 2\mathbb{A}$ | (ii)($\gamma$) | 1
B6 | $\langle 1,3,3^{2}\rangle$ | $\langle 1,3,2^{2}\rangle$ | (i)($\beta$) | 1
B7 | $\langle 1,3,3^{2}\rangle$ | $\langle 1,2^{2},3\cdot 2^{2}\rangle$ | (i)($\alpha$) | 1
B8 | $\langle 1,3^{2},3^{3}\rangle$ | $\langle 1\rangle\perp 2\mathbb{A}$ | (ii)($\gamma$) | 1
B9 | $\langle 1,3,3^{2}\rangle$ | $\langle 1\rangle\perp 2^{3}\mathbb{A}$ | (ii)($\gamma$) | 3
B10 | $\langle 1,3,3^{2}\rangle$ | $\langle 5\rangle\perp 2^{3}\mathbb{H}$ | (ii)($\gamma$) | 3
B11 | $\langle 1,3,3^{2}\rangle$ | $\langle 1,2^{4},3\cdot 2^{4}\rangle$ | (i)($\alpha$) | 3
B12 | $\langle 1,3^{2},3^{3}\rangle$ | $\langle 1\rangle\perp 2^{3}\mathbb{A}$ | (ii)($\gamma$) | 3
Table 4. Data for B1–B12

######  Proposition 5.1.

Let $f$ be one of the forms B1–B12, and let $n$ be a positive integer. Then
$n$ is represented everywhere locally, but not globally, by $f$ if and only if
$n$ lies in:

  * (i)

$M_{3}^{2}$ for B1, B2, B5, B6, B7, B8;

  * (ii)

$3M_{3}^{2}$ for B3;

  * (iii)

$M_{3}^{2},4M_{3}^{2}$ for B4, B9, B11, B12;

  * (iv)

$4M_{3}^{2}$ for B10.

###### Proof.

First consider the lattice $L$ corresponding to the form B1. Assume first that
$n\rightarrow\text{gen}\mspace{1.0mu}L$ but $n\not\rightarrow L$. Then
$n\not\rightarrow\text{spn}\mspace{1.0mu}L$, since $L$ is spinor regular. So
$n$ is a spinor exceptional integer for $\text{gen}\mspace{1.0mu}L$. From
Theorem 3.3, Lemma 3.4, the computation of $\theta(O^{+}(L_{2}))$, and Lemma
3.2, it follows that $\operatorname{ord}_{p}(n)$ is even for all $p\neq 3$. So
$\text{sqf}(n\Delta)=1$ or $3$. Since $\theta(O^{+}(L_{3}))\not\subseteq
N_{3}(-1)$, it must be that $\text{sqf}(n\Delta)=3$. So $n=2^{t}3^{s}w^{2}$
with $s,t$ even and $w\in M_{3}$, by Lemma 3.6. By Lemma 3.7 and Lemma 3.8,
$\operatorname{ord}_{p}(n)\leq 1$ for $p=2,3$. Hence, $s=t=0$ and the
conclusion follows.

For the reverse implication, assume that $n=w^{2}$ with $w\in M_{3}$. The
lattice corresponding to $(1,1,36,0,0,1)$ lies in SGII and represents $1$, and
so $n\rightarrow SGII$; in particular,
$n\rightarrow\text{gen}\mspace{1.0mu}L$. Since $\Delta=2^{4}3^{3}$, we have
$\text{sqf}(n\Delta)=3$; thus, $N_{p}(-n\Delta)=N_{p}(-3)$ and
$M_{n\Delta}=M_{3}$. Since $\operatorname{ord}_{p}(n\Delta)$ is even for
$p\neq 2,3$, it follows from Lemma 3.4 that $\theta(O^{+}(L_{p}))\subseteq
N_{p}(-n\Delta)$, and, by Corollary 3.6, $\theta(L_{p},n)=N_{p}(-n\Delta)$ for
all $p\neq 2,3$. For the primes $2$ and $3$, the direct computations show that
$\theta(O^{+}(L_{2}))=\mathbb{Z}_{2}^{\times}\dot{\mathbb{Q}}_{2}^{2}=N_{2}(-3)=N_{2}(-n\Delta)$
and
$\theta(O^{+}(L_{3}))=\\{1,3\\}\dot{\mathbb{Q}}_{3}^{2}=N_{3}(-3)=N_{3}(-n\Delta)$.
It then follows by Corollary 3.9 that $\theta(L_{p},n)=N_{p}(-n\Delta)$. So
the criteria of Theorem 3.3 are met and $n$ is a spinor exceptional integer
for $\text{gen}\mspace{1.0mu}L$. So $n\not\rightarrow SGI$ and, in particular,
$n\not\rightarrow L$. This completes the proof for B1.

The proofs for the forms B2, B5, B6, B7 and B8 proceed in exactly the same
way. For B4, B9, B11 and B12, the proof is analogous except that $\lambda=3$
for these cases. Thus the conditions for $\theta(L_{2},n)=N_{2}(-n\Delta)$ in
Lemma 3.8 hold also with $t=2$, and integers of the type $4M_{3}^{2}$ are also
spinor exceptional integers for these genera.

This leaves two remaining cases, B3 and B10. The only difference for the proof
of B3 is that $\operatorname{ord}_{3}(\Delta)$ is even. Since the other
aspects of the proof remain the same, including that $\text{sqf}(n\Delta)=1$,
we see in this case that if $n$ is a spinor exceptional integer for
$\text{gen}\mspace{1.0mu}L$, then $n=2^{t}3^{s}w^{2}$, where $t$ is even, $s$
is odd, and $w\in M_{3}$. As $\lambda=1$, it must be that $t=0$, and it
follows from Lemma 3.7 that $s=1$; so $n=3w^{2}$, $w\in M_{3}$. In this case,
$3\rightarrow SGII$, so we conclude that no elements of the type $3M_{3}^{2}$
are represented by $L$.

For the case B10, the proof of the forward implication proceeds as before up
to the point where we conclude that $n=2^{t}w^{2}$ with $t$ even and $w\in
M_{3}$. In this case, $L_{2}\cong\langle 5\rangle\perp 2^{3}\mathbb{A}$. So
the assumption that $n\rightarrow L_{2}$ implies that $t\neq 0$. Hence, $n\in
4M_{3}^{2}$, as claimed. For the reverse implication, observe that
$4\rightarrow SGII$, and so $4M_{3}^{2}\rightarrow SGII$. The remainder of the
argument then proceeds as before. ∎

######  Remark.

The set $M_{3}$ appearing in the statement of the proposition is generated by
1 and the primes congruent to 1 modulo 3.

##  6\. Discriminants divisible by 2 and 7

The spinor regular ternaries with discriminants divisible by 2 and 7 are
labelled as C1–C4 in [1]. Table 5 contains a list of representatives for all
of the classes of forms in the genus of each of these forms, along with their
separation into spinor genara and their discriminant. In each case, as for the
previous two groupings, the genus splits into two spinor genera, which are
labelled as before.

# | Spinor Genus I | Spinor Genus II | $\Delta$ | $L^{\prime}_{2}$
---|---|---|---|---
C1 | (2,7,8,7,1,0) | (1,7,14,7,0,0) | $2^{2}7^{3}$ | $\mathbb{H}\perp\langle 2\rangle$
|  | (1,2,49,0,0,1) |  |
C2 | (7,8,9,6,7,0) | (4,7,15,-7,4,0) | $2^{4}7^{3}$ | $\mathbb{A}\perp\langle 5\cdot 2^{3}\rangle$
|  | (1,7,51,-7,-1,0) |  |
C3 | (8,9,25,2,4,8) | (1,28,56,-28,0,0) | $2^{6}7^{3}$ | $\langle 1\rangle\perp 2\mathbb{H}$
|  | (4,8,49,0,0,4) |  |
C4 | (29,32,36,32,12,24) | (4,29,197,-2,-4,4) | $2^{10}7^{3}$ | $\langle 5\rangle\perp 2^{3}\mathbb{A}$
|  | (16,32,53,-8,16,-16) |  |
Table 5. Genera containing spinor regular ternaries with discriminant
divisible by 2 and 7

For each lattice $L$ corresponding to one of the forms C1–C4, we have
$L_{7}\cong\langle 1,7,7^{2}\rangle$, and it follows from [13, Satz 3] that

$\theta(O^{+}(L_{7}))=\\{1,7\\}\dot{\mathbb{Q}}_{7}^{2}=N_{7}(-7).$

The $2$-adic splittings for these lattices are given in the last column of
Table 5. The forms C1 and C2 are non-classic and the splittings listed for
them are for the lattice $L^{\prime}_{2}$ obtained from $L_{2}$ by scaling by
2; for C3 and C4, which are classic, the splittings listed are for
$L^{\prime}_{2}=L_{2}$. In all cases, it follows from [8, Theorem 3.14(i)]
that

$\theta(O^{+}(L_{2}))=\mathbb{Z}_{2}^{\times}\dot{\mathbb{Q}}_{2}^{2}.$

######  Proposition 6.1.

Let $f$ be one of the forms C1–C4, and let $n$ be a positive integer. Then $n$
is represented everywhere locally, but not globally, by $f$ if and only if $n$
lies in:

  * (i)

$M_{7}^{2}$ for C1, C2, C3;

  * (ii)

$4M_{7}^{2}$ for C4.

###### Proof.

Let $L$ be a lattice corresponding to one of the forms C1-C3. Assume first
that $n\rightarrow\text{gen}\mspace{1.0mu}L$ but $n\not\rightarrow L$. As
before, $n$ is a spinor exceptional integer for $\text{gen}\mspace{1.0mu}L$.
From Theorem 3.3, Lemma 3.4 and the computation of $\theta(O^{+}(L_{2}))$, it
follows that $\operatorname{ord}_{p}(n\Delta)$ is even for all $p\neq 7$. So
$\text{sqf}(n\Delta)=1$ or $7$. Since $\theta(O^{+}(L_{7}))\not\subseteq
N_{7}(-1)$, it must be that $\text{sqf}(n\Delta)=7$. So $n=2^{s}7^{t}w^{2}$
with $s,t$ even and $w\in M_{7}$, by Lemma 3.6. By Lemma 3.7,
$\operatorname{ord}_{7}(n)\leq 1$ and so $t=0$. Since $2\in M_{7}$, it follows
that $n\in M_{7}^{2}$ as claimed.

For the reverse implication, assume that $n=w^{2}$ with $w\in M_{7}$. It can
be seen from the representatives of SGII that $1\rightarrow SGII$, and so
$n\rightarrow SGII$; in particular, $n\rightarrow\text{gen}\mspace{1.0mu}L$.
Since $\operatorname{ord}_{2}(\Delta)$ is even and
$\operatorname{ord}_{7}(\Delta)$ is odd, we have $\text{sqf}(n\Delta)=7$;
thus, $N_{p}(-n\Delta)=N_{p}(-7)$ for all $p$, and $M_{n\Delta}=M_{7}$. Since
$\operatorname{ord}_{p}(n\Delta)$ is even for $p\neq 2,7$, it follows from
Lemma 3.4 that $\theta(O^{+}(L_{p}))\subseteq N_{p}(-n\Delta)$, and from
Corollary 3.6, that $\theta(L_{p},n)=N_{p}(-n\Delta)$ for all $p\neq 2,7$.
Also, $\theta(O^{+}(L_{7}))=N_{7}(-7)=N_{7}(-n\Delta)$ and
$\operatorname{ord}_{7}(n)=0$ gives $\theta(L_{7},n)=N_{7}(-n\Delta)$, by
Corollary 3.9. Since $-7\in\dot{\mathbb{Q}}_{2}^{2}$, we have
$\theta(O^{+}(L_{2}))=\mathbb{Z}_{2}^{\times}\dot{\mathbb{Q}}_{2}^{2}\subseteq
N_{2}(-7)=N_{2}(-n\Delta)$, and, by Lemma 3.1,
$\theta(L_{7},n)=N_{2}(-n\Delta)$. So the criteria of Theorem 3.3 are met and
$n$ is a spinor exceptional integer for $\text{gen}\mspace{1.0mu}L$. So
$n\not\rightarrow SGI$ and, in particular, $n\not\rightarrow L$. This
completes the proof for C1-C3.

For the case C4, the proof of the forward implication proceeds as above to the
point where we conclude that $n=2^{s}w^{2}$ with $s$ even and $w\in M_{7}$. In
this case, $L_{2}\cong\langle 5\rangle\perp 2^{3}\mathbb{A}$. So the
assumption that $n\rightarrow L_{2}$ implies that $s\neq 0$. Hence, $n\in
4M_{7}^{2}$, as claimed. For the reverse implication, observe that
$4\rightarrow SGII$, and so $4M_{7}^{2}\rightarrow SGII$. The remainder of the
argument then proceeds as before. ∎

######  Remark.

The set $M_{7}$ appearing in the statement of the proposition is generated by
1 and the primes congruent to 1, 2 or 4 modulo 7.

##  7\. Forms not alone in their spinor genus

For each of the 27 spinor regular ternaries that are alone in their spinor
genus, Aygin et al [1] summarize in Table A.17 the list of all positive
integers not represented by the form. As detailed in that paper, the results
for several of these forms appeared earlier in work of Lomadze [12] and
Berkovich [3]. However, for the remaining two forms which are not alone in
their spinor genus, the method of [1] yields only the even integers that fail
to be represented. For completeness, we will state here the full results for
these two forms, which are B4 and B11 in the list. Before stating the results,
we will establish three lemmas needed to analyze the local obstructions to
representation by these forms.

######  Lemma 7.1.

Let $L$ be a ternary quadratic $\mathbb{Z}$-lattice such that $L_{2}\cong
M\perp\langle 16\rangle$, where $M$ is a lattice corresponding to
$x^{2}+xy+y^{2}$, and let $n$ be a positive integer. Then $n\not\rightarrow
L_{2}$ if and only if $n=2+4\ell$ or $8+16\ell$, for some nonnegative integer
$\ell$.

###### Proof.

Since $\mathbb{Z}_{2}^{\times}\rightarrow M$, it follows that
$\mathbb{Z}_{2}^{\times},4\mathbb{Z}_{2}^{\times}\rightarrow L_{2}$. Also, for
$x,y\in\mathbb{Z}_{2}$, $x^{2}+xy+y^{2}$ lies in either
$\mathbb{Z}_{2}^{\times}$ or $4\mathbb{Z}_{2}$. So all odd integers and
integers of the type $4n_{0}$ with $n_{0}$ odd are represented by $L_{2}$, and
no integer of the type $2+4\ell$ can be represented by $L_{2}$.

Consider $n=4n_{0}$ with $n_{0}$ even. If there exist $x,y\in\mathbb{Z}_{2}$
such that

(7.1) $n=x^{2}+xy+y^{2}+16z^{2},$

then $x,y\in 2\mathbb{Z}_{2}$ and the righthand side of Equation (7.1) is in
$4\mathbb{Z}_{2}$. Hence, no element of the type $8+16\ell$ is represented by
$L_{2}$.

Finally, let $n=16n_{0}$, $n_{0}\in\mathbb{Z}$. Since
$\mathbb{Z}_{2}^{\times}\rightarrow M$, either $n_{0}$ or $n_{0}-1$ is
represented by $M$. So there exist $x_{0},y_{0},z\in\mathbb{Z}_{2}$ such that

(7.2) $n_{0}=x_{0}^{2}+x_{0}y_{0}+y_{0}^{2}+z^{2}.$

It follows that Equation (7.1) is satisfied with $x=4x_{0},y=4y_{0}$. This
completes the proof. ∎

######  Lemma 7.2.

Let $L$ be a ternary quadratic $\mathbb{Z}$-lattice such that
$L_{2}\cong\langle 1,16,48\rangle$, and let $n$ be a positive integer. Then
$n\not\rightarrow L_{2}$ if and only if $n=5+8\ell$, $2+4\ell$, $3+4\ell$,
$8+16\ell$, or $12+16\ell$, for some nonnegative integer $\ell$.

###### Proof.

Here $n\rightarrow L_{2}$ if and only if there exist $x,y,z\in\mathbb{Z}_{2}$
such that

(7.3) $n=x^{2}+16y^{2}+48z^{2}.$

If $2\nmid n$, then Equation (7.3) is solvable if and only if $n\equiv
1\,(\text{mod}\,8)$, thus ruling out integers of the type $5+8\ell$ and
$3+4\ell$. If $2\mid n$, then $x=2x_{0}$ for some $x_{0}\in\mathbb{Z}_{2}$, so

(7.4) $n=4x_{0}^{2}+16y^{2}+48z^{2}.$

So it must be that $4\mid n$, ruling out integers of the type $2+4\ell$. Write
$n=4n_{0}$, with $n_{0}\in\mathbb{Z}$. Dividing Equation (7.4) through by 4
then gives

(7.5) $n_{0}=x_{0}^{2}+4y^{2}+12z^{2}.$

If $2\nmid n_{0}$, then Equation (7.5) is solvable if and only if $n_{0}\equiv
1\,(\text{mod}\,4)$. This rules out integers of the type
$4(3+4\ell)=12+16\ell$. If $2\mid n_{0}$ then $x_{0}\in 2\mathbb{Z}_{2}$, and
so the righthand side of Equation (7.5) is in $4\mathbb{Z}_{2}$. This rules
out integers of the type $8+16\ell$. Finally, the lattice $\langle
1,1,3\rangle$ is isotropic over $\mathbb{Z}_{2}$ and is therefore
$\mathbb{Z}_{2}$-universal by [6, Proposition 4.1]. Hence, all positive
integers divisible by 16 are represented by $L_{2}$. This completes the proof.
∎

######  Lemma 7.3.

Let $L$ be a ternary quadratic $\mathbb{Z}$-lattice such that
$L_{3}\cong\langle 1,3,9\rangle$, and let $n$ be a positive integer. Then
$n\not\rightarrow L_{3}$ if and only if $n=2+3\ell$ or $n=9^{k}(6+9\ell)$, for
some nonnegative integers $k,\ell$.

###### Proof.

Let $V$ denote the underlying quadratic space. By a computation of Hasse
symbols, it follows that $V_{3}$ is anisotropic; hence $\alpha\not\rightarrow
V_{3}$ for any $\alpha\in-dV=-3\dot{\mathbb{Q}}_{3}^{2}$ (see, e.g., [6, Lemma
2.2]). As $9^{k}(6+9\ell)=3^{2k}\cdot
3(2+3\ell)\in-3\dot{\mathbb{Q}}_{3}^{2}$, no such integer can be represented
by $L_{3}$. Integers of the type $n=2+3\ell$ are ruled out for representation
by $L_{3}$ by the Local Square Theorem [10, 63:1].

It remains to show that all other integers are represented by $L_{3}$. Write
$n=3^{t}n_{0}$, with $t$ a nonnegative integer and $n_{0}\equiv
1,2\,(\text{mod}\,3)$. If $n_{0}\equiv 1\,(\text{mod}\,3)$, then there exists
$\lambda\in\mathbb{Z}_{3}^{\times}$ such that $n_{0}=\lambda^{2}$. So if $t=0$
and $n_{0}\equiv 1\,(\text{mod}\,3)$ then $n=\lambda^{2}\rightarrow L_{3}$. If
$t=2k+1$ is odd and $n_{0}\equiv 1\,(\text{mod}\,3)$, then
$n=3(3^{k}\lambda)^{2}\rightarrow L_{3}$. By [10, 92:1b],
$\mathbb{Z}_{3}^{\times}\rightarrow\langle 1,1\rangle$. So
$3^{2}\mathbb{Z}_{3}^{\times}\rightarrow\langle 1,3^{2}\rangle$. It then
follows that $n\rightarrow L_{3}$ whenever $t$ is even and $t\geq 2$. This
exhausts all cases and completes the proof. ∎

######  Proposition 7.4.

The form $3x^{2}+7y^{2}+7z^{2}+3xy+3xz+5yz$ represents a positive integer $n$
if and only if $n$ is not of the type

$2+3\ell,9^{k}(6+9\ell),4^{a}(2+4\ell)\,\,\text{or}\,\,4^{a}M_{3}^{2},$

where $k,\ell$ are nonnegative integers and $a\in\\{0,1\\}$.

###### Proof.

This is the form B4. Let $L$ be a lattice corresponding to this form and let
$n$ be a positive integer. By [10, 92:1b], $n\rightarrow L_{p}$ for all primes
$p\neq 2,3$. For $p=2$ and $p=3$, the conditions for representation by $L_{p}$
are given in Lemmas 7.1 and 7.3, respectively. Consequently,
$n\rightarrow\text{gen}\mspace{1.0mu}L$ if and only if $n$ is not of one of
the types $2+3\ell$, $9^{k}(6+9\ell)$ or $4^{a}(2+4\ell)$ for some nonnegative
integers $k,\ell$ and $a\in\\{0,1\\}$. If $n$ is not of one of these types,
then $n\rightarrow L$ if and only if $n$ does not lie in $M_{3}^{2}$ or
$4M_{3}^{2}$, by Proposition 5.1(iii). ∎

######  Proposition 7.5.

The form $9x^{2}+16y^{2}+48z^{2}$ represents a positive integer $n$ if and
only if $n$ is not of the type

$5+8\ell,4^{a}(2+4\ell),4^{a}(3+4\ell)\,\,\text{or}\,\,4^{a}M_{3}^{2},$

where $\ell$ is a nonnegative integer and $a\in\\{0,1\\}$.

###### Proof.

This is the form B11. Let $L$ be a lattice corresponding to this form and let
$n$ be a positive integer. By [10, 92:1b], $n\rightarrow L_{p}$ for all primes
$p\neq 2,3$. For $p=2$ and $p=3$, the conditions for representation by $L_{p}$
are given in Lemmas 7.2 and 7.3, respectively. Consequently,
$n\rightarrow\text{gen}\mspace{1.0mu}L$ if and only if $n$ is not of one of
the types $5+8\ell$, $4^{a}(2+4\ell)$ or $4^{a}(3+4\ell)$ for some nonnegative
integers $\ell$ and $a\in\\{0,1\\}$. If $n$ is not of one of these types, then
$n\rightarrow L$ if and only if $n$ does not lie in $M_{3}^{2}$ or
$4M_{3}^{2}$, by Proposition 5.1(iii). ∎

## References

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