# Even Unimodular Lattices from Quaternion Algebras00footnotetext: At the time
of carrying out most of this work, L. Amorós was with the Helsinki Institute
of Information Technology, Aalto University, and is now with the Finnish
Meteorological Institute; M. T. Damir was with the Department of Mathematics
and Systems Analysis, Aalto University, and is currently with the Department
of Computer Science, University of Helsinki. C. Hollanti is with the
Department of Mathematics, Aalto University. M. T. Damir would like to thank
Aalto Science Institute for partial funding towards his research. C.
Hollanti’s work was supported in part by the Academy of Finland grant #336005,
which is gratefully acknowledged.
Emails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Laia Amorós1,2 1Aalto University, 2Finnish Meteorological Institute,
3University of Helsinki  M. Taoufiq Damir1,3 1Aalto University, 2Finnish
Meteorological Institute, 3University of Helsinki  Camilla Hollanti1 1Aalto
University, 2Finnish Meteorological Institute, 3University of Helsinki

###### Abstract

We review a lattice construction arising from quaternion algebras over number
fields and use it to obtain some known extremal and densest lattices in
dimensions 8 and 16. The benefit of using quaternion algebras over number
fields is that the dimensionality of the construction problem is reduced by
3/4. We explicitly construct the $E_{8}$ lattice (resp. $E_{8}^{2}$ and
$\Lambda_{16}$) from infinitely many quaternion algebras over real quadratic
(resp. quartic) number fields and we further present a density result on such
number fields. By relaxing the extremality condition, we also provide a source
for constructing even unimodular lattices in any dimension multiple of $8$.

##  1 Introduction

Lattices are objects that naturally arise in many areas of mathematics and
have been used in a wide range of applications, including communications and
cryptography. A lattice $\Lambda$ is a discrete additive subgroup of
$\mathbb{R}^{n}$, generated by all integral combinations of $k$ linearly
independent vectors $\\{v_{1},\ldots,v_{k}\\}$ of $\mathbb{R}^{n}$. If $k=n$
we say that $\Lambda$ is a full-rank lattice. In this paper we are interested
in lattices arising from number theoretic constructions. Lattices arising from
ideals in number fields, known as ideal lattices, have been extensively
studied [BF99, BF02, BFN05]. A natural extension of ideal lattices is the
construction of lattices arising from orders in division algebras, and in
particular in quaternion algebras.

A lattice $\Lambda$ is said to be integral if the squared norms of its vectors
are integers. If these norms are even, then the lattice is called even. An
integral lattice of volume one is said to be unimodular. A key property in the
study of unimodular lattices is based on the fact that their associated theta
series are modular forms. This played an important role in deriving results
about such lattices using the analytic theory of modular forms. A well known
result in this direction shows that even unimodular lattices exist only in
dimensions multiple of $8$. Moreover, a celebrated theorem of Siegel shows
that the length of a shortest vector in an even unimodular lattice of
dimension $n$ is at most $\lfloor\frac{n}{24}\rfloor+1$. Lattices achieving
this bound are called extremal.

Classifying extremal even unimodular lattices in every dimension is a long
standing problem in the theory of lattices. Unfortunately, this classification
is only known up to dimension $24$, namely, in dimensions $8,16$ and $24$. It
is worth mentioning that the optimal lattice sphere packings in dimensions $8$
and $24$ are extremal even unimodular. Currently, the exact number of such
lattices is still an open problem in higher dimensions. For example, we only
know of a single one in dimension $72$ [Neb16] and four in dimension $80$.

In this paper we consider lattices arising from maximal orders in quaternion
algebras over real number fields. Under some verifiable conditions (see
Section 5), we obtain even (integral) lattices of dimension $4n$, where $n$ is
the degree of the underlying number field. A related study appears in the work
of Bayer and Martinet [BFM94], where the authors study quadratic forms related
to semi-simple algebras. We extend this study by giving explicit constructions
of the extremal even unimodular lattices in dimensions $8$ and $16$. Moreover
we give a counting estimate on the number of number fields giving rise to such
lattices. Namely, we show that a lattice constructed from a maximal order in a
quaternion algebra over a real number field is even unimodular with a high
probability. Unfortunately, this result cannot be extended to dimensions
$24,32$ and $40$ (see [BFM94]), but by relaxing the extremality condition, we
can still provide a source for constructing even unimodular lattices in any
dimension multiple of $8$.

###  1.1 Related work and contributions

We consider lattices arising from quaternion algebras over totally real number
fields. This is the same setting as the one taken in [TY12] and [BABC20] via a
twisted embedding of quaternion algebras. In [TY12] the authors use the
symmetric bilinear form associated to the trace form of the quaternion algebra
to obtain a formula for the determinant of the associated lattice. Then they
use this formula to give some examples for constructions of the lattices
$E_{8}$, $K_{12}$, $\Lambda_{16}$, $\Lambda_{24}$ and a densest 32-dimensional
lattice. This construction is also the one considered in [BFM94], where the
authors show that many well-known lattices can be constructed using this
setting. Their method, however, does not provide a constructive way of
obtaining a generator matrix of the lattices obtained.

In [BABC20] the authors describe how to construct $4n$-dimensional lattices
from ideals in maximal orders from quaternion algebras over totally real
number fields of degree $n$. They use this to construct a specific family of
lattices arising from the quaternion algebras
$B=\left(\frac{-1,-1}{F}\right)$, where $F$ denotes a totally real maximal
subfield of $\mathbb{Q}(\zeta_{2^{r}})$, of the form
$F=\mathbb{Q}(\zeta_{2^{r}}+\zeta_{2^{r}}^{-1})$ , where $r\geq 2$ and
$\zeta_{2^{r}}$ denotes a $2^{r}$-th root of unity. The lattices they obtain
include the densest packings known in dimension 1,2,4,8 and 16. Once one has a
basis for the maximal order, their construction allows to explicitly compute a
generator matrix and the minimum distance of the lattice. One of the main
difficulties in their method is to find a maximal order for a fixed quaternion
algebra. There is no known efficient algorithm for finding maximal orders in
quaternion algebras over number fields other than $\mathbb{Q}$. In [BABC20]
the authors find an infinite family of maximal orders in the quaternion
algebras mentioned above. More precisely, for each quaternion algebra
$B=\left(\frac{-1,-1}{F}\right)$, with
$F=\mathbb{Q}(\zeta_{2^{r}}+\zeta_{2^{r}}^{-1})$, they give a basis for a
maximal order in $B$. Here, we extend this work by providing infinitely many
explicit maximal orders _per dimension_ in dimensions 8 and 16, hence
obtaining a huge variety of candidate lattices all providing a dense lattice
packing, as explained below in more detail.

In this paper we use the same underlying lattice construction as in [BABC20],
and we use the associated lattice volume formula to ensure that our lattices
are unimodular. We consider quaternion algebras of the form
$B=\left(\frac{-1,-1}{F}\right)$, where $F$ is a totally real number field of
degree $n>1$. Then for $n=2,4$ we construct an infinite family of totally real
number fields $F$ of degree $n$ for which we control the discriminant, and we
find a totally real element $\alpha\in F$ in such a way that, for any maximal
order $\mathcal{O}$ in $B$, the associated lattice
$\Lambda_{(\mathcal{O},\alpha)}$ is an extremal even unimodular lattice or, in
some cases, a densest lattice packing. That is, for $n=2$ we find infinitely
many quaternion algebras from which we can construct the lattice $E_{8}$, and
for $n=4$ we find infinitely many quaternion algebras from which we obtain
either $\Lambda_{16}$ (densest) or $E_{8}^{2}$ (extremal). Our method can be
easily extended to different values of $n$. It is interesting to notice that
different maximal orders in the quaternion algebras we consider give different
even unimodular lattices, whose extremality can be checked and recognised by
computing their center density. The constructions are explicit as we are also
able to characterise a maximal order basis for the quaternion algebras we
consider.

The known constructions of even unimodular lattices in a given dimension $n$
are often based on algebraic structures of the same dimension (number fields,
codes). In the present paper, we first benefit from the structure of
quaternion algebras over number fields by reducing the dimensionality of the
problem by $3/4$. In other words, we will use a number field of degree $n$ to
construct even unimodular lattices of dimension $4n$. Our aim is to further
illustrate that such constructions also provide a significant freedom on the
choice of the underlying number field, and we present density results for the
obtained sets of even unimodular lattices.

###  1.2 Organization

The paper is organised as follows. In Section 2 we review the necessary
background on lattices arising from number fields and we introduce the reader
to quaternion algebras. In Section 3 we study in more detail definite
quaternion algebras of the form $\left(\frac{-1,-1}{F}\right)$, where $F$
denotes a totally real number field. We characterise bases of maximal orders
over real quadratic fields and over a family of quartic fields called the
simplest quartic fields, that will allow us to construct infinite families of
lattices. In Section 4 we describe the lattice construction from quaternion
algebras over totally real number fields and we introduce the necessary
notation that we will use afterwards. In Section 5 we turn to extremal even
unimodular/dense lattices. We give constructive results that provide an
infinite family of lattices similar to $E_{8}$ and an infinite family of
lattices similar to $\Lambda_{16}$ obtained from infinitely many quaternion
algebras. We also provide a density result showing that a large proportion of
the quaternion algebras we consider give rise to these lattices. In Section 6
we explain how the lattices we construct can have potential interest in
applications such as code design for wireless communications and physical
layer security and hint to possible future work. Finally we include some Magma
code to generate our lattices in the Appendix.

### Notation

Throughout the paper, $F$ will denote a number field, $\mathbb{Z}_{F}$ will
denote the ring of integers of $F$ and $d_{F}$ the discriminant of $F$. The
norm of an element $x$ in $F$ is denoted by $\mathrm{Nm}_{F/\mathbb{Q}}(x)$
and the trace of $x$ as $\mathrm{Tr}_{F/\mathbb{Q}}(x)$. We denote a
quaternion algebra by $B$, and $\mathcal{O}$ denotes an order in $B$.

##  2 Background

In this section we will introduce the necessary notions of quaternion algebras
and lattice theory that we will need in what follows.

###  2.1 Lattices from number fields

Let $\Lambda$ be a full-rank lattice in $\mathbb{R}^{n}$, with $n\geq 2$. The
minimum of $\Lambda$ is defined as

$\lambda_{1}(\Lambda):=\min\\{||x||^{2}:x\in\Lambda\setminus\\{0\\}\\}.$

The centre density of $\Lambda$ is defined as

$\delta_{\Lambda}:=\frac{||\lambda_{1}(\Lambda)||^{n/2}}{2^{n}\sqrt{\det\Lambda}}.$

Let $F$ be a number field of degree $n$ over $\mathbb{Q}$. Let $r$ denote the
number of real embedding $\\{\sigma_{1},\ldots,\sigma_{r}\\}$ and $2s$ the
number of complex embeddings
$\\{\sigma_{r+1},\overline{\sigma_{r+1}},\ldots,\sigma_{r+s},\overline{\sigma_{r+s}}\\}$
of $F$. We have a canonical embedding

$\begin{array}[]{rrcl}\sigma_{F}:&F&\rightarrow&\mathbb{R}^{r}\times\mathbb{C}^{s}\\\
&x&\mapsto&(\sigma_{1}(x),\ldots,\sigma_{r+s}(x)).\end{array}$

We can identify $\mathbb{R}^{r}\times\mathbb{C}^{s}$ with $\mathbb{R}^{n}$,
using the basis $\\{1,i\\}$ for $\mathbb{C}$.

Let $\mathcal{I}$ be a nonzero ideal in $\mathbb{Z}_{F}$ with integral basis
$\left\\{\alpha_{1},\ldots,\alpha_{n}\right\\}$. Then
$\sigma_{F}(\mathcal{I})$ is a lattice with generator matrix whose $i$-th row
is

$(\sigma_{1}(\alpha_{i}),\ldots,\sigma_{r}(\alpha_{i}),\mathrm{Re}(\sigma_{r+1}(\alpha_{i})),\mathrm{Im}(\sigma_{r+1}(\alpha_{i})),\ldots,\mathrm{Re}(\sigma_{r+s}(\alpha_{i})),\mathrm{Im}(\sigma_{r+s}(\alpha_{i}))).$

We call $\mathcal{I}$ an ideal lattice.

###  2.2 Arithmetic of quaternion algebras

######  Definition 2.1.

A quaternion algebra over $F$ is a central simple algebra that has dimension
$4$ over $F$. For $a,b\in\ F\setminus\\{0\\}$ we denote by
$\left(\frac{a,b}{F}\right)$ the $F$-algebra generated by a basis
$\\{1,i,j,k\\}$ such that $i^{2}=a,j^{2}=b,$ and $ij=-ji=k.$

Every quaternion algebra $B$ over $F$ is provided with an $F$-endomorphism
called conjugation and denoted by $\beta\mapsto\overline{\beta}$. If
$\beta=x+yi+zj+tk\in B$, with $x,y,z,t\in F$, then $\overline{\beta}=x-yi-zj-
tk$. The reduced trace of $\beta$ is defined as
$\mathrm{Tr}_{B}(\beta)=\beta+\overline{\beta}=2x$ and the reduced norm of
$\beta$ is defined as
$\mathrm{Nm}_{B}(\beta)=\beta\overline{\beta}=x^{2}-ay^{2}-bz^{2}+abt^{2}$.

According to the Wedderburn–Artin theorem [Vig80], a quaternion algebra is
either a division algebra (a skew field) or a matrix algebra. We will only be
interested in division algebras. For each place $\mathfrak{p}$ of $F$,
$B_{\mathfrak{p}}:=F_{\mathfrak{p}}\otimes B$ is a quaternion algebra, where
$F_{\mathfrak{p}}$ denotes the localization of $F$ at $\mathfrak{p}$. We say
that $B$ is ramified at $\mathfrak{p}$ if $B_{\mathfrak{p}}$ is a division
algebra. Otherwise we say that $B$ is non-ramified or split at $\mathfrak{p}$.
As usual, the archimedean places play a distinguished role. Let $n$ denote the
degree of $F$. The Wedderburn–Artin theorem tells us that we have an
isomorphism

$B\otimes_{\mathbb{Q}}\mathbb{R}\simeq\mathrm{M}_{2}(\mathbb{R})^{s}\times\mathbb{H}^{n-s},$

for some $0\leq s\leq n$, where
$\mathbb{H}=\left(\frac{-1,-1}{\mathbb{R}}\right)$ denotes the real Hamilton
quaternion algebra. We say that $B$ is a (totally) definite quaternion algebra
if $B$ is ramified at all the archimedean places, _i.e._ , if $s=0$. Otherwise
we say that $B$ is (totally) indefinite.

######  Definition 2.2.

The reduced discriminant $D_{B}$ of a quaternion algebra $B$ over $F$ is the
integral ideal of $\mathbb{Z}_{F}$ given by the product of prime ideals of
$\mathbb{Z}_{F}$ that ramify in $B$.

Let $B$ be quaternion algebra over a number field $F$. An _ideal_
$\mathcal{I}$ of $B$ is a $\mathbb{Z}_{F}$-lattice of $B$ of rank $4$. The
reduced norm $\mathrm{Nm}_{B}(\mathcal{I})$ of an ideal $\mathcal{I}$ is
$\text{gcd}\\{\mathrm{Nm}_{B}(\beta):\beta\in\mathcal{I}\\}$. An _order_
$\mathcal{O}$ of $B$ is an ideal which is also a subring. A _maximal order_ is
an order that is not properly contained in another order.

The discriminant $D_{\mathcal{O}}$ of a quaternion order
$\mathcal{O}=\mathbb{Z}_{F}[\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}]$ in
$B$ can be computed as the ideal
$D_{\mathcal{O}}^{2}=\det(\mathrm{Tr}_{B}(\alpha_{i}\alpha_{j}))_{1\leq
i,j\leq 4}\mathbb{Z}_{F}$. Check Def. 1.31 in [AB04] for the precise
definition. To check the maximality of $\mathcal{O}$ one can use the fact that
an order $\mathcal{O}$ in $B$ is maximal if and only if
$D_{\mathcal{O}}=D_{B}$ (see [AB04], Prop. 1.50). Note that, unlike in number
fields, maximal orders in quaternion algebras are not necessarily unique
[Vig80], but they all have the same discriminant, which coincides with the
discriminant of the algebra.

##  3 Computing discriminants and orders in quaternion algebras

In order to explicitly construct lattices from quaternion algebras, we will
need to be able to find bases of maximal orders explicitly. In this section we
will describe some infinite families of maximal orders in a given dimension
and construct an explicit basis for each one.

###  3.1 Computing the discriminant of definite quaternion algebras

We will study in detail the case of definite quaternion algebras of the form
$B=\left(\frac{-1,-1}{F}\right)$, with $F$ a totally real number field, as
this will allow us to do explicit computations in the sections that follow.
First we need to compute the reduced discriminant of $B$.

######  Definition 3.1.

Let $L$ be a local field. The Hilbert symbol over $L$ is defined as the
function $(\,\cdot\,,\,\cdot\,)_{L}:L^{\times}\times
L^{\times}\rightarrow\\{\pm 1\\}$ given by

$(a,b)_{L}=\left\\{\begin{array}[]{rl}1&\mbox{if}\ ax^{2}+by^{2}=z^{2}\
\mbox{has a non-zero solution}\ (x,y,z)\ \mbox{in}\ L^{3},\\\
-1&\mbox{otherwise}.\end{array}\right.$

The Hilbert symbol is closely related to the ramification of a quaternion
algebra $B=\left(\frac{a,b}{F}\right)$, with $a,b\in F^{\times}$. Indeed, $B$
is ramified at a place $\mathfrak{p}$ of $F$ if and only if
$(a,b)_{F_{\mathfrak{p}}}=-1$ (cf. [Vig80], Chapter III).

######  Proposition 3.2.

Let $F$ be a totally real Galois extension of $\mathbb{Q}$ of degree
$n=[F:\mathbb{Q}]$ and let $\mathbb{Z}_{F}$ denote its ring of integers. Let
$B=\left(\frac{-1,-1}{F}\right)$ be a quaternion algebra over $F$. Then all
the archimedean places are ramified, so $B$ is a definite quaternion algebra,
and

  * $(a)$

if $n=2$ and F has discriminant $d_{F}$, then

$D_{B}=\left\\{\begin{array}[]{lll}2\mathbb{Z}_{F}&\emph{if}&d_{F}\equiv
1\quad(\emph{mod}\ 8),\\\ \mathbb{Z}_{F}&\emph{if}&d_{F}\not\equiv
1\quad(\emph{mod}\ 8).\end{array}\right.$

  * $(b)$

if $n=2^{k}$, for some $k\in\mathbb{Z}$, then $D_{B}=\mathbb{Z}_{F}$.

  * $(c)$

if $n$ is odd, then $D_{B}=2\mathbb{Z}_{F}$.

  * $(d)$

if $F$ has degree $n=2^{k}m$, where $m>1$ is odd, let
$n_{2}=[F_{\mathfrak{p}}:\mathbb{Q}_{2}]$, where $\mathfrak{p}$ is a prime
above $2$. Then

$D_{B}=\left\\{\begin{array}[]{lll}2\mathbb{Z}_{F}&\emph{if}&n_{2}\
\emph{odd},\\\ \mathbb{Z}_{F}&\emph{if}&n_{2}\ \emph{even}.\end{array}\right.$

###### Proof.

The ramification of the archimedean places can be easily computed with the
Hilbert symbol. Indeed, the equation $-x^{2}-y^{2}=z^{2}$ has no solution in
$F_{\infty}\simeq\mathbb{R}$, since $F$ is totally real, thus
$(-1,-1)_{F_{\infty}}=-1$.

For every non-archimedean place $\mathfrak{p}$ of $F$ we want to compute the
Hilbert symbol $(-1,-1)_{F_{\mathfrak{p}}}$. Let $p$ the prime integer below
$\mathfrak{p}$, _i.e._ , $\mathfrak{p}\mid p$. Consider the decomposition of
$p$ in $\mathbb{Z}_{F}$:
$p\mathbb{Z}_{F}=\mathfrak{p}_{1}^{e_{1}}\ldots\mathfrak{p}_{g}^{e_{g}}$,
where $n=\sum_{i=1}^{g}e_{i}f_{i}$, $f_{i}$ is the inertia degree of the
$\mathfrak{p}_{i}$, and we set $\mathfrak{p}=\mathfrak{p}_{1}$. We have that
$[F_{\mathfrak{p}_{i}}:\mathbb{Q}_{p}]=e_{i}f_{i}$ (cf. [Jan96] Theorem 5.1).

We will prove first the quadratic case. If $F=\mathbb{Q}(\sqrt{D})$ is a
quadratic field, then $[F_{\mathfrak{p}}:\mathbb{Q}_{p}]=1$ if and only if
$g=2$, which means that $p$ splits completely and
$F_{\mathfrak{p}}=\mathbb{Q}_{p}$. In this case we can apply the well-known
formulas of the Hilbert symbol (see [Ser73], Chapter III) and obtain:

$(-1,-1)_{\mathbb{Q}_{p}}=\left\\{\begin{array}[]{rll}1&\mathrm{if}&p\neq
2,\\\ -1&\mathrm{if}&p=2.\end{array}\right.$

We only need to show when does 2 split completely in $F$. This happens if and
only if $\left(\frac{D}{2}\right)=1$, and this is true if and only if $D\equiv
1$ (mod $8$). So we have shown so far that if $D\equiv 1$ (mod $8$), then the
only primes that ramify are those above 2, and $D_{B}=2\mathbb{Z}_{F}$. Now if
$D\not\equiv 1$ (mod $8$), then $[F_{\mathfrak{p}_{i}}:\mathbb{Q}_{p}]=2$. By
[Jan96] Chapter II, Theorem 5.1 , we have that the field $F_{\mathfrak{p}}$ is
isomorphic to some finite extension $L=\mathbb{Q}_{p}(\beta)$ of
$\mathbb{Q}_{p}$. In this case, by [FV02] Chapter 5, we have that

$(-1,-1)_{F_{\mathfrak{p}}}=(-1,\mathrm{Nm}_{L/\mathbb{Q}_{p}}(-1))_{\mathbb{Q}_{p}}=(-1,1)_{\mathbb{Q}_{p}}=1.$

So $D_{B}=\mathbb{Z}_{F}$.

Let us next prove the case where $F$ is a number field of degree $n>2$. Again,
we denote by $L=\mathbb{Q}_{p}(\beta)$ the finite extension of
$\mathbb{Q}_{p}$ isomorphic to $F_{\mathfrak{p}}$. We have:

$(-1,-1)_{F_{\mathfrak{p}}}=(-1,\mathrm{Nm}_{L/\mathbb{Q}_{p}}(-1))_{\mathbb{Q}_{p}}=\left\\{\begin{array}[]{ll}(-1,1)_{\mathbb{Q}_{p}}=1&\mathrm{if}\
[L:\mathbb{Q}_{p}]\ \mathrm{even},\\\
(-1,-1)_{\mathbb{Q}_{p}}=\left\\{\begin{array}[]{rll}1&\mathrm{if}&p\neq 2,\\\
-1&\mathrm{if}&p=2.\end{array}\right.&\mathrm{if}\ [L:\mathbb{Q}_{p}]\
\mathrm{odd}.\end{array}\right.$

Notice that $[L:\mathbb{Q}_{p}]$ divides $[F:\mathbb{Q}]$. Fix $p=2$. If
$n=2^{k}$, for some $k$, then
$(-1)^{[L:\mathbb{Q}_{2}]}=(-1)^{[F:\mathbb{Q}]}=1$, and if $n$ is odd then
$(-1)^{[L:\mathbb{Q}_{2}]}=1$. ∎

###  3.2 Computing bases of maximal orders

####  3.2.1 Maximal orders over real quadratic fields

######  Lemma 3.3.

Let $D>0$ be a square-free integer with $D\equiv 3$ (mod $4$) and consider the
quaternion algebra $B=\left(\frac{-1,-1}{\mathbb{Q}(\sqrt{D})}\right)$. Then
$\mathcal{O}=\mathbb{Z}[\sqrt{D}][1,i,(\sqrt{D}i+j)/2,(\sqrt{D}+k)/2]$ is a
maximal order in $B$.

###### Proof.

First note that $(\sqrt{D}i+j)/2,(\sqrt{D}+k)/2\in\mathcal{O}$, since $D\equiv
3$ (mod $4$). It is easy to see that $\mathcal{O}$ is an order, one only needs
to check that for any
$\alpha=x+yi+z(\sqrt{D}i+j)/2+t(\sqrt{D}+k)/2\in\mathcal{O}$, with
$x,y,z,t\in\mathbb{Z}_{F}$ we have: $(i)$
$\mathrm{Tr}_{B}(\alpha),\mathrm{Nm}_{B}(\alpha)\in\mathbb{Z}_{F}$; $(ii)$
$\mathbb{Z}_{F}\subseteq\mathcal{O}$; and $(iii)$ $\mathcal{O}$ contains an
$F$-basis of $B$.

To check the maximality of $\mathcal{O}$ we use the fact that an order
$\mathcal{O}$ in $B$ is maximal if and only if
$\mathrm{disc}(\mathcal{O})=D_{B}$ (see [AB04], Prop. 1.50), where
$\mathrm{disc}(\mathcal{O})$ denotes the reduced discriminant of $\mathcal{O}$
(check Def. 1.31 in [AB04] for the precise definition). The reduced
discriminant of an order
$\mathcal{O}=\mathbb{Z}_{F}[\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4}]$ can
be computed as the ideal
$\mathrm{disc}(\mathcal{O})^{2}=\det(\mathrm{Tr}(\gamma_{i}\gamma_{j})_{i,j})\mathbb{Z}_{F}$.
So in our case we have

$\mathrm{disc}(\mathcal{O})^{2}=\det\left(\begin{smallmatrix}2&0&0&\sqrt{D}\\\
0&-2&-\sqrt{D}&0\\\ 0&-\sqrt{D}&-(D+1)/2&0\\\
\sqrt{D}&0&0&(D-1)/2\end{smallmatrix}\right)\mathbb{Z}_{F}=-1\mathbb{Z}_{F}=\mathbb{Z}_{F}.$

Thus, $\mathcal{O}$ is indeed a maximal order in $B$. ∎

######  Lemma 3.4.

Let $D>0$ be a square-free integer with $D\equiv 1$ (mod $4$) and consider the
quaternion algebra $B=\left(\frac{-1,-1}{\mathbb{Q}(\sqrt{D})}\right)$.

  * $(i)$

If $D\not\equiv 1$ (mod $8$), then
$\mathcal{O}=\mathbb{Z}_{F}\left[1,i,\frac{\sqrt{D}+1}{4}+\frac{\sqrt{D}+3}{4}i+\frac{j}{2},\frac{\sqrt{D}+3}{4}+\frac{\sqrt{D}+1}{4}i+\frac{k}{2}\right]$
is a maximal order in $B$.

  * $(ii)$

If $D\equiv 1$ (mod $8$), then
$\mathcal{O}=\mathbb{Z}_{F}[1,i,j,\frac{1+i+j+k}{2}]$ is a maximal order in
$B$.

###### Proof.

$(i)$ Suppose that $D\equiv 1$ (mod 4) and $D\not\equiv 1$ (mod $8$). Like in
Lemma 3.3, it is easy to check that $\mathcal{O}$ is an order. For the
maximality we compute its discriminant, which again gives
$\mathrm{disc}(\mathcal{O})=\mathbb{Z}_{F}$. By Proposition 3.2, in this case
$\mathrm{disc}(\mathcal{O})=D_{B}$, so $\mathcal{O}$ is maximal.

$(ii)$ Now suppose that $D\equiv 1$ (mod $8$). Again, checking that
$\mathcal{O}$ is an order is an easy task. In this case, the discriminant is

$\mathrm{disc}(\mathcal{O})^{2}=\det\left(\begin{smallmatrix}2&0&0&1\\\
0&-2&0&-1\\\ 0&0&-2&-1\\\
1&-1&-1&-1\end{smallmatrix}\right)\mathbb{Z}_{F}=-4\mathbb{Z}_{F},$

so $\mathrm{disc}(\mathcal{O})=2\mathbb{Z}_{F}=D_{B}$, according to Prop. 3.2.
∎

######  Lemma 3.5.

Consider the quaternion algebra
$B=\left(\frac{-1,-1}{\mathbb{Q}(\sqrt{2})}\right)$. Then
$\mathcal{O}=\mathbb{Z}[\sqrt{2}][1,\frac{1+i}{\sqrt{2}},\frac{1+j}{\sqrt{2}},\frac{1+i+j+k}{2}]$
is a maximal order in $B$.

###### Proof.

[Vig80] p. 141. ∎

####  3.2.2 Maximal ordes over totally real quartic fields

We consider the so called simplest quartic fields (more details are given in
Section 5.2). For any integer $m\in\mathbb{N}\setminus\\{3\\}$ such that the
odd part of $m^{2}+16$ is square-free, we consider the polynomial

$f_{m}(x)=x^{4}-mx^{3}-6x^{2}+mx+1$

of discriminant $d_{f_{m}}=4\Delta_{m}^{3}$, where $\Delta_{m}=m^{2}+16$. The
quartic number field $F_{m}$ defined by the polynomial $f_{m}$ is cyclic. The
fields $F_{m}$ are called the simplest quartic fields [Ste04]. Let $r_{m}$ be
the largest root of $f_{m}$. The root $r_{m}$ is explicitly given by [Ste04]
as

$r_{m}=\frac{1}{2}\left(\frac{m+\sqrt{\Delta_{m}}}{2}+\frac{\sqrt{\Delta_{m}+m\sqrt{\Delta_{m}}}}{2}\right).$

In this case we can write $F_{m}=\mathbb{Q}(r_{m})$.

######  Lemma 3.6.

Let $F_{m}$ be the quartic field defined by
$f_{m}(x)=x^{4}-mx^{3}-6x^{2}+mx+1\in\mathbb{Z}[x]$, with
$m\in\mathbb{N}\setminus\\{3\\}$ such that $m$ is even and the odd part of
$m^{2}+16$ is square-free. Consider the quaternion algebra
$B=\left(\frac{-1,-1}{F_{m}}\right)$. Then the discriminant of $B$ is
$\mathbb{Z}_{F_{m}}$ and
$\mathcal{O}=\mathbb{Z}_{F_{m}}\left[1,\frac{(1+r_{m})(1+i)}{2},\frac{(1+r_{m})(1+j)}{2},\frac{1+i+j+k}{2}\right]$
is a maximal order in $B$.

###### Proof.

An easy computation shows that
$\mathrm{disc}(\mathcal{O})=\frac{1}{2}\left(-\left(\frac{m}{2}+2\right)r_{m}^{3}-6r_{m}^{2}+\left(\frac{m}{2}-2\right)r_{m}\right)$,
which is a unit in $F_{m}$. ∎

##  4 Lattices from quaternion algebras over totally real number fields

We will see how to define a lattice from an ideal $\mathcal{I}$ of a
quaternion algebra $B$ defined over a totally real number field. Let $F$
denote a totally real field, let $\mathbb{Z}_{F}$ denote the ring of integers
of $F$ and let $\\{\alpha_{1},\ldots,\alpha_{n}\\}$ be a $\mathbb{Z}$-basis of
$\mathbb{Z}_{F}$. Let $\alpha$ denote a totally real element in $F$. We define
the twisted embedding

$\begin{array}[]{rccl}\sigma_{B,\alpha}:&B&\rightarrow&\mathbb{R}^{4n}\\\
&x+yi+zj+tk&\mapsto&(\sqrt{2\sigma_{1}(\alpha)}\sigma_{1}(x),\ldots,\sqrt{2\sigma_{n}(\alpha)}\sigma_{n}(x),\\\
&&&\ldots,\sqrt{2\sigma_{1}(\alpha)}\sigma_{1}(t),\ldots,\sqrt{2\sigma_{n}(\alpha)}\sigma_{n}(t)).\end{array}$

Let $\mathcal{I}$ be an integral ideal of $B$ with $\mathbb{Z}_{F}$-basis
$\left\\{\beta_{1},\beta_{2},\beta_{3},\beta_{4}\right\\}$.

######  Definition 4.1.

We define $\Lambda_{(\mathcal{I},\alpha)}:=\sigma_{B,\alpha}(\mathcal{I})$ to
be the lattice given by $\mathcal{I}$ and twisted by $\alpha$, or simply
twisted ideal lattice.

In [BABC20] the authors show that $\Lambda_{(\mathcal{I},\alpha)}$ is a
lattice with $\mathbb{Z}$-basis
$\\{\sigma_{B,\alpha}(\alpha_{r}\beta_{s})\\}$. Let
$\beta_{s}=\beta_{s1}+\beta_{s2}i+\beta_{s3}j+\beta_{s4}k$, for $1\leq s\leq
4$. The generator matrix of $\Lambda_{(\mathcal{I},\alpha)}$ is given by

$\displaystyle
M_{\alpha}=\left(\begin{smallmatrix}\sqrt{2\sigma_{1}(\alpha)}\sigma_{1}(\alpha_{1}\beta_{11})&\ldots&\sqrt{2\sigma_{n}(\alpha)}\sigma_{n}(\alpha_{1}\beta_{11})&\ldots&\sqrt{2\sigma_{1}(\alpha)}\sigma_{1}(\alpha_{1}\beta_{14})&\ldots&\sqrt{2\sigma_{n}(\alpha)}\sigma_{n}(\alpha_{1}\beta_{14})\\\
\vdots&\ddots&\vdots&&\vdots&&\vdots\\\
\sqrt{2\sigma_{1}(\alpha)}\sigma_{1}(\alpha_{n}\beta_{11})&\ldots&\sqrt{2\sigma_{n}(\alpha)}\sigma_{n}(\alpha_{n}\beta_{11})&\ldots&\sqrt{2\sigma_{1}(\alpha)}\sigma_{1}(\alpha_{n}\beta_{14})&\ldots&\sqrt{2\sigma_{n}(\alpha)}\sigma_{n}(\alpha_{n}\beta_{14})\\\
\vdots&&\vdots&\ddots&\vdots&&\vdots\\\
\sqrt{2\sigma_{1}(\alpha)}\sigma_{1}(\alpha_{1}\beta_{41})&\ldots&\sqrt{2\sigma_{n}(\alpha)}\sigma_{n}(\alpha_{1}\beta_{41})&\ldots&\sqrt{2\sigma_{1}(\alpha)}\sigma_{1}(\alpha_{1}\beta_{44})&\ldots&\sqrt{2\sigma_{n}(\alpha)}\sigma_{n}(\alpha_{1}\beta_{44})\\\
\vdots&&\vdots&&\vdots&\ddots&\vdots\\\
\sqrt{2\sigma_{1}(\alpha)}\sigma_{1}(\alpha_{n}\beta_{41})&\ldots&\sqrt{2\sigma_{n}(\alpha)}\sigma_{n}(\alpha_{n}\beta_{41})&\ldots&\sqrt{2\sigma_{1}(\alpha)}\sigma_{1}(\alpha_{n}\beta_{44})&\ldots&\sqrt{2\sigma_{n}(\alpha)}\sigma_{n}(\alpha_{n}\beta_{44})\end{smallmatrix}\right)$
(1)

In [BABC20] the authors prove the following formula for the volume of a
twisted ideal lattice.

######  Proposition 4.2.

Let $F$ be a totally real number field with discriminant $d_{F}$. Let $B$ be a
quaternion algebra over $F$ and $\mathcal{O}$ a maximal order in $B$. Let
$\mathcal{I}$ be a right ideal of $\mathcal{O}$ and $\alpha$ a totally
positive element in $F$. Then

$\displaystyle\det(\Lambda_{(\mathcal{I},\alpha)})=\mathrm{Nm}_{F/\mathbb{Q}}(\alpha)^{2}d_{F}^{2}\mathrm{Nm}_{F/\mathbb{Q}}(\mathrm{Nm}_{B}(\mathcal{I}))^{2}\mathrm{Nm}_{F/\mathbb{Q}}(D_{\mathcal{O}}).$
(2)

In [TY12], the authors give an equivalent construction to these ideal lattices
[BABC20]*Prop. 4.1, using a certain quadratic form associated to $\mathcal{I}$
and $\alpha$. However, one cannot obtain a generator matrix for the lattices
obtained from this construction. Instead, they show how to directly construct
the Gram matrix of the lattice without using the embedding
$\sigma_{B,\alpha}$.

The inner product $\langle\cdot,\cdot\rangle$ on the lattice
$\Lambda_{(\mathcal{I},\alpha)}$ is given by

$\displaystyle\langle
x,y\rangle=\mathrm{Tr}_{F/\mathbb{Q}}(\alpha\,\mathrm{Tr}_{B}(x\overline{y})),\quad\mbox{for}\
x,y\in\Lambda_{(\mathcal{I},\alpha)}.$ (3)

######  Proposition 4.3.

Let $B$ be a quaternion algebra over a totally real field $F$ of degree $n$,
$\mathcal{O}$ a maximal order in $B$ and $\mathcal{I}$ an integral right-ideal
of $\mathcal{O}$. The lattice given by $\mathcal{I}$ is an integral even
lattice of dimension $4n$.

###### Proof.

According to (3) we have $||x||^{2}=\langle
x,\overline{x}\rangle=\mathrm{Tr}_{F/\mathbb{Q}}(\alpha\mathrm{Tr}_{B}(\mathrm{Nm}_{B}(x)))=2\mathrm{Tr}_{F/\mathbb{Q}}(\alpha\mathrm{Nm}_{B}(x))$.
Note that since $\alpha\in F$,
$\mathrm{Tr}_{F/\mathbb{Q}}(\alpha\,\mathrm{Nm}_{B}(x))$ is not necessarily an
integer. Clearly, if $\alpha=1$, then
$\mathrm{Tr}_{F/\mathbb{Q}}(\mathrm{Nm}_{B}(x))\in\mathbb{Z}$. Hence,
$||x||^{2}\in 2\mathbb{Z}$ for any $x\in\Lambda_{\mathcal{I}}$. Indeed, this
property will hold for any $\alpha$ in the inverse of the different ideal (see
the next section). ∎

In [BFM94], Bayer and Martinet gave a characterization of the lattices arising
from quaternion algebras over totally real number fields.

######  Theorem 4.4.

Let $B$ be a quaternion algebra over a totally real number field $F$ with a
maximal order $\mathcal{O}$ and $\mathcal{I}$ a left ideal in $\mathcal{O}$.
The discriminants $\det(\Lambda_{(\mathcal{I},\alpha)})$ are of the form

$\displaystyle\mathrm{Nm}_{F/\mathbb{Q}}(\mathfrak{p}_{1}\dots\mathfrak{p}_{r})^{2}\mathrm{Nm}_{F/\mathbb{Q}}(\mathfrak{a})^{4},$
(4)

where $\mathfrak{p}_{1}\dots\mathfrak{p}_{r}$ are prime ideals in $F$ that are
ramified in $F/B$ and $\mathfrak{a}$ is an ideal in $F$ equivalent to the
different ideal in $F$ modulo squares in the narrow class group.

Theorem 4.4 gives a way to construct unimodular lattices from quaternion
algebras. In the next section we will further explore this result and present
explicit constructions of some known lattices over infinitely many totally
real number fields.

##  5 Extremal even unimodular lattices from infinitely many quaternion
algebras

In this section use the Landau symbols $O$ and $o$ with their usual meanings.
It is well known that even unimodular lattices (_i.e._ , integral even
lattices of determinant $1$) exist only in dimensions $8k$ where $k$ is a
positive integer. Furthermore, if $\Lambda$ is an $n$-dimensional even
unimodular lattice then

$\lambda_{1}(\Lambda)\leq 2+2\left\lfloor\frac{n}{24}\right\rfloor.$ (5)

Even unimodular lattices with the length of the shortest vector achieving the
bound in (5) are called extremal. The following table summarises some known
results about extremal even unimodular lattices.

$n$ | 8 | 16 | 24 | 32 | 40 | 48 | 72 | 80
---|---|---|---|---|---|---|---|---
$\lambda_{1}(\Lambda)$ | 2 | 2 | 4 | 4 | 4 | 6 | 8 | 8
| Number
---
of extremal
lattices
1 | 2 | 1 | $\geq 10^{7}$ | $\geq 10^{51}$ | $\geq 5$ | $\geq 1$ | $\geq 4$
Table 1: Extremal even unimodular lattices

In this section we will use the lattice construction from Section 4. Note that
the lattices constructed in Section 4 are of dimension $4n$, thus, the degree
$n$ should necessarily be even in order to obtain even unimodular lattices
using this method.

Before proceeding with our construction, we will need a few definitions from
algebraic number theory. We consider $F$ to be a totally real number field of
even degree $n$, discriminant $d_{F}$, $\alpha$ totally positive in $F$, and
$\mathcal{O}$ a maximal order in $B=\left(\frac{-1,-1}{F}\right)$. We define
the dual of $\mathbb{Z}_{F}$ to be

$\mathbb{Z}_{F}^{\vee}=\left\\{x\in
F~{}:~{}\mathrm{Tr}_{F/\mathbb{Q}}(x\mathbb{Z}_{F})\subset\mathbb{Z}\right\\}.$

By construction, if $\alpha\in\mathbb{Z}_{F}^{\vee}$, then the lattice
$\Lambda_{(\mathcal{O},\alpha)}$ is integral and consequently even. If we
additionally choose $\alpha$ such that
$\det(\Lambda_{(\mathcal{O},\alpha)})=1$, then the lattice
$\Lambda_{(\mathcal{O},\alpha)}$ will be clearly even unimodular. Using
Proposition 4.2, we get

$\det(\Lambda_{(\mathcal{O},\alpha)})=\mathrm{Nm}_{F/\mathbb{Q}}(\alpha)^{2}d_{F}^{2}\mathrm{Nm}_{F/\mathbb{Q}}(D_{\mathcal{O}}).$

Assume that $n>2$ is even and let $n_{2}=[F_{\mathfrak{p}}:\mathbb{Q}_{2}]$,
where $\mathfrak{p}$ is a prime above $2$. Then, by Proposition 3.2, we have
$D_{B}=\mathbb{Z}_{F}$ if and only if $n$ is a power of two or $n_{2}$ is
even. In this case,

$\det(\Lambda_{(\mathcal{O},\alpha)})=\mathrm{Nm}_{F/\mathbb{Q}}(\alpha)^{2}d_{F}^{2}.$

Hence, the lattice $\Lambda_{(\mathcal{O},\alpha)}$ is even unimodular if the
norm of $\alpha$ is $1/d_{F}$.

Indeed, Theorem 4 shows that it is necessary that $B$ is ramified at the
finite places of $F$ so that the lattices obtained from a maximal order in $B$
are unimodular. Note that this condition implies that $[F:\mathbb{Q}]$ is
even. Moreover, we need a totally positive element $\alpha\in F$ such that
$\alpha$ generates an ideal of the form
$\mathrm{Nm}_{B}(\mathcal{I})(\mathbb{Z}_{F}^{\vee})^{-1}$, where
$\mathcal{I}$ is a left ideal in $\mathcal{O}$. In the following, we will
investigate these conditions in order to obtain explicit unimodular lattices.

The dual $\mathbb{Z}_{F}^{\vee}$ is a fractional ideal in $F$ with
$\mathbb{Z}_{F}\subset\mathbb{Z}_{F}^{\vee}$. Thus, its inverse is an integral
ideal in $\mathbb{Z}_{F}$. The different ideal of $F$ is defined as the
integral ideal $\mathfrak{D}_{F}:=(\mathbb{Z}_{F}^{\vee})^{-1}$. An important
property of the different ideal is that
$\mathrm{Nm}_{F/\mathbb{Q}}(\mathfrak{D}_{F})=d_{F}$. Now let
$F=\mathbb{Q}(\beta)$ and $f$ be the minimal polynomial of $\beta$ in
$\mathbb{Q}[X]$. It is well known that $(f^{\prime}(\beta))$, the ideal
generated by the derivative of $f$, is a subset of $\mathfrak{D}_{F}$.
Furthermore,

$\mathrm{Nm}_{F/\mathbb{Q}}(f^{\prime}(\beta))=(-1)^{\frac{n(n-1)}{2}}[\mathbb{Z}_{F}:\mathbb{Z}[\beta]]^{2}d_{F}.$
(6)

Let us first investigate the case where
$[\mathbb{Z}_{F}:\mathbb{Z}[\beta]]=1$. In this case, the number field $F$ is
said to be monogenic. In other words, the basis
$\\{1,\beta,\beta^{2},\dots,\beta^{n-1}\\}$ generates $\mathbb{Z}_{F}$. Such a
basis is called a _power basis_. Quadratic and cyclotomic fields are examples
of such number fields. The literature on monogenic fields is extensive, and
deciding which fields are monogenic is a long standing problem in algebraic
number theory. For a detailed survey on the topic see [Gaá02]. A classical
argument shows that, if $F$ is monogenic with generator $\beta$ and minimal
polynomial $f$, then $\mathfrak{D}_{F}=(f^{\prime}(\beta))$. Consequently,
$\mathrm{Nm}_{F/\mathbb{Q}}(f^{\prime}(\beta))=d_{F}$. We will use this fact
to describe a strategy to construct even unimodular lattices from quaternion
algebras over totally real number fields of even degree such that
$\mathbb{Z}_{F}=\mathbb{Z}[\beta]$ for some $\beta\in\mathbb{Z}_{F}$ with
minimal polynomial $f$.

######  Theorem 5.1.

Let $F$ be a totally real monogenic number field of even degree $n=2^{t}$,
$t>0$, or $n_{2}=[F_{\mathfrak{p}}:\mathbb{Q}_{2}]$ even, where $\mathfrak{p}$
is a prime above $2$. Let $\varepsilon\in\mathbb{Z}_{F}$ be a unit such that
$\varepsilon\cdot f^{\prime}(\beta)$ is totally positive. Then the lattice
$\Lambda_{(\mathcal{O},\alpha)}$ is an even unimodular lattice for
$\alpha=\frac{1}{\varepsilon\cdot f^{\prime}(\beta)}$, where $\mathcal{O}$ is
any maximal order in $B=\left(\frac{-1,-1}{F}\right)$.

###### Proof.

Since $\mathbb{Z}_{F}$ is monogenic, we have that
$\mathfrak{D}_{F}=(f^{\prime}(\beta))$, where $f^{\prime}$ is the derivative
of $f$. If $[F:\mathbb{Q}]=2^{t}$, $t>0$, or
$n_{2}=[F_{\mathfrak{p}}:\mathbb{Q}_{2}]$ even, the discriminant of $B$ is
$\mathbb{Z}_{F}$ according to Proposition 3.2.

Using Proposition 4.2 we obtain
$\det(\Lambda_{(\mathcal{O},\alpha)})=d_{F}^{2}\mathrm{Nm}_{F/\mathbb{Q}}(\alpha)^{2}$.
In order to construct an unimodular lattice we need to find a totally positive
element $\alpha\in\mathbb{Z}_{F}^{\vee}$ such that
$\mathrm{Nm}_{F/\mathbb{Q}}(\alpha)=\frac{1}{d_{F}}$. By assumption, the
element $\frac{1}{\varepsilon\cdot f^{\prime}(\beta)}$ has these properties. ∎

######  Remark 5.2.

If $n_{2}$ is odd, then $D_{B}=2\mathbb{Z}_{F}$. In this case,
$\det(\Lambda_{(\mathcal{O},\alpha)})=2^{n}d_{F}^{2}\mathrm{Nm}_{F/\mathbb{Q}}(\alpha)^{2}$.
Consequently, to obtain an even integral lattice
$\Lambda_{(\mathcal{O},\alpha)}$ it is enough to find
$\alpha\in\mathbb{Z}_{F}$ such that
$\mathrm{Nm}_{F/\mathbb{Q}}(\alpha)=2^{\frac{n}{2}}$ and $\alpha\varepsilon
f(\beta)^{\prime}$ is totally positive, for some unit
$\varepsilon\in\mathbb{Z}_{F}$.

In the following, we will explicitly construct infinite families of even
unimodular lattices using the above construction. It is proved in [BFM94] that
it is impossible to construct the Leech using the same construction. In fact,
the authors proved the following theorem.

######  Theorem 5.3.

Even unimodular lattices of dimension $n<48$ obtained from quaternion algebras
are of minimal norm $2$. In particular, it is impossible to construct the
Leech lattice from a quaternion algebra over a totally real number field of
degree $6$.

###  5.1 Dimension 8

Lattices obtained from quaternion algebras that are equivalent to $E_{8}$ are
characterised in [BFM94]. In this section, we will present an infinite family
of lattices similar to $E_{8}$ from infinitely many quaternion algebras with
explicit bases for the underlying maximal orders.

Theorem 5.1 covers number fields of degree $n>2$. We will investigate
quadratic number fields and use a similar strategy to prove a density result
for the set of number fields for which $\Lambda_{(\mathcal{O},\alpha)}$ is the
$E_{8}$ lattice. It is well known [CS98]*Ch. 4 that the lattice $E_{8}$ is the
only even unimodular lattice of dimension $8$ up to isometry. Hence, to
construct $E_{8}$ it is enough to find a quadratic field $F$ with
$D_{B}=\mathbb{Z}_{F}$ and $F$ contains a totally positive element $\alpha$
such that $\mathrm{Nm}_{F/\mathbb{Q}}(\alpha)=\frac{1}{d_{F}}$.

######  Theorem 5.4.

Let $D\equiv 1\pmod{4}$ be a square-free positive integer of the form
$D=s^{2}+4$ and let $F=\mathbb{Q}(\sqrt{D})$ denote a real quadratic field.
Let $B=\big{(}\frac{-1,-1}{F}\big{)}$ denote the quaternion algebra over $F$
and let $\mathcal{O}$ denote the maximal order in $B$ given by Lemma 3.4. Let
$\alpha=\frac{\varepsilon}{\sqrt{D}}$, with
$\varepsilon=\frac{s+\sqrt{D}}{2}$. Then the lattice
$\Lambda_{(\mathcal{O},\alpha)}$ is similar to $E_{8}$ and its generator
matrix is given by $(\ref{gen_mat})$.

###### Proof.

Since $D\not\equiv 1\pmod{8}$, according to Prop. 3.2 we have
$D_{B}=\mathbb{Z}_{F}$, and according to Lemma 3.4, we have that

$\mathcal{O}=\mathbb{Z}_{F}\left[1,i,\frac{\sqrt{D}+1}{4}+\frac{\sqrt{D}+3}{4}i+\frac{j}{2},\frac{\sqrt{D}+3}{4}+\frac{\sqrt{D}+1}{4}i+\frac{k}{2}\right]$

is a maximal order in B. Using (2), the lattice
$\Lambda_{(\mathcal{O},\alpha)}$ twisted by some totally real $\alpha\in F$
has
$\det(\Lambda_{(\mathcal{O},\alpha)})=d_{F}^{4}Nm_{F/\mathbb{Q}}(\alpha)^{4}$,
where $d_{F}$ is the discriminant of $F$. After the previous remark, for the
lattice $\Lambda_{(\mathcal{O},\alpha)}$ to be similar to $E_{8}$, we need to
find a totally positive element $\alpha\in F$ of norm
$\mathrm{Nm}_{F/\mathbb{Q}}(\alpha)=\frac{1}{d_{F}}=\frac{1}{d}$ and with
$\mathrm{Tr}_{F/\mathbb{Q}}(\alpha\mathrm{Nm}_{B}(x))\in\mathbb{Z}$ for all
$x\in\mathcal{O}$.

Consider $\varepsilon=\frac{s+\sqrt{D}}{2}$. It is easy to check that
$\varepsilon$ is a unit in $\mathbb{Z}_{F}$ of norm $-1$. We can assume that
$\varepsilon>0$ without loss of generality, as we can replace $\varepsilon$ by
$-\varepsilon$ if necessary. Take $\alpha=\frac{\varepsilon}{\sqrt{D}}$. By
assumption we have that
$\mathrm{Nm}_{F}(\varepsilon)=\varepsilon\overline{\varepsilon}=-1$, thus
$\overline{\varepsilon}<0$ and
$\overline{\alpha}=\frac{\overline{\varepsilon}}{-\sqrt{D}}>0$. Hence,
$\alpha$ is totally positive and $\mathrm{Nm}_{F}(\alpha)=\frac{1}{D}$.
Furthermore, $\mathrm{Tr}_{F/\mathbb{Q}}(\alpha)=1$, so
$\mathrm{Tr}_{F/\mathbb{Q}}(\alpha\mathrm{Nm}_{B}(x))\in\mathbb{Z}$.

For the case $D|\not\equiv 1\pmod{4}$, we consider $D=s^{2}+1$,
$\varepsilon=s+\sqrt{D}$ and proceed in a similar way. ∎

It is shown in [Est31] that the set of integers $s$ such that $s^{2}+4$ is
square-free has a positive density. Hence the following corollary.

######  Corollary 5.5.

The set

$\mathcal{T}_{E_{8}}=\left\\{F\textrm{ quadratic totally real}\ \Big{|}\
\Lambda_{(\mathcal{O},\alpha)}\sim E_{8}\textrm{ where }\mathcal{O}\subseteq
B=\left(\frac{-1,-1}{F}\right)\textrm{ is a maximal order}\right\\}$

is infinite.

######  Remark 5.6.

The infinite set in Corollary 5.5 is explicitly given by
$\mathbb{Q}(\sqrt{D})$ and $\alpha$ as in the proof of Theorem 5.4.

We will further obtain a lower bound on the number of number fields $F$ from
which the $E_{8}$ lattice can be realised from a maximal order in
$B=\big{(}\frac{-1,-1}{F}\big{)}$. Let $F=\mathbb{Q}(\sqrt{D})$ be a real
quadratic field, with $D>1$ a positive square-free integer. We have
$\mathfrak{D}_{F}=(t)$, with $t=2\sqrt{D}$ if $D\not\equiv 1\pmod{4}$, or
$t=\sqrt{D}$ if $D\equiv 1\pmod{4}$. Clearly, $t$ is not totally positive in
both cases, thus, to apply a similar argument to the one in Theorem 5.1, we
need $F$ to contain a unit $\varepsilon$ of norm $-1$. This is equivalent to
the negative Pell equation

$x^{2}-d_{F}y^{2}=-1$ (7)

being solvable in $\mathbb{Z}^{2}$. Note that if $p\equiv 3\pmod{4}$ for some
$p\mid d_{F}$, then Equation (7) is not solvable, otherwise, $-1$ will be a
square modulo $p$. In the following, we will reduce the problem of counting
number fields $F$ such that $\Lambda_{(\mathcal{O},\alpha)}\sim E_{8}$ to
counting the number of discriminants $d_{F}$ in some given set. A folklore
theorem due to Hermite–Minkowski states that for a fixed degree $n$, there are
only finitely many number fields of discriminant $d_{F}\leq X$ for any $X>1$.
Thus, it is canonical to order number fields with respect to their
discriminant.

We denote by $\mathcal{N}^{-}$ the set of discriminants $d_{F}$ such that $F$
contains a fundamental unit of norm $-1$. Let

$\mathcal{D}_{2}=\\{d_{F}~{}|~{}F\textrm{ quadratic field, }p\not\equiv
3\pmod{4}\textrm{ for all }p|d_{F}\\}.$

Note that $\mathcal{D}_{2}$ is a disjoint union of the subsets

$\mathcal{D}_{\textrm{even}}=\\{d_{F}\in\mathcal{D}_{2}~{}|~{}d_{F}\equiv
0\pmod{8}\\}$

and

$\mathcal{D}_{\textrm{odd}}=\\{d_{F}\in\mathcal{D}_{2}~{}|~{}d_{F}\equiv
1\pmod{4}\\}.$

Using the above notation, we have the following lemma.

######  Lemma 5.7.

Let $F=\mathbb{Q}(\sqrt{D})$ be a real quadratic field such that
$d_{F}\in\mathcal{D}_{\textrm{even}}\cap\mathcal{N}^{-}$ or
$d_{F}\in\mathcal{D}_{\textrm{odd}}\cap\mathcal{N}^{-}$ with $d_{F}\not\equiv
1\pmod{8}$. Then there exists $\alpha\in F$ totally positive, such that
$\Lambda_{(\mathcal{O},\alpha)}\sim E_{8}$ for any maximal order $\mathcal{O}$
in $B=\left(\frac{-1,-1}{F}\right)$.

###### Proof.

Using Proposition 3.2 with $n=2$, we get that

$D_{B}=\left\\{\begin{array}[]{lll}2\mathbb{Z}_{F}&\mathrm{if}&D\equiv
1\quad(\mathrm{mod}\ 8),\\\ \mathbb{Z}_{F}&\mathrm{if}&D\not\equiv
1\quad(\mathrm{mod}\ 8).\end{array}\right.$

Clearly, if $d_{F}\in\mathcal{D}_{\textrm{even}}\cap\mathcal{N}^{-}$ or
$d_{F}\in\mathcal{D}_{\textrm{odd}}\cap\mathcal{N}^{-}$ with $d_{F}\not\equiv
1\pmod{8}$, then

$\mathrm{Nm}_{F/\mathbb{Q}}(D_{\mathcal{O}})=1.$

Consequently,

$\det(\Lambda_{(\mathcal{O},\alpha)})=\mathrm{Nm}_{F/\mathbb{Q}}(\alpha)^{2}d_{F}^{2}$

for any maximal order $\mathcal{O}\in B$.

On the other hand, we have $\mathfrak{D}_{F}=(t)$, with $t=2\sqrt{D}$ if
$D\not\equiv 1\pmod{4}$, or $t=\sqrt{D}$ if $D\equiv 1\pmod{4}$. By
assumption, $F$ contains a unit $\varepsilon$ of norm $-1$. If
$\varepsilon<0$, then its conjugate is positive, replacing $\varepsilon$ by
$-\varepsilon$ if necessary, we get $\varepsilon t$ is totally positive. Thus,
for $\alpha=\frac{1}{\varepsilon t}$, we have $\alpha\in\mathbb{Z}_{F}^{\vee}$
totally positive of norm $\frac{1}{d_{F}}$. Finally, for
$\alpha=\frac{1}{\varepsilon t}$, we have that
$\Lambda_{(\mathcal{O},\alpha)}$ is an even unimodular lattice of dimension
$8$. ∎

Denoting by

$\mathcal{T}_{E_{8}}(X)=\\#\left\\{F\mathrm{\ quadratic\ totally\ real}\
\Big{|}\ d_{F}\leq X\,,\Lambda_{(\mathcal{O},\alpha)}\sim E_{8},\textrm{ where
}\mathcal{O}\subseteq B=\left(\frac{-1,-1}{F}\right)\textrm{ is a maximal
order}\right\\}$

we obtain the following corollary.

######  Corollary 5.8.

For $X\rightarrow\infty$, we have the inequality

$(\tau-o(1))c_{1}\frac{X}{\sqrt{\log(X)}}\leq\mathcal{T}_{E_{8}}(X),$

where $c_{1}=\frac{9}{8\pi}\prod_{p\equiv 1\pmod{4}}(1-p^{-2})^{1/2}$ and
$\tau=\frac{1}{9}\prod_{j\textrm{ odd}}(1-2^{-j})=0,046602493\dots$

###### Proof.

Let $X>1$. We denote by $\mathcal{D}(X)$ the cardinality of
$\mathcal{D}\cap[1,X]$. It is known [Wei06] that

$\mathcal{D}_{2}(X)\sim c_{1}\frac{X}{\sqrt{\log(X)}},$
$\mathcal{D}_{\textrm{odd}}(X)\sim\frac{8}{9}c_{1}\frac{X}{\sqrt{\log(X)}},$

and

$\mathcal{D}_{\textrm{even}}(X)\sim\frac{1}{9}c_{1}\frac{X}{\sqrt{\log(X)}}.$
(8)

By a straightforward application of Lemma 5.7, we obtain

$(\mathcal{D}_{\textrm{even}}\cap\mathcal{N}^{-1})(X)\leq\mathcal{T}_{E_{8}}(X).$

It is proved in [FK10] that for $X\rightarrow\infty$, we have

$(\beta-o(1))(\mathcal{D}_{\textrm{even}}(X)\leq(\mathcal{D}_{\textrm{even}}\cap\mathcal{N}^{-1})(X)\leq\left(\frac{3}{2}+o(1)\right)\mathcal{D}_{\textrm{even}}(X)$

where

$\beta=\prod_{j\textrm{ odd}}(1-2^{-j}).$

The result now follows from (8). ∎

As a consequence of the previous results, we obtain an infinite family of
quaternion algebras from which one can construct the $E_{8}$ lattice.

######  Remark 5.9.

Corollary 5.8 gives a lower bound on the density of the set of quaternion
algebras $B$ where the $E_{8}$ lattice is realisable as an embedding of a
maximal order in $B$. Here we ordered the algebras with respect to their
underlying number fields. As mentioned previously, we order number fields with
respect to the discriminant. If we restrict our (discriminants) counting
function to $\mathcal{D}_{2}$ (instead of all integers), then the result means
that at least $49\%$ of the discriminants in $\mathcal{D}_{2}$ are
discriminants of number fields $F$ with $B=\left(\frac{-1,-1}{F}\right)$ in
$\mathcal{T}_{E_{8}}$. It is conjectured that this percentage is precisely
$58\%$ (see for instance [FK10]).

######  Example 5.10.

Let $F=\mathbb{Q}(\sqrt{5})$. We have $d_{F}=5$, clearly
$d_{F}\in\mathcal{D}_{\textrm{odd}}\cap\mathcal{N}^{-}$ with $d_{F}\not\equiv
1\pmod{8}$. In fact, $\varepsilon=2+\sqrt{5}$ has norm $-1$. Moreover,
$\mathbb{Z}_{F}$ is generated by $\frac{1+\sqrt{5}}{2}$, and
$\mathfrak{D}_{F}=(\sqrt{5})$. Furthermore,
$\alpha=\frac{1}{\varepsilon\sqrt{5}}$ is totally positive of norm
$\frac{1}{d_{F}}$. By Proposition 3.4,

$\mathcal{O}=\mathbb{Z}_{F}\left[1,i,\frac{\sqrt{5}+1}{4}+\frac{\sqrt{5}+3}{4}i+\frac{j}{2},\frac{\sqrt{5}+3}{4}+\frac{\sqrt{5}+1}{4}i+\frac{k}{2}\right]$

is a maximal order in $B=\left(\frac{-1,-1}{F}\right)$. Consequently, the
lattice $\Lambda_{(\mathcal{O},\alpha)}$ is even unimodular. Thus, it is
similar to $E_{8}$. Its associated Gram matrix is

$\left(\begin{smallmatrix}4&-2&0&0&-1&1&1&0\\\ -2&2&0&0&1&0&0&1\\\
0&0&4&-2&1&0&-1&1\\\ 0&0&-2&2&0&1&1&0\\\ -1&1&1&0&2&0&0&1\\\
1&0&0&1&0&2&1&1\\\ 1&0&-1&1&0&1&2&0\\\ 0&1&1&0&1&1&0&2\\\
\end{smallmatrix}\right).$

###  5.2 Dimension 16

The argument that we used to prove Theorem 5.8 is based on the monogenity of
$F$. Unfortunately, not all number fields have this property. Indeed, it was
proved in [Gra85] that with the exception of the maximal real subfields of
cyclotomic fields, abelian extensions of $\mathbb{Q}$ of prime degree greater
than or equal to $5$ are not monogenic. Moreover, Gras [Gra86] showed that
almost all abelian extensions of $\mathbb{Q}$ with degree coprime to $6$ are
not monogenic. In this section we will show a result analogous to Theorem 5.8
in dimension $16$ by investigating an infinite family of non-monogenic real
quartic number fields.

Note that in the previous section we used the fact that

$\mathfrak{D}_{F}=(f^{\prime}(\beta)).$

In general, we have that

$(f^{\prime}(\beta))\subset\mathfrak{D}_{F},$

where $F=\mathbb{Q}(\beta)$ and $\beta\in\mathbb{Z}_{F}$. Hence,

$\mathbb{Z}_{F}\subset\mathbb{Z}_{F}^{\vee}\subset\frac{1}{f^{\prime}(\beta)}\mathbb{Z}_{F}.$

Thus, assuming that $[\mathbb{Z}_{F}:\mathbb{Z}[\beta]]$ is a square, by
Proposition 6, the norm of

$\alpha=\frac{[\mathbb{Z}_{F}:\mathbb{Z}[\beta]]^{1/2}}{\varepsilon
f^{\prime}(\beta)}$

is

$|\mathrm{Nm}_{F/\mathbb{Q}}(\alpha)|=\frac{[\mathbb{Z}_{F}:\mathbb{Z}[\beta]]^{n/2-2}}{d_{F}}$

for any unit $\varepsilon\in\mathbb{Z}_{F}$. Consequently, if $n=4$, then
$\alpha\in\mathbb{Z}_{F}^{\vee}$. Hence, if $\varepsilon$ is chosen such that
$\alpha$ is totally positive, then the lattice
$\Lambda_{(\mathcal{O},\alpha)}$ is an even lattice of volume

$\det(\Lambda_{(\mathcal{O},\alpha)})=\mathrm{Nm}_{F/\mathbb{Q}}(D_{\mathcal{O}}).$

Table 1 shows that there are exactly two extremal even unimodular lattices (up
to isometry) in dimension $16$. In fact, it is well known that these lattices
are $E_{8}^{2}$ and $D_{16}$. We will use the above strategy to construct the
extremal lattice $E_{8}^{2}$ and the densest known lattice packing
$\Lambda_{16}$ from infinitely many quaternion algebras. For this purpose we
will use the so called simplest quartic fields. For any integer
$m\in\mathbb{N}\setminus\\{3\\}$ such that the odd part of $m^{2}+16$ is
square-free, we consider the polynomial

$f_{m}(x)=x^{4}-mx^{3}-6x^{2}+mx+1$

of discriminant $d_{f_{m}}=4\Delta_{m}^{3}$, where $\Delta_{m}=m^{2}+16$. The
polynomial $f_{m}$ defines different real cyclic quartic number fields $F_{m}$
for different values of $m$. The fields $F_{m}$ are called the simplest
quartic fields [Ste04]. Let $r_{m}$ be the largest root of $f_{m}$. Then we
can write $F_{m}=\mathbb{Q}(r_{m})$. Let $\sigma$ be a generator of
$\operatorname{Gal}(F_{m}/\mathbb{Q})$. One can show that [Ste04]

$\sigma(r_{m})=\frac{r_{m}-1}{r_{m}+1}\,,\
\sigma(r_{m})^{2}=-\frac{1}{r_{m}}\,,\ \textrm{ and }\
\sigma(r_{m})^{3}=-\frac{r_{m}+1}{r_{m}-1}.$

The monogenity problem for the fields $F_{m}$ was completely solved in [GP14].
In fact, the authors showed that, with the exceptions of $m=2,4$, the fields
$F_{m}$ are not monogenic. Moreover, the authors showed that
$[\mathbb{Z}_{F_{m}}:\mathbb{Z}[r_{m}]]=4,8$ for $v_{2}(m)=1,2$ respectively,
where $v_{2}(m)$ is the $2$-adic valuation of $m$. Furthermore, they present
an integral basis for $F_{m}$ for all $m$. More precisely, assuming that
$m^{2}+16$ is not divisible by an odd square, an integral basis of $F_{m}$ is
given by

$\displaystyle\left(1,r_{m},r_{m}^{2},\frac{1+r_{m}^{3}}{2}\right)~{}~{}$
$\displaystyle\mathrm{if}~{}v_{2}(m)=0,$
$\displaystyle\left(1,r_{m},\frac{1+r_{m}^{2}}{2},\frac{r_{m}+r_{m}^{3}}{2}\right)~{}~{}$
$\displaystyle\mathrm{if}~{}v_{2}(m)=1,$
$\displaystyle\left(1,r_{m},\frac{1+r_{m}^{2}}{2},\frac{1+r_{m}+r_{m}^{2}+r_{m}^{3}}{4}\right)~{}~{}$
$\displaystyle\mathrm{if}~{}v_{2}(m)=2,$
$\displaystyle\left(1,r_{m},\frac{1+2r_{m}-r_{m}^{2}}{4},\frac{1+r_{m}+r_{m}^{2}+r_{m}^{3}}{4}\right)~{}~{}$
$\displaystyle\mathrm{if}~{}v_{2}(m)\geq 3.$

It is proved in [GP14] that if $v_{2}(m)=1$, then
$[\mathbb{Z}_{F_{m}}:\mathbb{Z}[r_{m}]]=4$. We consider
$s=\frac{r_{m}^{3}-mr_{m}^{2}-5r_{m}+m}{2}$. Using the above we can see that
the element $\alpha=\frac{2}{sf_{m}(r_{m})^{\prime}}$ is totally positive of
norm $\frac{1}{d_{F_{m}}}$. Consequently, the lattice
$\Lambda_{(\mathcal{O},\alpha)}$ is even unimodular of dimension $16$ for any
maximal order in a quaternion algebra over $F_{m}$. We have the following
result.

######  Theorem 5.11.

$\mathcal{T}_{E_{8}^{2}}=\left\\{F\mathrm{\ quartic\ totally\ real}\ \Big{|}\
\Lambda_{(\mathcal{O},\alpha)}\sim E_{8}^{2},\textrm{ where
}\mathcal{O}\subseteq B=\left(\frac{-1,-1}{F}\right)\textrm{ is a maximal
order}\right\\}$

is infinite.

###### Proof.

Let $F_{m}$ denote a simplest quartic field as described at the beginning of
this section. We will show that, if $m=2k$ for some odd $k$, _i.e._ ,
$v_{2}(m)=1$, then $\Lambda_{(\mathcal{O},\alpha)}$ is even unimodular for
$\alpha=\frac{2}{sf_{m}(r_{m})^{\prime}}$. Moreover, we have
$\Delta_{m}=4(k^{2}+4)$. It is proved in [Est31] that the set of square-free
integers of the form $k^{2}+4$, with $k$ odd, has a positive density. We
consider the quaternion algebra $B=\left(\frac{-1,-1}{F_{m}}\right)$. The
basis

$\mathcal{B}=\left\\{1,\frac{(1+r_{m})(1+i)}{2},\frac{(1+r_{m})(1+j)}{2},\frac{1+i+j+k}{2}\right\\}$

is a basis of a maximal order in $B$ (indeed, the discriminant of the basis
$\mathcal{B}$ coincides with $D_{B}$). One can construct the lattice
associated to this order explicitly, using the integral basis for $F_{m}$ in
the case $v_{2}(m)=1$, and compare the densities of $E_{8}^{2}$ and
$\Lambda_{(\mathcal{O},\alpha)}$.

∎

If $v_{2}(m)=2$, then by equation 6, we have

$\mathrm{Nm}_{F_{m}/\mathbb{Q}}(f_{m}(r_{m})^{\prime})=2^{6}d_{F_{m}}.$ (9)

Hence,

$\mathrm{Nm}_{F_{m}/\mathbb{Q}}\left(\frac{f_{m}(r_{m})^{\prime}}{4}\right)=\frac{d_{F_{m}}}{4}.$

We consider $s=\frac{r_{m}^{3}-mr_{m}^{2}-5r_{m}+m}{4}$. Using the above we
can see that $\mathrm{Nm}_{F_{m}/\mathbb{Q}}(s)=1$. Moreover, the element
$\alpha=\frac{4}{sf_{m}(r_{m})^{\prime}}$ is totally positive of norm
$\frac{2^{4}}{d_{F_{m}}}$. Consequently, taking
$\alpha=\frac{2}{sf(r_{m})^{\prime}}$, the lattice
$\Lambda_{(\mathcal{O},\alpha)}$ is an even lattice of volume $2^{8}$ with
minimal norm $4$. A lattice with such properties is necessarily isometric to
the Barnes–Wall lattice $\Lambda_{16}$, which is the best known lattice sphere
packing in its respective dimension.

We can check that the basis $\mathcal{B}$ in the proof of Theorem 5.11
generates a maximal order in $\left(\frac{-1,-1}{F_{m}}\right)$. We arrive at
the following theorem, which provides an analogous result for the best known
sphere packing in dimension $16$.

######  Theorem 5.12.

The set

$\mathcal{T}_{\Lambda_{16}}=\left\\{F\mathrm{\ quartic~{}totally\ real}\
\Big{|}\ \Lambda_{(\mathcal{O},\alpha)}\sim\Lambda_{16},\textrm{ where
}\mathcal{O}\subseteq B=\left(\frac{-1,-1}{F}\right)\textrm{ is a maximal
order}\right\\}$

is infinite.

###### Proof.

The proof is analogous to the proof of Theorem 5.11. Note that if $m=4k$ with
$k$ odd, then we need the odd part in $\Delta_{m}=16(k^{2}+1)$ to be square-
free. Hence, the conclusion is implied by the fact that the set of integers
$k$ such that $k^{2}+1$ is square-free is of a positive density [Est31]. ∎

######  Example 5.13.

We consider $m=20$. The quartic number field $F_{20}$ is given by the
polynomial

$f_{20}(x)=x^{4}-10x^{3}-6x^{2}+20x+1.$

We have $F_{20}=\mathbb{Q}(r_{m})$ is of discriminant
$d_{F_{20}}=4(m^{2}+16)^{3}$, where

$r_{m}=\frac{1}{2}\left(\frac{m+\sqrt{\Delta_{m}}}{2}+\frac{\sqrt{\Delta_{m}+m\sqrt{\Delta_{m}}}}{2}\right).$

Taking $s=\frac{r_{m}^{3}-mr_{m}^{2}-5r_{m}+m}{2}$, the element
$\alpha=\frac{1}{sf_{m}(r_{m})^{\prime}}$ yields an even lattice
$\Lambda_{(\mathcal{O},\alpha)}$ similar to the lattice $\Lambda_{16}$, with
Gram matrix

$\setcounter{MaxMatrixCols}{16}\begin{pmatrix}4&0&0&2&2&2&0&4&2&2&0&4&2&0&0&1\\\
0&4&0&6&2&2&4&46&2&2&4&46&0&2&0&3\\\
0&0&4&42&0&4&42&448&0&4&42&448&0&0&2&21\\\
2&6&42&452&4&46&448&4786&4&46&448&4786&1&3&21&226\\\
2&2&0&4&4&4&4&50&2&2&2&25&2&2&0&4\\\
2&2&4&46&4&12&88&946&2&6&44&473&2&2&4&46\\\
0&4&42&448&4&88&892&9522&2&44&446&4761&0&4&42&448\\\
4&46&448&4786&50&946&9522&101660&25&473&4761&50830&4&46&448&4786\\\
2&2&0&4&2&2&2&25&4&4&4&50&2&2&0&4\\\
2&2&4&46&2&6&44&473&4&12&88&946&2&2&4&46\\\
0&4&42&448&2&44&446&4761&4&88&892&9522&0&4&42&448\\\
4&46&448&4786&25&473&4761&50830&50&946&9522&101660&4&46&448&4786\\\
2&0&0&1&2&2&0&4&2&2&0&4&4&0&0&2\\\ 0&2&0&3&2&2&4&46&2&2&4&46&0&4&0&6\\\
0&0&2&21&0&4&42&448&0&4&42&448&0&0&4&42\\\
1&3&21&226&4&46&448&4786&4&46&448&4786&2&6&42&452\end{pmatrix}\,.$

##  6 Potential applications and future work

While even unimodular lattices and dense lattice packings are interesting in
their own right, they can also be utilised in, _e.g._ , code design for
wireless communications and physical layer security [OV04, OBV07, VHLR09,
Set09, BO11, BO13, DKA+21]. A detailed introduction to the algebraic tools
used to design codes from number fields and cyclic division algebras and the
associated lattice code constructions is given in [OV04] for single-input
single-output communications (SISO) and in [OBV07] for multiple-input
multiple-output (MIMO) communications.

Typically, for SISO, vector lattices are utilised. Totally real number fields
and their full-diversity ideal lattices arising via the canonical embedding
have turned out to be beneficial in terms of reliable communications. In terms
of security, well-rounded lattices with good packing density have been
proposed for wiretap channels in [DKA+21] to enhance security on the physical
layer.

Motivated by the use of ideal lattices in the design of lattice codes for
reliability and security in fading channels, we are interested in lattices
arising from a similar structure, but which differ from ideals in number
fields in that they are non-commutative rings. Namely, we have resorted to
ideals in quaternion algebras, hence in a way using “MIMO machinery” [OBV07,
VHLR09, Set09] to construct lattices suitable for SISO communications.

Depending on the channel quality, different metrics become relevant as design
criteria. The lattices constructed in this paper are not of full diversity,
which means they are mainly suitable for low–to–mid-range channel quality,
where the density of the lattice is crucial. Such channel qualities are
present in, _e.g._ , long-distance or low-power communications, including
within the Internet of Things (IoT), 5G networks and beyond.

As future work, we are interested in exploring more dimensions where we can
potentially construct dense and interesting lattices from applications’ point
of view, to be utilised in, _e.g._ , coding theory, cryptography, or machine
learning. In order to obtain higher dimensional dense lattices with our
method, we aim to introduce some machine learning techniques to help us find
the elements that we need (_e.g._ , bases of maximal orders in quaternion
algebras or the twisting element $\alpha$). Similar approaches have been taken
recently to explore the use of machine learning algorithms to learn properties
about algebraic and number theoretic objects [ABH19], [HLO20], [He21].

## References

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## Appendix

In this section we provide the MAGMA source code for the construction used in
this paper.

    intrinsic basis_order(B :: AlgQuat) -> SeqEnum
    {Input: Quaternion algebra over real quadratic field and a=b=-1.
    Output: Basis of maximal order in B.}

        ZZ := Integers();
        F := BaseField(B); d := ZZ!(F.1^2);
        I := B.1; J := B.2; K := B.3;
        a := ZZ!(B.1^2); b := ZZ!(B.2^2);
        if d eq 2 then
            basis := [1, (1+I)*F.1/2, (1+J)*F.1/2, (1+I+J+K)/2];
        else if d mod 4 eq 3 then
            basis := [1, I, (F.1*I+J)/2, (F.1+K)/2];
        else
            if d mod 8 eq 1 then
                basis := [1,I,J,(1+I+J+K)/2];
            else
                basis := [1, I, (F.1+1)/4+(F.1+3)/4*I+J/2, (F.1+3)/4+(F.1+1)/4*I+K/2];
            end if;
        end if; end if;
        return basis;
    end intrinsic;

    intrinsic Z_basis_order(basis :: SeqEnum) -> SeqEnum
    {Input: basis of an integral ideal I in a maximal order in a quaternion algebra B
    over totally real field of degree d.
    Output: basis of I as a Z-basis of length 4d.}

        F := BaseRing(basis[1]);
        ZFbasis := IntegralBasis(F);
        Zbasis := [alpha*beta : alpha in ZFbasis, beta in basis];
        return Zbasis;
    end intrinsic;

    intrinsic Gram_matrix(basis:: SeqEnum: alpha:=1) -> AlgMatElt
    {Input: basis of an ideal I in a maximal order in a quaternion algebra B
    over totally real field F of degree d, and a totally real element alpha
    in F (alpha = 1 by default).
    Output: Gram matrix of the Z-lattice associated to I, alpha.}

        Zbasis := Z_basis_order(basis);
        QQ := Rationals();
        F := BaseRing(Zbasis[1]);
        rows := [QQ!(Trace(alpha*Trace(x*Conjugate(y)))) : x,y in Zbasis];
        G := Matrix(QQ, #Zbasis, rows);
        return G;
    end intrinsic;

    intrinsic Z_lattice_matrix(basis:: SeqEnum: alpha:=1) -> AlgMatElt
    {Input: basis of an ideal I in a maximal order in a quaternion algebra B
    over totally real field F of degree d, and a totally real element alpha in F
    (which is 1 by default).
    Output: generator matrix of the Z-lattice associated to I, alpha.}

        Zbasis := Z_basis_order(basis);
        RR := RealField(50);
        F := BaseRing(Zbasis[1]); d := Degree(F);
        n := #Zbasis;
        rows_short := [Eltseq(x) : x in Zbasis];

        if d eq 1 then
            rows :=  rows_short[1] cat rows_short[2] cat rows_short[3] cat rows_short[4];
            matrix := Matrix(Rationals(), 4, rows);
        else
            rows := [];
            if alpha in Rationals() then
                rows := [Sqrt(2*alpha)*RealEmbeddings(row[i])[j] :
                j in [1..d], i in [1..4], row in rows_short];
            else
                rows := [Sqrt(2*RealEmbeddings(alpha)[j])*RealEmbeddings(row[i])[j] :
                j in [1..d], i in [1..4], row in rows_short];
            end if;
            matrix := Matrix(RR, 4*d, 4*d, rows);
        end if;
        return matrix;
    end intrinsic;


# A Note on the Zeros of Jensen Polynomials

Young-One Kim and Jungseob Lee  Department of Mathematical Sciences and
Research Institute of Mathematics, Seoul National University, Seoul 08826,
Korea  kimyonesnu.ac.kr  Department of Mathematics, Ajou University, Suwon
16499, Korea  jsleeajou.ac.kr

(2021. 5. 12. )

A recent result of Griffin, Ono, Rolen and Zagier on Jensen polynomials
related with the Riemann zeta function is improved.

Zeros of polynomials and entire functions, Jensen polynomials, Riemann zeta
function

###### :

30C15, 30D15

††support: The first author was supported by the SNU Mathematical Sciences
Division for Creative Human Resources Development.

This note is concerned with the zeros of entire functions. Let $f$ be an
entire function. We denote the zero set of $f$ by $\Cal{Z}(f)$, that is,
$\Cal{Z}(f)=\\{z\in\mathbb{C}:f(z)=0\\}$; the order $\rho=\rho(f)$ of $f$ is
defined by

$\rho=\limsup_{r\to\infty}\frac{\log\log M(r;f)}{\log r},$

where $M(r;f)=\max_{|z|=r}|f(z)|$; and $f$ is said to be real if
$f^{(n)}(0)\in\mathbb{R}$ for all $n$.

Throughout this note $f$ denotes a real entire function of order $\rho<2$, and
in the case where $f$ is even $f_{0}$ denotes the function such that
$f(z)=f_{0}(z^{2})$ for all $z$. In this case, $f_{0}$ is a real entire
function of order $\rho/2<1$, $\Cal{Z}(f_{0})=\\{z^{2}:z\in\Cal{Z}(f)\\}$, and
we have

$f_{0}^{(k)}(0)=\frac{k!}{(2k)!}f^{(2k)}(0)\qquad(k=0,1,2,\dots).$

Since $f$ is a real entire function, we have $f(z)=0$ if and only if
$f(\bar{z})=0$; and since $f$ is of order less than $2$, it can be represented
in the form

$f(z)=cz^{m}e^{bz}\prod_{j}\left(1-\frac{z}{a_{j}}\right)e^{z/a_{j}},$

where $c,b\in\mathbb{R}$, $m$ is a non-negative integer and
$\sum|a_{j}|^{-2}<\infty$. As a consequence, there are real polynomials
$P_{1},P_{2},\dots$ such that $\Cal{Z}(P_{k})\subset\Cal{Z}(f)\cup\mathbb{R}$
for all $k$, and $P_{k}\to f$ uniformly on compact sets in the complex plane.
In particular, if $\Cal{Z}(f)\subset\mathbb{R}$ then
$\Cal{Z}(f^{(n)})\subset\mathbb{R}$ for every $n$ unless $f^{(n)}$ is
identically equal to zero. The Riemann Xi-function is given by

$\Xi(iz)=\frac{1}{2}\left(z^{2}-\frac{1}{4}\right)\pi^{-\frac{z}{2}-\frac{1}{4}}\Gamma\left(\frac{z}{2}+\frac{1}{4}\right)\zeta\left(\frac{1}{2}+z\right).$

It is an even real entire function of order $1$, its zero set is contained in
the infinite strip $\mathbb{S}=\\{z\in\mathbb{C}:|\operatorname{Im}z|<1/2\\}$,
and the Riemann hypothesis is the conjecture that
$\Cal{Z}(\Xi)\subset\mathbb{R}$, which is equivalent to that
$\Cal{Z}(\Xi_{0})\subset[0,\infty)$.

A real polynomial is said to be hyperbolic if it has real zeros only. The
Jensen polynomials $J(f;1),J(f;2),\dots$ of $f$ are defined by

$J(f;d)(z)=\sum_{k=0}^{d}\binom{d}{k}f^{(k)}(0)z^{k}\qquad(d=1,2,\dots).$

It is known [3, 8] that $\Cal{Z}(f)\subset\mathbb{R}$ if and only if all the
Jensen polynomials of $f$ are hyperbolic. Let $d$ be a positive integer. If
there is a non-negative integer $N$ such that $J(f^{(n)};d)$ is hyperbolic for
every $n\geq N$, we denote the smallest such integer by $N(f;d)$, otherwise we
write $N(f;d)=\infty$. One can easily check that the Riemann hypothesis holds
if and only if $N(\Xi_{0};d)=0$ for all $d$. Griffin et al. [2] showed that
$N(\Xi_{0};d)<\infty$ for all $d$ by establishing an asymptotic formula for
the sequence $\langle J(\Xi_{0}^{(n)};d)\rangle_{n=1}^{\infty}$ which relates
(for large $n$) the zeros of $J(\Xi_{0}^{(n)};d)$ with those of the $d$-th
Hermite polynomial. We improve the result by applying some general theorems on
real entire functions of order less than $2$ with restricted zeros. However it
should be remarked that our proof gives no information about the Jensen
polynomials considered, except that they are hyperbolic.

###### Theorem 1

Suppose that $f$ is a transcendental real entire function of order $\rho<2$
and $\Cal{Z}(f)\subset\mathbb{S}$. Then for every $c>\rho$ we have
$N(f;d)=O(d^{c/2})$ as $d\to\infty$. If $f$ is even, we also have
$N(f_{0};d)=O(d^{c/2})$ as $d\to\infty$ for every $c>\rho$.

Since $\Xi$ is of order $1$, it follows that $N(\Xi;d)=O(d^{c})$ and
$N(\Xi_{0};d)=O(d^{c})$ as $d\to\infty$ for every $c>1/2$. Theorem 1 is a
consequence of Theorems 2 and 3 below. Let
$\tilde{\mathbb{S}}=\\{z^{2}:z\in\mathbb{S}\\}=\\{z\in\mathbb{C}:(\operatorname{Im}z)^{2}-4^{-1}<\operatorname{Re}z\\}$.

###### Theorem 2

Suppose that $f$ is a transcendental real entire function of order $\rho<2$
and $\Cal{Z}(f)\subset\mathbb{S}$. Then for every $c>\rho$ there is a positive
integer $n_{1}$ such that

$\Cal{Z}(f^{(n)})\subset\left\\{z\in\mathbb{S}:|\operatorname{Re}z|\geq
n^{1/c}\right\\}\cup\mathbb{R}\qquad(n\geq n_{1}).$ $None$

If $f$ is even, then for every $c>\rho$ there is a positive integer $n_{2}$
such that

$\Cal{Z}(f_{0}^{(n)})\subset\left\\{z\in\tilde{\mathbb{S}}:\operatorname{Re}z\geq
n^{2/c}\right\\}\cup[0,\infty)\qquad(n\geq n_{2}).$

###### Demonstration Proof

The first part follows from Theorem 2 in [4], and the second part is a
consequence of Theorem 1 in [5] (with $k=1/2$) applied to $g$ given by
$g(z)=f_{0}(z-4^{-1})$. ∎

For $\delta>0$ we set
$S(\delta)=\\{z\in\mathbb{C}:|\operatorname{Im}z|\leq\delta|z|\\}$.

###### Theorem 3

Suppose that $P$ and $Q$ are real polynomials, $\delta>0$, $\Cal{Z}(P)\subset
S(\delta)$, $Q$ is hyperbolic and $\deg Q\leq\delta^{-2}$. Then the polynomial

$P(D)Q=\sum_{k=0}^{\deg P}\frac{P^{(k)}(0)}{k!}Q^{(k)}$

is hyperbolic.

###### Demonstration Proof

See [6]. ∎

###### Corollary

Suppose that $P$ is a real polynomial, $\delta>0$ and $\Cal{Z}(P)\subset
S(\delta)$. Then $J(P;d)$ is hyperbolic for $d\leq\delta^{-2}$.

###### Demonstration Proof

If $Q(z)=z^{d}$, then $J(P;d)(z)=z^{d}(P(D)Q)(z^{-1})$. ∎

###### Demonstration Proof of Theorem 1

Suppose that $c>\rho$. By Theorem 2, there is a positive integer $n_{1}$ such
that (1) holds. Let $d$ and $n$ be positive integers such that

$n\geq\max\\{n_{1},(d/4)^{c/2}\\},$

and put $\delta=2^{-1}n^{-1/c}$. Then $d\leq\delta^{-2}$, and from (1) we can
see that $|\operatorname{Im}z|\leq\delta|z|$ for all $z\in\Cal{Z}(f^{(n)})$.
Hence we have $\Cal{Z}(f^{(n)})\cup\mathbb{R}\subset S(\delta)$. Let
$P_{1},P_{2},\dots$ be real polynomials such that
$\Cal{Z}(P_{k})\subset\Cal{Z}(f^{(n)})\cup\mathbb{R}$ for all $k$, and
$P_{k}\to f^{(n)}$ uniformly on compact sets in the complex plane. Then
$J(P_{k};d)\to J(f^{(n)};d)$ uniformly on compact sets in the complex plane,
and the corollary to Theorem 3 implies that $J(P_{k};d)$ is hyperbolic for
every $k$. Therefore $J(f^{(n)};d)$ is hyperbolic, and we conclude that

$N(f;d)\leq\left\lceil\max\\{n_{1},(d/4)^{c/2}\\}\right\rceil\qquad(d=1,2,\dots).$

This proves the first part of the theorem. The second part is analogously
proved. ∎

###### Remark Remark

Since our argument deals with entire functions of order less than $2$, it
refines the proof of Theorem 3.6 in [1].

Suppose that $T>0$ and
$\Cal{Z}(f)\subset\mathbb{S}(T)=\\{z\in\mathbb{S}:|\operatorname{Re}z|\geq
T\\}\cup\mathbb{R}$. Then the same argument as in the proof of Theorem 1 shows
that $J(f;d)$ is hyperbolic whenever $d\leq 1+4T^{2}$, because we have
$\mathbb{S}(T)\subset S(\delta)$ with $\delta=(1+4T^{2})^{-1/2}$. Since the
condition $\Cal{Z}(f)\subset\mathbb{S}(T)$ does not imply
$\Cal{Z}(f^{\prime})\subset\mathbb{S}(T)$ in general, it is not clear whether
the polynomials $J(f^{(n)};d)$, $n=1,2,\dots$, are hyperbolic for $d\leq
1+4T^{2}$. If $f$ is even and $T\geq 1/2$, then it is the case.

###### Theorem 4

Suppose that $f$ is an even transcendental real entire function of order less
than $2$, $T\geq 1/2$ and $\Cal{Z}(f)\subset\mathbb{S}(T)$. Then
$J(f^{(n)};d)$ is hyperbolic for $d\leq 1+4T^{2}$ and for every $n$.

###### Demonstration Proof

Let $\delta=(1+4T^{2})^{-1/2}$,
$\tilde{\delta}=2\delta\sqrt{1-\delta^{2}}\,$($=T^{-1}(1+4^{-1}T^{-2})^{-1}$)
and $\tilde{S}=\\{z\in S(\tilde{\delta}):\operatorname{Re}z\geq 0\\}$. Then
$\mathbb{S}(T)\subset S(\delta)$, and $\tilde{S}$ is convex. Since $T\geq
1/2$, we have $0<\delta\leq 2^{-1/2}$, and it follows that $z\in S(\delta)$ if
and only if $z^{2}\in\tilde{S}$.

We prove the theorem by showing that $\Cal{Z}(f^{(n)})\subset S(\delta)$ for
every $n$. We have $\Cal{Z}(f)\subset\mathbb{S}(T)\subset S(\delta)$, and this
implies that $\Cal{Z}(f_{0})\subset\tilde{S}$. Since $f_{0}$ is of order less
than $1$ and $\tilde{S}$ is convex, the Gauss-Lucas theorem implies that
$\Cal{Z}(f_{0}^{\prime})\subset\tilde{S}$. Hence $\Cal{Z}(f^{\prime})\subset
S(\delta)$, because $f^{\prime}(z)=2zf_{0}^{\prime}(z^{2})$. We have

$f^{\prime\prime}(z)=2\left(f_{0}^{\prime}(z^{2})+2z^{2}f_{0}^{\prime\prime}(z^{2})\right).$

If we write $g(z)=z(f_{0}^{\prime}(z))^{2}$, then $g$ is of order less than
$1$ and $\Cal{Z}(g)\subset\tilde{S}$. Again, it follows from the Gauss-Lucas
theorem that $\Cal{Z}(g^{\prime})\subset\tilde{S}$, that is, all the roots of
the equation

$f_{0}^{\prime}(z)\left(f_{0}^{\prime}(z)+2zf_{0}^{\prime\prime}(z)\right)=0$

lie in $\tilde{S}$. Therefore $\Cal{Z}(f^{\prime\prime})\subset S(\delta)$.
Since $f^{\prime\prime}$ is even, the result follows from an induction. ∎

###### Remark Remark

The proof shows that $\Cal{Z}(f_{0}^{(n)})\subset\tilde{S}\subset
S(\tilde{\delta})$ for all $n$. Hence $J(f_{0}^{(n)};d)$ is hyperbolic for
$d\leq T^{2}(1+4^{-1}T^{-2})^{2}$ and for every $n$.

Currently, it is known [7] that $\Cal{Z}(\Xi)\subset\mathbb{S}(T)$ for some
$T\geq 3\cdot 10^{12}$. Thus $J(\Xi^{(n)};d)$ is hyperbolic for $d\leq
3.6\cdot 10^{25}$ and for every $n$, and $J(\Xi_{0}^{(n)};d)$ is hyperbolic
for $d\leq 9\cdot 10^{24}$ and for every $n$.

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