# Every finite abelian group is the group of rational points of an ordinary
abelian variety over $\mathbb{F}_{2}$, $\mathbb{F}_{3}$ and $\mathbb{F}_{5}$

Stefano Marseglia  Mathematical Institute, Utrecht University, P.O. Box 80010,
3508 TA, Utrecht, The Netherlands<EMAIL_ADDRESS>and  Caleb Springer
Department of Mathematics, University College London, Gower Street, London
WC1H 0AY, UK
and The Heilbronn Insitute for Mathematical Research, Bristol, UK
<EMAIL_ADDRESS>
###### Abstract.

We show that every finite abelian group occurs as the group of rational points
of an ordinary abelian variety over $\mathbb{F}_{2}$, $\mathbb{F}_{3}$ and
$\mathbb{F}_{5}$. We produce partial results for abelian varieties over a
general finite field $\mathbb{F}_{q}$. In particular, we show that certain
abelian groups cannot occur as groups of rational points of abelian varieties
over $\mathbb{F}_{q}$ when $q$ is large. Finally, we show that every finite
cyclic group arises as the group of rational points of infinitely many simple
abelian varieties over $\mathbb{F}_{2}$.

###### Key words and phrases:

Abelian variety, finite fields, group of rational points

###### 2020 Mathematics Subject Classification:

Primary: 14K15. Secondary: 14G15, 11G10

Marseglia is supported by NWO grant VI.Veni.202.107.

Springer is partially supported by National Science Foundation award
CNS-2001470 and by the Additional Funding Programme for Mathematical Sciences,
delivered by EPSRC (EP/V521917/1) and the Heilbronn Institute for Mathematical
Research.

##  1\. Introduction

Recently, Howe and Kedlaya [HK21] proved that every positive integer $m$ is
the order of the group of rational points of an ordinary abelian variety over
$\mathbb{F}_{2}$. Shortly afterwards, Van Bommel, Costa, Li, Poonen and Smith
[vBCL+21] extended the result to the finite fields $\mathbb{F}_{3}$ and
$\mathbb{F}_{5}$. Similar results, with some exceptions, are obtained for the
finite fields $\mathbb{F}_{4}$, $\mathbb{F}_{7}$ and, when $m$ is large
enough, for a general finite field $\mathbb{F}_{q}$. In another direction,
Kedlaya expanded the result of [HK21] to prove that every positive integer is
the order of the group of rational points of infinitely many (not necessarily
ordinary) simple abelian varieties over $\mathbb{F}_{2}$ [Ked21, Theorem 1.1].
The goal of this paper is to strengthen these results from statements
regarding cardinality to statements regarding groups.

###  1.1. Finite fields of small cardinality

For clarity, we start with a simplified statement of our first main result.
Recall that an abelian variety $A$ of dimension $g$ over a finite field of
characteristic $p$ is called ordinary, resp. almost ordinary, if the
$p$-torsion of $A(\overline{\mathbb{F}}_{p})$ consists of $p^{g}$ points,
resp. $p^{g-1}$.

######  Main Theorem 1.

Let $G$ be a finite abelian group. Then the following statements hold.

  1. (1)

Let $k$ be one of the finite fields $\mathbb{F}_{2}$, $\mathbb{F}_{3}$ or
$\mathbb{F}_{5}$. Then there is an ordinary abelian variety $A$ defined over
$k$, such that $A(k)\cong G$.

  2. (2)

Over $\mathbb{F}_{4}$, there is an abelian variety $B$ which is either
ordinary or almost ordinary such that $B(\mathbb{F}_{4})\cong G$.

In fact, over $\mathbb{F}_{2}$, the proof of this result also provides a way
to bound the dimension of the abelian variety $A$ appearing in the theorem.
Moreover, the non-ordinary abelian varieties used over $\mathbb{F}_{4}$ can be
described precisely. The version of the theorem including all details appears
in Section 3 as Theorem 3.3.

The outline of the proof of the first part of Main Theorem 1 is as follows. To
begin, we reduce to the case when the group $G$ is cyclic, and we focus our
attention on isogeny classes with the key property of being square-free; see
Definition 2.1. Square-free isogeny classes over prime fields $\mathbb{F}_{p}$
and square-free ordinary isogeny classes over any finite field
$\mathbb{F}_{q}$ are well-understood in terms of fractional ideals in étale
algebras over $\mathbb{Q}$; see Definition 2.2 and Proposition 2.5. Using this
description, we prove that in every such isogeny class there is an abelian
variety with cyclic group of points; see Proposition 2.7. Therefore, because
the number of rational points on an abelian variety is an isogeny invariant
[Tat66, Theorem 1.(c)], we simply require a square-free isogeny class of
abelian varieties with the correct number of points. The result of Howe and
Kedlaya [HK21] provides the necessary isogeny classes in the case of
$\mathbb{F}_{2}$, and the result of Van Bommel et al. [vBCL+21] does the same
for the remaining cases. See Theorems 3.1 and 3.2 for the precise statements
of their results. This allows us to conclude the proof of the first part of
Main Theorem 1. The second part requires a small modification of the argument
since there is no ordinary abelian variety over $\mathbb{F}_{4}$ with $3$
rational points; see [vBCL+21, Remark 1.16]. The argument shows that any
finite cyclic group is the group of rational points of a square-free ordinary
abelian variety over $\mathbb{F}_{2}$, $\mathbb{F}_{3}$ and $\mathbb{F}_{5}$;
see Corollary 3.5. The same is true over $\mathbb{F}_{4}$ and
$\mathbb{F}_{7}$, with some exceptions; see again Corollary 3.5 and Corollary
3.6.

###  1.2. Finite fields of arbitrary cardinality

We pause to stress that Propositions 2.5 and 2.7 work over any finite field
$\mathbb{F}_{q}$. Since [vBCL+21] shows that, for any prime power $q$, every
sufficiently large integer is the order of an abelian variety defined over
$\mathbb{F}_{q}$, this gives us the following result.

######  Main Theorem 2.

Let $q$ be a prime power. If $m_{1},\dots,m_{r}$ are integers satisfying
$m_{i}\geq q^{3\sqrt{q}\log q}$ for all $i$, then the group
$\mathbb{Z}/m_{1}\mathbb{Z}\leavevmode\nobreak\
\times\dots\times\leavevmode\nobreak\ \mathbb{Z}/m_{r}\mathbb{Z}$ is
isomorphic to the group of rational points of an ordinary abelian variety over
$\mathbb{F}_{q}$.

Main Theorem 2 is recalled in Section 4 as Theorem 4.4, together with further
discussion about general finite fields. For example, we show that abelian
groups of small exponent, regardless of cardinality, never appear as a group
of rational points for an abelian variety over $\mathbb{F}_{q}$ when $q$ is
large; see Proposition 4.2.

###  1.3. Infinitely many occurrences

We now return to the finite field $\mathbb{F}_{2}$. After proving that every
positive integer is the order of the group of rational points of infinitely
many simple abelian varieties over $\mathbb{F}_{2}$ [Ked21, Theorem 1.1],
Kedlaya suggested that it would be interesting to show an analogous statement
regarding groups. The results contained in Section 2 allow us to immediately
deduce that every finite cyclic group is the group of rational points of
infinitely many simple abelian varieties over $\mathbb{F}_{2}$; see
Proposition 5.2. Using this, we can prove the following stronger statement,
which will be recalled in Section 5 as Theorem 5.3.

######  Main Theorem 3.

For every $n\geq 1$, there is an infinite set of Weil $2$-polynomials
$\\{f_{n,j}(t)\\}_{j\geq 1}$ which are pairwise coprime and enjoy the
following property. For each $j$, every finite abelian group of cardinality
$n$ arises as the group of rational points of an abelian variety with Weil
polynomial $f_{n,j}(t)$.

However, as noted by Kedlaya, a result of Kadets [Kad21] shows that these
results are impossible over $\mathbb{F}_{q}$ for larger $q$. Indeed, if $q>2$,
then for each positive integer $m$ there are only finitely many simple abelian
varieties with $m$ rational points.

###  1.4. Related work

We conclude this section by outlining additional relevant literature. The
groups of points of elliptic curves have been studied extensively, in
particular in relation to their application to cryptography; see for example
[Rüc87], [TVN07, Theorem 3.3.15], and [Vol88]. The groups of points of abelian
surfaces have been studied by Xing in [Xin94] and [Xin96], Rybakov in [Ryb12],
and by David et al. in [DGS+14]. Such results were extended to dimension three
by Kotelnikova in [Kot19]. Giangreco-Maidana determined precisely when a given
Weil polynomial corresponds to a cyclic isogeny class (Definition 2.6) in
[GM19, GM20, GM21]. Rybakov provided classification results for the group of
points in [Ryb10] and [Ryb15] in terms of the Newton polygon of the
characteristic polynomial of Frobenius. The second author gave a
classification in terms of the endomorphism ring of the abelian variety in
[Spr21].

### Acknowledgements

The authors thank Bjorn Poonen, Valentijn Karemaker and Kiran Kedlaya for
their helpful comments.

##  2\. The square-free case

######  Definition 2.1.

An isogeny class of abelian varieties over $\mathbb{F}_{q}$ is called square-
free if the corresponding characteristic polynomial of Frobenius (also known
as its _Weil polynomial_) has no multiple complex roots. An abelian variety
$A$ over $\mathbb{F}_{q}$ is called square-free if it belongs to a square-free
isogeny class.

######  Definition 2.2.

Let $A$ be an abelian variety over a finite field $\mathbb{F}_{q}$. We say
that $A$ satisfies condition Ord if it is ordinary. We say that $A$ satisfies
condition CS if $q$ is a prime number and the Weil polynomial has no real
roots; such abelian varieties were studied by Centeleghe and Stix in [CS15].

Note that in [HK21], an isogeny class is called square-free if the isogeny
decomposition has no repeated factors. Their definition differs from ours in
general, so we provide the following lemma to disambiguate the use of
terminology. For example, we do not call an elliptic curve over
$\mathbb{F}_{p^{2}}$ with Weil polynomial $(x-p)^{2}$ square-free in this
paper even though it is simple.

######  Lemma 2.3.

Let $A$ be an abelian variety over $\mathbb{F}_{q}$. The following are
equivalent.

  1. (1)

$A$ has square-free Weil polynomial, i.e. $A$ is square-free.

  2. (2)

The endomorphism algebra
$\operatorname{End}_{\mathbb{F}_{q}}(A)\otimes\mathbb{Q}$ is commutative.

Additionally, these conditions imply the following.

  1. (3)

The isogeny decomposition of $A$ contains no repeated factors.

Moreover, if $A$ satisfies Ord or CS, then all three conditions are
equivalent. In particular, if $A$ satisfies Ord or CS, then $A$ is simple if
and only if its Weil polynomial is irreducible.

###### Proof.

The first equivalence is [Tat66, Theorem 2(c)]. The implication that the first
two conditions imply the third follows from [Tat66, Theorem 1(b)]. For
property Ord, it follows from Honda-Tate theory that an ordinary isogeny class
is simple if and only if its Weil polynomial is irreducible; see for example
[How95, Theorem 3.3]. For property CS, note that the Weil polynomial of any
simple abelian variety over a prime finite field $\mathbb{F}_{p}$ is
irreducible, unless the Weil polynomial has a real root; see for example
[Wat69, Theorem 6.1] or [Tat71, p.96]. ∎

######  Remark 2.4.

Over a prime field $\mathbb{F}_{p}$, there is only one simple isogeny class
whose Weil polynomial has real roots, namely $(x^{2}-p)^{2}$.

Square-free abelian varieties satisfying Ord or CS are well-understood in
terms of fractional ideals, thanks to certain equivalences of categories
proved by Deligne in [Del69] and Centeleghe-Stix in [CS15]. We summarize the
relevant results in the following Proposition; see [Mar21, Cor. 4.4 and Cor.
4.7] for the proofs. Let $f_{A}$ be the characteristic polynomial of Frobenius
for a square-free abelian variety $A$ satisfying Ord or CS. Let $K$ be the
étale algebra generated by the Frobenius endomorphism $\pi$, that is,
$K=\mathbb{Q}[\pi]=\mathbb{Q}[x]/(f_{A})$. Denote by
$\mathbb{Z}[\pi,\overline{\pi}]$ the order in $K$ generated by the Frobenius
and Verschiebung endomorphisms.

######  Proposition 2.5.

There is an equivalence111In the CS case, the functor is contravariant. This
detail is not needed in this paper. $\mathcal{F}$ between the category of
abelian varieties isogenous to $A$ (with $\mathbb{F}_{q}$-homomorphisms) and
the category of fractional $\mathbb{Z}[\pi,\overline{\pi}]$-ideals in $K$
(with $\mathbb{Z}[\pi,\overline{\pi}]$-linear morphisms). In particular, if
$\mathcal{F}(B)=I$ then we have an isomorphism of finite abelian groups

$B(\mathbb{F}_{q})\cong\frac{I}{(\pi-1)I}.$

Before applying this proposition and concluding this section, we recall one
more definition.

######  Definition 2.6.

An abelian variety $A$ over $\mathbb{F}_{q}$ is _cyclic_ if
$A(\mathbb{F}_{q})$ is a cyclic group. An isogeny class is cyclic if every
abelian variety in the isogeny class is cyclic.

Using Proposition 2.5, we can prove the existence of cyclic abelian varieties
within every isogeny class satisfying Ord or CS. The following proposition can
be viewed as a generalization of a result of Galbraith for elliptic curves
[Gal99, Lemma 1].

######  Proposition 2.7.

Every square-free isogeny class over $\mathbb{F}_{q}$ satisfying Ord or CS
contains a cyclic abelian variety

###### Proof.

We fix a square-free isogeny class over $\mathbb{F}_{q}$ satisfying Ord or CS.
By Proposition 2.5 there exists an abelian variety $A$ in such an isogeny
class such that $\mathcal{F}(A)=\mathbb{Z}[\pi,\overline{\pi}]$. Observe that
a natural surjective map

$\varphi:\frac{\mathbb{Z}[x,y]}{(f_{A}(x),xy-q,x-1)}\to\frac{\mathbb{Z}[\pi,\overline{\pi}]}{(\pi-1)}$

is induced by mapping $x$ to $\pi$ and $y$ to $\overline{\pi}$ because
$f_{A}(\pi)=0$ and $\pi\overline{\pi}=q$. The target of $\varphi$ is
isomorphic to the group of rational points $A(\mathbb{F}_{q})$ by Proposition
2.5. To conclude the proof, we will show that the domain is a cyclic group of
order $f_{A}(1)=\\#A(\mathbb{F}_{q})$, which implies that the natural
surjective map is actually an isomorphism. We pause to note that the closely-
related surjective map
$\mathbb{Z}[x,y]/(f_{A}(x),xy-q)\to\mathbb{Z}[\pi,\overline{\pi}]$ is _not_ an
isomorphism in general; see [CS15, Theorem 11] for the correct description of
$\mathbb{Z}[\pi,\overline{\pi}]$ which could be used in an alternate
presentation of this proof.

By the division algorithm we can write $f_{A}(x)=(x-1)p(x)+f_{A}(1)$, for some
polynomial $p(x)$. This relation together with $y-q=(xy-q)-y(x-1)$ shows that
we have the following equality of ideals of $\mathbb{Z}[x,y]$:

$(f_{A}(x),xy-q,x-1)=(f_{A}(1),x-1,y-q).$

Therefore the evaluation map $(x,y)\mapsto(1,q)$ induces an isomorphism

$\frac{\mathbb{Z}[x,y]}{(f_{A}(1),x-1,y-q)}\cong\frac{\mathbb{Z}}{(f_{A}(1))}.$

We conclude that $\varphi$ is an isomorphism and $A(\mathbb{F}_{q})$ is a
cyclic group. ∎

######  Remark 2.8.

We note that the same result can be deduced from [Ryb10], and in the simple
case, the result can also be deduced from [Spr21, Theorem 1.3]. Our proof
shows that we can choose the cyclic ordinary abelian variety $A$ to have
endomorphism ring isomorphic to $\mathbb{Z}[\pi,\overline{\pi}]$. In fact, any
abelian variety in the given isogeny class with minimal endomorphism ring is
cyclic because $\mathbb{Z}[\pi,\overline{\pi}]$ is Gorenstein; see [CS15,
Theorem 11].

##  3\. Proof of Main Theorem 1

We now focus on abelian varieties defined over $\mathbb{F}_{2}$. As indicated
in the introduction, Howe and Kedlaya proved the following theorem.

######  Theorem 3.1 (Theorem 1, [HK21]).

Let $m>0$ and $d>2$ be integers with $m<(4/3)2^{d}+1$. Then there is a square-
free ordinary abelian variety $A$ over $\mathbb{F}_{2}$ of dimension at most
$d$ with $m=\\#A(\mathbb{F}_{2})$.

Over $\mathbb{F}_{3},\mathbb{F}_{4}$ and $\mathbb{F}_{5}$, Van Bommel et al.
proved the following result.

######  Theorem 3.2 ([vBCL+21, Theorem 1.13(a), Remarks 1.16 and 1.18]).

Let $m$ be a positive integer and $k$ be $\mathbb{F}_{3},\mathbb{F}_{4}$ or
$\mathbb{F}_{5}$. Then there is a square-free abelian variety $A$ over $k$
with $m=\\#A(k)$. One can choose $A$ to be ordinary except in the case $m=3$
and $k=\mathbb{F}_{4}$.

We are now ready to complete the proof of Main Theorem 1.

######  Theorem 3.3.

Let

$G=\mathbb{Z}/n_{1}\mathbb{Z}\times\dots\times\mathbb{Z}/n_{r}\mathbb{Z}$

be a finite abelian group. The following statements hold.

  * •

Let $k$ be one of the finite fields
$\mathbb{F}_{2},\mathbb{F}_{3},\mathbb{F}_{5}$. Then there is an ordinary
abelian variety $A$ defined over $k$, such that $A(k)\cong G$;

  * •

There is an abelian variety $B$ over $\mathbb{F}_{4}$, such that
$B(\mathbb{F}_{4})\cong G$ and $B$ is either ordinary or of the form $B\cong
B^{\prime}\times E$ where $B^{\prime}$ is ordinary and $E$ belongs to the
unique isogeny class of supersingular elliptic curves over $\mathbb{F}_{4}$
with $3$ rational points; see the LMFDB label [LMF21, 1.4.ac]. The variety $B$
can be taken to be ordinary unless $G$ is a $3$-group such that
$G\cong(\mathbb{Z}/3\mathbb{Z})^{n_{1}}\times\prod_{j>1}(\mathbb{Z}/3^{j}\mathbb{Z})^{n_{j}}$
for $n_{1}$ odd.

Moreover, if $k=\mathbb{F}_{2}$ and $d_{1},\dots,d_{r}$ are integers
satisfying $n_{j}<(4/3)2^{d_{j}}+1$ and $d_{j}\geq 3$ for each $1\leq j\leq
r$, then there is an ordinary abelian variety $A$ defined over
$\mathbb{F}_{2}$ of dimension at most $d=d_{1}+\cdots+d_{r}$ such that
$A(\mathbb{F}_{2})\cong G$.

###### Proof.

Assume that $k$ is one of the finite fields $\mathbb{F}_{2}$,
$\mathbb{F}_{3}$, or $\mathbb{F}_{5}$. We can immediately reduce to the case
when $r=1$, that is, when the group $G$ is cyclic. By Theorems 3.1 and 3.2,
there exists a square-free ordinary isogeny class over $k$ with $|G|$ rational
points. By Proposition 2.7, we have an abelian variety $A$ within this isogeny
class with cyclic group of points, that is, we have $A(k)\cong G$.

We deal now with the finite field $\mathbb{F}_{4}$. Consider the simple
ordinary isogeny class of abelian surfaces over $\mathbb{F}_{4}$ with $9$
rational points given by the Weil polynomial $x^{4}-3x^{3}+7x^{2}-12x+16$; see
the LMFDB label [LMF21, 2.4.ad_h]. Let $K=\mathbb{Q}(\pi)$ be the endomorphism
algebra of this isogeny class and let $\mathcal{O}_{K}$ be the maximal order
of $K$. We observe that

$\dfrac{\mathcal{O}_{K}}{(\pi-1)\mathcal{O}_{K}}\cong\dfrac{\mathbb{Z}}{3\mathbb{Z}}\times\dfrac{\mathbb{Z}}{3\mathbb{Z}}.$

Hence by Proposition 2.5, in this isogeny class there is an abelian surface
$B_{0}$ with group of rational points isomorphic to
$(\mathbb{Z}/3\mathbb{Z})^{2}$. Let $E$ belong to the unique isogeny class of
supersingular elliptic curves over $\mathbb{F}_{4}$ with $3$ rational points;
see the LMFDB label [LMF21, 1.4.ac].

Write the group $G$ as a product of cyclic groups

$G\cong(\mathbb{Z}/3\mathbb{Z})^{2e+\delta}\times(\mathbb{Z}/s_{1}\mathbb{Z})\times\dots\times(\mathbb{Z}/s_{f}\mathbb{Z})$

where $\delta\in\\{0,1\\}$, $e\geq 0$ and each $s_{j}$ is a prime power
satisfying either $s_{j}=2$ or $s_{j}>3$ for all $j$. Using Theorem 3.2 with
the method from above, we can find an ordinary abelian variety $B_{1}$ over
$\mathbb{F}_{4}$ whose group of rational points is isomorphic to
$(\mathbb{Z}/s_{1}\mathbb{Z})\times\dots\times(\mathbb{Z}/s_{f}\mathbb{Z})$.
If $G$ is not a $3$-group then there exists an $s_{j}$ which is not divisible
by $3$, say $s_{1}$. Again, the above construction using Theorem 3.2 gives us
an ordinary abelian variety $B^{\prime}_{1}$ over $\mathbb{F}_{4}$ whose group
of rational points is isomorphic to
$(\mathbb{Z}/3s_{1}\mathbb{Z})\times\dots\times(\mathbb{Z}/s_{f}\mathbb{Z})$.

With this set up, we now distinguish three cases. If $\delta=0$ then we set
$B=B_{0}^{e}\times B_{1}$. If $\delta=1$ and $G$ is not a $3$-group then set
$B=B_{0}^{e}\times B^{\prime}_{1}$. Finally, if $\delta=1$ and $G$ is a
$3$-group then we set $B=E^{\delta}\times B_{0}^{e}\times B_{1}$. In all three
cases, we have that $B(\mathbb{F}_{4})\cong G$ by construction and the Chinese
Remainder Theorem. In the first two cases, the variety $B$ is ordinary, while
in the last one $B$ is almost ordinary. ∎

######  Remark 3.4.

We stress that Theorem 3.3 does not exclude the existence of ordinary abelian
varieties over $\mathbb{F}_{4}$ with group of points isomorphic to
$(\mathbb{Z}/3\mathbb{Z})^{n_{1}}\times\prod_{j>1}(\mathbb{Z}/3^{j}\mathbb{Z})^{n_{j}}$
for $n_{1}$ odd. Indeed, the LMFDB [LMF21] contains ordinary abelian varieties
over $\mathbb{F}_{4}$ with group of points
$\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/9\mathbb{Z}$; for example, see the
isogeny class with label [LMF21, 2.4.b_f]. However, the LMFDB does not
currently contain any ordinary abelian varieties over $\mathbb{F}_{4}$ with a
group of rational points isomorphic to $(\mathbb{Z}/3\mathbb{Z})^{2e+1}$ for
$e\geq 1$. Nonexistence when $e=0$ follows from [Kad21, Theorem 3.2]; see
[vBCL+21, Remark 1.16]. Proving more general existence or nonexistence results
will require additional understanding of which groups can occur as the group
of rational points of an abelian variety in a given isogeny class.

From the proof of Theorem 3.3 we immediately deduce the following special
case.

######  Corollary 3.5.

Let $k$ be one of the finite fields $\mathbb{F}_{2}$, $\mathbb{F}_{3}$ or
$\mathbb{F}_{5}$. Let $G$ be a finite cyclic abelian group. Then there are

  * •

a square-free ordinary abelian variety $A$ over $k$ with $A(k)\cong G$; and

  * •

a square-free abelian variety $B$ over $\mathbb{F}_{4}$ with
$B(\mathbb{F}_{4})\cong G$, which we can choose to be ordinary if
$G\neq\mathbb{Z}/3\mathbb{Z}$.

In a similar fashion, using [vBCL+21, Theorem 1.13(a), Remarks 1.17 and 1.18],
we can achieve a result analogous to Corollary 3.5 for abelian varieties over
the finite field $\mathbb{F}_{7}$.

######  Corollary 3.6.

Let $G$ be a finite cyclic abelian group with
$|G|\not\in\\{2,8,14,16,17,73\\}$. Then there is a square-free ordinary
abelian variety $A$ over $\mathbb{F}_{7}$ with $A(\mathbb{F}_{7})\cong G$.

######  Remark 3.7.

Using Proposition 2.7 or Remark 2.8, one can construct square-free abelian
varieties over $\mathbb{F}_{7}$, necessarily non-ordinary, with group of
rational points isomorphic to $\mathbb{Z}/8\mathbb{Z}$ and
$\mathbb{Z}/73\mathbb{Z}$.

##  4\. Proof of Main Theorem 2

For large $q$, the Weil bounds prohibit the existence of abelian varieties
over $\mathbb{F}_{q}$ with a relatively small number of points; see [Wei48]
and [Kad21]. However, in [vBCL+21] it is proven that:

######  Theorem 4.1 ([vBCL+21, Theorem 1.13(b), Remarks 1.16 and 1.18]).

For an arbitrary prime power $q$, every integer $m\geq q^{3\sqrt{q}\log q}$
arises as $m=\\#A(\mathbb{F}_{q})$ for some ordinary square-free abelian
variety $A$ over $\mathbb{F}_{q}$.

Still, Theorem 4.1 does not imply that every abelian group of sufficiently
large order arises as the group of points of an abelian variety over
$\mathbb{F}_{q}$, as shown by the following proposition, which was suggested
by Bjorn Poonen.

######  Proposition 4.2.

Let $m>1$ be an integer and $q$ a power of a prime $p$. Suppose there exists
an abelian variety $A$ over $\mathbb{F}_{q}$ such that $A(\mathbb{F}_{q})$ is
a group of exponent $m$. Then we have the following.

  1. (1)

Unconditionally, $q\leq(m+1)^{2}$;

  2. (2)

If $m$ is also a power of $p$, then we have $q\leq(\sqrt{m}+1)^{2}$.

###### Proof.

Suppose that $g$ is the dimension of $A$. Then, using the Weil bounds and the
structure of torsion subgroups, we get

$(\sqrt{q}-1)^{2g}\leq\\#A(\mathbb{F}_{q})\leq\\#A(\overline{\mathbb{F}}_{q})[m]\leq
m^{2g}.$

Rearranging, we obtain $q\leq(m+1)^{2}$. The stricter upper bound follows
similarly by using the structure of the $p^{e}$-torsion of an abelian variety
in characteristic $p$. ∎

######  Corollary 4.3.

If $q>9$ or $q=8$, then the group $(\mathbb{Z}/2\mathbb{Z})^{r}$ for $r\geq 1$
does not arise as the group of rational points for any abelian variety over
$\mathbb{F}_{q}$.

The following theorem, whose proof is exactly like that of Theorem 3.3, is the
best that can be obtained with our current methods. The nonexistence results
above show that restrictions on the sizes of the cyclic factors are necessary.
We observe that there are some finite abelian groups which are not outlawed by
Proposition 4.2 but which are also not realized in Theorem 4.4. For general
$q$, determining which of these groups arise as the group of rational points
of an ordinary abelian variety over $\mathbb{F}_{q}$ remains an open question
for future research.

######  Theorem 4.4.

Let $q$ be a prime power. If $m_{1},\dots,m_{r}$ are integers satisfying
$m_{i}\geq q^{3\sqrt{q}\log q}$ for all $i$, then the group
$\mathbb{Z}/m_{1}\mathbb{Z}\leavevmode\nobreak\
\times\dots\times\leavevmode\nobreak\ \mathbb{Z}/m_{r}\mathbb{Z}$ is
isomorphic to the group of rational points of an ordinary abelian variety over
$\mathbb{F}_{q}$.

In the previous section, we deduced that certain cyclic groups can be realized
as the group of rational points of a square-free abelian variety over
$\mathbb{F}_{q}$ for various small values of $q$; see Corollaries 3.5 and 3.6.
For arbitrary $q$, we prove an analogous corollary concerning square-free
abelian varieties with an explicit bound for the size of the group.

######  Corollary 4.5.

Fix a prime power $q$ and let $m$ be an integer satisfying $m\geq
q^{3\sqrt{q}\log q}$. Then there is a square-free ordinary abelian variety
over $\mathbb{F}_{q}$ whose group of rational points is isomorphic to
$\mathbb{Z}/m\mathbb{Z}$.

Alternatively, we prove the following result concerning _geometrically simple_
abelian varieties. As a trade-off for this stronger condition on the abelian
variety, there is no effective lower bound for the size of the group in the
following theorem.

######  Theorem 4.6.

Fix a prime power $q$ and let $n$ be sufficiently large with respect to $q$.
There is a geometrically simple ordinary abelian variety $A$ over
$\mathbb{F}_{q}$ with $A(\mathbb{F}_{q})\cong\mathbb{Z}/n\mathbb{Z}$.

###### Proof.

When $n$ is sufficiently large, another result [vBCL+21, Corollary 1.3] from
the same paper as Theorem 4.1 proves that there is a geometrically simple
ordinary abelian variety $A_{0}$ over $\mathbb{F}_{q}$ with
$|A_{0}(\mathbb{F}_{q})|=n$. By Proposition 2.7, there is an abelian variety
$A$ isogenous to $A_{0}$ which is cyclic, which concludes the theorem. ∎

##  5\. Proof of Main Theorem 3

The goal of this section is to prove Main Theorem 3. As a foundation, we have
the following result of Kedlaya regarding cardinality.

######  Theorem 5.1 (Theorem 1.1, [Ked21]).

Every positive integer is equal to the order of the group of rational points
of infinitely many simple abelian varieties (of various dimensions) over
$\mathbb{F}_{2}$.

Using this result together with Lemma 2.3, Remark 2.4 and Proposition 2.7, we
can immediately prove the following result.

######  Proposition 5.2.

Let $n\geq 1$ be a positive integer. There are infinitely many simple abelian
varieties $A$ over $\mathbb{F}_{2}$ with
$A(\mathbb{F}_{2})\cong\mathbb{Z}/n\mathbb{Z}$.

When searching for examples of (possibly non-simple) abelian varieties $A$
over $\mathbb{F}_{2}$ such that $A(\mathbb{F}_{2})$ is isomorphic to a given
finite abelian group $G$, there is one way to “cheat” to find an infinite
number of examples: Find one example $A_{0}$ with $A_{0}(\mathbb{F}_{2})\cong
G$ using Theorem 3.3, then consider the infinite set $\\{A_{0}\times
B\mid\\#B(\mathbb{F}_{2})=1\\}$. The fact that there are infinitely many
abelian varieties $B$ over $\mathbb{F}_{2}$ with only one point is a special
case of Kedlaya’s theorem which was originally proven by Madan and Pal [MP77].
In order to prove the existence of genuinely interesting infinite storehouses
of examples, we find examples with pairwise coprime Weil polynomials. This
corresponds to finding examples whose isogeny classes share no simple factors
in common. We now prove Main Theorem 3.

######  Theorem 5.3.

For every $n\geq 1$, there is an infinite set of Weil $2$-polynomials
$\\{f_{n,j}(t)\\}_{j\geq 1}$ which are pairwise coprime and enjoy the
following property. For each $j$, every finite abelian group of cardinality
$n$ arises as the group of rational points of an abelian variety with Weil
polynomial $f_{n,j}(t)$.

###### Proof.

Write $n=\ell_{1}\dots\ell_{r}$, where $\ell_{1},\dots,\ell_{r}$ are primes
that are not necessarily distinct. By Proposition 5.2, for each ${1\leq i\leq
r}$, there are infinitely many simple abelian varieties $A_{i}$ over
$\mathbb{F}_{2}$ such that
$A_{i}(\mathbb{F}_{2})\cong\mathbb{Z}/\ell_{i}\mathbb{Z}$. By Remark 2.4,
there is only one simple isogeny class over $\mathbb{F}_{2}$ whose Weil
polynomial has real roots; see [LMF21, 2.2.a_ae]. Hence, by Lemma 2.3, we can
choose the $A_{i}$ to have distinct irreducible Weil polynomial. Let
$\mathcal{C}$ be the isogeny class of $A_{1}\times\dots\times A_{r}$. Note
that $\mathcal{C}$ is square-free by construction.

Because each sub-product $\prod_{j\in S}A_{j}$ for $S\subset\\{1,\dots,r\\}$
is also square-free, we can apply Proposition 2.7 to get a cyclic isogenous
variety $A_{S}$ satisfying $A_{S}(\mathbb{F}_{2})\cong\mathbb{Z}/(\prod_{j\in
S}\ell_{j})\mathbb{Z}$. In this way, we obtain all abelian groups of
cardinality $n$ as the group of rational points of an abelian variety in
$\mathcal{C}$. ∎

######  Remark 5.4.

With the currently technology we are not able to determine whether given an
arbitrary finite abelian group $G$ there are infinitely many _simple_ abelian
varieties over $\mathbb{F}_{2}$ with group of rational points isomorphic to
$G$. A possible approach to this questions is to study which groups can occur
in a given isogeny class.

## References

  * [vBCL+21] Raymond van Bommel, Edgar Costa, Wanlin Li, Bjorn Poonen, and Alexander Smith, _Abelian varieties of prescribed order over finite fields_ , arXiv e-prints (2021), arXiv:2106.13651.
  * [CS15] Tommaso Giorgio Centeleghe and Jakob Stix, _Categories of abelian varieties over finite fields, I: Abelian varieties over $\mathbb{F}_{p}$_, Algebra Number Theory 9 (2015), no. 1, 225–265. MR 3317765
  * [Del69] Pierre Deligne, _Variétés abéliennes ordinaires sur un corps fini_ , Invent. Math. 8 (1969), 238–243. MR 0254059
  * [DGS+14] Chantal David, Derek Garton, Zachary Scherr, Arul Shankar, Ethan Smith, and Lola Thompson, _Abelian surfaces over finite fields with prescribed groups_ , Bull. Lond. Math. Soc. 46 (2014), no. 4, 779–792. MR 3239616
  * [Gal99] Steven D. Galbraith, _Constructing isogenies between elliptic curves over finite fields_ , LMS J. Comput. Math. 2 (1999), 118–138. MR 1728955
  * [GM19] Alejandro J. Giangreco-Maidana, _On the cyclicity of the rational points group of abelian varieties over finite fields_ , Finite Fields Appl. 57 (2019), 139–155. MR 3921286
  * [GM20] by same author, _Local cyclicity of isogeny classes of abelian varieties defined over finite fields_ , Finite Fields Appl. 62 (2020), 101628, 9. MR 4053144
  * [GM21] by same author, _Corrigendum to “Local cyclicity of isogeny classes of abelian varieties defined over finite fields” [Finite Fields Appl. 62 (2020) 101628]_ , Finite Fields Appl. 69 (2021), 101703, 2. MR 4183334
  * [HK21] Everett W. Howe and Kiran S. Kedlaya, _Every positive integer is the order of an ordinary abelian variety over $\mathbb{F}_{2}$_, Research in Number Theory 7 (2021), no. 4, 59.
  * [How95] Everett W. Howe, _Principally polarized ordinary abelian varieties over finite fields_ , Trans. Amer. Math. Soc. 347 (1995), no. 7, 2361–2401. MR 1297531
  * [Kad21] Borys Kadets, _Estimates for the number of rational points on simple abelian varieties over finite fields_ , Math. Z. 297 (2021), no. 1-2, 465–473. MR 4204701
  * [Ked21] Kiran S. Kedlaya, _Abelian varieties over $\mathbb{F}_{2}$ of prescribed order_, arXiv e-prints (2021), arXiv:2107.12453.
  * [Kot19] Yulia Kotelnikova, _Groups of points on abelian threefolds over finite fields_ , Finite Fields Appl. 58 (2019), 177–199. MR 3947815
  * [LMF21] The LMFDB Collaboration, _The L-functions and modular forms database_ , http://www.lmfdb.org, 2021, [Online; accessed 20 July 2021].
  * [Mar21] Stefano Marseglia, _Computing square-free polarized abelian varieties over finite fields_ , Math. Comp. 90 (2021), no. 328, 953–971. MR 4194169
  * [MP77] Manohar L. Madan and Sat Pal, _Abelian varieties and a conjecture of R. M. Robinson_ , J. Reine Angew. Math. 291 (1977), 78–91. MR 439848
  * [Rüc87] Hans-Georg Rück, _A note on elliptic curves over finite fields_ , Math. Comp. 49 (1987), no. 179, 301–304. MR 890272
  * [Ryb10] Sergey Rybakov, _The groups of points on abelian varieties over finite fields_ , Cent. Eur. J. Math. 8 (2010), no. 2, 282–288. MR 2610753
  * [Ryb12] by same author, _The groups of points on abelian surfaces over finite fields_ , Arithmetic, geometry, cryptography and coding theory, Contemp. Math., vol. 574, Amer. Math. Soc., Providence, RI, 2012, pp. 151–158. MR 2961407
  * [Ryb15] by same author, _On classification of groups of points on abelian varieties over finite fields_ , Mosc. Math. J. 15 (2015), no. 4, 805–815. MR 3438835
  * [Spr21] Caleb Springer, _The structure of the group of rational points of an abelian variety over a finite field_ , European Journal of Mathematics (2021), 1–13.
  * [Tat66] John Tate, _Endomorphisms of abelian varieties over finite fields_ , Invent. Math. 2 (1966), 134–144. MR 0206004
  * [Tat71] by same author, _Classes d’isogénie des variétés abéliennes sur un corps fini (d’après T. Honda)_ , Séminaire Bourbaki. Vol. 1968/69: Exposés 347–363, Lecture Notes in Math., vol. 175, Springer, Berlin, 1971, pp. Exp. No. 352, 95–110. MR 3077121
  * [TVN07] Michael Tsfasman, Serge Vlăduţ, and Dmitry Nogin, _Algebraic geometric codes: basic notions_ , Mathematical Surveys and Monographs, vol. 139, American Mathematical Society, Providence, RI, 2007. MR 2339649
  * [Vol88] J. F. Voloch, _A note on elliptic curves over finite fields_ , Bull. Soc. Math. France 116 (1988), no. 4, 455–458 (1989). MR 1005390
  * [Wat69] William C. Waterhouse, _Abelian varieties over finite fields_ , Ann. Sci. École Norm. Sup. (4) 2 (1969), 521–560. MR 0265369
  * [Wei48] André Weil, _Sur les courbes algébriques et les variétés qui s’en déduisent_ , Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1041, Hermann et Cie., Paris, 1948, Publ. Inst. Math. Univ. Strasbourg 7 (1945). MR 0027151
  * [Xin94] Chaoping Xing, _The structure of the rational point groups of simple abelian varieties of dimension two over finite fields_ , Arch. Math. (Basel) 63 (1994), no. 5, 427–430. MR 1300737
  * [Xin96] by same author, _On supersingular abelian varieties of dimension two over finite fields_ , Finite Fields Appl. 2 (1996), no. 4, 407–421. MR 1409453


# Generating Ray Class Fields of Real Quadratic Fields
via Complex Equiangular Lines

Marcus Appleby  Centre for Engineered Quantum Systems, School of Physics,
University of Sydney<EMAIL_ADDRESS>,  Steven Flammia  Centre
for Engineered Quantum Systems, School of Physics, University of Sydney
<EMAIL_ADDRESS>,  Gary McConnell  Controlled Quantum Dynamics
Theory Group, Imperial College, London<EMAIL_ADDRESS>and  Jon
Yard  Inst. for Quantum Computing, Dept. of Combinatorics & Optimization,
University of Waterloo and Perimeter Institute for Theoretical Physics,
Waterloo, ON<EMAIL_ADDRESS>
###### Abstract.

Let $K$ be a real quadratic field. For certain $K$ with sufficiently small
discriminant we produce explicit unit generators for specific ray class fields
of $K$ using a numerical method that arose in the study of complete sets of
equiangular lines in $\mathbb{C}^{d}$ (known in quantum information as
symmetric informationally complete measurements or SICs). The construction in
low dimensions suggests a general recipe for producing unit generators in
infinite towers of ray class fields above arbitrary $K$ and we summarise this
in a conjecture. Such explicit generators are notoriously difficult to find,
so this recipe may be of some interest.

In a forthcoming paper we shall publish promising results of numerical
comparisons between the logarithms of these canonical units and the values of
$L$-functions associated to the extensions, following the programme laid out
in the Stark Conjectures.

Let $D\geq 2$ be a square-free integer with corresponding real quadratic field
$K=\mathbb{Q}(\sqrt{D})$. Using class field theory, many questions about the
abelian extensions of $K$ can be reduced to arithmetic in the corresponding
ring of integers $\mathbb{Z}_{K}$. But in contrast to the abelian extensions
of $\mathbb{Q}$, of imaginary quadratic fields [5], or more generally, of CM
fields [16], it is still not known how to construct explicit generators for
the abelian extensions of an arbitrary real quadratic field.

Let $d\geq 4$ be an integer such that $D$ is the square-free part of
$(d-1)^{2}-4=(d+1)(d-3)$. By Lemma 3.1 below, there are infinitely many such
$d$ for any given $D$, and conversely, every $d\geq 4$ corresponds to a non-
trivial value of $D$. Let $d^{\prime}=2d$ if $d$ is even and $d^{\prime}=d$ if
$d$ is odd. This paper gives evidence that certain configurations of complex
lines in $\mathbb{C}^{d}$ yield natural generators for the ray class field
$\mathfrak{R}$ of $K$ modulo $d^{\prime}$ with ramification allowed at both
infinite places ${\infty_{1}}$ and ${\infty_{2}}$ of $K$, where we abbreviate
the ideal $d^{\prime}\mathbb{Z}_{K}$ to $d^{\prime}$.

Equipping $\mathbb{C}^{d}$ with the usual hermitian inner product
$(v,w)=v^{\dagger}w$, we consider symmetric configurations of complex lines,
pairwise equiangular as measured by the hermitian angle

$\cos^{-1}\left(\frac{|(v,w)|}{\lVert v\rVert\lVert
w\rVert}\right)=\cos^{-1}\left(\sqrt{\operatorname{Tr}\Pi_{v}\Pi_{w}}\right)$

between lines $\mathbb{C}v$, $\mathbb{C}w$, where
$\Pi_{v}=\frac{vv^{\dagger}}{(v,v)}$ is the rank-1 hermitian projector onto
$\mathbb{C}v$. It is easily shown [6] that $\mathbb{C}^{d}$ can contain at
most $d^{2}$ equiangular lines. Following established terminology in quantum
information theory [13], we say that a complete set of $d^{2}$ equiangular
lines is _Symmetric and Informationally Complete_ (_SIC_), or a _SIC_. The
common angle of a SIC is necessarily
$\cos^{-1}\bigl{(}\frac{1}{\sqrt{d+1}}\bigr{)}$ [6].

Explicit constructions of SICs have been found in dimensions $d=1$–21, 24, 28,
30, 35, 39 and 48. Numerical evidence suggests that they exist for all $d\leq
121$, and they are conjectured to exist in every finite dimension [13, 14, 15,
21]. With one exception [7], every known construction is unitarily equivalent
to an orbit of the group $(\mathbb{Z}/d\mathbb{Z})^{2}$ under the projective
unitary representation taking $j$ to the image of $X_{d}^{j_{1}}Z_{d}^{j_{2}}$
in the projective unitary group $\mathrm{PU}(d)$, where

(1) $X_{d}=\left(\begin{array}[]{cccccc}0&0&0&\ldots&0&1\\\
1&0&0&\ldots&0&0\\\ 0&1&0&\ldots&0&0\\\ 0&0&1&\vspace{-2.5pt}\ldots&0&0\\\
\vdots&\vdots&\vdots&\ddots&\ddots&\vdots\\\
0&0&0&\ldots&1&0\end{array}\right),\quad
Z_{d}=\left(\begin{array}[]{cccccc}1&0&0&0&\ldots&0\\\
0&\zeta_{d}&0&0&\ldots&0\\\ 0&0&\zeta_{d}^{2}&0&\ldots&0\\\
0&0&0&\zeta_{d}^{3}&\ldots&0\\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&0&0&\ldots&\zeta_{d}^{d-1}\end{array}\right)$

and where $\zeta_{d}=e^{2\pi i/d}$ is a primitive $d$th root of unity. In what
follows, all SICs considered will be orbits of $(\mathbb{Z}/d\mathbb{Z})^{2}$
under the action of this representation. If $v\in\mathbb{C}^{d}$ is such that
$\mathbb{C}v$ is contained in some SIC, we call $v$ and its corresponding
projector $\Pi_{v}$ _fiducial_.

Given a nonzero $v\in\mathbb{C}^{d}$, let
$\mathbb{Q}([v])=\mathbb{Q}(v_{i}/v_{j}:v_{j}\neq 0)$ be the minimal field of
definition of the corresponding point $[v]\in\mathbb{P}^{d-1}$ in projective
space. For a set $E$ of lines in $\mathbb{C}^{d}$, let $\mathbb{Q}(E)$ be the
minimal field of definition of the corresponding subset of $\mathbb{P}^{d-1}$,
i.e. the smallest subfield of $\mathbb{C}$ containing the fields
$\mathbb{Q}([v])$ for every $\mathbb{C}v\in E$. Then we have the following
result and subsequent natural conjecture:

######  Proposition 1.1.

Let $d$ be any one of the following dimensions:

(2) $4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,24,28,30,35,39,48.$

Then $\mathbb{C}^{d}$ contains one or more unitarily inequivalent SICs with at
least one orbit $E$ satisfying $\mathbb{Q}(E)=\mathfrak{R}$. Moreover, for
every other known orbit $E$, $\mathbb{Q}(E)/\mathbb{Q}$ is a finite Galois
extension containing $\mathfrak{R}$.

######  Conjecture 1.2.

The statement of Proposition 1.1 holds for every dimension $d\geq 4$.

These statements are a substantial strengthening of results and conjectures in
[2]. Short calculations reveal that the field $\mathbb{Q}(\Pi_{v})$ generated
by the matrix elements $\frac{v_{j}v^{*}_{k}}{(v,v)}$ of $\Pi_{v}$ is equal to
the closure $\mathbb{Q}([v],[\bar{v}])$ of $\mathbb{Q}([v])$ under complex
conjugation for each dimension in (2). Because the above fields
$\mathbb{Q}(E)$ are Galois and are thus closed under complex conjugation, they
are also equal to the fields generated by the matrix elements $\Pi_{v}$ for
all $\mathbb{C}v\in E$.

After introducing the necessary geometric and number-theoretical apparatus in
the next two sections, we establish some properties of the ray class fields in
§4. In §5 and §6 we relate invariants of the geometric objects to arithmetic
quantities in the ray class fields and to associated canonical units. Section
7 outlines our computational verification of Proposition 1.1. We illustrate
the main results in the appendix by focussing on the particular case of
dimension 19.

##  2\. SIC orbits under the extended Weil representation

A subgroup $H\subset\mathrm{U}(d)$ sitting in a central extension

(3) $1\rightarrow\mathrm{U}(1)\rightarrow
H\rightarrow(\mathbb{Z}/d\mathbb{Z})^{2}\rightarrow 0$

canonically determines a projective unitary representation
$(\mathbb{Z}/d\mathbb{Z})^{2}\rightarrow\mathrm{PU}(d)$ via
$j\mapsto[\Delta_{j}]$, where $\Delta_{j}\leftmapsto j$ is any choice of
section, and where $[U]\in\mathrm{PU}(d)$ denotes the image of
$U\in\mathrm{U}(d)$ in the projective unitary group. The $\Delta_{j}$
determine a 2-cocycle $(j,k)\mapsto c_{j,k}\in\mathrm{U}(1)$ via
$\Delta_{j}\Delta_{k}=c_{j,k}\Delta_{j+k}$, by definition satisfying
$c_{k,\ell}c_{j,k+\ell}=c_{j+k,\ell}c_{j,k}$. In turn, this cocycle determines
a skew-multiplicative pairing $(j,k)\mapsto e_{j,k}=c_{j,k}/c_{k,j}$, which
satisfies $\Delta_{j}\Delta_{k}\Delta_{j}^{-1}\Delta_{k}^{-1}=e_{j,k}I$, where
$e_{j,k}\in\langle\zeta_{d}\rangle$ for every $j,k$. It is independent of the
choice of section, hence an invariant of the extension. The extension (3) is a
_Heisenberg group_ if $e$ is nondegenerate (i.e. induces an isomorphism
between $\\{(j,0):j\in\mathbb{Z}/d\mathbb{Z}\\}$ and
$\\{(0,j):j\in\mathbb{Z}/d\mathbb{Z}\\})$, whence
$e((\mathbb{Z}/d\mathbb{Z})^{2})=\langle\zeta_{d}\rangle$.

From now on let $H=\big{\langle}\phi
X_{d}^{j_{1}}Z_{d}^{j_{2}}:\phi\in\mathrm{U}(1),j\in(\mathbb{Z}/d\mathbb{Z})^{2}\big{\rangle}$.
Then $H\subset\mathrm{U}(d)$ is a Heisenberg group and the _Stone-von Neumann
theorem_ [17, 18] asserts that every unitary representation
$\rho:H\rightarrow\mathrm{U}(d)$ acting as the identity on scalars has the
form $\rho(h)=U_{\rho}hU_{\rho}^{-1}$, for some $U_{\rho}\in\mathrm{U}(d)$
that is uniquely determined up to scalars. Thus follows the existence of the
_Weil representation_ [19], a canonical projective unitary representation of
the group $\operatorname{Aut}^{0}(H)$ of automorphisms of $H$ fixing its
center, taking $t\in\operatorname{Aut}^{0}(H)$ to $[U_{t}]$ for any $U_{t}$
satisfying $t(h)=U_{t}^{\vphantom{\dagger}}hU_{t}^{\dagger}$ for every $h\in
H$. In quantum computing, $\operatorname{Aut}^{0}(H)$ is known as the
_projective generalized Clifford group_ and the normalizer of $H$ in
$\mathrm{U}(d)$ is known as the _generalized Clifford group_. This group sits
in an extension [12]

(4)
$0\rightarrow(\mathbb{Z}/d\mathbb{Z})^{2}\rightarrow\operatorname{Aut}^{0}(H)\rightarrow\mathrm{Sp}_{2}(\mathbb{Z}/d\mathbb{Z})\rightarrow
1$

of the subgroup
$\mathrm{Sp}_{2}(\mathbb{Z}/d\mathbb{Z})=\mathrm{SL}_{2}(\mathbb{Z}/d\mathbb{Z})$
of $\mathrm{GL}_{2}(\mathbb{Z}/d\mathbb{Z})$ fixing $e$. The action of
$t\in\operatorname{Aut}^{0}(H)$ on a section $\Delta_{j}\leftmapsto j$ is
given by [12]

(5) $t(\Delta_{j})=\varphi_{t}(j)\Delta_{F_{t}j},$

where $\varphi_{t}\colon(\mathbb{Z}/d\mathbb{Z})^{2}\rightarrow\mathrm{U}(1)$
is a function and $t\mapsto F_{t}$ is a surjective homomorphism. The injection
in (4) is defined by $j\mapsto(h\mapsto\Delta_{j}h\Delta_{-j})$ and gives an
action of $(\mathbb{Z}/d\mathbb{Z})^{2}$ that depends only on the pairing:

$j(\Delta_{k})=\Delta_{j}\Delta_{k}\Delta_{j}^{-1}=e_{j,k}\Delta_{k}.$

For the section $j\mapsto\Delta_{j}=\phi_{j}X_{d}^{j_{1}}Z_{d}^{j_{2}}\in H$
determined by phases $\phi_{j}\in U(1)$, the cocycle is given by
$c_{j,k}=\phi_{j}\phi_{k}\phi_{j+k}^{-1}\zeta_{d}^{j_{2}k_{1}}$, while the
pairing is independent of the phases:
$e_{j,k}=c_{j,k}/c_{k,j}=\zeta_{d}^{k_{1}j_{2}-k_{2}j_{1}}$. The specific
choice $\phi_{j}=(-\zeta_{2d})^{j_{1}j_{2}}$ of phases gives the section

(6) $\Delta_{j}=(-\zeta_{2d})^{j_{1}j_{2}}X_{d}^{j_{1}}Z_{d}^{j_{2}}$

used in [2, 14] and in §5 and §6, which satisfies
$\Delta_{j}^{-1}=\Delta_{-j}$ and
$c_{j,k}=(-\zeta_{2d})^{j_{2}k_{1}-j_{1}k_{2}}\in\langle\zeta_{d^{\prime}}\rangle$.
When $d$ is odd, then $-\zeta_{2d}=\zeta_{d}^{2^{-1}\textrm{ mod }d}$, in
which case the following hold [12]: $e$ and $c$ have the same isometry group,
the phase in (5) is a homomorphism
$\varphi_{t}\colon(\mathbb{Z}/d\mathbb{Z})^{2}\rightarrow\langle\zeta_{d}\rangle$
and the extension (4) splits as a semidirect product
$\operatorname{Aut}^{0}(H)\simeq(\mathbb{Z}/d\mathbb{Z})^{2}\rtimes\mathrm{Sp}_{2}(\mathbb{Z}/d\mathbb{Z})$
via the injective homomorphism taking
$F\in\mathrm{Sp}_{2}(\mathbb{Z}/d\mathbb{Z})$ to the automorphism of $H$
defined by $F(\Delta_{j})=\Delta_{Fj}$. This splitting identifies each
automorphism $t\in\operatorname{Aut}^{0}(H)$ with a pair $t=(j_{t},F_{t})$. We
henceforth identify $(\mathbb{Z}/d\mathbb{Z})^{2}$ and
$\mathrm{Sp}_{2}(\mathbb{Z}/d\mathbb{Z})$ with their images $j\mapsto(j,1)$
and $(0,F)\leftmapsto F$ in $\operatorname{Aut}^{0}(H)$ under this splitting.
For general $d$, there is a surjective homomorphism
$(\mathbb{Z}/d\mathbb{Z})^{2}\rtimes\mathrm{Sp}_{2}(\mathbb{Z}/d^{\prime}\mathbb{Z})\rightarrow\operatorname{Aut}^{0}(H)$
that is an isomorphism when $d$ is odd, but has kernel
$\simeq(\mathbb{Z}/2\mathbb{Z})^{3}\simeq\ker(\mathrm{SL}_{2}(\mathbb{Z}/d^{\prime}\mathbb{Z})\rightarrow\mathrm{SL}_{2}(\mathbb{Z}/d\mathbb{Z}))$
when $d$ is even [1].

Let $\operatorname{EAut}^{0}(H)$ be the group of automorphisms of $H$ acting
either as the identity or by complex conjugation on the center. It may be
viewed as the projectivization of the _extended Clifford group_ , defined in
[1] as the normalizer of $H$ in the group $\mathrm{EU}(d)$ of all unitary and
anti-unitary operators in dimension $d$ [20, Chap. 26]. It is generated by
$\operatorname{Aut}^{0}(H)$ together with complex conjugation in the standard
basis. It lies in an extension

$0\rightarrow(\mathbb{Z}/d\mathbb{Z})^{2}\rightarrow\operatorname{EAut}^{0}(H)\rightarrow\mathrm{ESp}_{2}(\mathbb{Z}/d\mathbb{Z})\rightarrow
1$

of the determinant $\pm 1$ subgroup
$\mathrm{ESp}_{2}(\mathbb{Z}/d\mathbb{Z})\subset\mathrm{GL}_{2}(\mathbb{Z}/d\mathbb{Z})$,
with complex conjugation mapping to ${\bigl{(}\begin{smallmatrix}1&0\\\
0&-1\end{smallmatrix}\bigr{)}}$. There is similarly a surjective homomorphism
$(\mathbb{Z}/d\mathbb{Z})^{2}\rtimes\mathrm{ESp}_{2}(\mathbb{Z}/d^{\prime}\mathbb{Z})\rightarrow\operatorname{EAut}^{0}(H)$.
Because we only consider SICs that are orbits of
$(\mathbb{Z}/d\mathbb{Z})^{2}$, the set of all SICs decomposes into orbits of
$\mathrm{ESp}_{2}(\mathbb{Z}/d^{\prime}\mathbb{Z})$, and aside111In $d=3$
there is an uncountably infinite pencil of inequivalent SICs with
(necessarily) transcendental fields of definition apart from special points
[2, §10],[23]. For an in-depth look at the case $d=3$, see also [8]. from
dimension $3$, there are only finitely many such orbits in every known case.

We define the _stabilizer group_ of a fiducial as the subgroup of
$(\mathbb{Z}/d\mathbb{Z})^{2}\rtimes\mathrm{ESp}_{2}(\mathbb{Z}/d^{\prime}\mathbb{Z})$
leaving it invariant. Every known orbit contains a fiducial such that

  * •

The stabilizer group $\boldsymbol{S}_{0}$ is contained in
$\mathrm{ESp}_{2}(\mathbb{Z}/d^{\prime}\mathbb{Z})$.

  * •

$\boldsymbol{S}_{0}$ contains an order-3 element of
${\mathrm{Sp}_{2}}(\mathbb{Z}/d^{\prime}\mathbb{Z})$ with trace $\equiv-1\bmod
d$.

We refer to such a fiducial as _centred_ (in [2] such fiducials were called
“simple”). As observed in [14], the order-3 element can always be chosen to be
conjugate to either $F_{z}={\bigl{(}\begin{smallmatrix}0&d-1\\\
d+1&d-1\end{smallmatrix}\bigr{)}}$ or
$F_{a}={\bigl{(}\begin{smallmatrix}1&d+3\\\
4d/3-1&d-2\end{smallmatrix}\bigr{)}}$, the latter possibility only occurring
when $d\equiv 3\bmod 9$. We note that $\boldsymbol{S}_{0}$ is in fact always
cyclic in the known instances for $d\geq 4$.

In the enumeration scheme of [14] the orbits for $d\neq 3$ are specified by
the dimension followed by a letter, and for future reference we also list here
the type of stabilizer ($F_{z}$ or $F_{a}$). For dimensions in the list (2),
the orbits that are known to generate $\mathfrak{R}$ over the rationals are as
follows:

(7) $\displaystyle[\text{$F_{z}$-type}]$ $\displaystyle
4a,5a,6a,7b,8b,9ab,10a,11c,13ab,14ab,15d,16ab,17c,18ab,19e,20ab,24c,28c,35j,$
$\displaystyle[\text{$F_{a}$-type}]$ $\displaystyle 12b,21e,30d,39ij,48g.$

Here, two or more letters indicates several orbits—for instance $13ab$ is
shorthand for $13a$, $13b$ and $39a\text{-}f$ means $39a,\ldots,39f$. The
field generated is known to contain, but to be strictly larger than
$\mathfrak{R}$ for the remaining orbits:

(8) $\displaystyle[\text{$F_{z}$-type}]$ $\displaystyle
7a,8a,11ab,12a,15abc,17ab,19abcd,21abcd,24ab,28ab,35a\text{-}i,39a\text{-}f,48abcdf,$
$\displaystyle[\text{$F_{a}$-type}]$ $\displaystyle 39gh,48e.$

See [2, 3] for a detailed analysis of the number fields associated to the
orbits in the two lists. Scott and Grassl [14] have, with high probability,
identified every orbit for $d\leq 50$; however, many of them are only known
numerically. The above two lists contain every orbit for which an exact
solution is known. In the first list (7) the number of orbits in each
dimension is the same as the class number of $K$ for that dimension. This
might not be an accident, as we discuss in §5.

##  3\. Pell’s equation and towers of ray class fields over $K$

We now prove the assertions made in the introduction about infinite sequences
of dimensions $d_{1},d_{2},\ldots$ all of which give rise to the same value of
$D$. Conjecturally these give us ‘towers’ of ray class fields lying above each
quadratic field $K$.

Fix $D$, $K={\mathbb{Q}({{\sqrt{D}}})}$ as above and write ${\infty_{1}}$ for
the place associated to the embedding of $K$ into $\mathbb{R}$ under which
${\sqrt{D}}$ is sent to the positive square root of $D$. Let
$u_{f}\in\mathbb{Z}_{K}^{\times}$ be the fundamental unit of $K$ which is $>1$
under ${\infty_{1}}$. If the norm of $u_{f}$ is $-1$ then set
$u_{D}=u_{f}^{2}$; otherwise set $u_{D}=u_{f}$. In other words, $u_{D}$ is the
first power of $u_{f}$ of norm 1.

######  Lemma 3.1.

The following are equivalent.

  * (i)

$d$ is a positive integer such that the square-free part of $(d-1)^{2}-4$ is
equal to $D$

  * (ii)

$\frac{d-1}{2}$ is the rational part of $u_{D}^{r}$ for some $r\in\mathbb{N}$.

###### Proof.

Statement (i) is equivalent to saying there exists $y\in\mathbb{Z}$ such that
$\left(\tfrac{d-1}{2}\right)^{2}-D(\tfrac{y}{2})^{2}=1$, which in turn is
equivalent to $\frac{d-1}{2}+\frac{y}{2}{\sqrt{D}}$ being an element of $K$ of
norm 1. But it is also a root of the monic integral polynomial
$X^{2}-(d-1)X+1$, hence a unit. Therefore [5, §11B] it is a power of $u_{D}$,
proving (ii). The converse is just a restatement of the definitions. ∎

######  Corollary 3.2.

For every square-free value of $D\geq 2$ there exist infinitely many values of
$d$ such that the square-free part of $(d-1)^{2}-4=(d+1)(d-3)$ is equal to
$D$.∎

######  Definition 3.3.

All $d$ yielding the same value of $D$ are indexed by the appropriate power of
$u_{D}$:

(9) $d_{r}=1+u_{D}^{r}+u_{D}^{-r}.$

In particular, $d_{1}$ is the smallest dimension corresponding to $D$. We
shall generally omit reference to the underlying $D$ but  _the notation
$d_{r}$ always applies with reference to a specific fixed value of $D$_. Note
that the rational part of $u_{D}^{r}$ is given by
$\tfrac{u_{D}^{r}+u_{D}^{-r}}{2}=\cosh{(r\log u_{D})}$. The Chebyshev
polynomials of the first kind $T_{n}$ tell us how to go from the rational part
of $u_{D}^{r}$ to that of $u_{D}^{rs}$:

(10) $\displaystyle T_{s}\left({\tfrac{d_{r}-1}{2}}\right)\ =\
T_{s}\left({\cosh{(r\log u_{D})}}\right)\ =\ \cosh{(sr\log u_{D})}\ =\
\tfrac{d_{rs}-1}{2}\ =\ T_{r}\left({\tfrac{d_{s}-1}{2}}\right).$

Let us define a shifted version of the functions $T_{n}$ by

$T^{*}_{n}\left({x}\right)=1+2T_{n}\left({\tfrac{x-1}{2}}\right),$

and for convenience extend it to negative $n$ by defining
$T^{*}_{-n}=T^{*}_{n}$. Equation (10) may be rephrased in terms of the new
functions $T^{*}_{n}$ and rearranged to read:

(11)
$T^{*}_{r}\left({T^{*}_{s}\left({d_{1}}\right)}\right)=T^{*}_{r}\left({d_{s}}\right)=d_{rs}=d_{sr}=T^{*}_{s}\left({d_{r}}\right)=T^{*}_{s}\left({T^{*}_{r}\left({d_{1}}\right)}\right).$

It is also easy to show that $d_{1},d_{2},d_{3},\ldots$ is a strictly
increasing sequence of positive integers.

######  Proposition 3.4.

Within each sequence $d_{1},d_{2},\dotsc$ of dimensions corresponding to a
fixed $D$, there are infinitely many distinct infinite subsequences
$d_{k_{1}},d_{k_{2}},\dotsc$ with the property that $d_{k_{1}}\mid
d_{k_{2}}\mid\cdots$.

###### Proof.

Fix $D$. The defining recursion

$T_{0}\left({x}\right)=1,\,\,\,T_{1}\left({x}\right)=x,\,\,\,T_{n}\left({x}\right)=2xT_{n-1}\left({x}\right)-T_{n-2}\left({x}\right)$

for the Chebyshev polynomials yields the following recursion for the shifted
version:

$T^{*}_{0}\left({x}\right)=3,\,\,\,T^{*}_{\pm
1}\left({x}\right)=x,\,\,\,T^{*}_{n}\left({x}\right)=xT^{*}_{n-1}\left({x}\right)-xT^{*}_{n-2}\left({x}\right)+T^{*}_{n-3}\left({x}\right).$

If $x\in\mathbb{Z}$ then all terms are in $\mathbb{Z}$, so for the particular
case of a positive integer $d$,

(12) $T^{*}_{n}\left({d}\right)\equiv T^{*}_{n-3}\left({d}\right)\bmod d,$

and in fact for any positive integer $r$ we have $d_{nr}\equiv d_{(n-3)r}\bmod
d_{r}$. We now have three cases according to the congruence class of $n$
modulo 3:

  * •

C0: $n\equiv 0\ \bmod 3$: $T^{*}_{0}\left({d_{r}}\right)=3$, implying
$d_{nr}-3$ is a multiple of $d_{r}$.

  * •

C1: $n\equiv 1\ \bmod 3$: $T^{*}_{1}\left({d_{r}}\right)=d_{r}$, implying
$d_{nr}$ is a multiple of $d_{r}$.

  * •

C2: $n\equiv 2\ \bmod 3$: $T^{*}_{2}\left({d_{r}}\right)=d_{r}(d_{r}-2)$,
implying $d_{nr}$ is a multiple of $d_{r}$.

So if $(i_{j})_{j\geq 1}$ is any increasing sequence of integers coprime to
$3$ such that $i_{1}\mid i_{2}\mid i_{3}\mid\cdots$, it follows that
$d_{i_{1}}\mid d_{i_{2}}\mid d_{i_{3}}\mid\cdots$. There are infinitely many
such dimension towers because $d_{s}>d_{r}>3\ \forall\ s>r$, so by C0,
$n>m\implies d_{3^{m}i_{k}}\nmid d_{3^{n}i_{j}}$ for any $j,k$. ∎

######  Corollary 3.5.

If Conjecture 1.2 is true, then for every square-free $D\geq 2$, there exist
infinitely many infinite ray class field towers above each
${\mathbb{Q}({{\sqrt{D}}})}$ whose successive generators may be found by
constructing SICs in the $\mathbb{C}^{d_{k_{i}}}$. ∎

##  4\. Properties of the ray class fields

Fix $D\geq 2$ and an associated dimension $d\geq 4$. Recall that
$d^{\prime}=d$ if $d$ is odd or $2d$ if $d$ is even, and that
$\zeta_{d^{\prime}}$ is an arbitrary primitive $d^{\prime}$-th root of unity.
Write ${\mathfrak{d^{\prime}}}$ for the modulus
$d^{\prime}{\infty_{1}}{\infty_{2}}$ and let
$\\!\\!\\!\\!\mod^{\\!\\!\\!\times}\\!$ denote multiplicative congruence. Let
${\sigma_{D}}$ be the unique non-trivial element of
${\mathrm{Gal}({K/\mathbb{Q}})}$. Write $\mathfrak{R}=\mathfrak{R}_{1,2}$ for
the ray class field of $K$ modulo $d^{\prime}$ with ramification allowed at
both infinite primes; similarly we write $\mathfrak{R}_{0},\mathfrak{R}_{1}$
and $\mathfrak{R}_{2}$ when ramification is allowed respectively at no
infinite primes, at ${\infty_{1}}$ and at ${\infty_{2}}$.

######  Proposition 4.1.

  * (i)

$\mathfrak{R}$ contains the $d^{\prime}$-th roots of unity $\mu_{d^{\prime}}$,
and ${\zeta_{d^{\prime}}+\zeta_{d^{\prime}}^{-1}\in\mathfrak{R}_{0}}$.

  * (ii)

$\mathfrak{R}_{0}$ is a non-trivial extension of $K$. Hence so are
$\mathfrak{R}_{1},\mathfrak{R}_{2},\mathfrak{R}$.

  * (iii)

$\mathfrak{R}_{0}$ and $\mathfrak{R}$ are Galois over $\mathbb{Q}$.

  * (iv)

$\mathfrak{R}_{1}$ and $\mathfrak{R}_{2}$ have isomorphic Galois groups over
$K$.

  * (v)

$\mathfrak{R}$ is the compositum $\mathfrak{R}_{1}\mathfrak{R}_{2}$, and
$\mathfrak{R}_{0}=\mathfrak{R}_{1}\bigcap\mathfrak{R}_{2}$.

###### Proof.

$\mathfrak{R}$ is the fixed field of the Artin symbols
${\left((\alpha),\mathfrak{R}/K\right)}$ attached to the integral principal
ideals $(\alpha)$ where
$\alpha\equiv{1}\\!\\!\\!\mod^{\\!\\!\\!\times}{{\mathfrak{d^{\prime}}}}$. The
extension ${K(\mu_{d^{\prime}})/K}$ is unramified outside primes dividing
${\mathfrak{d^{\prime}}}$, so to show that
${\mu_{d^{\prime}}\subset\mathfrak{R}}$ it suffices to show that these Artin
symbols fix $\zeta_{d^{\prime}}$. This action translates [10, X §1] via the
restriction map from ${K(\mu_{d^{\prime}})/K}$ down to
${\mathbb{Q}(\mu_{d^{\prime}})/\mathbb{Q}}$, to raising $\zeta_{d^{\prime}}$
to the power ${\mathrm{N}_{K/\mathbb{Q}}\left(\alpha\right)}$. But
${\mathfrak{d^{\prime}}}$ is fixed by ${\sigma_{D}}$ and so (i) follows from:

$\alpha\equiv{1}\\!\\!\\!\mod^{\\!\\!\\!\times}{{\mathfrak{d^{\prime}}}}\implies\alpha^{\sigma_{D}}\equiv{1}\\!\\!\\!\mod^{\\!\\!\\!\times}{{\mathfrak{d^{\prime}}}}\implies\alpha\alpha^{\sigma_{D}}\equiv{1}\\!\\!\\!\mod^{\\!\\!\\!\times}{{\mathfrak{d^{\prime}}}}\implies\mathrm{N}_{K/\mathbb{Q}}\left(\alpha\right)\equiv
1\ (d^{\prime}).$

As a direct consequence,
${\zeta_{d^{\prime}}+\zeta_{d^{\prime}}^{-1}\in\mathfrak{R}_{0}}$. So to prove
(ii) we only need show that
${\zeta_{d^{\prime}}+\zeta_{d^{\prime}}^{-1}\not\in{\mathbb{Q}({{\sqrt{D}}})}}$,
which follows from the definitions of $D,d^{\prime}$ and basic cyclotomic
theory.

Now lift ${\sigma_{D}}$ to some
${{\hat{\sigma}}_{D}\in{\mathrm{Gal}({\overline{\mathbb{Q}}/\mathbb{Q}})}}$.
In terms of moduli, ${\hat{\sigma}}_{D}$ fixes $d^{\prime}$ and
${\mathfrak{d^{\prime}}}$ and interchanges $d^{\prime}{\infty_{1}}$ and
$d^{\prime}{\infty_{2}}$. The field
${{\hat{\sigma}}_{D}\left(\mathfrak{R}\right)}$ sits inside a normal closure
of ${\mathfrak{R}/\mathbb{Q}}$: it is an abelian extension of $K$ with
conductor ${\mathfrak{d^{\prime}}}$ and with Galois group
${{\hat{\sigma}}_{D}{\mathrm{Gal}({\mathfrak{R}/K})}{\hat{\sigma}}_{D}^{-1}}$
of order ${[\mathfrak{R}:K]}$. So
${{\hat{\sigma}}_{D}\left(\mathfrak{R}\right)=\mathfrak{R}}$ and
${\mathfrak{R}/\mathbb{Q}}$ is Galois. Similarly for $\mathfrak{R}_{0}$.
Finally (iv), (v) follow from a consideration of the moduli under the map
${\hat{\sigma}}_{D}$. ∎

Interestingly, $\mathfrak{R}_{0}$ is abelian over $\mathbb{Q}$ in dimensions
$d=4,5,7,8$. However this is never true of $\mathfrak{R}$. Moreover, we always
have
${\mathrm{Gal}({\mathfrak{R}/\mathfrak{R}_{0}})}\simeq\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$,
as we now show.

######  Proposition 4.2.

The groups ${\mathrm{Gal}({\mathfrak{R}/\mathfrak{R}_{1}})}$,
${\mathrm{Gal}({\mathfrak{R}_{1}/\mathfrak{R}_{0}})}$,
${\mathrm{Gal}({\mathfrak{R}/\mathfrak{R}_{2}})}$,
${\mathrm{Gal}({\mathfrak{R}_{2}/\mathfrak{R}_{0}})}$ all have order 2.

We shall prove this after pointing out some consequences for the structure of
the SIC fields. Write ${\gamma}_{j}$ for the unique non-trivial element of
${{\mathrm{Gal}({\mathfrak{R}_{j}/\mathfrak{R}_{0}})}}$ for $j=1,2$: that is,
${{\gamma}_{j}\in{\mathrm{Gal}({\mathfrak{R}_{j}/K})}}$ is complex conjugation
under every complex embedding of ${\mathfrak{R}_{j}}$. Let ${\Gamma_{12}}$ be
the subgroup of ${\mathrm{Gal}({\mathfrak{R}/K})}$ of order 2 generated by the
product ${\gamma}_{1}{\gamma}_{2}$, and recall the notation $M^{G}$ for the
elements of a $G$-module $M$ fixed by the action of $G$. From Proposition 4.1
it is clear that any lifting ${\hat{\sigma}}_{D}$ of ${\sigma_{D}}$ swaps
${\gamma}_{1}$ and ${\gamma}_{2}$.

######  Corollary 4.3.

_(i)_ $\mathfrak{R}$ is non-abelian over $\mathbb{Q}$.

_(ii)_ ${\mathfrak{R}_{1},\mathfrak{R}_{2}}$ are non-Galois over $\mathbb{Q}$,
having ${[\mathfrak{R}_{1}:K]}$ real places and
${\frac{1}{2}[\mathfrak{R}_{1}:K]}$ pairs of complex places.

_(iii)_
${\mathfrak{R}=\mathfrak{R}_{1}(\zeta_{k})=\mathfrak{R}_{2}(\zeta_{k})}$ for
any $k>2$ which is a divisor of $d^{\prime}$.

_(iv)_ ${\mathfrak{R}_{0}(\zeta_{k})=\mathfrak{R}^{\Gamma_{12}}}$ is a CM-
field. ${\mathfrak{R}_{1},\mathfrak{R}_{2}}$ and
${\mathfrak{R}_{0}(\zeta_{k})}$ are the 3 quadratic fields between
${\mathfrak{R}_{0}}$ and ${\mathfrak{R}}$.

###### Proof.

(of Corollary). If ${\mathfrak{R}/\mathbb{Q}}$ were abelian then the inner
automorphism ${\hat{\sigma}}_{D}$ would be trivial. But ${\mathfrak{R}}$ is
the compositum of ${\mathfrak{R}_{1}}$ and
${\mathfrak{R}_{2}={\mathfrak{R}_{1}}^{{\hat{\sigma}}_{D}}}$, so
${\mathfrak{R}=\mathfrak{R}_{1}=\mathfrak{R}_{2}}$, contradicting Proposition
4.2. This proves (i). For (ii): ${\mathfrak{R}_{1}}$ is Galois over $K$ and
${\mathfrak{R}_{1}/\mathfrak{R}_{0}}$ is non-trivial; so it has
${\frac{1}{2}[\mathfrak{R}_{1}:K]}$ pairs of complex places over
${\infty_{1}}$. The corresponding ${[\mathfrak{R}_{1}:K]}$ places over
${\infty_{2}}$ are clearly real. This argument is symmetric in
${\mathfrak{R}_{1},\mathfrak{R}_{2}}$ by Proposition 4.1. For (iii) just
combine the facts that ${\mathfrak{R}_{1}}$ is not totally complex,
${\mathfrak{R}/\mathfrak{R}_{1}}$ has degree 2, and
${K(\mu_{d^{\prime}})\subseteq\mathfrak{R}}$. Since
${\zeta_{d^{\prime}}+\zeta_{d^{\prime}}^{-1}\in\mathfrak{R}_{0}}$ this also
implies (iv) . ∎

###### Proof.

(of Proposition 4.2). Let $h_{K}$ denote the class number of $\mathbb{Z}_{K}$.
If ${\mathfrak{m}}$ is any modulus of $K$ we denote by
$U_{{\mathfrak{m}}}^{1}$ the subgroup of $U_{K}=\mathbb{Z}_{K}^{\times}$
consisting of units
$\equiv{1}\\!\\!\\!\mod^{\\!\\!\\!\times}{{\mathfrak{m}}}$. This has finite
index $[U_{K}:U_{{\mathfrak{m}}}^{1}]$ in $U_{K}$. Express
${\mathfrak{m}}={\mathfrak{m}}_{0}{\mathfrak{m}}_{\infty}$ as a product of its
finite and infinite parts respectively. We denote by $\Phi$ the generalized
Euler totient function, so $\Phi({\mathfrak{m}}_{0})$ is the order of the
multiplicative group of the ring $\mathbb{Z}_{K}/{\mathfrak{m}}_{0}$. By
analogy we write $\Phi({\mathfrak{m}}_{\infty})$ for the size of the signature
group $\\{\pm 1\\}^{r_{{\mathfrak{m}}}}$ where $r_{{\mathfrak{m}}}$ is the
number of real places included in ${\mathfrak{m}}_{\infty}$. The formula [10,
VI §1] for the order $h_{{\mathfrak{m}}}$ of the ray class group of $K$ of
modulus ${\mathfrak{m}}$ is:

(13)
$h_{{\mathfrak{m}}}=\frac{h_{K}\Phi({\mathfrak{m}}_{0})\Phi({\mathfrak{m}}_{\infty})}{[U_{K}:U_{{\mathfrak{m}}}^{1}]}\
.$

Let $u_{f}$ and $u_{D}$ be defined as in section 3. Since $K$ is a real
quadratic field,

$U_{{\mathfrak{d^{\prime}}}}^{1}\subseteq
U_{d^{\prime}{\infty_{1}}}^{1}\subseteq
U_{d^{\prime}}^{1}\subseteq\mathbb{Z}_{K}^{\times}=u_{f}^{\mathbb{Z}}\times\\{\pm
1\\}$

with each inclusion being of finite index; and similarly with
$U_{d^{\prime}{\infty_{2}}}^{1}$ replacing $U_{d^{\prime}{\infty_{1}}}^{1}$.
So for our purposes it suffices to show that
$U_{{\mathfrak{d^{\prime}}}}^{1}=U_{d^{\prime}{\infty_{1}}}^{1}=U_{d^{\prime}{\infty_{2}}}^{1}=U_{d^{\prime}}^{1},$
or equivalently:

(14) $U_{{\mathfrak{d^{\prime}}}}^{1}=U_{d^{\prime}}^{1},$

for then the signature factor $\Phi({\mathfrak{m}}_{\infty})$ in (13) will
tell the whole story of the growth in the size of the ray class groups as we
successively add real places into the modulus.

Now (14) says that every unit in $U_{d^{\prime}}^{1}$ is totally positive.
Since $d^{\prime}\in\mathbb{Z}$, $U_{d^{\prime}}^{1}$ is closed under the
action of ${\sigma_{D}}$. Hence
$\mathrm{N}_{K/\mathbb{Q}}\left(U_{d^{\prime}}^{1}\right)\subseteq
U_{d^{\prime}}^{1}$. But $-1\notin U_{d^{\prime}}^{1}$ because $d^{\prime}>2$.
So $U_{d^{\prime}}^{1}$ is a rank 1 torsion-free abelian group, all of whose
elements have absolute norm $+1$. In particular their signature must either be
$(+,+)$ or $(-,-)$, so we are reduced to showing that  _no unit in
$U_{d^{\prime}}^{1}$ can be totally negative_.

By construction $u_{D}$ is the first totally positive power of $u_{f}$, so any
totally negative unit in $U_{d^{\prime}}^{1}$ must be of the form $-u_{D}^{r}$
for some $r\in\mathbb{Z}$. Suppose that such a unit exists and that $r\geq 1$
is minimal. Let $j$ be the positive integer such that $d=d_{j}$, in the sense
of (9). Since both $u_{D}^{r}$ and $u_{D}^{-r}\equiv-1\bmod d^{\prime}_{j}$ it
follows that

(15) $d_{r}=u^{r}_{D}+u^{-r}_{D}+1\equiv-1\bmod d^{\prime}_{j}.$

There are two cases. If $3\nmid j$, then the congruences C1, C2 imply
$d_{1}\mid d^{\prime}_{j}$ and consequently $u_{D}^{r}\equiv-1\bmod d_{1}$.
Moreover (15) shows $d_{r}\equiv-1\bmod d_{1}$, so a second application of C0,
C1, C2 forces $d_{1}=4$ and $3\mid r$. We therefore have $D=5$ and
$u_{D}=u_{f}^{2}=\frac{3+\sqrt{5}}{2}$, which has order 3 inside
$\mathbb{Z}_{K}/(4)$. Since $3|r$ this means $u_{D}^{r}\equiv 1\bmod d_{1}$, a
contradiction.

On the other hand, if $3\mid j$ then write $j=3^{t}q$ for $t,q\geq 1$ and
$3\nmid q$. It follows from C1, C2 that $d_{3^{t}}\mid d^{\prime}_{j}$,
implying $u^{r}_{D}\equiv-1\bmod d_{3^{t}}$. So if such a totally negative
unit exists for $j=3^{t}q$ then it must also exist for $j=3^{t}$. Redefine $r$
to be minimal for  _this_ relation: then the order of $u_{D}$ modulo
$d_{3^{t}}$ is $2r$, which is even. But
$d_{3c}=T^{*}_{3}(d_{c})=d^{2}_{c}(d_{c}-3)+3$ for any integer $c$ and so
$d_{3^{t}}$ is odd, which by Lemma 4.4 below implies that the order of $u_{D}$
modulo $d_{3^{t}}$ is also odd, a contradiction. ∎

Finally we prove the technical lemma used in the last proof.

######  Lemma 4.4.

The order of $u_{D}$ modulo $d_{r}^{\prime}$ is
$3r\frac{d_{r}^{\prime}}{d_{r}}$ (i.e. it is $3r$ if $d_{r}$ is odd and $6r$
if $d_{r}$ is even).

###### Proof.

Consider the absolute minimal polynomial for $u_{D}^{r}$:

(16) $X^{2}-(d_{r}-1)X+1=(X-u_{D}^{r})(X-u_{D}^{-r}).$

Multiplying by $(X-1)$ shows that $u_{D}^{3r}\equiv 1\bmod d_{r}$ for all $r$.
When $d_{r}$ is even, we know from C2 that $d_{2r}$ is divisible by
$d_{r}^{\prime}=2d_{r}$ and thus:

$(X-u_{D}^{2r})(X-u_{D}^{-2r})=X^{2}-(d_{2r}-1)X+1\equiv X^{2}+X+1\bmod
2d_{r}.$

Once again, multiplying by $(X-1)$ yields $u_{D}^{6r}\equiv 1\bmod
d_{r}^{\prime}$, for any $r$.

It remains to show that the power $3r\frac{d_{r}^{\prime}}{d_{r}}$ is minimal
in each case. We first show that $3r$ is minimal modulo $d_{r}$ for all $r$.
Let $q\in\mathbb{N}$ be minimal such that $u_{D}^{q}\equiv 1\bmod d_{r}$. It
is easy to see that $q\mid 3r$. Now the minimal polynomial of $u_{D}^{q}$ is
$X^{2}-(d_{q}-1)X+1$, so this must vanish modulo $d_{r}$ at $X=1$. In other
words $d_{r}$ divides into $d_{q}-3$, proving that $r<q$ since $(d_{j})_{j\geq
1}$ is a strictly increasing sequence. So $3\mid q$, since otherwise $q\mid
3r\implies q\mid r\implies q\leq r$. Writing $q=3q_{0}$ we see that $q_{0}\mid
r$ and so $q_{0}\leq r<3q_{0}=q$. This forces $q_{0}=r$ or $q_{0}=r/2$ and we
now show in fact $q_{0}=r$. If $r$ is odd we are done. If $r$ is even it
follows from equation (16) that
${u_{D}^{-\frac{r}{2}}(u_{D}^{2r}+u_{D}^{r}+1)\equiv 0\bmod d_{r}}$, which
translates via equation (9) to ${u_{D}^{\frac{3r}{2}}\equiv
1-d_{\frac{r}{2}}\bmod d_{r}}$. But clearly $d_{\frac{r}{2}}\not\equiv 0\bmod
d_{r}$, so again $q_{0}=r$.

Finally assume that $d_{r}$ is even and let $q^{\prime}\in\mathbb{N}$ be the
minimal integer such that $u_{D}^{q^{\prime}}\equiv 1\bmod
d_{r}^{\prime}=2d_{r}$. Again we must have $q^{\prime}\mid 6r$; moreover
$q^{\prime}<3r$ would (by reduction modulo $d_{r}$) contradict the proof above
for $d_{r}$. So either $q^{\prime}=3r$ or $q^{\prime}=6r$. Using (9), (16) as
above $u_{D}^{3r}\equiv 1-d_{r}\bmod d_{2r}$, $d_{r}\not\equiv 0\bmod 2d_{r}$
and the proof is complete again observing that $d_{r}$ even $\implies
2d_{r}\mid d_{2r}$. ∎

##  5\. Linking unitary geometry and arithmetic via subgroups of
${\mathrm{GL}_{2}}(\mathbb{Z}/d^{\prime}\mathbb{Z})$

Let $v$ be a fiducial vector taken from one of the orbits in the list (7) that
is _centred_ as in §2. This section and the next are devoted to a study of the
function
$f_{v}\colon(\mathbb{Z}/d^{\prime}\mathbb{Z})^{2}\rightarrow\mathfrak{R}$
defined by

$f_{v}(j)=\frac{\left(v,\Delta_{j}v\right)}{(v,v)}=\operatorname{Tr}\Pi_{v}\Delta_{j},$

where the $\Delta_{j}$ are defined as in (6). Unlike the components of the
fiducial vector the values of the function $f_{v}$ are independent of both the
scaling and the basis. It follows from Proposition 1.1 and (6) that the values
$f_{v}(j)$ for $j\nequiv\binom{0}{0}\bmod d$ are all elements of
$\mathfrak{R}$ with modulus $\tfrac{1}{\sqrt{\\!d\\!+\\!1}}$. In fact we have
found empirically that it is possible to choose $v$ so that the field
$\mathbb{Q}(f_{v})$ obtained by adjoining to $\mathbb{Q}$ all of the values of
$f_{v}$ is equal to $\mathfrak{R}_{1}$. We say that a centred fiducial vector
for which that is true is _strongly centred_. Such fiducials may be
characterized as follows:

  * •

If $3\nmid d$ every centred fiducial is strongly centred.

  * •

If $3\mid d$ let $n=d/3$ and $B=\Delta_{n,2n}$. Then for each centred fiducial
$v$ exactly one of the three vectors $v,Bv,B^{2}v$ is strongly centred.

From now on it will be assumed without comment that fiducials are strongly
centred.

Let $L$ be the smallest subfield of $\mathbb{Q}(\mathfrak{R}_{1})$ fixing
$f_{v}((\mathbb{Z}/d^{\prime}\mathbb{Z})^{2})$ setwise. In all of our
examples, $L\supset K$ with index $[L:K]$ equal to the number of
$\mathrm{Sp}_{2}(\mathbb{Z}/d^{\prime}\mathbb{Z})$-orbits of fiducials [2,
§7]; see the remark at the end of this section. In addition to the Galois
action, ${\mathrm{GL}_{2}}(\mathbb{Z}/d^{\prime}\mathbb{Z})$ acts naturally on
functions
$f\colon(\mathbb{Z}/d^{\prime}\mathbb{Z})^{2}\rightarrow\mathfrak{R}_{1}$ by
translation via $(gf)(j)=f(gj)$. What follows is a tantalizing observation
linking these two distinct actions, strengthening a statement in §7 of [2].
Define

$\boldsymbol{S}=\\{(\det F)F\colon F\in\boldsymbol{S}_{0}\\},$

where
$\boldsymbol{S}_{0}\subset\mathrm{ESp}_{2}(\mathbb{Z}/d^{\prime}\mathbb{Z})$
is the stabilizer defined in Sec. 2. As shown in [2], $\boldsymbol{S}$ is the
stabilizer of $f_{v}$, i.e.
$\boldsymbol{S}=\\{g\in{\mathrm{ESp}_{2}}(\mathbb{Z}/d^{\prime}\mathbb{Z}):gf_{v}=f_{v}\\}$.

######  Proposition 5.1.

For all orbits listed in  _( 7)_ $L$ is the Hilbert class field of $K$, and

(17) $\mathbf{M}\\!\left(\boldsymbol{S}\right)/\,\boldsymbol{S}\ \ \simeq\
{\mathrm{Gal}({\mathfrak{R}_{1}/L})},$

where $\mathbf{M}\\!\left(\boldsymbol{S}\right)$ is a maximal abelian subgroup
of ${\mathrm{GL}_{2}}(\mathbb{Z}/d^{\prime}\mathbb{Z})$ containing
$\boldsymbol{S}$. For the orbits of $F_{z}$-type in  _( 7)_
$\mathbf{M}\\!\left(\boldsymbol{S}\right)$ is in fact
$\mathbf{C}\\!\left(\boldsymbol{S}\right)$, the centralizer of
$\boldsymbol{S}$ inside ${\mathrm{GL}_{2}}(\mathbb{Z}/d^{\prime}\mathbb{Z})$;
for the orbits of $F_{a}$-type $\mathbf{M}\\!\left(\boldsymbol{S}\right)$ is
conjugate to one of three possible subgroups that are characterized in Ref.
[3].

We are grateful to John Coates for the following observation, which we hope to
address in a forthcoming paper:

######  Remark 5.2.

The appearance of the Hilbert class field in (17) cannot be a coincidence. The
LHS is ostensibly a geometrically defined abelian subgroup of
${\mathrm{GL}_{2}}(\mathbb{Z}/d^{\prime}\mathbb{Z})$. On the other hand the
RHS contains information about a subgroup of a ray class group, and therefore
also potentially about the ideal class group $\mathcal{C}_{K}$ of $K$. But
this cannot be true in general, since the structure of $\mathcal{C}_{K}$ is
very erratic as $D$ varies. We can therefore be confident that $L$ will
typically contain the Hilbert class field.

Our empirical observations suggest that $L$ may be identical with the Hilbert
class field in all dimensions, not just those in (2). It follows that the
${\mathrm{Gal}({L/K})}$-set of distinct
$\mathrm{Sp}_{2}(\mathbb{Z}/d^{\prime}\mathbb{Z})$-orbits may actually be a
$\mathcal{C}_{K}$-set. Preliminary numerical results communicated to us by
Andrew Scott for certain higher dimensions, wherein $\mathcal{C}_{K}$ is much
larger than for the dimensions in (2), provide additional evidence for this
speculation.

##  6\. Canonical units associated to the ray class fields

We now proceed to link invariants of the SICs with canonical units associated
to the ray class fields. The condition of equiangularity means that the inner
products in $\mathcal{J}$ for which $j\nequiv\binom{0}{0}\bmod d$ are all of
the form $\frac{\mathrm{e}^{i\theta(j)}}{\scriptstyle{\sqrt{d\\!+\\!1}}}$ with
the $\theta(j)\in\mathbb{R}$. Let $\mathcal{F}$ denote
$\mathfrak{R}_{1}(\small{\hbox{$\sqrt{d\\!+\\!1}$}})$, with ring of integers
$\mathbb{Z}_{\mathcal{F}}$. Empirically, the  _normalized inner products_
$\mathrm{e}^{i\theta(j)}\in U(1)$ display the following remarkable properties:

######  Proposition 6.1.

Fix any $d$ in  _( 2)_ and consider a SIC on an
$\mathrm{ESp}(\mathbb{Z}/d^{\prime}\mathbb{Z})$-orbit.

  * (i)

Every one of the normalized inner products lies in the unit group
$\mathbb{Z}_{\mathcal{F}}^{\times}$.

Let $u_{d}$ be a normalized inner product of _maximal_ degree $n_{d}$ over
$K(\small{\hbox{$\sqrt{d\\!+\\!1}$}})$.

  * (ii)

$u_{d}$ generates222Except in orbit 12b where we also need to adjoin
$\sqrt{3}$. $\mathcal{F}$ over $\mathbb{Q}(\small{\hbox{$\sqrt{d\\!+\\!1}$}})$
and therefore over $K(\small{\hbox{$\sqrt{d\\!+\\!1}$}})$, and all of its
conjugates appear as normalized inner products for the same SIC. In
particular, $[\mathcal{F}:K(\small{\hbox{$\sqrt{d\\!+\\!1}$}})]=n_{d}$.

  * (iii)

Given any complex embedding of $\mathcal{F}$, the
${{\mathrm{Gal}({\mathcal{F}/K})}}$-conjugates of $u_{d}$ all lie on $U(1)$.

  * (iv)

The minimal polynomial $f_{d}(x)$ of $u_{d}$ is a  _reciprocal_ polynomial:
that is, $x^{n_{d}}f_{d}(\frac{1}{x})=f_{d}(x)$. ∎

######  Remark 6.2.

We have {(i),(ii)}$\implies$(iii)$\implies$(iv), because the abelian structure
imposes a unique complex conjugation operator
$\sigma_{c}\in{{\mathrm{Gal}({\mathcal{F}/K})}}$ upon every complex embedding.
So for any $g\in{{\mathrm{Gal}({\mathcal{F}/K})}}$,

$\sigma_{c}(g(u_{d}))=g(\sigma_{c}(u_{d}))=g(u_{d}^{-1})=(g(u_{d}))^{-1}.$

Since ${{\mathrm{Gal}({\mathcal{F}/K})}}$ is transitive on the roots, $f_{d}$
is a product of reciprocal polynomials of the form $x^{2}-(u+u^{-1})x+1$,
hence is itself reciprocal. For more on the Galois theory of fields generated
by reciprocal polynomials see [9].

The units may be calculated (with some considerable work) from the set of
inner products using the fiducial vectors in [2] and (6). In Appendix A we
give a worked example for the case $d=19$.

For reference we now prove some criteria for when
$\small{\hbox{$\sqrt{d\\!+\\!1}$}}\in\mathfrak{R}_{1}$, _i.e._ when
$\mathcal{F}=\mathfrak{R}_{1}$. Observe at the outset that
$\small{\hbox{$\sqrt{d\\!+\\!1}$}}\in K$ if and only if either $d+1$ or $d-3$
is a perfect square.

######  Proposition 6.3.

Let $d\geq 4$ and assume that neither $d+1$ nor $d-3$ is a perfect square.
Then

  * (i)

If $d$ is even then $\small{\hbox{$\sqrt{d\\!+\\!1}$}}\in\mathfrak{R}_{0}$.

  * (ii)

If $d\equiv 1\ (4)$ then
$\small{\hbox{$\sqrt{d\\!+\\!1}$}}\not\in\mathfrak{R}_{0}$.

  * (iii)

If $d\equiv 3\ (4)$ then
$\small{\hbox{$\sqrt{d\\!+\\!1}$}}\in\mathfrak{R}_{0}$ iff the extension
$K(\small{\hbox{$\sqrt{d\\!+\\!1}$}})/K$ is everywhere unramified. In
particular $\small{\hbox{$\sqrt{d\\!+\\!1}$}}\not\in\mathfrak{R}_{0}$ if the
class number of $K$ is 1.

###### Proof.

By assumption ${K(\small{\hbox{$\sqrt{d\\!+\\!1}$}})}$ is a totally real
biquadratic extension of $\mathbb{Q}$, with exactly three distinct proper
quadratic subextensions
${K={\mathbb{Q}({{\sqrt{D}}})},\mathbb{Q}(\small{\hbox{$\sqrt{d\\!+\\!1}$}})}$
and ${\mathbb{Q}(\small{\hbox{$\sqrt{d\\!-\\!3}$}})}$. If $p$ is an odd prime
divisor of the square-free part of $d+1$ then $p$ will ramify in
${K/\mathbb{Q}}$ but not in ${K(\small{\hbox{$\sqrt{d\\!+\\!1}$}})/K}$. So
${K(\small{\hbox{$\sqrt{d\\!+\\!1}$}})/K}$ is unramified at every place not
lying over the prime 2.

When $d$ is even, ${d^{\prime}=2d}$ is divisible by at least $2^{2}$ whereas
${d+1}$ is odd, so the 2-primary part of the conductor of
${K(\small{\hbox{$\sqrt{d\\!+\\!1}$}})/K}$ divides that of $d^{\prime}$ [5,
§17F]. So $\small{\hbox{$\sqrt{d\\!+\\!1}$}}\in\mathfrak{R}_{0}$, proving (i).

When $d$ is odd, so $d^{\prime}=d$, the only cases where $\sqrt{d\\!+\\!1}$
can possibly lie in the ray class field $\mathfrak{R}_{0}$ are those where 2
does not ramify in the extension ${K(\small{\hbox{$\sqrt{d\\!+\\!1}$}})/K}$.
This proves assertion (iii) for  _any_ odd $d$, with the consequence that the
Hilbert class field of $K$ must be non-trivial if
$\small{\hbox{$\sqrt{d\\!+\\!1}$}}\in\mathfrak{R}_{0}$.

To prove (ii), if $d\equiv 1\ \bmod 4$ then we claim that the unique prime of
$K$ above 2 always ramifies again in
${K(\small{\hbox{$\sqrt{d\\!+\\!1}$}})/K}$: meaning that $\sqrt{d\\!+\\!1}$
can never lie in $\mathfrak{R}_{0}$. First of all,
${\mathbb{Q}(\small{\hbox{$\sqrt{d\\!+\\!1}$}})/\mathbb{Q}}$ and
${\mathbb{Q}(\small{\hbox{$\sqrt{d\\!-\\!3}$}})/\mathbb{Q}}$ are each ramified
over 2, since both ${d+1}$ and ${d-3\equiv 2\ \bmod 4}$. Also
${(d+1)(d-3)\equiv 12\ \bmod 16}$ and since ${2^{2}\mid(d+1)(d-3)}$ it follows
that ${D\equiv 3\ \bmod 4}$. So 2 is ramified in all three quadratic
subextensions ${{\mathbb{Q}({{\sqrt{D}}})},\
\mathbb{Q}(\small{\hbox{$\sqrt{d\\!+\\!1}$}}),\
\mathbb{Q}(\small{\hbox{$\sqrt{d\\!-\\!3}$}})}$. Therefore [5, §17F] the
inertia group at 2 must be the whole Galois group
${\mathrm{Gal}({K({\scriptstyle{\sqrt{d\\!+\\!1}}})/\mathbb{Q}})}$, proving
the claim. ∎

######  Corollary 6.4.

For the dimensions in list (2) it follows that
$\small{\hbox{$\sqrt{d\\!+\\!1}$}}\in\mathfrak{R}_{0}$ except for
$d=5,9,11,13,17,21$. _(_ The case $d=11$ follows from the fact that the class
number of $\mathbb{Q}(\sqrt{6})$ is 1 _)_. ∎

The final result of this paper is striking but the size of the necessary
calculations has prevented us from testing dimensions $16,17,18,20,21,30$ and
$39$ in list (2). Let ${\mathbb{U}}$ denote the group of units of the ring of
integers of the field ${\mathcal{F}}$ under any one of its complex embeddings,
and let ${\mathbb{V}}$ be the subgroup generated by the normalized inner
products. By construction ${\mathbb{V}}$ is contained in the unit circle
subgroup ${\mathbb{U}\cap\mathbb{U}(1)}$ of ${\mathbb{U}}$. We would like to
compare the relative ranks of ${\mathbb{V}}$ and
${\mathbb{U}\cap\mathbb{U}(1)}$. Let ${N_{d}=[\,\mathcal{F}:K\,]}$, so
${N_{d}=n_{d}}$ or ${2n_{d}}$ according to whether
${\small{\hbox{$\sqrt{d\\!+\\!1}$}}\in\mathfrak{R}_{0}}$ or not. Part (ii) of
the Corollary to Proposition 4.2 implies that ${\mathcal{F}}$ has ${N_{d}}$
real places and ${\tfrac{N_{d}}{2}}$ pairs of complex places; so by
Dirichlet’s unit theorem ${\mathbb{U}}$ is an abelian group of rank
${\tfrac{3N_{d}}{2}-1}$. By the same Proposition
${\mathfrak{R}_{0}(\small{\hbox{$\sqrt{d\\!+\\!1}$}})}$ is the maximal real
subfield of ${\mathcal{F}}$ and has index 2: so it is a consequence of the
lemma in §5 of [11] that the unit circle subgroup
${\mathbb{U}\cap\mathbb{U}(1)}$ of ${\mathbb{U}}$ has ${\mathbb{Z}}$-rank
${\tfrac{N_{d}}{2}}$.

######  Proposition 6.5.

$\operatorname{rank}(\mathbb{V})=\begin{cases}\operatorname{rank}(\mathbb{U}\cap\mathbb{U}(1)),&d=7,15,19,35;\\\
\frac{1}{2}\operatorname{rank}(\mathbb{U}\cap\mathbb{U}(1)),&d=4,5,6,8,9,10,11,12,13,14,24,28,48.\end{cases}$

In dimensions $4$–$8$, $12$ and $19$, where we have been able to
calculate333The calculations were performed using the function
IndependentUnits in _Magma_ , which assumes the validity of the generalized
Riemann Hypothesis. $\mathbb{U}$, $\mathbb{V}$ is a direct summand of
$\mathbb{U}\cap\mathbb{U}(1)$. In particular,
$\mathbb{U}\cap\mathbb{U}(1)=\mathbb{V}$ in dimensions $7$ and $19$. ∎

######  Proposition 6.6.

In dimensions $4$–$15$, $19$, $24$, $28$, $35$ and $48$, the group
$\mathbb{V}$ is generated by the ${{\mathrm{Gal}({\mathcal{F}/K})}}$-orbit of
$u_{d}$. ∎

##  7\. Verification of Proposition 1.1

We give a very cursory outline of the steps necessary in verifying the claim
for each dimension in Proposition 1.1. Let $d$ be one of the dimensions in (2)
and let ${D,K={\mathbb{Q}({{\sqrt{D}}})}}$ be as above. For each orbit in the
list (7), choose a fiducial vector $v$. Let ${E}$ be the corresponding SIC and
let ${S(E)}$ be the set of all matrix elements of the projectors in ${E}$ in
the distinguished basis. As in the introduction, write ${K(E)}$ for
${K\bigl{(}S(E)\bigr{)}}$. The degrees
${\left[K(E):K\right],\left[\mathfrak{R}:K\right]}$ can be calculated in MAGMA
[4]. We verify that they are equal and that the defining polynomials for
${\mathfrak{R}}$ factorise completely over ${K(E)}$.

We illustrate these calculations for $d=19$ in Appendix A.

### Acknowledgements

It is a pleasure to thank John Coates and Andrew Scott for their continuing
support and guidance. Also thanks to Steve Donnelly for his help with the
MAGMA code and with interpreting the number-theoretic results; to James McKee
and Chris Smyth for explaining various aspects of the theory of reciprocal
units; and to Brian Conrad for comments on an earlier draft.

This research was supported by the Australian Research Council via EQuS
project number CE11001013, and SF acknowledges support from an Australian
Research Council Future Fellowship FT130101744 and JY from National Science
Foundation Grant No. 116143.

##  Appendix A Example of a unit generator for
$\mathfrak{R}_{1}:\mathbb{Q}(\sqrt{D})$ in dimension 19

We illustrate the foregoing for the case $d=19$. So $D=5$ and
$\small{\hbox{$\sqrt{d\\!+\\!1}$}}=2\sqrt{5}\in K=\mathbb{Q}(\sqrt{5})$. There
are $19\\!\times\\!\\!18$ values of inner products which upon normalisation by
$\sqrt{d\\!+\\!1}$ yield a set of 21 units falling into three Galois orbits,
as expressed by their minimal polynomials:

$f_{1}(y)=y-1$, $f_{2}(y)=y^{2}+\frac{1}{2}(\sqrt{5}-5)y+1,$

$f_{3}(y)=y^{18}+\frac{1}{2}(5\sqrt{5}-5)y^{17}+(5\sqrt{5}-6)y^{16}+(-57\sqrt{5}+134)y^{15}+(41\sqrt{5}-83)y^{14}\\\
+\frac{1}{2}(1031\sqrt{5}-2285)y^{13}+(-445\sqrt{5}+1004)y^{12}+(-2130\sqrt{5}+4769)y^{11}\\\
+\frac{1}{2}(1757\sqrt{5}-3917)y^{10}+(4297\sqrt{5}-9602)y^{9}+\frac{1}{2}(1757\sqrt{5}-3917)y^{8}\\\
+(-2130\sqrt{5}+4769)y^{7}+(-445\sqrt{5}+1004)y^{6}+\frac{1}{2}(1031\sqrt{5}-2285)y^{5}\\\
+(41\sqrt{5}-83)y^{4}+(-57\sqrt{5}+134)y^{3}+(5\sqrt{5}-6)y^{2}+\frac{1}{2}(5\sqrt{5}-5)y+1.$

The polynomials $f_{2},f_{3}$ illustrate the  _reciprocal_ property referred
to above. All three polynomials split inside the field
$\mathbb{Z}_{K}[y]/(f_{3}(y))$. Separately, the defining polynomials for
$\mathfrak{R}_{1}=\mathcal{F}$ over $\mathbb{Q}(\sqrt{5})$ are
$y^{2}+2\sqrt{5}+1$ and

$y^{9}-9747y^{7}+136458y^{6}+25001055y^{5}-320013504y^{4}-24511034322y^{3}+97474113234y^{2}+9503726040315y+66721030508673.$

These generate abelian extensions of degree 2 and 9 respectively and so
together generate a Galois extension of degree 18, as expected. So it remains
to check that they split over the field $\mathbb{Z}_{K}[y]/(f_{3}(y))$, which
is straightforward using a program like MAGMA [4].

Write $t=\cos\frac{\pi}{19}$. An example of one of the roots of $f_{3}$ is

$\frac{i}{19}\
\left[(-192\sqrt{5}+704)t^{7}+(48\sqrt{5}-176)t^{6}+(312\sqrt{5}-1144)t^{5}+(-28\sqrt{5}+204)t^{4}\right.\\\
+\left.(-150\sqrt{5}+550)t^{3}+(-14\sqrt{5}-50)t^{2}+(21\sqrt{5}-77)t+(4\sqrt{5}-2)\right]\sqrt{2\sqrt{5}+1}\\\
+(16\sqrt{5}-16)t^{6}+(-8\sqrt{5}+8)t^{5}+(-20\sqrt{5}+20)t^{4}+(10\sqrt{5}-10)t^{3}+(6\sqrt{5}-6)t^{2}+(-3\sqrt{5}+3)t+\frac{1}{2}(-\sqrt{5}+1).$

This unit generates $\mathfrak{R}_{1}\big{/}K$. Together with its
${\mathrm{Gal}({\mathfrak{R}_{1}/K})}$-conjugates it generates a subgroup of
the unit group $\mathbb{U}$ of $\mathbb{Z}_{\mathcal{F}}^{\times}$ of rank
$\frac{N_{19}}{2}=9$, which is also the rank of the subgroup
$\mathbb{U}\cap\mathbb{U}(1)$ consisting of all units of
$\mathfrak{R}_{1}=\mathcal{F}$ of modulus 1 [11, §5].

## References

  * [1] D. M. Appleby, _Symmetric Informationally Complete Positive Operator Valued Measures and the Extended Clifford Group_ , J. Math. Phys. 46, 052107 (2005).
  * [2] D. M. Appleby, H. Yadsan-Appleby and G. Zauner, _Galois Automorphisms of a Symmetric Measurement_ , Quantum Information & Computation 13, 672–720 (2013).
  * [3] Marcus Appleby, Steven T. Flammia, Tuan-Yow Chien and Shayne Waldron _Systematically Constructing Exact SIC-POVMs from Numerics_ , in preparation.
  * [4] Wieb Bosma, John Cannon, and Catherine Playoust, _The Magma algebra system. I. The user language_ , J. Symbolic Comput. 24, 235–265 (1997).
  * [5] Harvey Cohn, _A Classical Invitation to Algebraic Numbers and Class Fields_ , Universitext, Springer-Verlag NY, second printing (1988).
  * [6] P. Delsarte, J. M. Goethals, and J. J. Seidel, _Bounds for systems of lines and Jacobi polynomials_ , Philips Res. Rep. 30, 91 (1975).
  * [7] Stuart G. Hoggar, _64 lines from a quaternionic polytope_ , Geom. Dedicata 69 (3), 287–289 (1998).
  * [8] Lane P. Hughston and Simon M. Salamon, _Surveying points in the complex projective plane_ , Advances in Mathematics 286, 1017 (2016).
  * [9] Franck Lalande, _Corps de nombres engendrés par un nombre de Salem_ , Acta Arithmetica LXXXVIII.2, 191–200 (1999).
  * [10] Serge Lang, _Algebraic Number Theory_ , GTM 110, Springer-Verlag New York (1986).
  * [11] Noboru Nakahata, _On Units of Galois Extensions over $\mathbb{Q}$_, Proceedings of the Faculty of Science of Tokai University 15, 23–27 (1980).
  * [12] Amritanshu Prasad, _On character values and decomposition of the Weil representation associated to a finite abelian group_ , The Journal of Analysis, volume 17, 73–85, (2009).
  * [13] J. M. Renes, R. Blume-Kohout, A. J. Scott and C. M. Caves, _Symmetric Informationally Complete Quantum Measurements_ , J. Math. Phys. 45, 2171–2180 (2004).
  * [14] A. J. Scott and M. Grassl, _SIC-POVMs: A New Computer Study_ , J. Math. Phys. 51, 042203 (2010).
  * [15] Andrew Scott, _private communication_ , (2015).
  * [16] Goro Shimura and Yutaka Taniyama, _Complex multiplication of abelian varieties and its applications to number theory_ , Publications of the Mathematical Society of Japan, 6 The Mathematical Society of Japan, Tokyo (1961).
  * [17] M. H. Stone, _Linear transformations in Hilbert space. III. Operational methods and group theory_ , Proc. Natl. Acad. Sci. USA, 16 172–175 (1930).
  * [18] J. von Neumann, _Die Eindeutigkeit der Schrödingerschen Operatoren_ , Math. Ann., 104 570–578 (1931).
  * [19] A. Weil, _Sur certains groupes d’opérateurs unitaires_ , Acta Math., 111, 143–211 (1964).
  * [20] Eugene Wigner, _Group theory, and its Application to the Quantum Mechanics of Atomic Spectra_ , Academic Press, (1959).
  * [21] G. Zauner, _Quantendesigns. Grundzüge einer nichtkommutativen Designtheorie_ , PhD thesis, University of Vienna (1999). English translation: _Quantum Designs: Foundations of a Non-Commutative Design Theory_ , Int. J. Quantum Inf. 9, 445–507 (2011).
  * [22] Gerhard Zauner personal website, http://www.gerhardzauner.at/sicfiducials.html.
  * [23] Huangjun Zhu, _SIC POVMs and Clifford groups in prime dimensions_ , J. Phys. A: Math. Theor. 43 (2010).