# Frobenius distributions of low dimensional abelian varieties over finite
fields

Santiago Arango-Piñeros  Department of Mathematics, Emory University, Atlanta,
GA 30322, USA<EMAIL_ADDRESS>https://sarangop1728.github.io/ ,
Deewang Bhamidipati  Mathematics Department, University of California, Santa
Cruz, CA 95064, USA<EMAIL_ADDRESS>https://bdeewang.com/ and  Soumya
Sankar  542 Math Tower, 231 West 18th Avenue, Columbus, OH 43210-1174, USA
<EMAIL_ADDRESS>https://sites.google.com/site/soumya3sankar/

###### Abstract.

Given a $g$-dimensional abelian variety $A$ over a finite field
$\mathbf{F}_{q}$, the Weil conjectures imply that the normalized Frobenius
eigenvalues generate a multiplicative group of rank at most $g$. The
Pontryagin dual of this group is a compact abelian Lie group that controls the
distribution of high powers of the Frobenius endomorphism. This group, which
we call the Serre–Frobenius group, encodes the possible multiplicative
relations between the Frobenius eigenvalues. In this article, we classify all
possible Serre–Frobenius groups that occur for $g\leq 3$. We also give a
partial classification for simple ordinary abelian varieties of prime
dimension $g>3$.

###### Key words and phrases:

Abelian varieties over finite fields, Frobenius traces, Equidistribution

###### 2020 Mathematics Subject Classification:

11G10, 11G25, 11M38, 14K02, 14K15

##  1\. Introduction

Let $E$ be an elliptic curve over a finite field $\mathbf{F}_{q}$ of
characteristic $p>0$. The zeros $\alpha_{1},\overline{\alpha}_{1}$ of the
characteristic polynomial of Frobenius acting on the Tate module of $E$ are
complex numbers of absolute value $\sqrt{q}$. Consider
$u_{1}\colonequals\alpha_{1}/\sqrt{q}$ and $\overline{u}_{1}$ the normalized
zeros in the unit circle $\mathsf{U}(1)$. The curve $E$ is _ordinary_ if and
only if $u_{1}$ is not a root of unity, and in this case, the sequence
$(u_{1}^{r})_{r=1}^{\infty}$ is equidistributed in $\mathsf{U}(1)$. Further,
the normalized Frobenius traces $x_{r}\colonequals
u_{1}^{r}+\overline{u}_{1}^{r}$ are equidistributed on the interval $[-2,2]$
with respect to the pushforward of the probability Haar measure on
$\mathsf{U}(1)$ via $u\mapsto u+\overline{u}$, namely

(1.1) $\lambda_{1}(x)\colonequals\frac{\mathrm{d}x}{\pi\sqrt{4-x^{2}}},$

where $\,\mathrm{d}x$ is the restriction of the Lebesgue measure to $[-2,2]$
(see [8, Proposition 2.2]).

In contrast, if $E$ is supersingular, the sequence
$(u_{1}^{r})_{r=1}^{\infty}$ generates a finite cyclic subgroup of order $m$,
$C_{m}\subset\mathsf{U}(1)$. In this case, the normalized Frobenius traces are
equidistributed with respect to the pushforward of the uniform measure on
$C_{m}$.

This dichotomy branches out in an interesting way for abelian varieties of
higher dimension $g>1$: potential non-trivial multiplicative relations between
the Frobenius eigenvalues
$\alpha_{1},\overline{\alpha}_{1},\dots,\alpha_{g},\overline{\alpha}_{g}$
increase the complexity of the problem of classifying the distribution of
normalized traces of high powers of Frobenius,

(1.2)
$x_{r}\colonequals(\alpha_{1}^{r}+\overline{\alpha}_{1}^{r}+\cdots+\alpha_{g}^{r}+\overline{\alpha}_{g}^{r})/q^{r/2}\in[-2g,2g],\text{
for }r\geq 1.$

In analogy with the case of elliptic curves, we identify a compact abelian
subgroup of $\mathsf{U}(1)^{g}$ controlling the distribution of Sequence (1.2)
via pushforward of the Haar measure. In this article, we provide a complete
classification of this subgroup, which we call the _Serre–Frobenius group_ ,
for abelian varieties of dimension up to 3. We do this by classifying the
possible multiplicative relations between the Frobenius eigenvalues. This
classification provides a description of all the possible distributions of
Frobenius traces in these cases (see Corollary 1.1.1). We also provide a
partial classification for simple ordinary abelian varieties of odd prime
dimension.

######  Definition 1.0.1 (Serre–Frobenius group).

Let $A$ be an abelian variety of dimension $g$ over $\mathbf{F}_{q}$. Let
$\alpha_{1},\alpha_{2}\ldots,\alpha_{g},\overline{\alpha}_{1},\overline{\alpha}_{2}\ldots\overline{\alpha}_{g}$
denote the eigenvalues of Frobenius, ordered such that
$\arg(\alpha_{i})\geq\arg(\alpha_{j})$ if $g\geq i>j\geq 1$. Let
$u_{i}=\alpha_{i}/\sqrt{q}$ denote the normalized Frobenius eigenvalues. The
Serre–Frobenius group of $A$, denoted by $\mathsf{SF}(A)$, is the closure of
the subgroup of $\mathsf{U}(1)^{g}$ generated by the vector
$\mathbf{u}\colonequals(u_{1},\dots,u_{g})$.

We classify the Serre–Frobenius groups of abelian varieties of dimension
$g\leq 3$.

######  Theorem A (Elliptic curves).

Let $E$ be an elliptic curve defined over $\mathbf{F}_{q}$. Then

  1. (1)

$E$ is ordinary if and only if $\mathsf{SF}(E)=\mathsf{U}(1)$.

  2. (2)

$E$ is supersingular if and only if
$\mathsf{SF}(E)\in\mathopen{}\mathclose{{}\left\\{C_{1},C_{3},C_{4},C_{6},C_{8},C_{12}}\right\\}$.

Moreover, each one of these groups is realized for some prime power $q$.

We note that the classification of supersingular Serre–Frobenius groups of
elliptic curves follows from Deuring [3] and Waterhouse’s [31] classification
of Frobenius traces (see also [20, Section 14.6] and [25, Theorem 2.6.1]).

######  Theorem B (Abelian surfaces).

Let $S$ be an abelian surface over $\mathbf{F}_{q}$. Then, $S$ has
Serre–Frobenius group according to Figure 4. The possible options for the
connected component of the identity, $\mathsf{SF}(S)^{\circ}$, and the size of
the cyclic component group $\mathsf{SF}(S)/\mathsf{SF}(S)^{\circ}$ are given
below. Further, each one of these groups is realized for some prime power $q$.

$\mathsf{SF}(S)^{\circ}$ | $\\#\mathsf{SF}(S)/\mathsf{SF}(S)^{\circ}$
---|---
$1$ | 1,2,3,4,5,6,8,10,12,24
$\mathsf{U}(1)$ | 1,2,3,4,6,8,12
$\mathsf{U}(1)^{2}$ | 1

######  Theorem C (Abelian threefolds).

Let $X$ be an abelian threefold over $\mathbf{F}_{q}$. Then, $X$ has
Serre–Frobenius group according to Figure 10. The possible options for the
connected component of the identity, $\mathsf{SF}(X)^{\circ}$, and the size of
the cyclic component group $\mathsf{SF}(X)/\mathsf{SF}(X)^{\circ}$ are given
below. Further, each one of these groups is realized for some prime power $q$.

$\mathsf{SF}(X)^{\circ}$ | $\\#\mathsf{SF}(X)/\mathsf{SF}(X)^{\circ}$
---|---
$1$ | 1,2,3,4,5,6,7,8,9,10,12,14,15,18,20,24,28,30,36
$\mathsf{U}(1)$ | 1,2,3,4,5,6,7,8,10,12,24
$\mathsf{U}(1)^{2}$ | 1,2,3,4,6,8,12,24
$\mathsf{U}(1)^{3}$ | 1

If $g$ is an odd prime, we have the following classification for simple
ordinary abelian varieties. In the following theorem, we say that an abelian
variety $A$ splits over a field extension $\mathbf{F}_{q^{m}}$ if $A$ is
isogenous over $\mathbf{F}_{q^{m}}$ to a product of proper abelian
subvarieties.

######  Theorem D (Prime dimension).

Let $A$ be a simple ordinary abelian variety defined over $\mathbf{F}_{q}$ of
prime dimension $g>2$. Then, exactly one of the following conditions holds.

  1. (1)

$A$ is absolutely simple.

  2. (2)

$A$ splits over a degree $g$ extension of $\mathbf{F}_{q}$ as a power of an
elliptic curve, and $\mathsf{SF}(A)\cong\mathsf{U}(1)\times C_{g}$.

  3. (3)

$2g+1$ is prime (i.e., $g$ is a Sophie Germain prime) and $A$ splits over a
degree $2g+1$ extension of $\mathbf{F}_{q}$ as a power of an elliptic curve,
and $\mathsf{SF}(A)\cong\mathsf{U}(1)\times C_{2g+1}$.

Key to our results is the relation between the Serre–Frobenius group and the
multiplicative subgroup of $U_{A}\subset\mathbf{C}^{\times}$ generated by the
normalized eigenvalues $u_{1},\dots,u_{g}$. Indeed, an equivalent definition
of the former is via the Pontryagin dual of the latter (see Lemma 2.3.1). The
rank of the group $U_{A}$ is called the angle rank of the abelian variety and
the order of the torsion subgroup is called the angle torsion order. The
relation between $\mathsf{SF}(A)$ and the group generated by the normalized
eigenvalues gives us the following structure theorem.

######  Theorem E.

Let $A$ be an abelian variety defined over $\mathbf{F}_{q}$. Then

$\mathsf{SF}(A)\cong\mathsf{U}(1)^{\delta}\times C_{m},$

where $\delta=\delta_{A}$ is the angle rank and $m=m_{A}$ is the angle torsion
order. Furthermore, the connected component of the identity is

$\mathsf{SF}(A)^{\circ}=\mathsf{SF}(A_{\mathbf{F}_{q^{m}}}).$

###  1.1. Application to distributions of Frobenius traces

Our results can be applied to understanding the distribution of Frobenius
traces of an abelian variety over $\mathbf{F}_{q}$ as we range over finite
extensions of the base field. Indeed, for each integer $r\geq 1$, we may
rewrite Equation (1.2) as

$x_{r}=u_{1}^{r}+\overline{u}_{1}^{r}+\cdots+u_{g}^{r}+\overline{u}_{g}^{r}\in[-2g,2g]$

denote the normalized Frobenius trace of the base change of an abelian variety
$A$ to $\mathbf{F}_{q^{r}}$.

In [1], the authors study Jacobians of smooth projective genus $g$ curves with
maximal angle rank111In their notation, this is the condition that the
Frobenius angles are linearly independent modulo 1. and show that the sequence
$(x_{r}/2g)_{r=1}^{\infty}$ is equidistributed on $[-1,1]$ with respect to an
explicit measure. The Serre–Frobenius group enables us to remove the
assumption of maximal angle rank.

######  Corollary 1.1.1.

Let $A$ be a $g$-dimensional abelian variety defined over $\mathbf{F}_{q}$.
Then, the sequence $(x_{r})_{r=1}^{\infty}$ of normalized traces of Frobenius
is equidistributed in $[-2g,2g]$ with respect to the pushforward of the Haar
measure on $\mathsf{SF}(A)\subseteq\mathsf{U}(1)^{g}$ via

(1.3)
$\mathsf{SF}(A)\subseteq\mathsf{U}(1)^{g}\to[-2g,2g],\quad(z_{1},\dots,z_{g})\mapsto
z_{1}+\overline{z}_{1}+\cdots+z_{g}+\overline{z}_{g}.$

The classification of the Serre–Frobenius groups in our theorems can be used
to distinguish between the different Frobenius trace distributions occurring
in each dimension.

######  Example 1.1.2.

Let $S$ be a simple abelian surface over $\mathbf{F}_{q}$ with Frobenius
eigenvalues
$R_{S}=\mathopen{}\mathclose{{}\left\\{\alpha_{1},\alpha_{2},\overline{\alpha}_{1},\overline{\alpha}_{2}}\right\\}$
and suppose that $S_{(2)}\colonequals
S\times_{\mathbf{F}_{q}}\mathbf{F}_{q^{2}}$ is isogenous to $E^{2}$ for some
ordinary elliptic curve $E/\mathbf{F}_{q^{2}}$. In this case,
$\mathopen{}\mathclose{{}\left\\{\alpha_{1}^{2},\overline{\alpha}_{1}^{2}}\right\\}=R_{E}=\mathopen{}\mathclose{{}\left\\{\alpha_{2}^{2},\overline{\alpha}_{2}^{2}}\right\\}$.
Normalizing, and using the fact the $S$ is simple, we see that either (1)
$u_{2}=-u_{1}$, or (2) $u_{2}=-\overline{u}_{1}$. The Serre–Frobenius groups
in these cases can be calculated as follows.

  1. (1)

When $u_{2}=-u_{1}$, the vector of normalized eigenvalues
$\mathbf{u}=(u_{1},u_{2})=(u_{1},-u_{1})$ generates the group

$\mathsf{SF}(S)=\overline{\mathopen{}\mathclose{{}\left\\{(u_{1}^{m},-u_{1}^{m})\colon
m\in\mathbf{Z}}\right\\}}=\mathopen{}\mathclose{{}\left\\{(u,-u):u\in\mathsf{U}(1)}\right\\}\subset\mathsf{U}(1)^{2}.$

Extending scalars to $\mathbf{F}_{q^{2}}$, we get:

$\mathsf{SF}(S_{(2)})=\overline{\mathopen{}\mathclose{{}\left\\{(u_{1}^{2m},(-u_{1})^{2m}):m\in\mathbf{Z}}\right\\}}={\mathopen{}\mathclose{{}\left\\{(u,u)\colon
u\in\mathsf{U}(1)}\right\\}}\subset\mathsf{U}(1)^{2}.$

  2. (2)

When $u_{2}=-\overline{u}_{1}$, the vector of normalized eigenvalues
$\mathbf{u}=(u_{1},u_{2})=(u_{1},-u_{1}^{-1})$ generates the group
$\mathsf{SF}(S)=\mathopen{}\mathclose{{}\left\\{(u,-u^{-1}):u\in\mathsf{U}(1)}\right\\}\subset\mathsf{U}(1)^{2}$.
Similar to the case above,
$\mathsf{SF}(S_{(2)})=\mathopen{}\mathclose{{}\left\\{(u,u^{-1}):u\in\mathsf{U}(1)}\right\\}$.

In both cases, the sequence of normalized traces is given by

$\displaystyle
x_{r}=u_{1}^{r}+\overline{u}_{1}^{r}+(-1)^{r}\overline{u}_{1}^{r}+(-1)^{r}u_{1}^{r}\in[-4,4].$

In particular, $x_{r}=0$ when $r$ is odd, and
$x_{r}=2u_{1}^{r}+2\overline{u}_{1}^{r}$ when $r$ is even. Extending the base
field to $\mathbf{F}_{q^{2}}$ yields the sequence of normalized traces
$x_{r}(S_{(2)})=x_{2r}(S)=2x_{r}(E)$. The equality of the trace distributions
is a consequence of the fact the $\mathsf{SF}(S)$ in both cases is isomorphic
to $\mathsf{U}(1)\times C_{2}$. The data of the embedding
$\mathsf{SF}(S)\subseteq\mathsf{U}(1)^{2}$ precisely captures the (non-
trivial) multiplicative relations between the Frobenius eigenvalues.

In both cases (1) and (2), the normalized traces $x_{r}(S)$ are
equidistributed with respect to the pushforward of the Haar measure under the
map $\mathsf{SF}(S)\subseteq\mathsf{U}(1)^{2}\to[-4,4]$ given by
$(z_{1},z_{2})\mapsto z_{1}+\overline{z}_{1}+z_{2}+\overline{z}_{2}$. This can
be computed explicitly as

(1.4)
$\tfrac{1}{2}\delta_{0}+\frac{\,\mathrm{d}x}{2\pi\sqrt{16-x^{2}}}\quad\text{
and }\quad\frac{\,\mathrm{d}x}{\pi\sqrt{16-x^{2}}}$

for $S$ and $S_{(2)}$ respectively, where $\,\mathrm{d}x$ is the restriction
of the Haar measure to $[-4,4]$, and $\delta_{0}$ is the Dirac measure
supported at $0$.

For instance, choose the surface $S$ to be in the isogeny class with LMFDB
label222Recall the labelling convention for isogeny classes of abelian
varieties over finite fields in the LMFDB: g.q.iso where g is the dimension, q
is the cardinality of the base field, and iso specifies the isogeny class by
writing the coefficients of the Frobenius polynomial in base 26. 2.5.a_ab and
Weil polynomial $P(T)=T^{4}-T^{2}+25$. This isogeny class is ordinary and
simple, but not geometrically simple. Indeed, $S_{(2)}$ is in the isogeny
class $\texttt{1.25.ab}^{2}=$ 2.25.ac_bz corresponding to the square of an
ordinary elliptic curve. The corresponding $a_{1}$-histograms describing the
frequency of the sequence $(x_{r})_{r=1}^{\infty}$ are depicted in Figure 1.
Each graph represents a histogram of $16^{6}=16777216$ samples placed into
$4^{6}=4096$ buckets partitioning the interval $[-2g,2g]$. The vertical axis
has been suitably scaled, with the height of the uniform distribution, $1/4g$,
indicated by a gray line.

Figure 1. $a_{1}$-histograms for 2.5.a_ab and 2.25.ac_bz.

###  1.2. Relation to other work

The reason for adopting the name “Serre-Frobenius group” is that the Lie group
$\mathsf{SF}(A)$ is closely related to Serre’s Frobenius torus [24], as
explained in Remark 2.3.3.

####  1.2.1. Angle rank

In this article, we study multiplicative relations between Frobenius
eigenvalues, a subject studied extensively by Zarhin [35, 36, 15, 37, 38]. Our
classification relies heavily on being able to understand multiplicative
relations in low dimension, and we use results of Zarhin in completing parts
of it. The number of multiplicative relations is quantified by the angle rank,
an invariant studied in [6], [5] for absolutely simple abelian varieties by
elucidating its interactions with the Galois group and Newton polygon of the
Frobenius polynomial. We study the angle rank as a stepping stone to
classifying the full Serre–Frobenius group. While our perspective differs from
that in [6], the same theme is continued here: the Serre–Frobenius groups
depend heavily on the Galois group of the Frobenius polynomial. It is worth
noting that here that the results about the angle rank in the non-absolutely
simple case cannot be pieced together by knowing the results in the absolutely
simple cases (see for instance, see Zywina’s exposition of Shioda’s example
[40, Remark 1.16]).

####  1.2.2. Sato–Tate groups

The Sato–Tate group of an abelian variety defined over a number field controls
the distribution of the Frobenius of the reduction modulo prime ideals, and it
is defined via it’s $\ell$-adic Galois representation (see [27, Section 3.2]).
The Serre–Frobenius group can also be defined via $\ell$-adic representations
in an analogous way: it is conjugate to a maximal compact subgroup of the
image of Galois representation
$\rho_{A,\ell}\colon\operatorname{Gal}(\overline{\mathbf{F}}_{q}/\mathbf{F}_{q})\to\operatorname{Aut}(V_{\ell}A)\otimes\mathbf{C}$,
where $V_{\ell}A$ is the $\ell$-adic Tate vector space. Therefore it is
natural to expect that the Sato–Tate and the Serre–Frobenius group are related
to each other. The following observations support this claim:

  * •

Assuming standard conjectures, the connected component of the identity of the
Sato–Tate group can be recovered from knowing the Frobenius polynomial at two
suitably chosen primes ([40, Theorem 1.6]).

  * •

Several abelian Sato–Tate groups (see [9, 10]) appear as Serre–Frobenius
groups of abelian varieties over finite fields. The ones with maximal angle
rank are:

    * –

$\mathsf{U}(1)$ is the Sato–Tate group of an elliptic curve with complex
multiplication over any number field that contains the CM field (see
1.2.B.1.1a). It is also the Serre–Frobenius group of any ordinary elliptic
curve (see Figure 2), and the $a_{1}$-moments coincide.

    * –

$\mathsf{U}(1)^{2}$ is the Sato–Tate group of weight 1 and degree 4 (see
1.4.D.1.1a). It is also the Serre–Frobenius group of an abelian surface with
maximal angle rank (see Figure 7), and the $a_{1}$-moments coincide.

    * –

$\mathsf{U}(1)^{3}$ is the Sato–Tate group of weight 1 and degree 6 (see
1.6.H.1.1a). It is also the Serre-Frobenius group of abelian threefolds with
maximal angle rank (see Figure 11), and the $a_{1}$-moments coincide.

This is not unexpected, since $\mathsf{U}(1)^{g}$ embeds into
$\mathrm{USp}_{2g}(\mathbf{C})$ and composition with the trace map gives the
normalized traces $(x_{r})_{r=1}^{\infty}$.

###  1.3. Outline

In Section 2, we give some background on abelian varieties over finite fields,
expand on the definition of the Serre–Frobenius group, and describe how it
controls the distribution of traces of high powers of Frobenius. In Section 3,
we prove some preliminary results on the geometric isogeny types of abelian
varieties of dimension $g\leq 3$ and $g$ odd prime. We also recall some
results about Weil polynomials of supersingular abelian varieties, and
Zarhin’s notion of neatness. In Sections 4, 5, and 6, we give a complete
classification of the Serre–Frobenius group for dimensions 1, 2, and 3
respectively. In Section 7, we discuss the case of simple ordinary abelian
varieties of odd prime dimension. A list of tables containing different pieces
of the classification follows this section.

###  1.4. Notation

Throughout this paper, $A$ will denote a $g$-dimensional abelian variety over
a finite field $\mathbf{F}_{q}$ of characteristic $p$. The polynomial
$P_{A}(T)=\sum_{i=1}^{2g}a_{i}T^{2g-i}$ will denote the characteristic
polynomial of Frobenius acting on the Tate module of $A$, and $h_{A}(T)$ its
minimal polynomial. The set of roots of $P_{A}(T)$ is denoted by $R_{A}$. We
usually write
$\alpha_{1},\overline{\alpha}_{1}\dots,\alpha_{g},\overline{\alpha}_{g}\in
R_{A}$ for the Frobenius eigenvalues. In the case that $P_{A}(T)$ is a power
of $h_{A}(T)$, we will denote by $e_{A}$ this power (See 2.1). The subscript
$(\cdot)_{(r)}$ will denote the base change of any object or map to
$\mathbf{F}_{q^{r}}$. The group $U_{A}$ will denote the multiplicative group
generated by the normalized eigenvalues of Frobenius, $\delta_{A}$ its rank
and $m_{A}$ the order of its torsion subgroup. The group $\Gamma_{A}$ will
denote the multiplicative group generated by
$\\{\alpha_{1},\alpha_{2}\ldots\alpha_{g},q\\}$. In Section 5, $S$ will be
used to denote an abelian surface, while in Section 6, $X$ will be used to
denote a threefold.

###  1.5. Acknowledgements

We would like to thank David Zureick-Brown, Kiran Kedlaya, Francesc Fité,
Brandon Alberts, Edgar Costa and Andrew Sutherland for useful conversations
about this paper. We thank Yuri Zarhin for providing us with useful
references. We would also like to thank Everett Howe for helping us with a
missing piece of the puzzle in Theorem 3.1.1. This project started as part of
the Rethinking Number Theory workshop in 2021. We would like to thank the
organizers of the workshop for giving us the opportunity and space to
collaborate, and the funding sources for the workshop, AIM, the Number Theory
Foundation, and the University of Wisconsin-Eau Claire Department of
Mathematics. We would also like to thank Rachel Pries for her guidance at the
beginning of the workshop, which helped launch this project.

###### Contents

  1. 1 Introduction
  2. 2 Frobenius multiplicative groups
  3. 3 Preliminary Results
  4. 4 Elliptic Curves
  5. 5 Abelian Surfaces
  6. 6 Abelian Threefolds
  7. 7 Simple ordinary abelian varieties of odd dimension

###### List of Tables

  1. 1 Minimal polynomial of a supersingular $q$-Weil number $\alpha$.
  2. 2 Serre–Frobenius groups of elliptic curves.
  3. 3 Serre–Frobenius groups of simple ordinary surfaces.
  4. 4 Serre–Frobenius groups of non-simple ordinary surfaces.
  5. 5 Serre–Frobenius groups of simple supersingular surfaces.
  6. 6 Serre–Frobenius groups of non-simple ordinary threefolds.
  7. 7 Serre–Frobenius groups of simple almost ordinary threefolds.
  8. 8 Serre–Frobenius groups of abelian threefolds of $p$-rank 1.
  9. 9 Serre–Frobenius groups of absolutely simple abelian threefolds of $p$-rank $0$.
  10. 10 Serre–Frobenius groups of simple supersingular threefolds.

###### List of Figures

  1. 1 $a_{1}$-histograms for 2.5.a_ab and 2.25.ac_bz.
  2. 2 $a_{1}$-distribution for ordinary elliptic curves.
  3. 3 $a_{1}$-histograms of supersingular elliptic curves $E/\mathbf{F}_{q}$.
  4. (a) $\mathsf{SF}(E)=C_{1}$.
  5. (b) $\mathsf{SF}(E)=C_{3}$.
  6. (c) $\mathsf{SF}(E)=C_{4}$
  7. (d) $\mathsf{SF}(E)=C_{6}$
  8. (e) $\mathsf{SF}(E)=C_{8}$
  9. (f) $\mathsf{SF}(E)=C_{12}$.
  10. 4 Theorem B: Classification in dimension 2.
  11. 5 $a_{1}$-histograms for simple ordinary abelian surfaces.
  12. (a)  2.2.ab_b.
  13. (b)  2.2.a_ad.
  14. (c)  2.2.ab_ab.
  15. (d)  2.3.ac_c.
  16. (e) 2.2.ad_f.
  17. 6 $a_{1}$-histograms of non-simple ordinary abelian surfaces.
  18. (a) 2.2.ac_f.
  19. (b)  2.2.a_d.
  20. (c)  2.7.af_s
  21. (d)  2.5.ag_s
  22. (e)  2.7.aj_bi.
  23. 7 $a_{1}$-histogram of simple almost ordinary abelian surface 2.2.ab_a.
  24. 8 Roots of Z-3 type normalized Frobenius polynomials in Table 6.
  25. (a) $\Psi_{5,1}(T)$.
  26. (b) $\Psi_{5,1}(-T)$.
  27. (c) $\Psi_{2,3}(T)$.
  28. (d) $\Psi_{2,3}(-T)$.
  29. 9 Sets $M$ of possible orders of the Serre–Frobenius groups of non-simple supersingular surfaces as a function of $p$ and $q=p^{d}$.
  30. (a) When $q$ is a square.
  31. (b) When $q$ is not a square.
  32. 10 Theorem C: Classification in dimension 3
  33. 11 $a_{1}$-distributions for simple ordinary threefolds.
  34. (a) $a_{1}$-histogram of 3.2.ad_f_ah.
  35. (b) $a_{1}$-histogram of 3.2.a_a_ad.
  36. (c) $a_{1}$ histogram of 3.2.ae_j_ap.
  37. 12 $a_{1}$-distributions for non-simple ordinary abelian threefolds of splitting (6.3-a).
  38. (a)  3.2.ad_j_an.
  39. (b)  3.2.ab_f_ad.
  40. (c)  3.2.a_a_af
  41. (d)  3.5.ak_bv_afc
  42. (e)  3.7.ao_di_alk.
  43. 13 $a_{1}$-distributions for non-simple ordinary abelian threefolds of splitting (6.3-b).
  44. (a)  3.3.af_r_abi.
  45. (b) 3.2.ab_b_b
  46. (c) 3.3.ad_d_ac
  47. (d)  3.3.af_p_abg
  48. (e) 3.2.ae_k_ar
  49. 14 Serre–Frobenius groups of non-simple almost ordinary threefolds.
  50. 15 $q$-Newton polygon of $p$-rank $1$ abelian threefolds.
  51. 16 $q$-Newton polygon of $p$-rank $0$ non-supersingular abelian threefolds.
  52. 17 $a_{1}$-distribution for $p$-rank 0 non-supersingular threefolds of type (6.7-a).
  53. (a)  3.8.ag_bk_aea.
  54. (b)  3.2.a_a_ac.
  55. (c)  3.8.ai_bk_aeq.
  56. 18 Roots of Z-3 type normalized Frobenius polynomials in Table 11.
  57. (a) $h_{7,1}(T)$.
  58. (b) $h_{7,1}(-T)$.
  59. (c) $h_{3,3}(T)$.
  60. (d) $h_{3,3}(-T)$.

##  2\. Frobenius multiplicative groups

In this section we introduce the Serre–Frobenius group of $A$ and explain how
it is related to Serre’s theory of Frobenius tori [24]. We do this from the
perspective of the theory of algebraic groups of multiplicative type, as in
[16, Chapter 12]. We start by recalling some facts about abelian varieties
over finite fields.

###  2.1. Background on Abelian varieties over finite fields

Fix $A$ a $g$ dimensional abelian variety over $\mathbf{F}_{q}$. A $q$-Weil
number is an algebraic integer $\alpha$ such that $|\phi(\alpha)|=\sqrt{q}$
for every embedding $\phi\colon\mathbf{Q}(\alpha)\to\mathbf{C}$. Let
$P_{A}(T)$ denote the characteristic polynomial of the Frobenius endomorphism
acting on the $\ell$-adic Tate module of $A$. The polynomial $P_{A}(T)$ is
monic of degree $2g$, and Weil [32] showed that its roots are $q$-Weil
numbers; we denote the set of roots of $P_{A}(T)$ by
$R_{A}\colonequals\\{\alpha_{1},\alpha_{2}\ldots,\alpha_{g},\alpha_{g+1},\dots,\alpha_{2g}\\}$
with $\alpha_{g+j}=q/\alpha_{j}$ for
$j\in\mathopen{}\mathclose{{}\left\\{1,\dots,g}\right\\}$. We index the first
$g$ roots according to non-decreasing angles; that is
$\arg(\alpha_{j})\leq\arg(\alpha_{i})$ if $j<i$. The seminal work of Honda
[12] and Tate [29] [30] classifies the isogeny decomposition type of $A$ in
terms of the factorization of $P_{A}(T)$. In particular, if $A$ is simple, we
have that $P_{A}(T)=h_{A}(T)^{e_{A}}$ where $h_{A}(T)$ is the minimal
polynomial of the Frobenius endomorphism and $e_{A}$ is the degree, i.e., the
square root of the dimension, of the central simple algebra
$\mathrm{End}^{0}(A)\colonequals\mathrm{End}(A)\otimes\mathbf{Q}$ over its
center. The Honda–Tate theorem gives a bijective correspondence between
isogeny classes of simple abelian varieties over $\mathbf{F}_{q}$ and
conjugacy classes of $q$-Weil numbers, sending the isogeny class determined by
$A$ to the set of roots $R_{A}$. Further, if $A\sim A_{1}\times
A_{2}\ldots\times A_{k}$, then $P_{A}(T)=\prod_{i=1}^{k}P_{A_{i}}(T)$.

Writing $P_{A}(T)=\sum_{i=0}^{2g}a_{i}T^{2g-i}$, the $q$-Newton polygon of $A$
is the lower convex hull of the set of points
$\\{(i,\nu(a_{i}))\in\mathbf{R}^{2}:a_{i}\neq 0\\}$ where $\nu$ is the
$p$-adic valuation normalized so that $\nu(q)=1$. The Newton polygon is
isogeny invariant. Define the $p$-rank of $A$ as the number of slope $0$
segments of the Newton polygon. An abelian variety is called ordinary if it
has maximal $p$-rank, i.e. its $p$-rank is equal to $g$. It is called
supersingular if all the slopes of the Newton polygon are equal to $1/2$. The
field $L=L_{A}\colonequals\mathbf{Q}(\alpha_{1},\dots,\alpha_{g})$ is the
splitting field of the Frobenius polynomial. By definition, the Galois group
$\operatorname{Gal}(L/\mathbf{Q})$ acts on the roots $R_{A}$ by permuting
them.

######  Notation.

Whenever $A$ is fixed or clear from context, we will omit the subscript
corresponding to it from the notation described above. In particular, we will
use $P(T),h(T)$ and $e$ instead of $P_{A}(T),h_{A}(T)$ and $e_{A}$.

###  2.2. Angle groups

Denote by $\Gamma\colonequals\Gamma_{A}$ the multiplicative subgroup of
$\mathbf{C}^{\times}$ generated by the set of Frobenius eigenvalues $R_{A}$,
and let $\Gamma_{(r)}\colonequals\Gamma_{A_{(r)}}$ for every $r\geq 1$. Since
$\alpha\mapsto q/\alpha$ is a permutation of $R_{A}$, the set
$\mathopen{}\mathclose{{}\left\\{\alpha_{1},\dots,\alpha_{g},q}\right\\}$
generates $\Gamma$ is a set of generators for $\Gamma$; that is, every
$\gamma\in\Gamma$ can be written as

(2.1) $\gamma=q^{k}\prod_{j=1}^{g}\alpha_{j}^{k_{j}}$

for a some $(k,k_{1},\dots,k_{g})\in\mathbf{Z}^{g+1}$.

Since $\Gamma$ is a subgroup of $\overline{\mathbf{Q}}^{\times}$, it is
naturally a $\operatorname{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$-module.
However, this perspective is not necessary for our applications. This group is
denoted as $\Phi_{A}$ in [40].

######  Definition 2.2.1.

We define the angle group of $A$ to be $U\colonequals U_{A}$, the
multiplicative subgroup of $\mathsf{U}(1)$ generated by the unitarized
eigenvalues $\\{u_{j}\colonequals\alpha_{j}/\sqrt{q}:j=1,\dots,g$}. When $A$
is fixed, for every $r\geq 1$ we abbreviate $U_{(r)}\colonequals U_{A_{(r)}}$.

######  Definition 2.2.2.

The angle rank of an abelian variety $A/\mathbf{F}_{q}$ is the rank of the
finitely generated abelian group $U_{A}$. It is denoted by
$\delta_{A}\colonequals\operatorname{rk}U_{A}$. The angle torsion order
$m_{A}$ is the order of the torsion subgroup of $U_{A}$, so that
$U_{A}\cong\mathbf{Z}^{\delta_{A}}\oplus\mathbf{Z}/m_{A}\mathbf{Z}$.

The angle rank $\delta$ is by definition an integer between $0$ and $g$. When
$\delta=g$, there are no multiplicative relations among the normalized
eigenvalues. In other words, there are no additional relations among the
generators of $\Gamma_{A}$ apart from the ones imposed by the Weil
conjectures. If $A$ is absolutely simple, the maximal angle rank condition
also implies that the Tate conjecture holds for all powers of $A$ (see Remark
1.3 in [6]). On the other extreme, $\delta=0$ if and only if $A$ is
supersingular (See Example 5.1 [5]).

######  Remark 2.2.3.

The angle rank is invariant under base extension: $\delta(A)=\delta(A_{(r)})$
for every $r\geq 1$. Indeed, any multiplicative relation between
$\mathopen{}\mathclose{{}\left\\{u_{1}^{r},\dots,u_{g}^{r}}\right\\}$ is a
multiplicative relation between
$\mathopen{}\mathclose{{}\left\\{u_{1},\dots,u_{g}}\right\\}$. We have that
$U_{A}/\mathrm{Tors}(U_{A})\cong U_{A_{(r)}}/\mathrm{Tors}(U_{A_{(r)}})$ for
every positive integer $r$. In particular, $U_{A}/\mathrm{Tors}(U_{A})\cong
U_{A_{(m)}}$ where $m=m_{A}$ is the angle torsion order of $A$.

######  Example 2.2.4 (Extension and restriction of scalars).

Let $A/\mathbf{F}_{q}$ be an abelian variety with Frobenius polynomial
$P_{A}(T)=\prod(T-\alpha)\in\mathbf{C}[T]$ and circle group $U_{A}=\langle
u_{1},\dots,u_{g}\rangle$. Then, the extension of scalars $A_{(r)}$ has
Frobenius polynomial $P_{(r)}(T)=\prod(T-\alpha^{r})$ and circle group
$U_{A_{(r)}}=\langle u_{1}^{r},\dots,u_{g}^{r}\rangle\subset U_{A}$. On the
other hand, if $B/\mathbf{F}_{q^{r}}$ is an abelian variety for some $r\geq
1$, and $A/\mathbf{F}_{q}$ is the Weil restriction of $B$ to $\mathbf{F}_{q}$,
then $P_{A}(T)=P_{B}(T^{r})$ and $U_{A}=\langle U_{B},\zeta_{r}\rangle\supset
U_{B}$. See [7].

###  2.3. The Serre–Frobenius group

For every locally compact abelian group $G$, denote by $\widehat{G}$ its
Pontryagin dual; this is the topological group of continuous group
homomorphisms $G\to\mathsf{U}(1)$. It is well known that $G\mapsto\widehat{G}$
gives an anti-equivalence of categories from the category of locally compact
abelian groups to itself. Moreover, this equivalence preserves exact
sequences, and every such $G$ is canonically isomorphic to its double dual via
the evaluation isomorphism. See [21] for the original reference and [18] for a
gentle introduction.

Recall that we defined the Serre–Frobenius group of $A$ as the topological
group generated by the vector $\mathbf{u}=(u_{1},\dots,u_{g})$ of normalized
eigenvalues (see Definition 1.0.1). This explicit description of the group is
practical for calculating examples, but the following equivalent definition is
conceptually advantageous.

######  Lemma 2.3.1.

The Serre–Frobenius group of an abelian variety $A$ has character group
$U_{A}$. In particular, $\mathsf{SF}(A)\cong\widehat{U}_{A}$ canonically via
the evaluation isomorphism.

###### Proof.

We have an injection $U_{A}\to\widehat{\mathsf{SF}(A)}$ given by mapping
$\gamma$ to the character $\phi_{\gamma}$ that maps $\mathbf{u}$ to $\gamma$.
To see that this map is surjective, observe that by the exactness of
Pontryagin duality, the inclusion
$\mathsf{SF}(A)\hookrightarrow\mathsf{U}(1)^{g}$ induces a surjection
$\mathbf{Z}^{g}=\widehat{\mathsf{U}(1)}^{g}\to\widehat{\mathsf{SF}(A)}$.
Explicitly, this tells us that every character of $\mathsf{SF}(A)$ is given by
$\phi(z_{1},\dots,z_{g})=z_{1}^{m_{1}}\dots z_{g}^{m_{g}}$ for some
$(m_{1},\cdots,m_{g})\in\mathbf{Z}^{g}$. By continuity, every character $\phi$
of $\mathsf{SF}(A)$ is completely determined by $\phi(\mathbf{u})$. In
particular, we have that $\phi(\mathbf{u})=u_{1}^{m_{1}}\dots u_{g}^{m_{g}}\in
U_{A}$. ∎

The following theorem should be compared to [27, Theorem 3.12]

######  Theorem 2.3.2 (Theorem E).

Let $A$ be an abelian variety defined over $\mathbf{F}_{q}$. Then

$\mathsf{SF}(A)\cong\mathsf{U}(1)^{\delta}\times C_{m},$

where $\delta=\delta_{A}$ is the angle rank and $m=m_{A}$ is the angle torsion
order. Furthermore, the connected component of the identity is

$\mathsf{SF}(A)^{\circ}=\mathsf{SF}(A_{(m)}).$

###### Proof.

Since every finite subgroup of $\mathsf{U}(1)$ is cyclic, the torsion part of
the finitely generated group $U_{A}$ is generated by some primitive $m$-th
root of unity $\zeta_{m}$. The group $U_{(m)}$ is torsion free by Remark
2.2.3. We thus have the split short exact sequence

(2.2)
${1}$${\langle\zeta_{m}\rangle}$${U_{A}}$${U_{(m)}}$${1.}$$\scriptstyle{u\mapsto
u^{m}}$

After dualizing, we get:

(2.3)
${1}$${\mathsf{SF}(A_{(m)})}$${\mathsf{SF}(A)}$${\langle\zeta_{m}\rangle}$${1.}$

We conclude that $\mathsf{SF}(A)^{\circ}=\mathsf{SF}(A_{(m)})$ and
$\mathsf{SF}(A)/\mathsf{SF}(A)^{\circ}\cong\langle\zeta_{m}\rangle$. ∎

######  Remark 2.3.3.

By definition, $U_{A}$ is the image of $\Gamma_{A}$ under the radial
projection $\psi\colon\mathbf{C}^{\times}\to\mathsf{U}(1),z\mapsto z/|z|$.
Thus, we have a short exact sequence

(2.4)
${1}$${\Gamma_{A}\cap\mathbf{R}_{>0}}$${\Gamma_{A}}$${U_{A}}$${1,}$$\scriptstyle{\psi|_{\Gamma}}$

which is split by the section $u_{j}\mapsto\alpha_{j}$. The kernel
$\Gamma\cap\mathbf{R}_{>0}$ is free of rank $1$ and contains the group
$q^{\mathbf{Z}}$. The relation between the Serre–Frobenius group
$\mathsf{SF}(A)$ and Serre’s Frobenius Torus (see [24], [2, Section 3]) can be
understood via their character groups.

  * •

The (Pontryagin) character group of $\mathsf{SF}(A)$ is $U_{A}$.

  * •

The (algebraic) character group of the Frobenius torus of $A$ is the torsion
free part of $\Gamma_{A}$.

###  2.4. Equidistribution results

Let $(Y,\mu)$ be a measure space in the sense of Serre (see Appendix A.1 in
[23]). Recall that a sequence $(y_{r})_{r=1}^{\infty}\subset Y$ is
$\mu$-equidistributed if for every continuous function $f\colon
Y\to\mathbf{C}$ we have that

(2.5) $\int_{Y}f\mu=\lim_{n\to\infty}\frac{1}{n}\sum_{r=1}^{n}f(y_{r}).$

In our setting, $Y$ will be a compact abelian Lie group with probability Haar
measure $\mu$. We have the following lemma.

######  Lemma 2.4.1.

Let $G$ be a compact group, and $h\in G$. Let $H$ be the closure of the group
generated by $h$. Then, the sequence $(h^{r})_{r=1}^{\infty}$ is
equidistributed in $H$ with respect to the Haar measure $\mu_{H}$.

###### Proof.

For a non-trivial character $\phi\colon H\to\mathbf{C}^{\times}$, the image of
the generator $\phi(h)=u\in\mathsf{U}(1)$ is not trivial. We see that

(2.6)
$\lim_{n\to\infty}\frac{1}{n}\sum_{r=1}^{n}\phi(h^{r})=\lim_{n\to\infty}\frac{1}{n}\sum_{r=1}^{n}u^{r}=0,$

both when $u$ has finite or infinite order. The latter case follows form
Weyl’s equidistribution theorem in $\mathsf{U}(1)$. The result follows from
Lemma 1 in [23, I-19] and the Peter–Weyl theorem. ∎

######  Corollary 2.4.2 (Corollary 1.1.1).

Let $A$ be a $g$-dimensional abelian variety defined over $\mathbf{F}_{q}$.
Then, the sequence $(x_{r})_{r=1}^{\infty}$ of normalized traces of Frobenius
is equidistributed in $[-2g,2g]$ with respect to the pushforward of the Haar
measure on $\mathsf{SF}(A)\subseteq\mathsf{U}(1)^{g}$ via

$\mathsf{SF}(A)\subseteq\mathsf{U}(1)^{g}\to[-2g,2g],\quad(z_{1},\dots,z_{g})\mapsto
z_{1}+\overline{z}_{1}+\cdots+z_{g}+\overline{z}_{g}.$

###### Proof.

By Lemma 2.4.1, the sequence $(\mathbf{u}^{r})_{r=1}^{\infty}$ is
equidistributed in $\mathsf{SF}(A)$ with respect to the Haar measure
$\mu_{\mathsf{SF}(A)}$. By definition, the sequence $(x_{r})_{r=1}^{\infty}$
is equidistributed with respect to the pushforward measure. ∎

######  Remark 2.4.3 (Maximal angle rank).

When $A$ has maximal angle rank $\delta=g$, the Serre–Frobenius group is the
full torus $\mathsf{U}(1)^{g}$, and the sequence of normalized traces of
Frobenius is equidistributed with respect to the pushforward of the measure
$\mu_{\mathsf{U}(1)^{g}}$; which we denote by $\lambda_{g}(x)$ following the
notation333Beware of the different choice of normalization. We chose to use
the interval $[-2g,2g]$ instead of $[-1,1]$ to be able to compare our
distributions with the Sato–Tate distributions of abelian varieties defined
over number fields. in [1].

##  3\. Preliminary Results

For this entire section, we let $A$ be an abelian variety over
$\mathbf{F}_{q}$, where $q=p^{d}$ for some prime $p$.

###  3.1. Splitting of simple ordinary abelian varieties of odd prime
dimension

Recall from Section 1 that an abelian variety $A$ splits over a field
extension $\mathbf{F}_{q^{m}}$ if $A\sim_{(m)}A_{1}\times A_{2}$ and $\dim
A_{1},\dim A_{2}<\dim A$, i.e., if $A$ obtains at least one isogeny factor
when base-changed to $\mathbf{F}_{q^{m}}$. We say that $A$ splits completely
over $\mathbf{F}_{q^{m}}$ if $A_{(m)}\sim A_{1}\times A_{2}\times\ldots\times
A_{k}$, where each $A_{i}$ is an absolutely simple abelian variety defined
over $\mathbf{F}_{q^{m}}$. In other words, $A$ acquires its geometric isogeny
decomposition over $\mathbf{F}_{q^{m}}$.

In this section, we analyze the splitting behavior of simple ordinary abelian
varieties of _prime dimension_ $g>2$. Our first result is analogous to [13,
Theorem 6] for odd primes.

######  Theorem 3.1.1 (Theorem D).

Let $A$ be a simple ordinary abelian variety defined over $\mathbf{F}_{q}$ of
prime dimension $g>2$. Then, exactly one of the following conditions holds.

  1. (1)

$A$ is absolutely simple.

  2. (2)

$A$ splits over a degree $g$ extension of $\mathbf{F}_{q}$ as a power of an
elliptic curve, and $\mathsf{SF}(A)\cong\mathsf{U}(1)\times C_{g}$.

  3. (3)

$2g+1$ is prime (i.e., $g$ is a Sophie Germain prime) and $A$ splits over a
degree $2g+1$ extension of $\mathbf{F}_{q}$ as a power of an elliptic curve,
and $\mathsf{SF}(A)\cong\mathsf{U}(1)\times C_{2g+1}$.

###### Proof.

Let $\alpha=\alpha_{1}$ be a Frobenius eigenvalue of $A$, and denote by
$K=\mathbf{Q}(\alpha)\cong\mathbf{Q}[T]/P(T)$ the number field generated by
$\alpha$. Since $A$ is ordinary, $\mathbf{Q}(\alpha^{n})\neq\mathbf{Q}$ is a
CM-field over $\mathbf{Q}$ for every positive integer $n$, and $P(T)$ is
irreducible and therefore $[\mathbf{Q}(\alpha):\mathbf{Q}]=2g$. Suppose that
$A$ is not absolutely simple, and let $m$ be the smallest positive integer
such that $A_{(m)}$ splits; by [13, Lemma 4] this is also the smallest $m$
such that $\mathbf{Q}(\alpha^{m})\subsetneq\mathbf{Q}(\alpha)$. Since
$\mathbf{Q}(\alpha^{m})$ is also a CM field, it is necessarily a quadratic
imaginary number field.

Observe first that $m$ must be odd. Indeed, if $m$ was even, then
$\mathbf{Q}(\alpha^{m/2})=\mathbf{Q}(\alpha)$ and
$[\mathbf{Q}(\alpha^{m/2}):\mathbf{Q}(\alpha^{m})]=2$. This contradicts the
fact that $[\mathbf{Q}(\alpha):\mathbf{Q}]=2g$, since $g$ is an odd prime. By
[13, Lemma 5], there are two possibilities:

  1. (i)

$P(T)\in\mathbf{Q}[T^{m}]$,

  2. (ii)

$K=\mathbf{Q}(\alpha^{m},\zeta_{m})$.

If (i) holds and $P(T)=T^{2m}+bT^{m}+q^{g}$, we conclude that $m=g$ and
$b=a_{g}$. In this case, the minimal polynomial of $\alpha^{g}$ has degree 2
and is of the form $h_{(g)}(T)=(T-\alpha^{g})(T-\overline{\alpha}^{g})$. Note
that $\alpha^{g}$ and $\overline{\alpha}^{g}$ are distinct, since $A$ is
ordinary. Thus, $P_{g}(T)=h_{(g)}(T)^{g}$ and $A$ must split over a degree $g$
extension.

If (ii) holds, we have that $\varphi(m)\mid 2g$. Since $m>1$ is odd and
$\varphi(m)$ takes even values, we have two possible options: either
$\varphi(m)=2$ or $\varphi(m)=2g$. If $\varphi(m)=2$, then
$[K:\mathbf{Q}(\alpha^{m})]\leq 2$ which contradicts the fact that
$\mathbf{Q}(\alpha)$ is a degree $2g$ extension of $\mathbf{Q}$. Therefore,
necessarily, $\varphi(m)=2g$, and $\mathbf{Q}(\alpha)=\mathbf{Q}(\zeta_{m}).$
Recall from elementary number theory that the solutions to this equation are
$(m,g)=(9,3)$ or $(m,g)=(2g+1,g)$ for $g$ a Sophie Germain prime.

  * •

($g>3$) In this case, (ii) only occurs when $2g+1$ is prime.

  * •

($g=3$) In this case, either $m=7$ or $m=9$. To conclude the proof, we show
that $m=9$ does not occur. More precisely, we will show that if $A$ splits
over a degree $9$ extension, it splits over a degree $3$ extension as well. In
fact, suppose that $K=\mathbf{Q}(\zeta)=\mathbf{Q}(\alpha)$ for some primitive
$9$th root of unity. The subfield $F=\mathbf{Q}(\zeta^{3})$ is the only
quadratic imaginary subfield of $K$, so if a power of $\alpha$ does not
generate $K$, it must lie in $F$. Suppose $\alpha^{9}$ lies in $F$. Let
$\sigma$ be the generator of $\operatorname{Gal}(K/F)$ sending $\zeta$ to
$\zeta^{4}$. The minimal polynomial of $\alpha$ over $F$ divides
$T^{9}-\alpha^{9}$, so $\sigma(\alpha)=\alpha\cdot\zeta^{j}$ for some $j$, and
$\sigma^{2}(\alpha)=\alpha\zeta^{5j}$. Since the product of three conjugates
of $\alpha$ over $F$ must lie in $F$, we have that
$\alpha^{3}\cdot\zeta^{6j}=(\alpha)(\alpha\cdot\zeta^{j})(\alpha\cdot\zeta^{5j})\in
F$, which implies that $\alpha^{3}\in F$ and we conclude that $A$ splits over
a degree-$3$ extension of the base field.

∎

We thank Everett Howe for explaining to us why the case $m=9$ above does not
occur.

###  3.2. Zarhin’s notion of neatness

In this section we discuss Zarhin’s notion of _neatness_ , a useful technical
definition closely related to the angle rank. Define

(3.1)
$R_{A}^{\prime}\colonequals\mathopen{}\mathclose{{}\left\\{u_{j}^{2}:\alpha_{j}\in
R_{A}}\right\\}.$

Note that according to our numbering convention, we have that
$u_{j}^{-1}=\overline{u}_{j}=u_{j+g}$ for every
$j\in\mathopen{}\mathclose{{}\left\\{1,\dots,g}\right\\}$.

######  Definition 3.2.1 (Zarhin).

Let $A$ be an abelian variety defined over $\mathbf{F}_{q}$. We say that $A$
is neat if it satisfies the following conditions:

  1. (Na)

$\Gamma_{A}$ is torsion free.

  2. (Nb)

For every function $e\colon R_{A}^{\prime}\to\mathbf{Z}$ satisfying

$\prod_{\beta\in R_{A}^{\prime}}\beta^{e(\beta)}=1,$

then $e(\beta)=e(\beta^{-1})$ for every $\beta\in R_{A}^{\prime}$.

######  Remarks 3.2.2.

  1. (3.2.2.a)

If $A$ is supersingular and $\Gamma_{A}$ is torsion free, then $A$ is neat.
Indeed, in this case we have that
$R_{A}^{\prime}=\mathopen{}\mathclose{{}\left\\{1}\right\\}$ and condition
(Nb) is trivially satisfied.

  2. (3.2.2.b)

Suppose that the Frobenius eigenvalues of $A$ are distinct and not
supersingular. Some base extension of $A$ is neat if and only if $A$ has
maximal angle rank.

  3. (3.2.2.c)

In general, maximal angle rank always implies neatness.

###  3.3. Behavior of Serre–Frobenius groups in products

We begin by stating an important lemma, attributed to Bjorn Poonen in [14].

######  Lemma 3.3.1 (Poonen).

If $E_{1},\dots,E_{n}$ are $n$ pairwise absolutely non-isogenous elliptic
curves over $\mathbf{F}_{q}$, then their eigenvalues of Frobenius
$\alpha_{1},\dots,\alpha_{n}$ are multiplicatively independent.

In fact, for abelian varieties that split completely as products of elliptic
curves, we can give an explicit description of the Serre–Frobenius group.

######  Proposition 3.3.2.

Let $A$ be a $g$-dimensional abelian variety over $\mathbf{F}_{q}$ that splits
completely as a product of elliptic curves. Let $r$ be the degree of the
smallest extension such that $A\sim_{(r)}A_{1}\times B_{1}\times
B_{2}\ldots\times B_{s}$, satisfying

  1. (i)

$A_{1}$ is supersingular or trivial,

  2. (ii)

each $B_{j}$ splits over $\mathbf{F}_{q^{rm_{j}}}$ as the power of an ordinary
elliptic curve $E_{j}/\mathbf{F}_{q^{rm_{j}}}$, and

  3. (iii)

$E_{j}$ is not geometrically isogenous to $E_{i}$ for $i\neq j$.

Let $n_{1}\geq 1$ be the smallest integer such that $A_{1}$ is isogenous to a
power of an elliptic curve $E$ over $\mathbf{F}_{q^{rn_{1}}}$. Then,
$\mathsf{SF}(A)=\mathsf{U}(1)^{s}\times C_{m_{A}}$, where

$m_{A}=r\operatorname{lcm}\mathopen{}\mathclose{{}\left(n_{1}m_{E},m_{1},m_{2},\ldots,m_{s}}\right).$

The proof of this proposition follows from the following lemmas.

######  Lemma 3.3.3.

Let $B/\mathbf{F}_{q}$ be an abelian variety such that $B$ splits completely
over $\mathbf{F}_{q^{m}}$ as a power of an ordinary elliptic curve, for some
$m\geq 1$. Then, $\mathsf{SF}(B)=\mathsf{U}(1)\times C_{m}$.

###### Proof.

Angle rank is invariant under base change, so $\delta_{B}=\delta_{E^{g}}=1$.
It remains to show that the angle torsion order $m_{B}$ is equal to $m$. Since
$B_{(m)}\sim E^{g}$, we have that $P_{B,(m)}(T)=P_{E}(T)^{g}$. If we denote by
$\gamma_{1},\overline{\gamma}_{1},\ldots\gamma_{g},\overline{\gamma}_{g}$ and
$\pi_{1},\overline{\pi}_{1}$ the Frobenius eigenvalues of $B$ and $E$
respectively, we have that
$\mathopen{}\mathclose{{}\left\\{\gamma_{1}^{m},\overline{\gamma}_{1}^{m},\dots,\gamma_{g}^{m},\overline{\gamma}_{g}^{m}}\right\\}=\mathopen{}\mathclose{{}\left\\{\pi_{1},\overline{\pi}_{1}}\right\\}$.
Possibly after relabelling, we have that
$\gamma_{j}=\zeta_{m}^{\nu_{j}}\gamma_{1}$ for $j=1,\dots,g$ and at least one
$\zeta_{m}^{\nu_{j}}$ is a primitive $m$-th root. This shows that
$C_{m}\subset U_{B}$, so that $m\mid m_{B}$. On the other hand, we have that
$\mathsf{SF}(B_{(m)})=\mathsf{SF}(E^{g})\cong\mathsf{U}(1)$ is connected. This
implies that $m_{B}\mid m$ and the result follows. ∎

######  Lemma 3.3.4.

Let $A=A_{1}\times B$ be an abelian variety over $\mathbf{F}_{q}$ such that
$A_{1}$ is supersingular with angle torsion order $m_{A_{1}}=m_{1}$ and $B$ is
simple and splits completely over $\mathbf{F}_{q^{m}}$ as the power of an
ordinary elliptic curve. Then, $\mathsf{SF}(A)^{\circ}\cong\mathsf{U}(1)$ and
$m_{A}=\operatorname{lcm}(m_{1},m)$.

###### Proof.

From the discussion above, we see that
$U_{A}=\langle\zeta_{m_{1}},\zeta_{m},v_{1}\rangle$, where $v_{1}=$
$\gamma_{1}/\sqrt{q}$ and all the other roots $\gamma_{j}$ can be written as
$\zeta_{m}^{\nu_{j}}\gamma_{1}$ with at least one $\zeta_{m}^{\nu_{j}}$
primitive. It follows that $U_{A}=C_{\operatorname{lcm}(m_{1},m)}\oplus\langle
v_{1}\rangle$ so that $\delta_{A}=1$ and $m_{A}=\operatorname{lcm}(m_{1},m)$.
∎

######  Lemma 3.3.5.

If $B/\mathbf{F}_{q}$ is an ordinary abelian variety such that
$B\sim_{(r)}B_{1}\times\cdots\times B_{s}$ and satisfying

  1. (i)

each $B_{j}$ splits over $\mathbf{F}_{q^{m_{j}}}$ the power of an ordinary
elliptic curve $E_{j}/\mathbf{F}_{q^{m_{j}}}$, and

  2. (ii)

$E_{j}$ is not geometrically isogenous to $E_{i}$ for $i\neq j$.

then $\mathsf{SF}(B)\cong\mathsf{U}(1)^{s}\times C_{m_{B}}$ with
$m_{B}=r\operatorname{lcm}(m_{1},\dots,m_{s})$.

###### Proof.

This follows from combining Lemma 3.3.1 with the fact that the Serre–Frobenius
group of $B$ is connected over an extension of degree
$\operatorname{lcm}(m_{1},m_{2},\ldots,m_{s})$. The proof then proceeds as in
Lemma 3.3.3. ∎

###  3.4. Supersingular Serre–Frobenius groups

Recall that a $q$-Weil number $\alpha$ is called supersingular if
$\alpha/\sqrt{q}$ is a root of unity. In [39, Proposition 3.1], Zhu classified
the minimal polynomials $h(T)$ of supersingular $q$-Weil numbers. Let
$\Phi_{r}(T)$ denote the $r$th cyclotomic polynomial,
$\varphi(r)\colonequals\deg\Phi_{r}(T)$ the Euler totient function, and
$\mathopen{}\mathclose{{}\left(\tfrac{a}{b}}\right)$ the Jacobi symbol. Then
the possibilities for the minimal polynomials of supersingular $q$-Weil
numbers are given in Table 2.

Table 1. Minimal polynomial of a supersingular $q$-Weil number $\alpha$. Type | $d$ |  | $h(T)$ | Roots
---|---|---|---|---
Z-1 | Even | - | $\Phi_{m}^{[\sqrt{q}]}(T)\colonequals\sqrt{q}^{\varphi(m)}\Phi_{m}(T/\sqrt{q})$ | $\zeta_{m}^{j}\sqrt{q}\text{ for }j\in(\mathbf{Z}/m\mathbf{Z})^{\times}$
Z-2 | Odd | $\mathbf{Q}(\alpha)\neq\mathbf{Q}(\alpha^{2})$ | $\Phi_{n}^{[q]}(T^{2})\colonequals q^{\varphi(n)}\Phi_{n}(T^{2}/q)$ | $\pm\zeta_{2n}^{j}\sqrt{q}\text{ for }j\in(\mathbf{Z}/n\mathbf{Z})^{\times}$
Z-3 | Odd | $\mathbf{Q}(\alpha)=\mathbf{Q}(\alpha^{2})$ | $\displaystyle\prod_{\begin{subarray}{c}1\leq j\leq n\\\ \gcd(j,n)=1\end{subarray}}\mathopen{}\mathclose{{}\left(T-\mathopen{}\mathclose{{}\left(\dfrac{q}{j}}\right)\zeta_{m}^{\nu j}\sqrt{q}}\right)$ | $\mathopen{}\mathclose{{}\left(\dfrac{q}{j}}\right)\zeta_{m}^{\nu j}\sqrt{q}\text{ for }j\in(\mathbf{Z}/n\mathbf{Z})^{\times}$

######  Notation (Table 2).

In case (Z-1), $m$ is any positive integer. In cases (Z-2) and (Z-3), $m$
additionally satisfies $m\not\equiv 2\text{ mod }4$, and $n\colonequals
m/\gcd(2,m)$. The symbol $\zeta_{m}$ denotes the primitive $m$-th root of
unity given by $e^{\frac{2\pi i}{m}}$, and $\zeta_{m}^{\nu}$ is also
primitive. Note that in this case, $\varphi(n)=\varphi(m)/\gcd(2,m)$.
Following the notation in [26], given a polynomial $f(T)\in K[T]$ for some
field $K$, and a constant $a\in K^{\times}$, let

$f^{[a]}(T)\colonequals a^{\deg f}f(T/a).$

Given any supersingular abelian variety $A$ defined over $\mathbf{F}_{q}$, the
Frobenius polynomial $P_{A}(T)$ is a power of the minimal polynomial
$h_{A}(T)$, and this minimal polynomial is of type (Z-1), (Z-2), or (Z-3) as
above. We say that $A$ is of type Z-i if the minimal polynomial $h_{A}(T)$ is
of type (Z-i) for $i=1,2,3$.

Since $U_{A}$ is finite in the supersingular case, we have that
$\mathsf{SF}(A)\cong U_{A}$. In particular, we can read off the character
group $U_{A}$ from the fourth column in Table 2. For instance, if $m=3$ and
$d$ is even, then we have a polynomial of type Z-1, and the Serre–Frobenius
group is isomorphic to $C_{3}$. On the other hand, if $m=3$ and we have a
polynomial of type Z-2, then the Serre–Frobenius group is isomorphic to
$C_{6}$. Given a $q$-Weil polynomial $f(T)\in\mathbf{Q}[T]$ with roots
$\alpha_{1},\cdots,\alpha_{2n}$, the associated normalized polynomial
$\tilde{f}(T)\in\mathbf{R}[T]$ is the monic polynomial with roots
$u_{1}=\alpha_{1}/\sqrt{q},\dots,u_{2n}=\alpha_{2n}/\sqrt{q}$. Table 2 allows
us to go back and forth between $q$-Weil polynomials $f(T)$ and the normalized
polynomials $\tilde{f}(T)$.

  * •

If $h(T)$ is the minimal polynomial of a supersingular $q$-Weil number of type
Z-1, the normalized polynomial $\tilde{h}(T)$ is the cyclotomic polynomial
$\Phi_{m}(T)$. Conversely, we have that $h(T)=\tilde{h}^{[\sqrt{q}]}(T)$.

  * •

If $h(T)$ is the minimal polynomial of a supersingular $q$-Weil number of type
Z-2, the normalized polynomial $\tilde{h}(T)$ is the polynomial
$\Phi_{n}(T^{2})$. Conversely, $h(T)=\tilde{h}^{[q]}(T)$.

##  4\. Elliptic Curves

The goal of this section is to prove Theorem A. Furthermore, we give a
thorough description of the set of possible orders $m$ for the supersingular
Serre–Frobenius groups $\mathsf{SF}(E)=C_{m}$ in terms of $p$ and $q=p^{d}$.

The isogeny classes of elliptic curves over $\mathbf{F}_{q}$ were classified
by Deuring [3] and Waterhouse [31, Theorem 4.1]. Writing the characteristic
polynomial of Frobenius as $P(T)=T^{2}+a_{1}T+q$, the Weil bounds give
$|a_{1}|\leq 2\sqrt{q}$. Conversely, the integers $a$ in the interval $|a|\leq
2\sqrt{q}$ corresponding to the isogeny class of an elliptic curve are the
following.

######  Theorem 4.0.1 ([25, Theorem 2.6.1]).

Let $p$ be a prime and $q=p^{d}$. Let $a\in\mathbf{Z}$ satisfy $|a|\leq
2\sqrt{q}$.

  1. (1)

If $p\nmid a$, then $a$ is the trace of Frobenius of an elliptic curve over
$\mathbf{F}_{q}$. This is the ordinary case.

  2. (2)

If $p\mid a$, then $a$ is the trace of Frobenius of an elliptic curve over
$\mathbf{F}_{q}$ if and only if one of the following holds:

    1. (i)

$d$ is even and $a=\pm 2\sqrt{q}$,

    2. (ii)

$d$ is even and $a=\sqrt{q}$ with $p\not\equiv 1\text{ mod }3$,

    3. (iii)

$d$ is even and $a=-\sqrt{q}$ with $p\not\equiv 1\text{ mod }3$,

    4. (iv)

$d$ is even and $a=0$ with $p\not\equiv 1\text{ mod }4$,

    5. (v)

$d$ is odd and $a=0$,

    6. (vi)

$d$ is odd, $a=\pm\sqrt{2q}$ with $p=2$.

    7. (vii)

$d$ is odd, $a=\pm\sqrt{3q}$ with $p=3$.

This is the supersingular case.

$\sqrt{q}$$-a/2$$\alpha_{1}$$0$$\vartheta_{1}$

In the ordinary case, the normalized Frobenius eigenvalue $u_{1}$ is not a
root of unity, and thus $\mathsf{SF}(E)=\mathsf{U}(1)$. In the supersingular
case, the normalized Frobenius eigenvalue $u_{1}$ is a root of unity, and thus
$\mathsf{SF}(E)=C_{m}$ is cyclic, with $m$ equal to the order of $u_{1}$. For
each value of $q$ and $a$ in Theorem 4.0.1 part (2), we get a right triangle
of hypotenuse of length $\sqrt{q}$ and base $-a/2$, from which we can deduce
the angle $\vartheta_{1}$ and thus the order $m$ of the corresponding root of
unity $u_{1}$. We thus obtain the following restatement of Theorem 4.0.1 in
terms of the classification of Serre–Frobenius groups for elliptic curves.

Table 2. Serre–Frobenius groups of elliptic curves. Thm. 4.0.1 | $p$ | $d$ | $a$ | $\mathsf{SF}(E)$
---|---|---|---|---
(1) | - | - | $\gcd(a,p)=1$ | $\mathsf{U}(1)$
2-2(i) | - | Even | $\pm 2\sqrt{q}$ | $C_{1}$
2-2(iii) | $p\not\equiv 1\text{ mod }3$ | Even | $-\sqrt{q}$ | $C_{3}$
2-2(iv) | $p\not\equiv 1\text{ mod }4$ | Even | $0$ | $C_{4}$
2-2(v) | - | Odd | $0$ | $C_{4}$
2-2(ii) | $p\not\equiv 1\text{ mod }3$ | Even | $\sqrt{q}$ | $C_{6}$
2-2(vi) | 2 | Odd | $\pm\sqrt{2q}$ | $C_{8}$
2-2(vii) | 3 | Odd | $\pm\sqrt{3q}$ | $C_{12}$

There are seven Serre–Frobenius groups for elliptic curves, and they
correspond to seven possible Frobenius distributions of elliptic curves over
finite fields. For ordinary elliptic curves (as explained in Section 1), the
sequence of normalized traces $(x_{r})_{r=1}^{\infty}$ is equidistributed in
the interval $[-2,2]$ with respect to the measure $\lambda_{1}(x)$ (Equation
1.1) obtained as the pushforward of the Haar measure $\mu_{\mathsf{U}(1)}$
under $z\mapsto z+\overline{z}$. See Figure 2.

Figure 2. $a_{1}$-distribution for ordinary elliptic curves.

The remaining six Serre–Frobenius groups are finite and cyclic; they
correspond to supersingular elliptic curves. For a given
$C_{m}=\langle\zeta_{m}\rangle\subset\mathsf{U}(1)$, denote by $\delta_{m}$
the measure obtained by pushforward along $z\mapsto z+\overline{z}$ of the
normalized counting measure,

(4.1) $\mu_{C_{m}}(f)\colonequals\int
f\,\mu_{C_{m}}\colonequals\frac{1}{m}\sum_{j=1}^{m}f(\zeta_{m}^{j}).$

(a) $\mathsf{SF}(E)=C_{1}$.

(b) $\mathsf{SF}(E)=C_{3}$.

(c) $\mathsf{SF}(E)=C_{4}$

.

(d) $\mathsf{SF}(E)=C_{6}$

(e) $\mathsf{SF}(E)=C_{8}$

(f) $\mathsf{SF}(E)=C_{12}$.

Figure 3. $a_{1}$-histograms of supersingular elliptic curves
$E/\mathbf{F}_{q}$.

##  5\. Abelian Surfaces

The goal of this section is to classify the possible Serre–Frobenius groups of
abelian surfaces (Theorem B). The proof is a careful case-by-case analysis,
described by Flowchart 4.

Figure 4. Theorem B: Classification in dimension 2.

We separate our cases first according to $p$-rank, and then according to
simplicity. In the supersingular and almost ordinary cases this stratification
is enough. In the ordinary case, we have to further consider the geometric
isogeny type of the surface.

###  5.1. Simple ordinary surfaces

We restate a theorem of Howe and Zhu in our notation.

######  Theorem 5.1.1 ([13, Theorem 6]).

Suppose that $P(T)=T^{4}+a_{1}T^{3}+a_{2}T^{2}+qa_{1}T+q^{2}$ is the Frobenius
polynomial of a simple ordinary abelian surface $S$ defined over
$\mathbf{F}_{q}$. Then, exactly one of the following conditions holds:

  1. (a)

$S$ is absolutely simple.

  2. (b)

$a_{1}=0$ and $S$ splits over a quadratic extension.

  3. (c)

$a_{1}^{2}=q+a_{2}$ and $S$ splits over a cubic extension.

  4. (d)

$a_{1}^{2}=2a_{2}$ and $S$ splits over a quartic extension.

  5. (e)

$a_{1}^{2}=3a_{2}-3q$ and $S$ splits over a sextic extension.

######  Lemma 5.1.2 (Node S-A in Figure 4).

Let $S$ be a simple ordinary abelian surface over $\mathbf{F}_{q}$. Then,
exactly one of the following conditions holds:

  1. (a)

$S$ is absolutely simple and $\mathsf{SF}(S)\cong\mathsf{U}(1)^{2}$.

  2. (b)

$S$ splits over a quadratic extension and
$\mathsf{SF}(S)\cong\mathsf{U}(1)\times C_{2}$.

  3. (c)

$S$ splits over a cubic extension and $\mathsf{SF}(S)\cong\mathsf{U}(1)\times
C_{3}$.

  4. (d)

$S$ splits over a quartic extension and
$\mathsf{SF}(S)\cong\mathsf{U}(1)\times C_{4}$.

  5. (e)

$S$ splits over a sextic extension and $\mathsf{SF}(S)\cong\mathsf{U}(1)\times
C_{6}$.

Table 3. Serre–Frobenius groups of simple ordinary surfaces. Splitting type | $\mathsf{SF}(S)$ | Example
---|---|---
Absolutely simple | $\mathsf{U}(1)^{2}$ | 2.2.ab_b
Splits over quadratic extension | $\mathsf{U}(1)\times C_{2}$ | 2.2.a_ad
Splits over cubic extension | $\mathsf{U}(1)\times C_{3}$ | 2.2.ab_ab
Splits over quartic extension | $\mathsf{U}(1)\times C_{4}$ | 2.3.ac_c
Splits over sextic extension | $\mathsf{U}(1)\times C_{6}$ | 2.2.ad_f

###### Proof.

  1. (a)

From [38, Theorem 1.1], we conclude that some finite base extension of an
absolutely simple abelian surface is neat and therefore has maximal angle rank
by Remark (3.2.2.c). Alternatively, this also follows from the proof of [1,
Theorem 2] for Jacobians of genus 2 curves, which generalizes to any abelian
surface. Theorem E then implies that $\mathsf{SF}(S)=\mathsf{U}(1)^{2}$.

  2. (b,c,d,e)

Denote by $m$ the smallest degree of the extension
$\mathbf{F}_{q^{m}}\supset\mathbf{F}_{q}$ over which $S$ splits. By Theorem
5.1.1 we know that $m\in\mathopen{}\mathclose{{}\left\\{2,3,4,6}\right\\}$.
Let
$\alpha\in\mathopen{}\mathclose{{}\left\\{\alpha_{1},\overline{\alpha}_{1},\alpha_{2},\overline{\alpha}_{2}}\right\\}$
be a Frobenius eigenvalue of $S$. From [13, Lemma 4] and since $S$ is
ordinary, we have that
$[\mathbf{Q}(\alpha):\mathbf{Q}(\alpha^{m})]=[\mathbf{Q}(\alpha^{m}):\mathbf{Q}]=2$.
In particular, the minimal polynomial $h_{(m)}(T)$ of $\alpha^{m}$ is
quadratic, and $P_{(m)}(T)=h_{(m)}(T)^{2}$. This implies that
$\mathopen{}\mathclose{{}\left\\{\alpha_{1}^{m},\overline{\alpha}_{1}^{m}}\right\\}=\mathopen{}\mathclose{{}\left\\{\alpha_{2}^{m},\overline{\alpha}_{2}^{m}}\right\\}$,
so that there is a primitive $m$-th root of unity $\zeta$ giving one of the
following multiplicative relations:

$\alpha_{2}=\zeta\alpha_{1},\qquad\alpha_{2}=\zeta\overline{\alpha}_{1}.$

We note here that $\zeta$ must be a primitive $m$-th root, since otherwise,
$P_{n}(T)$ would split for some $n\leq m$, contradicting the minimality of
$m$. If $\alpha_{2}=\zeta\alpha_{1}$, then

$\mathsf{SF}(S)=\overline{\langle(u_{1},\zeta
u_{1})\rangle}=\mathopen{}\mathclose{{}\left\\{(u,\zeta^{k}u):u\in\mathsf{U}(1),k\in\mathbf{Z}/m\mathbf{Z}}\right\\}\cong\mathsf{U}(1)\times
C_{m}$

and $\mathsf{SF}(S)^{\circ}$ embeds diagonally in $\mathsf{U}(1)^{2}$.
Similarly, if $\alpha_{2}=\zeta\overline{\alpha}_{1}$, then
$\mathsf{SF}(S)\cong\mathsf{U}(1)\times C_{m}$ with embedding
$\mathsf{SF}(S)=\mathopen{}\mathclose{{}\left\\{(u,\zeta^{k}u^{-1}):u\in\mathsf{U}(1),k\in\mathbf{Z}/m\mathbf{Z}}\right\\}\subset\mathsf{U}(1)^{2}$.

∎

(a) 2.2.ab_b.

(b) 2.2.a_ad.

(c) 2.2.ab_ab.

(d) 2.3.ac_c.

(e) 2.2.ad_f.

Figure 5. $a_{1}$-histograms for simple ordinary abelian surfaces.

###  5.2. Non-simple ordinary surfaces

Let $S$ be a non-simple ordinary abelian surface defined over
$\mathbf{F}_{q}$. Then, $S$ is isogenous to a product of two ordinary elliptic
curves $E_{1}\times E_{2}$. As depicted in Figure 4, we consider two cases:

  1. (S-B)

$E_{1}$ and $E_{2}$ are not isogenous over $\overline{\mathbf{F}}_{q}$.

  2. (S-C)

$E_{1}$ and $E_{2}$ become isogenous over some base extension
$\mathbf{F}_{q^{m_{1}}}\supseteq\mathbf{F}_{q}$, for $m_{1}\geq 1$.

######  Lemma 5.2.1 (Node S-B in Figure 4).

Let $S$ be an abelian surface defined over $\mathbf{F}_{q}$ such that $S$ is
isogenous to $E_{1}\times E_{2}$, for $E_{1}$ and $E_{2}$ absolutely non-
isogenous ordinary elliptic curves. Then $S$ has maximal angle rank $\delta=2$
and $\mathsf{SF}(S)=\mathsf{U}(1)^{2}$.

The proof is a straightforward application of Lemma 3.3.1.

######  Lemma 5.2.2 (Node S-C in Figure 4).

Let $S$ be an abelian surface defined over $\mathbf{F}_{q}$ such that $S$ is
isogenous to $E_{1}\times E_{2}$, for $E_{1}$ and $E_{2}$ absolutely isogenous
ordinary elliptic curves. Then $S$ has angle rank $\delta=1$ and
$\mathsf{SF}(S)=\mathsf{U}(1)\times C_{m}$ for
$m\in\mathopen{}\mathclose{{}\left\\{1,2,3,4,6}\right\\}$. Furthermore, $m$ is
precisely the degree of the extension of $\mathbf{F}_{q}$ over which $E_{1}$
and $E_{2}$ become isogenous.

###### Proof.

Let $\alpha_{1},\overline{\alpha}_{1}$ and $\alpha_{2},\overline{\alpha}_{2}$
denote the Frobenius eigenvalues of $E_{1}$ and $E_{2}$ respectively. Let
$m_{1}$ be the smallest positive integer such that $E_{1}\sim_{(m_{1})}E_{2}$.
From Proposition 3.3.2, we immediately have that
$\mathsf{SF}(S)\cong\mathsf{U}(1)\times C_{m}$, where $m=m_{1}$. In order to
find the value of $m$, observe that
$\mathopen{}\mathclose{{}\left\\{\alpha_{1}^{m},\overline{\alpha}_{1}^{m}}\right\\}=\mathopen{}\mathclose{{}\left\\{\alpha_{2}^{m},\overline{\alpha}_{2}^{m}}\right\\}$,
from which we get one of the following multiplicative relations:

(5.1)
$\alpha_{2}=\zeta\alpha_{1},\qquad\alpha_{2}=\zeta\overline{\alpha}_{1},$

for some primitive $m$-th root of unity $\zeta$. Since the curves $E_{1}$ and
$E_{2}$ are ordinary, the number fields $\mathbf{Q}(\alpha_{1})$ and
$\mathbf{Q}(\alpha_{2})$ are imaginary quadratic and
$\mathbf{Q}(\alpha_{1})=\mathbf{Q}(\alpha_{1}^{m})=\mathbf{Q}(\alpha_{2}^{m})=\mathbf{Q}(\alpha_{2})$.
Hence, $\zeta\in\mathbf{Q}(\alpha_{1})$ and thus
$\varphi(m)=[\mathbf{Q}(\zeta):\mathbf{Q}]\in\mathopen{}\mathclose{{}\left\\{1,2}\right\\}$;
therefore $m\in\mathopen{}\mathclose{{}\left\\{1,2,3,4,6}\right\\}$. Depending
on whether $\alpha_{2}=\zeta\alpha_{1}$ or
$\alpha_{2}=\zeta\overline{\alpha}_{1}$, the group
$\mathsf{SF}(S)=\mathsf{U}(1)\times C_{m}$ embeds in $\mathsf{U}(1)^{2}$ as
$(u,\zeta^{r})\mapsto(u,\zeta^{r}u)$ or
$(u,\zeta^{r})\mapsto(u,\zeta^{r}u^{-1})$. ∎

Table 4. Serre–Frobenius groups of non-simple ordinary surfaces. $m$ such that $E_{1}\sim_{(m)}E_{2}$ | $\mathsf{SF}(E_{1}\times E_{2})$ | Example
---|---|---
1 | $\mathsf{U}(1)$ | 2.2.ac_f
2 | $\mathsf{U}(1)\times C_{2}$ | 2.2.a_d
3 | $\mathsf{U}(1)\times C_{3}$ | 2.7.af_s
4 | $\mathsf{U}(1)\times C_{4}$ | 2.5.ag_s
6 | $\mathsf{U}(1)\times C_{6}$ | 2.7.aj_bi

(a) 2.2.ac_f.

(b) 2.2.a_d.

(c) 2.7.af_s

(d) 2.5.ag_s

(e) 2.7.aj_bi.

Figure 6. $a_{1}$-histograms of non-simple ordinary abelian surfaces.

###  5.3. Simple almost ordinary surfaces

An abelian variety is called almost ordinary if the set of slopes of the
Newton polygon is $\mathopen{}\mathclose{{}\left\\{0,1/2,1}\right\\}$ and the
slope $1/2$ has length $2$. In [15] Lenstra and Zarhin carried out a careful
study of the multiplicative relations of Frobenius eigenvalues of simple
almost ordinary varieties, which was later generalized in [6]. In particular,
they prove that even-dimensional simple almost ordinary abelian varieties have
maximal angle rank ([15, Theorem 5.8]). Since every abelian surface of
$p$-rank $1$ is almost ordinary, their result allows us to deduce the
following:

######  Lemma 5.3.1 (Node S-D in Figure 4).

Let $S$ be a simple and almost ordinary abelian surface defined over
$\mathbf{F}_{q}$. Then, $S$ has maximal angle rank $\delta=2$ and
$\mathsf{SF}(S)=\mathsf{U}(1)^{2}$.

Figure 7. $a_{1}$-histogram of simple almost ordinary abelian surface
2.2.ab_a.

###  5.4. Non-simple almost ordinary surfaces

If $S$ is almost ordinary and not simple, then $S$ is isogenous to the product
of an ordinary elliptic curve $E_{1}$ and a supersingular elliptic curve
$E_{2}$.

######  Lemma 5.4.1 (Node S-E in Figure 4).

Let $S$ be a non-simple almost ordinary abelian surface defined over
$\mathbf{F}_{q}$. Then, $S$ has angle rank $\delta=1$ and
$\mathsf{SF}(S)\cong\mathsf{U}(1)\times C_{m}$ for some
$m\in\mathopen{}\mathclose{{}\left\\{1,3,4,6,8,12}\right\\}$.

###### Proof.

Let $E_{1}$ be an ordinary elliptic curve and $E_{2}$ a supersingular elliptic
curve such that $S\sim E_{1}\times E_{2}$. By Proposition 3.3.2,
$\mathsf{SF}(S)=\mathsf{SF}(E_{1})\times\mathsf{SF}(E_{2})\cong\mathsf{U}(1)\times
C_{m}$ with $m$ in the list of possible orders of Serre–Frobenius groups of
supersingular elliptic curves. ∎

###  5.5. Simple supersingular surfaces

Since every supersingular abelian variety is geometrically isogenous to a
power of an elliptic curve, the Serre–Frobenius group only depends on the
extension over which this occurs (Proposition 3.3.2). We separate our analysis
into the simple and non-simple cases.

The classification of Frobenius polynomials of supersingular abelian surfaces
over finite fields was completed by Maisner and Nart [17, Theorem 2.9]
building on work of Xing [34] and Rück [22]. Denoting by $(a_{1},a_{2})$ the
isogeny class of abelian surfaces over $\mathbf{F}_{q}$ with Frobenius
polynomial $P_{S}(T)=T^{4}+a_{1}T^{3}+a_{2}T^{2}+qa_{1}T+q^{2}$, the following
lemma gives the classification of Serre–Frobenius groups of simple
supersingular surfaces.

######  Lemma 5.5.1 (Node S-F in Table 4).

Let $S$ be a simple supersingular abelian surface defined over
$\mathbf{F}_{q}$. The Serre–Frobenius group of $S$ is classified according to
Table 6.

Table 5. Serre–Frobenius groups of simple supersingular surfaces. $(a_{1},a_{2})$ | $p$ | $d$ | $e$ | Type | $\tilde{h}(T)$ | $\mathsf{SF}(S)$
---|---|---|---|---|---|---
$(0,0)$ | $\not\equiv 1\text{ mod }8$ | even | $1$ | Z-1 | $\Phi_{8}(T)$ | $C_{8}$
$(0,0)$ | $\neq 2$ | odd | $1$ | Z-2 | $\Phi_{8}(T)$ | $C_{8}$
$(0,q)$ | - | odd | $1$ | Z-2 | $\Phi_{3}(T^{2})$ | $C_{6}$
$(0,-q)$ | $\not\equiv 1\text{ mod }12$ | even | $1$ | Z-1 | $\Phi_{12}(T)$ | $C_{12}$
$(0,-q)$ | $\neq 3$ | odd | $1$ | Z-2 | $\Phi_{6}(T^{2})=\Phi_{12}(T)$ | $C_{12}$
$(\sqrt{q},q)$ | $\not\equiv 1\text{ mod }5$ | even | $1$ | Z-1 | $\Phi_{5}(T)$ | $C_{5}$
$(-\sqrt{q},q)$ | $\not\equiv 1\text{ mod }5$ | even | $1$ | Z-1 | $\Phi_{10}(T)=\Phi_{5}(-T)$ | $C_{10}$
$(\sqrt{5q},3q)$ | $=5$ | odd | $1$ | Z-3 | $\Psi_{5,1}(T)$ | $C_{10}$
$(-\sqrt{5q},3q)$ | $=5$ | odd | $1$ | Z-3 | $\Psi_{5,1}(-T)$ | $C_{10}$
$(\sqrt{2q},q)$ | $=2$ | odd | $1$ | Z-3 | $\Psi_{2,3}(T)$ | $C_{24}$
$(-\sqrt{2q},q)$ | $=2$ | odd | $1$ | Z-3 | $\Psi_{2,3}(-T)$ | $C_{24}$
$(0,-2q)$ | - | odd | $2$ | Z-2 | $\Phi_{1}(T^{2})$ | $C_{2}$
$(0,2q)$ | $\equiv 1\text{ mod }4$ | even | $2$ | Z-1 | $\Phi_{4}(T)$ | $C_{4}$
$(2\sqrt{q},3q)$ | $\equiv 1\text{ mod }3$ | even | $2$ | Z-1 | $\Phi_{3}(T)$ | $C_{3}$
$(-2\sqrt{q},3q)$ | $\equiv 1\text{ mod }3$ | even | $2$ | Z-1 | $\Phi_{6}(T)=\Phi_{3}(-T)$ | $C_{6}$

The notation for polynomials of type Z-3 is taken from [26], where the authors
classify simple supersingular Frobenius polynomials for $g\leq 7$. We have

(5.2)
$\Psi_{5,1}(T)\colonequals\prod_{a\in(\mathbf{Z}/5)^{\times}}\mathopen{}\mathclose{{}\left(T-\mathopen{}\mathclose{{}\left(\tfrac{a}{5}}\right)\zeta_{5}^{a}}\right)=T^{4}+\sqrt{5}T^{3}+3T^{2}+\sqrt{5}T+1,$

and

(5.3)
$\Psi_{2,3}(T)\colonequals\prod_{a\in(\mathbf{Z}/3)^{\times}}\mathopen{}\mathclose{{}\left(T-\zeta_{8}\zeta_{3}^{a}}\right)\mathopen{}\mathclose{{}\left(T-\overline{\zeta}_{8}\zeta_{3}^{a}}\right)=T^{4}+\sqrt{2}T^{3}+T^{2}+\sqrt{2}T+1.$

We exhibit the proof of the second line in Table 6 for exposition. The
remaining cases can be checked similarly. If $(a_{1},a_{2})=(0,0)$, $p\neq 2$
and $q$ is an odd power of $p$: then,
$P(T)=T^{4}+q^{2}=\sqrt{q}^{4}\Phi_{8}(T/\sqrt{q})=q^{2}\Phi_{4}(T^{2}/q)$ and
$\tilde{h}(T)=\Phi_{8}(T)$. Thus $U_{S}$ is generated by a primitive 8th root
of unity.

(a) $\Psi_{5,1}(T)$.

(b) $\Psi_{5,1}(-T)$.

(c) $\Psi_{2,3}(T)$.

(d) $\Psi_{2,3}(-T)$.

Figure 8. Roots of Z-3 type normalized Frobenius polynomials in Table 6.

###  5.6. Non-simple supersingular surfaces

If $S$ is a non-simple supersingular surface, then $S$ is isogenous to a
product of two supersingular elliptic curves $E_{1}$ and $E_{2}$. If
$m_{E_{1}}$ and $m_{E_{2}}$ denote the torsion orders of $E_{1}$ and $E_{2}$
respectively, then the extension over which $E_{1}$ and $E_{2}$ become
isogenous is precisely $\operatorname{lcm}(m_{E_{1}},m_{E_{2}})$. Thus, by
Proposition 3.3.2, we have the following result, depending on the values of
$q=p^{d}$ as in Table 3.

######  Lemma 5.6.1 (Node S-G in Figure 4).

Let $S$ be a non-simple supersingular abelian surface defined over
$\mathbf{F}_{q}$. Then, $S$ has angle rank $\delta=0$ and
$\mathsf{SF}(S)=C_{m}$ for $m$ in the set $M=M(p,d)$ described in Figure 9.

(a) When $q$ is a square.

(b) When $q$ is not a square.

Figure 9. Sets $M$ of possible orders of the Serre–Frobenius groups of non-
simple supersingular surfaces as a function of $p$ and $q=p^{d}$.

##  6\. Abelian Threefolds

In this section, we classify the Serre–Frobenius groups of abelian threefolds
(see Figure 10). Let $X$ be an abelian variety of dimension $3$ defined over
$\mathbf{F}_{q}$. For our analysis, we will first stratify the cases by
$p$-rank and then by simplicity. Before we proceed, we make some observations
about simple threefolds that will be useful later.

Figure 10. Theorem C: Classification in dimension 3

###  6.1. Simple abelian threefolds

If $X$ is a simple abelian threefold, there are only two possibilities for the
Frobenius polynomial $P_{X}(T)=h_{X}(T)^{e}$:

(6.1) $\displaystyle P_{X}(T)=$ $\displaystyle\,h_{X}(T)$ (6.2) $\displaystyle
P_{X}(T)=$ $\displaystyle\,h_{X}(T)^{3}.$

Indeed, if $h_{X}(T)$ were a linear or cubic polynomial, it would have a real
root, $\pm\sqrt{q}$. By an argument of Waterhouse ([31, Chapter 2]), the
$q$-Weil numbers $\pm\sqrt{q}$ must come from simple abelian varieties of
dimension 1 or 2. Further, Xing [33] showed that 6.2 can only happen in very
special cases (see also [11, Proposition 1.2]).

######  Theorem 6.1.1 ([33], [11, Prop 1.2]).

Let $X$ be a simple abelian threefold over $\mathbf{F}_{q}$. Then,
$P_{X}(T)=h_{X}(T)^{3}$ if and only if $3$ divides $\log_{p}(q)$ and
$h_{X}(T)=T^{2}+aq^{1/3}T+q$ with $\gcd(a,p)=1$.

Note that in this case, $X$ is non-supersingular and has Newton Polygon as in
Figure 16.

Further, putting these observations together gives us that every simple
abelian threefold is either absolutely simple or is isogenous over an
extension to the cube of an elliptic curve. Thus, we have the following fact.

######  Fact 6.1.2.

If $X$ is an abelian threefold defined over $\mathbf{F}_{q}$ that is not
ordinary or supersingular, then $X$ is simple if and only if it is absolutely
simple.

###  6.2. Simple ordinary threefolds

In this section, $X$ will denote a simple ordinary threefold defined over
$\mathbf{F}_{q}$. As a corollary to Theorem 3.1.1, we have the following.

######  Proposition 6.2.1.

Let $X$ be a simple ordinary abelian threefold defined over $\mathbf{F}_{q}$.
Then, exactly one of the following conditions is satisfied.

  1. (1)

$X$ is absolutely simple.

  2. (2)

$X$ splits over a degree $3$ extension and $P_{X}(T)=T^{6}+a_{3}T^{3}+q^{3}$.

  3. (3)

$X$ splits over a degree $7$ extension and the number field of $P_{X}(T)$ is
$\mathbf{Q}(\zeta_{7})$.

######  Lemma 6.2.2 (Node X-A in Figure 10).

Let $X$ be an absolutely simple abelian threefold defined over
$\mathbf{F}_{q}$. Then $X$ has maximal angle rank $\delta=3$ and
$\mathsf{SF}(X)=\mathsf{U}(1)^{3}$.

###### Proof.

Let $m=m_{X}$ be the order of the torsion subgroup of $\Gamma_{X}$. By [38,
Theorem 1.1], we have that $X_{(m)}$ is neat. Since $X_{(m)}$ is ordinary and
simple, its Frobenius eigenvalues are distinct and non-real. Remark (3.2.2.b)
implies that $X_{(m)}$ has maximal angle rank. Since angle rank is invariant
under base extension (Remark 2.2.3) we have that $\delta(X)=\delta(X_{(m)})=3$
as we wanted to show. ∎

######  Lemma 6.2.3.

Let $X$ be a simple ordinary abelian threefold over $\mathbf{F}_{q}$ that is
not absolutely simple. Then $X$ has angle rank $1$ and

  1. (a)

$\mathsf{SF}(X)=\mathsf{U}(1)\times C_{3}$ if $X$ splits over a degree $3$
extension, or

  2. (b)

$\mathsf{SF}(X)=\mathsf{U}(1)\times C_{7}$ if $X$ splits over a degree $7$
extension.

###### Proof.

From the proof of Theorem 3.1.1, we have that the torsion free part of $U_{X}$
is generated by a fixed normalized root $u_{1}=\alpha_{1}/\sqrt{q}$, and all
other roots $u_{j}$ for $1<j\leq g$ are related to $u_{1}$ by a primitive root
of unity of order $3$ or $7$ respectively. ∎

######  Example 6.2.4.

The isogeny class 3.2.ad_f_ah is ordinary and absolutely simple. According to
Lemma 6.2.3, its Serre–Frobenius group is the full torus $\mathsf{U}(1)^{3}$
and the following histogram approximates the distribution corresponding to the
measure $\lambda_{3}$.

######  Example 6.2.5.

The isogeny class 3.2.a_a_ad is ordinary and simple, but it splits over a
degree $3$ extension as $\texttt{1.8.ad}^{3}$. According to Lemma 6.2.3, its
Serre–Frobenius group is $\mathsf{U}(1)\times C_{3}$, and the histogram
corresponding to this group is the following.

######  Example 6.2.6.

The isogeny class 3.2.ae_j_ap is ordinary and simple, but it splits over a
degree $7$ extension as $\texttt{1.128.an}^{3}$. According to Lemma 6.2.3, its
Serre–Frobenius group is $\mathsf{U}(1)\times C_{7}$, and the histogram
corresponding to this group is the following.

(a) $a_{1}$-histogram of 3.2.ad_f_ah.

(b) $a_{1}$-histogram of 3.2.a_a_ad.

(c) $a_{1}$ histogram of 3.2.ae_j_ap.

Figure 11. $a_{1}$-distributions for simple ordinary threefolds.

###  6.3. Non-simple ordinary threefolds

Let $X$ be a non-simple ordinary threefold defined over $\mathbf{F}_{q}$. Then
$X$ is isogenous to a product $S\times E$, for some ordinary surface $S$ and
some ordinary elliptic curve $E$.

The Frobenius polynomial of $X$ is the product of the Frobenius polynomials of
$S$ and $E$. Further, exactly one of the following is true for $S$: either it
is absolutely simple, or it is simple and geometrically isogenous to the power
of a single elliptic curve, or it is not simple (see observation after 5.1.2).
The Serre–Frobenius group of $X$ depends its geometric isogeny decomposition,
of which there are five possibilities:

  1. (6.3-a)

$X$ is geometrically isogenous to $E^{3}$.

  2. (6.3-b)

$X$ is geometrically isogenous to $E_{1}^{2}\times E$, for some ordinary
elliptic curve $E_{1}$, with $E_{1}\not\sim_{\overline{\mathbf{F}}_{q}}E$.

  3. (6.3-c)

$X$ is geometrically isogenous to $E_{1}\times E_{2}\times E$, for ordinary
and pairwise geometrically non-isogenous elliptic curves $E_{1},E_{2}$ and
$E$.

  4. (6.3-d)

$X$ is geometrically isogenous to $S\times E$ for an absolutely simple
ordinary surface $S$ and an ordinary elliptic curve $E$.

######  Lemma 6.3.1.

Let $X$ be a non-simple ordinary abelian threefold over $\mathbf{F}_{q}$. The
Serre–Frobenius group of $X$ is given by Table 7.

Table 6. Serre–Frobenius groups of non-simple ordinary threefolds. Geometric isogeny type | $\delta_{X}$ | $M$
---|---|---
(6.3-a) | $1$ | $\mathopen{}\mathclose{{}\left\\{1,2,3,4,6}\right\\}$
(6.3-b) | $2$ | $\mathopen{}\mathclose{{}\left\\{1,2,3,4,6}\right\\}$
(6.3-c) | $3$ | $\mathopen{}\mathclose{{}\left\\{1}\right\\}$
(6.3-d) | $3$ | $\mathopen{}\mathclose{{}\left\\{1}\right\\}$

###### Proof.

Recall that $X\sim S\times E$ over $\mathbf{F}_{q}$.
(6.3-a) If $X$ is geometrically isogenous to $E^{3}$, then $S$ is
geometrically isogenous to $E^{2}$. By Proposition 3.3.2
$\mathsf{SF}(X)=\mathsf{U}(1)\times C_{m}$, where $m$ is the smallest
extension over which $S\sim_{(m)}E^{2}$. By [13, Theorem 6], we have that
$m\in\\{1,2,3,4,6\\}$.
(6.3-b)

In this case, by Proposition 3.3.2, $\mathsf{SF}(X)=\mathsf{U}(1)^{2}\times
C_{m}$, where $m$ is the smallest extension over which $S\sim_{(m)}E_{1}^{2}$.
As in the previous case, $m\in\\{1,2,3,4,6\\}$.

(6.3-c) In this case $S\sim E_{1}\times E_{2}$ over the base field. By Lemma
3.3.1 we conclude that $\delta_{X}=3$.
(6.3-d) In this case, $X\sim S\times E$ with $S$ absolutely simple. By [38,
Theorem 1.1], we know that $X$ is neat. Since $X$ is ordinary and $S$ is
simple, all Frobenius eigenvalues are distinct and not supersingular. By
Remark (3.2.2.b), we conclude that $\delta_{X}=3$. ∎

######  Example 6.3.2 (Non-simple ordinary threefolds of splitting type
(6.3-a)).

  1. ($m=1$)

The isogeny class 3.2.ad_j_an is isogenous over the field of definition to
$\texttt{1.2.ab}^{3}$.

  2. ($m=2$)

The base change of 3.2.ab_f_ad over a quadratic extension is
$\texttt{1.4.d}^{3}$.

  3. ($m=3$)

The base change of 3.2.a_a_af over a cubic extension is $\texttt{1.8.af}^{3}$.

  4. ($m=4$)

The base change of 3.5.ak_bv_afc over a quartic extension is
$\texttt{1.625.o}^{3}$.

  5. ($m=6$)

The base change of 3.7.ao_di_alk over a degree $6$ extension is
$\texttt{1.117649.la}^{3}$.

(a) 3.2.ad_j_an.

(b) 3.2.ab_f_ad.

(c) 3.2.a_a_af

(d) 3.5.ak_bv_afc

(e) 3.7.ao_di_alk.

Figure 12. $a_{1}$-distributions for non-simple ordinary abelian threefolds of
splitting (6.3-a).

######  Example 6.3.3 (Non-simple ordinary threefolds of splitting type
(6.3-b)).

  1. ($m=1$)

The isogeny class 3.3.af_r_abi is isogenous to
$\texttt{1.3.ac}^{2}\times\texttt{1.3.ab}$.

  2. ($m=2$)

The base change of 3.2.ab_b_b over a quadratic extension is
$\texttt{1.4.ab}^{2}\times\texttt{1.4.d}$.

  3. ($m=3$)

The base change of 3.3.ad_d_ac over a cubic extension is
$\texttt{1.27.ai}^{2}\times\texttt{1.27.k}$.

  4. ($m=4$)

The base change of 3.3.af_p_abg over a quartic extension is
$\texttt{1.81.ao}^{2}\times\texttt{1.81.ah}$.

  5. ($m=6$)

The base change of 3.2.ae_k_ar over a sextic extension is
$\texttt{1.64.l}^{2}\times\texttt{1.64.aj}$.

(a) 3.3.af_r_abi.

(b) 3.2.ab_b_b

(c) 3.3.ad_d_ac

(d) 3.3.af_p_abg

(e) 3.2.ae_k_ar

Figure 13. $a_{1}$-distributions for non-simple ordinary abelian threefolds of
splitting (6.3-b).

###  6.4. Simple almost ordinary threefolds

Let $X$ be a simple and almost ordinary abelian threefold over
$\mathbf{F}_{q}$. Recall that $X$ is in fact absolutely simple, so that the
Frobenius polynomial $P_{(r)}(T)$ is irreducible for every positive integer
$r$.

Table 7. Serre–Frobenius groups of simple almost ordinary threefolds. Neat | $\sqrt{q}\in\mathbf{Q}(\alpha)$ | $\delta_{X}$ | $M$
---|---|---|---
Yes | - | $3$ |  $\\{$1$\\}$
No | Yes | $2$ |  $\\{$1, 2, 3, 4, 6$\\}$
No | No | $2$ |  $\\{$1, 2, 3, 4, 6, 8, 12$\\}$

######  Lemma 6.4.1.

Let $X$ be a simple almost ordinary abelian threefold over $\mathbf{F}_{q}$.
The Serre–Frobenius group of $X$ can be read from Table 8.

###### Proof.

Let $m\colonequals m_{X}$ be the torsion order of $U_{X}$, and consider the
base extension $Y\colonequals X_{(m)}$. By [15, Theorem 5.7], we know that
$\delta_{X}=\delta_{Y}\geq 2$. Furthermore, since $Y$ is absolutely simple, by
the discussion in Section 6.1, the roots of $P_{Y}(T)=P_{(m)}(T)$ are distinct
and non-supersingular. If $Y$ is neat, Remark (3.2.2.b) implies that
$\delta_{X}=\delta_{Y}=3$. Assume then that $Y$ is not neat, so that
$\delta_{X}=2$. Let $\alpha=\alpha_{1}$ be a Frobenius eigenvalue of $X$. By
[38, Theorem 1.1] and the discussion thereafter, we have that the sextic CM-
field $\mathbf{Q}(\alpha)=\mathbf{Q}(\alpha^{m})$ contains a quadratic
imaginary field $B$, and
$(u_{1}u_{2}u_{3})^{2m}=\mathrm{Norm}_{\mathbf{Q}(\alpha)/B}(u_{1}^{2m})=1$.
Since $U_{Y}$ has no torsion, this implies that $(u_{1}u_{2}u_{3})^{m}=1$.
Moreover, this means that $u_{1}u_{2}u_{3}=\zeta$ for some primitive444The
primitivity of $\zeta$ follows from the fact that $m$ is the minimal positive
integer such that $U_{(m)}$ is torsion free. $m$-th root of unity $\zeta$.
Therefore,

(6.3) $\zeta^{2}=\mathrm{Norm}_{\mathbf{Q}(\alpha)/B}(u_{1}^{2})\in B.$

If $m$ is odd, $\zeta^{2}$ is also primitive, so that $\varphi(m)\leq 2$ and
$m\in\mathopen{}\mathclose{{}\left\\{1,3}\right\\}$. If $m$ is even, then we
may distinguish between two cases. If $\sqrt{q}\in\mathbf{Q}(\alpha)$, we know
that $u_{1}\in\mathbf{Q}(\alpha)$ so that in fact
$\pm\zeta=\mathrm{Norm}_{\mathbf{Q}(\alpha)/B}(u_{1})\in B$ and
$\varphi(m)\leq 2$ implies that
$m\in\mathopen{}\mathclose{{}\left\\{2,4,6}\right\\}$. If
$\sqrt{q}\not\in\mathbf{Q}(\alpha)$, then $\zeta^{2}$ is a primitive
$m/2$-root of unity and
$m/2\in\mathopen{}\mathclose{{}\left\\{1,2,3,4,6}\right\\}$. ∎

###  6.5. Non-simple almost ordinary threefolds

Since $X$ is not simple, we have that $X\sim S\times E$ for some surface $S$
and some elliptic curve $E$. For this section, we let
$\pi_{1},\overline{\pi}_{1},\pi_{2},\overline{\pi}_{2}$ and
$\alpha,\overline{\alpha}$ be the Frobenius eigenvalues of $S$ and $E$
respectively. The normalized eigenvalues will be denoted by
$u_{1}\colonequals\pi_{1}/\sqrt{q},u_{2}=\pi_{2}/\sqrt{q}$ and
$u\colonequals\alpha/\sqrt{q}$. Instead of paragraph below: if $X$ has a
geometric supersingular factor, by Honda–Tate theory, it must have a
supersingular factor over the base field; and without loss of generality we
may assume that this factor is $E$.

######  Lemma 6.5.1.

Let $X\sim S\times E$ be a non-simple almost ordinary abelian threefold over
$\mathbf{F}_{q}$. The Serre–Frobenius group of $X$ can be read from Flowchart
14. In particular, if $X$ has no supersingular factor, then $\delta_{X}=3$. If
$E$ is supersingular, then
$\delta_{X}\in\mathopen{}\mathclose{{}\left\\{1,2}\right\\}$ and
$m_{X}=\operatorname{lcm}(m_{S},m_{E})$. The list of possible torsion orders
$m_{X}$ in this case is given by:

  1. (1)

$\delta_{X}=1$, $d$ even:
$M(p,d)=\mathopen{}\mathclose{{}\left\\{1,2,3,4,6,12}\right\\}$.

  2. (2)

$\delta_{X}=1$, $d$ odd:
$M(p,d)=\mathopen{}\mathclose{{}\left\\{4,12,24}\right\\}$.

  3. (3)

$\delta_{X}=2$: All possible orders in Table 3.

###### Proof.

First, suppose that $X$ has no supersingular factor. Thus $E$ is ordinary and
$S$ is almost ordinary and absolutely simple. This implies that
$\mathbf{Q}(\pi_{1}^{r})$ and $\mathbf{Q}(\alpha^{r})$ are CM-fields of
degrees $4$ and $2$ respectively, for every positive integer $r$. In
particular,
$\\#\mathopen{}\mathclose{{}\left\\{\pi_{1}^{r},\overline{\pi}_{1}^{r},\pi_{2}^{r},\overline{\pi}_{2}^{r},\alpha^{r},\overline{\alpha}^{r}}\right\\}$
$=6$ for every $r$. Let $m=m_{X}$ and consider the base extension $X_{(m)}$.
Since $X_{(m)}$ is not simple, [38, Theorem 1.1] implies that $X_{(m)}$ is
neat. The eigenvalues of $X_{(m)}$ are all distinct and not supersingular, so
that $\delta(X)=\delta(X_{(m)})=3$ by Remark (3.2.2.b).

Now, suppose that $X$ does have a supersingular factor, namely $E$. This
implies that $\delta_{X}\leq 2$ since $u=\alpha/\sqrt{q}=\zeta_{m_{E}}$ is a
root of unity. Since $S$ is ordinary in this case, we have that the sets
$\mathopen{}\mathclose{{}\left\\{u_{1},u}\right\\}$ and
$\mathopen{}\mathclose{{}\left\\{u_{2},u}\right\\}$ are multiplicatively
independent, so that $\delta_{X}=1\text{ or }2$ depending on the rank of the
subgroup $U_{S}\subset U_{X}$. Similarly, we see that
$U_{X}[\mathrm{tors}]=\langle\zeta_{m_{S}},\zeta_{m_{E}}\rangle$ and
$m_{X}=\operatorname{lcm}(m_{S},m_{E})$. If $S$ is simple, the result follows
from Lemma 5.1.2. If $S$ is not simple, the result follows from 5.2.1. ∎

Figure 14. Serre–Frobenius groups of non-simple almost ordinary threefolds.

###  6.6. Abelian threefolds of K3-type

In this section $X$ will be an abelian threefold defined over $\mathbf{F}_{q}$
of $p$-rank $1$. The $q$-Newton polygon of such a variety is give in Figure
15. This is the three-dimensional instance of abelian varieties of K3 type,
which were studied by Zarhin in [36] and [35].

Figure 15. $q$-Newton polygon of $p$-rank $1$ abelian threefolds.

######  Definition 6.6.1.

An abelian variety $A$ defined over $\mathbf{F}_{q}$ is said to be of K3-type
if the set of slopes is either $\mathopen{}\mathclose{{}\left\\{0,1}\right\\}$
or $\mathopen{}\mathclose{{}\left\\{0,1/2,1}\right\\}$, and the segments of
slope $0$ and $1$ have length one.

By [35, Theorem 5.9], simple abelian varieties of K3-type have maximal angle
rank. As a corollary, we have another piece of the classification.

######  Lemma 6.6.2 (Node X-F in Figure 10).

Let $X$ be a simple abelian threefold over $\mathbf{F}_{q}$ of $p$-rank $1$.
Then $X$ has maximal angle rank and $\mathsf{SF}(X)\cong\mathsf{U}(1)^{3}$.

Now assume that $X$ is not simple, so that $X\sim S\times E$ for some surface
$S$ and elliptic curve $E$.

######  Lemma 6.6.3 (Node X-G in Figure 10).

Let $X\sim S\times E$ be a non-simple abelian threefold over $\mathbf{F}_{q}$
of $p$-rank $1$. The Serre–Frobenius group of $X$ is given by Table 8.

###### Proof.

As in Section 6.3, we let
$\pi_{1},\overline{\pi}_{1},\pi_{2},\overline{\pi}_{2}$ and
$\alpha,\overline{\alpha}$ be the Frobenius eigenvalues of $S$ and $E$
respectively. Denote the normalized eigenvalues by
$u_{1}\colonequals\pi_{1}/\sqrt{q},u_{2}=\pi_{2}/\sqrt{q}$ and
$u\colonequals\alpha/\sqrt{q}$. We consider three cases:

  1. (6.6.3-a)

$S$ is simple and almost ordinary, and $E$ is supersingular.

  2. (6.6.3-b)

$S$ is non-simple and almost ordinary, and $E$ is supersingular.

  3. (6.6.3-c)

$S$ is supersingular and $E$ is ordinary.

Type | $\delta_{X}$ | $M$
---|---|---
(6.6.3-a) | $2$ | $\mathopen{}\mathclose{{}\left\\{1,3,4,6,8,12}\right\\}$
(6.6.3-b) | $1$ | Diagram 9.
(6.6.3-c) | $1$ | $\mathopen{}\mathclose{{}\left\\{1,2,3,4,5,6,8,10,12,24}\right\\}$
Table 8. Serre–Frobenius groups of abelian threefolds of $p$-rank 1.

Suppose first that $X$ is of type (6.6.3-a). By Lemma 5.3.1, the set
$\mathopen{}\mathclose{{}\left\\{u_{1},u_{2}}\right\\}$ is multiplicatively
independent. Since $u$ is a root of unity, $U_{X}=\langle
u_{1},u_{2},u\rangle=U_{S}\oplus U_{E}\cong\mathbf{Z}^{2}\oplus C_{m}$ for
$m\in M=\mathopen{}\mathclose{{}\left\\{1,3,4,6,8,12}\right\\}$ the set of
possible torsion orders for supersingular elliptic curves. Thus,
$\mathsf{SF}(X)\cong\mathsf{U}(1)^{2}\times C_{m}$ in this case.

If $X$ is of type (6.6.3-b), then $S\sim E_{1}\times E_{2}$ with $E_{1}$
ordinary and $E_{2}$ supersingular. By Proposition 3.3.2,
$\mathsf{SF}(X)\cong\mathsf{U}(1)\times C_{m}$, with $m$ in the set of
possible torsion orders of non-simple supersingular surfaces.

If $X$ is of type (6.6.3-c), we have $U_{X}=U_{E}\oplus
U_{S}\cong\mathbf{Z}\oplus C_{m}$ for $m$ in the set
$M=\mathopen{}\mathclose{{}\left\\{1,2,3,4,5,6,8,10,12,24}\right\\}$ of
possible torsion orders of supersingular surfaces from Lemmas 5.5.1 and 5.6.1.
∎

###  6.7. Absolutely simple p-rank 0 threefolds

In this section, $X$ will be a non-supersingular $p$-rank $0$ abelian
threefold over $\mathbf{F}_{q}$. From the $q$-Newton polygon of the Frobenius
polynomial $P(T)=P_{X}(T)$ (see Figure 16) we see that $X$ is absolutely
simple, since the slope $1/3$ does not occur for abelian varieties of smaller
dimension. Let $e_{r}^{2}$ denote the dimension of $\mathrm{End}^{0}(X_{(r)})$
over its center. We consider two cases:

  1. (6.7-a)

There exists $r\geq 1$ such that $e_{r}=3$. In this case we have
$P_{(r)}(T)=h_{(r)}(T)^{3}$ and $h_{(r)}(T)$ is as in Theorem 6.1.1, so that
$3$ divides $r\cdot\log_{p}(q)$.

  2. (6.7-b)

$e_{r}=1$ for every positive integer $r$.

Figure 16. $q$-Newton polygon of $p$-rank $0$ non-supersingular abelian
threefolds.

######  Lemma 6.7.1.

Let $X$ be an absolutely simple abelian threefold of $p$-rank $0$ defined over
$\mathbf{F}_{q}$. Then, the Serre–Frobenius group of $X$ is classified
according to Table 10. Furthermore, $X$ is of type (6.7-a), $m_{X}$ is the
smallest positive integer $r$ such that $e_{r}=3$.

Table 9. Serre–Frobenius groups of absolutely simple abelian threefolds of $p$-rank $0$. Type | $q=p^{d}$ | $\delta_{X}$ | $M$
---|---|---|---
(6.7-a) | $3\mid m_{X}\cdot d$ | $1$ | $\mathopen{}\mathclose{{}\left\\{1,3,7}\right\\}$
(6.7-b) | - | 3 | $\mathopen{}\mathclose{{}\left\\{1}\right\\}$

######  Remark 6.7.2.

The techniques for proving the Generalized Lenstra–Zarhin result in [6,
Theorem 1.5], cannot be applied to this case. Thus, even the angle rank
analysis in this case is particularly interesting.

###### Proof.

Suppose first that $X$ is of type (6.7-a), and let $m$ be the minimal positive
integer such that $e_{m}=3$. Maintaining previous notation,
$P_{(m)}(T)=h_{(m)}(T)^{3}$ implies that $\alpha_{2}=\zeta\cdot\alpha_{1}$ and
$\alpha_{3}=\xi\cdot\alpha_{1}$ for primitive $m$-th roots of unity $\zeta$
and $\xi$. By Proposition 2.3.1, this implies that
$\mathsf{SF}(X)\cong\mathsf{U}(1)\times C_{m}$. We conclude that
$\delta_{X}=1$ and $m=m_{X}$. To calculate the set $M$ of possible torsion
orders, assume that $m_{X}=m>1$. Then $\mathbf{Q}(\alpha_{1}^{m})$ is a
quadratic imaginary subextension of $\mathbf{Q}(\alpha_{1})\supset\mathbf{Q}$,
and we can argue as in the proof of Theorem 3.1.1 (with $\ell=3$) to conclude
that $m\in\mathopen{}\mathclose{{}\left\\{3,7}\right\\}$.

Assume now that $X$ is of type (6.7-b). This implies that
$\mathbf{Q}(\alpha_{1}^{r})$ is a degree $6$ CM-field for every positive
integer $r$. If $m:=m_{X}$, the base extension $X_{(m)}$ is neat and the
Frobenius eigenvalues are distinct and not supersingular. By Remark (3.2.2.b)
we have that $\delta_{X}=3$ and $m=1$. ∎

######  Example 6.7.3 (Histograms for X of type (6.7-a)).

  * ($m_{X}=1$)

The isogeny class 3.8.ag_bk_aea satisfies $m_{X}=1$. Note that $3$ divides
$m_{X}\cdot\log_{2}(8)$.

  * ($m_{X}=3$)

The isogeny class 3.2.a_a_ac has angle rank $1$ and irreducible Frobenius
polynomial $P(T)=T^{6}-2T^{3}+8$. The cubic base extension gives the isogeny
class 3.8.ag_bk_aea with reducible Frobenius polynomial
$P_{(3)}(T)=(T^{6}-2T^{3}+8)^{3}$. Note that $3$ divides
$m_{X}\cdot\log_{2}(2)$.

  * ($m_{X}=7$)

The isogeny class 3.8.ai_bk_aeq has angle rank $1$ and irreducible Frobenius
polynomial $P(T)=T^{6}-8T^{5}+36T^{4}-120T^{3}+288T^{2}-512T+512$. It’s base
change over a degree $m_{X}=7$ extension is the isogeny class
3.2097152.ahka_bfyoxc_adesazpwa with Frobenius polynomial

$P_{(7)}(T)=(T^{2}-1664T+2097152)^{3}.$

In this example, $q=8$, so that $3$ divides $m_{X}\cdot\log_{2}(8)$.

(a) 3.8.ag_bk_aea.

(b) 3.2.a_a_ac.

(c) 3.8.ai_bk_aeq.

Figure 17. $a_{1}$-distribution for $p$-rank 0 non-supersingular threefolds of
type (6.7-a).

###  6.8. Simple supersingular threefolds

Nart and Ritzensthaler [19] showed that the only degree $6$ supersingular
$q$-Weil numbers are the conjugates of:

$\displaystyle\pm\sqrt{q}\zeta_{7},\pm\sqrt{q}\zeta_{9},\quad\text{ when
}q\text{ is a square, and}$ $\displaystyle
7^{d/2}\zeta_{28},3^{d/2}\zeta_{36},\quad\text{ when }q\text{ is not a
square.}$

Building on their work, Haloui [11, Proposition 1.5] completed the
classification of simple supersingular threefolds. This classification is also
discussed in [26]; and we adapt their notation for the polynomials of Z-3
type. Denoting by $(a_{1},a_{2},a_{3})$ the isogeny class of abelian
threefolds over $\mathbf{F}_{q}$ with Frobenius polynomial
$P_{X}(T)=T^{6}+a_{1}T^{5}+a_{2}T^{4}+a_{3}T^{3}+qa_{2}T^{2}+q^{2}a_{1}T+q^{3}$,
the following lemma gives the classification of Serre–Frobenius groups of
simple supersingular threefolds, which is a corollary of Haloui’s result.

######  Lemma 6.8.1 (Node X-F in Figure 10).

Let $X$ be a simple supersingular abelian threefold defined over
$\mathbf{F}_{q}$. The Serre–Frobenius group of $X$ is classified according to
Table 11.

Table 10. Serre–Frobenius groups of simple supersingular threefolds. $(a_{1},a_{2},a_{3})$ | $p$ | $d$ | Type | $\tilde{h}(T)$ | $\mathsf{SF}(X)$
---|---|---|---|---|---
$(\sqrt{q},q,q\sqrt{q})$ | $7\nmid(p^{3}-1)$ | even | Z-1 | $\Phi_{7}(T)$ | $C_{7}$
$(-\sqrt{q},q,-q\sqrt{q})$ | $7\nmid(p^{3}-1)$ | even | Z-1 | $\Phi_{14}(T)$ | $C_{14}$
$(0,0,q\sqrt{q})$ | $\not\equiv 1\text{ mod }3$ | even | Z-1 | $\Phi_{9}(T)$ | $C_{9}$
$(0,0,-q\sqrt{q})$ | $\not\equiv 1\text{ mod }3$ | even | Z-1 | $\Phi_{18}(T)$ | $C_{18}$
$(\sqrt{7q},3q,q\sqrt{7q})$ | $=7$ | odd | Z-3 | $h_{7,1}(T)$ | $C_{28}$
$(-\sqrt{7q},3q,-q\sqrt{7q})$ | $=7$ | odd | Z-3 | $h_{7,1}(-T)$ | $C_{28}$
$(0,0,q\sqrt{3q})$ | $=3$ | odd | Z-3 | $h_{3,3}(T)$ | $C_{36}$
$(0,0,-q\sqrt{3q})$ | $=3$ | odd | Z-3 | $h_{3,3}(-T)$ | $C_{36}$

###### Proof.

By Xing’s theorem 6.1.1, we know that the Frobenius polynomial of all
supersingular threefolds $P_{X}(T)$ coincides with the minimal polynomial
$h_{X}(T)$ and $e=1$ in every row of the table.

The first four rows of Table 11 correspond to isogeny classes of type (Z-1).
By the discussion in Section 3.4, the minimal polynomials are of the
form555Recall that $f^{[a]}(T)\colonequals a^{\deg f}f(T/a)$.
$\Phi_{m}^{[\sqrt{q}]}(T)$ and the normalized polynomials are just the
cyclotomic polynomials $\Phi_{m}(T)$.

The last four rows of Table 11 correspond to isogeny classes of type (Z-3).
The normalized Frobenius polynomials are $h_{7,1}(\pm
T)=\,T^{6}\pm\sqrt{7}T^{5}+3T^{4}\pm\sqrt{7}T^{3}+3T^{2}\pm\sqrt{7}T+1$, and
$h_{3,3}(\pm T)=\,T^{6}\pm\sqrt{3}T^{3}+1.$ Noting that
$h_{7,1}(T)h_{7,1}(-T)=\Phi_{28}(T)$ and $h_{3,3}(T)h_{3,3}(-T)=\Phi_{36}(T)$
we conclude that the unit groups $U_{X}$ are generated by $\zeta_{28}$ and
$\zeta_{36}$ respectively. ∎

(a) $h_{7,1}(T)$.

(b) $h_{7,1}(-T)$.

(c) $h_{3,3}(T)$.

(d) $h_{3,3}(-T)$.

Figure 18. Roots of Z-3 type normalized Frobenius polynomials in Table 11.

###  6.9. Non-simple supersingular threefolds

If $X$ is a non-simple supersingular threefold over $\mathbf{F}_{q}$, then
there are two cases:

  1. (6.9.0-a)

$X\sim S\times E$, with $S$ a simple supersingular surface over
$\mathbf{F}_{q}$ and $E$ a supersingular elliptic curve.

  2. (6.9.0-b)

$X\sim E_{1}\times E_{2}\times E_{3}$, where each $E_{i}$ is a supersingular
elliptic curve.

The classification of the Serre–Frobenius group in these cases can be
summarized in the following lemma.

######  Lemma 6.9.1 (Node X-J in 10).

If $X$ is a non-simple supersingular threefold as in Case (6.9.0-a), then
$\mathsf{SF}(X)\cong C_{m}$, for $m\in M(p,d)$, where

  * •

If $d$ is even, $M(p,d)=\\{3,4,5,6,8,10,12,15,20,24,30\\}$,

  * •

If $d$ is odd, $M(p,d)=\\{{4,8,12,20,24}\\}$.

###### Proof.

In this case, $m=\operatorname{lcm}(m_{S},m_{E})$, since this is the degree of
the smallest extension over which the Serre–Frobenius group becomes connected.
The list of values for $m_{E}$ and $m_{S}$ come from Tables 3 and 6. ∎

######  Lemma 6.9.2 (Node X-J in 10).

If $X$ is a non-simple supersingular threefold as in Case (6.9.0-b), then
$\mathsf{SF}(X)\cong C_{m}$, for $m\in M(p,d)$, where

  * •

If $d$ is even, $M(p,d)=\\{1,3,4,6,12\\}$,

  * •

If $d$ is odd, $M(p,d)=\\{4,8,12\\}$.

###### Proof.

By Proposition 3.3.2, $m$ is the degree of the extension over which all the
elliptic curve factors $E_{i}$ become isogenous. This is precisely the least
common multiple of the $m_{E_{i}}$’s. From Table 3, we can calculate the
various possibilities for the $\operatorname{lcm}$’s depending on the parity
of $d$. ∎

##  7\. Simple ordinary abelian varieties of odd dimension

We conclude this article with a corollary of Theorem 3.1.1.

######  Theorem 7.0.1 (Restatement of Theorem 3.1.1).

Let $g>2$ be prime, and let $A$ be a simple ordinary abelian variety of
dimension $g$ over $\mathbf{F}_{q}$ that is not absolutely simple. Then $A$
has angle rank $1$ and

  1. (a)

$A$ splits over a degree $g$ extension and
$\mathsf{SF}(A)/\mathsf{SF}(A)^{\circ}\cong C_{g}$, or

  2. (b)

$2g+1$ is prime, $A$ splits over a degree $2g+1$ extension and
$\mathsf{SF}(A)/\mathsf{SF}(A)^{\circ}\cong C_{2g+1}$.

The proof of this lemma is the same as the proof of Lemma 6.2.3, so we do not
repeat it here. However, it would be interesting to have a more complete
result for simple ordinary abelian varieties of prime dimension; that is,
whether every ordinary absolutely simple abelian variety of prime dimension
$g>3$ has maximal angle rank. Tankeev [28] showed that the angle rank of any
absolutely simple abelian variety of prime dimension lies in
$\mathopen{}\mathclose{{}\left\\{1,g-1,g}\right\\}$. We also know from [6]
that a necessary condition for $\delta_{A}=g$ is that the _code_ is trivial.
Furthermore, the answer is negative when the dimension is not prime (see [4]).

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