# Abelian varieties over finite fields and their groups of rational points

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

We study the groups of rational points of abelian varieties defined over a
finite field $\mathbb{F}_{q}$ whose endomorphism rings are commutative, or,
equivalently, whose isogeny classes are determined by squarefree
characteristic polynomials. When $\mathrm{End}(A)$ is locally Gorenstein, we
show that the group structure of $A(\mathbb{F}_{q})$ is determined by
$\mathrm{End}(A)$. Moreover, we prove that the same conclusion is attained if
$\mathrm{End}(A)$ has local Cohen-Macaulay type at most $2$, under the
additional assumption that $A$ is ordinary or $q$ is prime. The result in the
Gorenstein case is used to characterize squarefree cyclic isogeny classes in
terms of conductor ideals. Going in the opposite direction, we characterize
squarefree isogeny classes of abelian varieties with $N$ rational points in
which every abelian group of order $N$ is realized as a group of rational
points. Finally, we study when an abelian variety $A$ over $\mathbb{F}_{q}$
and its dual $A^{\vee}$ succeed or fail to satisfy several interrelated
properties, namely $A\cong A^{\vee}$, $A(\mathbb{F}_{q})\cong
A^{\vee}(\mathbb{F}_{q})$, and $\mathrm{End}(A)=\mathrm{End}(A^{\vee})$. In
the process, we exhibit a sufficient condition for $A\not\cong A^{\vee}$
involving the local Cohen-Macaulay type of $\mathrm{End}(A)$. In particular,
such an abelian variety $A$ is not a Jacobian, or even principally
polarizable.

###### Key words and phrases:

Abelian variety, finite fields, group of rational points

###### 2020 Mathematics Subject Classification:

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

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

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

##  1\. Introduction

The groups of rational points of abelian varieties defined over a finite field
$\mathbb{F}_{q}$ have recently received a considerable amount of attention.
For example, in [HK21], Howe and Kedlaya showed that every positive integer
occurs as the order of the group of rational points of an abelian variety over
$\mathbb{F}_{2}$. In [vCL+21], Van Bommel, Costa, Li, Poonen and Smith prove,
among other results, a version of this statement over arbitrary finite fields
$\mathbb{F}_{q}$, although only sufficiently large orders are realizable if
$q\geq 7$.

These statement are, in fact, results about isogeny classes. Indeed, two
abelian varieties $A$ and $B$ are isogenous over $\mathbb{F}_{q}$ if and only
if $\\#A(\mathbb{F}_{q^{n}})=\\#B(\mathbb{F}_{q^{n}})$ for all $n\geq 1$, or
equivalently, if $h_{A}(x)=h_{B}(x)$, where $h_{A},h_{B}\in\mathbb{Z}[x]$ are
the characteristic polynomials of the Frobenius endomorphisms of $A$ and $B$,
respectively; see [Tat66, Thm. 1.(c)]. Moreover, one has
$\\#A(\mathbb{F}_{q})=h_{A}(1)$.

The recent results mentioned above concerning cardinalities were upgraded to
statements about finite abelian groups in [MS22]: for each $q$ in
$\\{2,3,4,5\\}$, we showed that every finite abelian group is isomorphic to
the group of rational points of some abelian variety over $\mathbb{F}_{q}$.
This upgrade, however, does not work on the level of isogeny classes. Indeed,
the group structure of $A(\mathbb{F}_{q})$ is not uniquely determined by its
isogeny class. This phenomenon is observed even for elliptic curves. In this
paper, we seek to understand and describe this extra level of structure.

### Notation and conventions

Before presenting our main results, we set some notation and conventions.
Throughout the paper, all isogenies and morphisms between abelian varieties
over $\mathbb{F}_{q}$ are defined over the base field $\mathbb{F}_{q}$. In
particular, given an abelian variety $A$ over $\mathbb{F}_{q}$ with
characteristic polynomial $h$, we denote its ($\mathbb{F}_{q}$-rational)
endomorphism ring by $\operatorname{End}(A)$ and its
($\mathbb{F}_{q}$-)isogeny class by $\mathcal{I}_{h}$. We say that $A$ and
$\mathcal{I}_{h}$ are _squarefree_ if $h$ is a squarefree polynomial. An
equivalent definition of squarefree is given by requiring the endomorphism
algebra $\operatorname{End}(A)\otimes_{\mathbb{Z}}\mathbb{Q}$ to be
commutative; see [Tat66, Thm. 2.(c)]. See also [MS22, Lem. 2.3] for a
comparison with other notions of squarefree.

Given a squarefree isogeny class $\mathcal{I}_{h}$ over $\mathbb{F}_{q}$, set
$K=\mathbb{Q}[x]/(h)$ and let $\pi$ be the class of $x$ in $K$. We will denote
by $\mathcal{O}_{K}$ the maximal order of $K$. For every $A$ in the isogeny
class, as in [Wat69, §3.1], we fix an isomorphism
$\operatorname{End}(A)\otimes_{\mathbb{Z}}\mathbb{Q}\cong K$ which sends the
Frobenius endomorphism of $A$ to $\pi$. From now on, we will identify
$\operatorname{End}(A)$ with its image inside $K$, which is an order. Under
this identification, the Rosati involution of $A$ acts as the complex
conjugation $x\mapsto\overline{x}$ in $K$. Note that $\overline{\pi}=q/\pi$.
In particular, if $\operatorname{End}(A)=S\subset K$ then
$\operatorname{End}(A^{\vee})=\overline{S}$, where $A^{\vee}$ denotes the dual
abelian variety of $A$.

###  1.1. Groups of rational points and endomorphism rings

As noted above, given an abelian variety $A$ over a finite field
$\mathbb{F}_{q}$, the sequence $(\\#A(\mathbb{F}_{q^{n}}))_{n\geq 1}$ of point
counts is an isogeny invariant, but the group structure of $A(\mathbb{F}_{q})$
is not. Classifying abelian varieties over finite fields up to isogeny is the
same as classifying them according to their endomorphism algebra. A more
precise classification is given by the endomorphism ring, which is an order in
the endomorphism algebra. When $E$ is an elliptic curve over $\mathbb{F}_{q}$,
the group structure of $E(\mathbb{F}_{q^{n}})$ is uniquely determined for all
$n\geq 1$ by the endomorphism ring $\operatorname{End}(E)$; see [Len96, Thm.
1].

In Main Theorem 1, we exhibit a similar result for abelian varieties of
arbitrary dimension, under certain hypotheses on the endomorphism ring which
are automatically satisfied in the case of elliptic curves. In particular, we
use the notion of the (Cohen-Macaulay) type of an order $S$ at a prime
$\mathfrak{p}$. This type, denoted $\operatorname{type}_{\mathfrak{p}}(S)$, is
defined as the minimal number of generators of
$(S^{t})_{\mathfrak{p}}=S^{t}\otimes_{S}S_{\mathfrak{p}}$, where
$S_{\mathfrak{p}}$ is the localization of $S$ at $\mathfrak{p}$ and $S^{t}$ is
the trace dual ideal of $S$. The order $S$ is Gorenstein at $\mathfrak{p}$ if
$\operatorname{type}_{\mathfrak{p}}(S)=1$. We say that $S$ is Gorenstein, if
it so at every prime. See Section 2 for definitions and details. Recall that
an abelian variety $A$ over $\mathbb{F}_{q}$ is called _ordinary_ if the
coefficient of $x^{\dim A}$ in the characteristic polynomial $h(x)$ of $A$ is
coprime to $q$.

######  Main Theorem 1.

Let $A$ be an abelian variety in a squarefree isogeny class $\mathcal{I}_{h}$
over $\mathbb{F}_{q}$. Write $S=\operatorname{End}(A)$ and fix $n\geq 1$.

  1. (a)

If $S$ is Gorenstein at all prime ideals containing $(1-\pi^{n})$, then

$A(\mathbb{F}_{q^{n}})\cong\frac{S}{(1-\pi^{n})S}\leavevmode\nobreak\ $

are isomorphic as $S$-modules.

  2. (b)

Assume $\mathcal{I}_{h}$ is ordinary _( Ord)_ or $q=p$ is prime _( CS)_. If
$\operatorname{type}_{\mathfrak{p}}(S)\leq 2$ for every prime $\mathfrak{p}$
of $S$ above $(1-\pi^{n})$, then

$A(\mathbb{F}_{q^{n}})\cong\frac{S}{(1-\pi^{n})S}$

are isomorphic as $\mathbb{Z}$-modules.

In contrast, in Example 6.7, we show that it is possible to have
$A(\mathbb{F}_{q})\not\cong B(\mathbb{F}_{q})$ for isogenous abelian varieties
$A$ and $B$ over $\mathbb{F}_{q}$, even if
$\operatorname{End}(A)=\operatorname{End}(B)$. This example, like all the
others in this paper, has been computed with the help of Magma [BCP97].

Part a, appearing in the text as Corollary 3.3, is proven by generalizing the
methods of [Len96] and [Spr21]. Specifically, we view the group of rational
points $A(\mathbb{F}_{q})$ of an abelian variety $A$ over $\mathbb{F}_{q}$ as
a module over the endomorphism ring $\operatorname{End}(A)$ and use the
Gorenstein property to describe the module, as desired.

For Part b, we use the additional assumptions to consider the abelian variety
$A$ itself to “be” a module over $\operatorname{End}(A)$, as we now explain.
In [Del69], Deligne constructed an equivalence between the category of
ordinary abelian varieties over a finite field $\mathbb{F}_{q}$ and the
category of free $\mathbb{Z}$-modules with a “Frobenius”-like endomorphism. In
[CS15], Centeleghe-Stix extended Deligne’s result, using a different functor,
to the category of abelian varieties over a prime field $\mathbb{F}_{p}$ whose
characteristic polynomial does not have real roots.

Given a squarefree isogeny class $\mathcal{I}_{h}$ over $\mathbb{F}_{q}$, we
write Ord for the condition that $\mathcal{I}_{h}$ is ordinary, and CS for the
condition that $q=p$ is prime. Note that, if $\mathcal{I}_{h}$ is squarefree,
then the characteristic polynomial $h$ does not have real roots. If we
restrict the functor of Deligne (resp. Centeleghe-Stix) to a particular
squarefree isogeny class $\mathcal{I}_{h}$ satisfying Ord (resp. CS), then the
modules in the image of the functor are precisely the fractional
$\mathbb{Z}[\pi,\overline{\pi}]$-ideals in the endomorphism algebra
$K=\mathbb{Q}[x]/(h)$. See Section 6 below, or [Mar21] for a detailed account.
In particular, this lends itself to another route for describing groups of
rational points in terms of orders and fractional ideals in the endomorphism
algebra; see Theorem 6.2. This description allows us to deduce Main Theorem
1.b, written as Proposition 6.5 below.

Beyond Main Theorem 1, the techniques and perspectives introduced in this
section continue to be used extensively throughout the paper. In Section 2, we
recall the necessary background and prove some foundational results regarding
orders in étale algebras, which we then use in the remainder of the paper.

###  1.2. Cyclicity

In Section 4, we study isogeny classes which are _cyclic_ , meaning that every
abelian variety in the isogeny class has a cyclic group of points. A criterion
for cyclicity which only involves the characteristic polynomial was given in
[GM19, Thm. 2.2]. Although this criterion applies _a priori_ to all isogeny
classes, we prove in Theorem 4.3 that an isogeny class over $\mathbb{F}_{q}$
is cyclic if and only if it contains a variety of the form
$A_{\text{sf}}\times A_{1}$ where $A_{\text{sf}}$ is squarefree and $A_{1}$
has only one rational point. Moreover, $A_{1}$ must be $0$-dimensional if
$q\geq 5$ by the Weil bounds.

We provide a new criterion for cyclicity, written below as Theorem 4.5.

######  Main Theorem 2.

Consider a squarefree isogeny class $\mathcal{I}_{h}$ of abelian varieties
over $\mathbb{F}_{q}$. Let $\pi$ be the class of $x$ in the endomorphism
algebra $K=\mathbb{Q}[x]/(h)$. The isogeny class $\mathcal{I}_{h}$ is cyclic
if and only if $(1-\pi)\mathbb{Z}[\pi,\overline{\pi}]$ is coprime to the
conductor $\mathfrak{f}=(\mathcal{O}_{K}:\mathbb{Z}[\pi,\overline{\pi}])$.

Rather than relating the property of cyclicity to the coefficients of the
characteristic polynomial as in [GM19], Main Theorem 2 relates cyclicity to
the algebraic properties of orders in the endomorphism algebra. In particular,
it shows that the property of cyclicity is equivalent to the local maximality
of the order $\mathbb{Z}[\pi,\overline{\pi}]$ generated by Frobenius and
Verschiebung at all primes over $(1-\pi)$. A key ingredient in our proof is
Main Theorem 1.a, applied to abelian varieties $A$ with maximal endomorphism
ring.

###  1.3. Richness and non-cyclic groups

In contrast to cyclic isogeny classes, we inspect the opposite extreme in
Section 5. We say that a squarefree isogeny class $\mathcal{I}_{h}$ is _rich_
if every abelian group of order $h(1)$ occurs as the group of rational points
of some abelian variety in $\mathcal{I}_{h}$. In previous work by the authors,
it was shown that, for each $N\geq 1$, there are infinitely many rich
squarefree isogeny classes $\mathcal{I}_{h}$ over $\mathbb{F}_{2}$ with
$N=h(1)$; see [MS22, Thm. 5.3]. These isogeny classes are built from Kedlaya’s
infinite sets of simple isogeny classes of abelian varieties over
$\mathbb{F}_{2}$ with prescribed numbers of points; see [Ked21, Thm. 1.1]. We
must use a different technique to find rich isogeny classes in general because
there are at most finitely many simple abelian varieties over $\mathbb{F}_{q}$
with a prescribed number of points $N$ when $q>2$ by [Kad21].

We present a criterion for richness in Main Theorem 3 which is easy to compute
using only the characteristic polynomial. An expanded statement is proved as
Theorem 5.7, and we compare the conditions of cyclicity and richness for
abelian varieties of small dimension over small finite fields in Example 5.9.

######  Main Theorem 3.

Consider a squarefree isogeny class $\mathcal{I}_{h}$ of abelian varieties
over $\mathbb{F}_{q}$ of dimension $g$. Let $K=\mathbb{Q}[x]/(h)$ be the
endomorphism algebra, and let $\pi$ be the class of $x$. Write
$N=h(1)=\prod_{j=1}^{s}\ell_{j}^{e_{j}}$ for the number of rational points on
each abelian variety in $\mathcal{I}_{h}$. The following are equivalent.

  1. (a)

$\mathcal{I}_{h}$ is rich, that is, every abelian group of order $N$ arises as
$A(\mathbb{F}_{q})$ for some $A\in\mathcal{I}_{h}$.

  2. (b)

For all $1\leq i\leq 2g$, we have

$\frac{h^{(i)}(1)}{i!}\cdot\ell_{1}^{i-e_{1}}\cdots\ell_{s}^{i-e_{s}}\in\mathbb{Z}.\leavevmode\nobreak\
$

To obtain this theorem, we first prove Lemma 5.4, generalizing a lemma of
Giangreco-Maidana [GM19, Lem. 2.1] which was originally used to study
cyclicity. We then deduce Main Theorem 3 by applying a result of Rybakov
[Ryb10, Thm. 1.1]. As a consequence, we also prove that a squarefree isogeny
class is rich if and only if its simple factors are rich; see Corollary 5.8.

We conclude Section 5 by proving the existence of some abelian varieties whose
groups of rational points have at least two generators. In particular, we show
in Corollary 5.11 that a squarefree isogeny class $\mathcal{I}_{h}$ over
$\mathbb{F}_{q}$ is non-cylic if $q$ is odd and $h(1)$ is divisible by $4$.
Finally, we prove the existence of ordinary abelian varieties over
$\mathbb{F}_{4}$ with certain prescribed non-cyclic groups of rational points
in Theorem 5.13, thereby improving [MS22, Thm. 3.3].

###  1.4. Duality

In Section 7, we turn our attention to the dual $A^{\vee}$ of an abelian
variety $A$. At the AMS MRC, _Explicit Methods in Arithmetic Geometry in
Characteristic $p$_, in June 2019, Bjorn Poonen suggested the problem of
finding an abelian variety $A$ defined over a finite field $\mathbb{F}_{q}$
such that $A(\mathbb{F}_{q})\not\cong A^{\vee}(\mathbb{F}_{q})$. In Example
7.2, we find such a variety by using Main Theorem 1.a. In this example, we
observe that $\operatorname{End}(A)$ is a Gorenstein order and
$\operatorname{End}(A)\neq\operatorname{End}(A^{\vee})$. In Example 7.3, we
show that these examples are not rare.

Section 7 concludes with a further investigation of the relationships between
these properties, along with the properties of being a Jacobian, principally
polarizable, or self-dual. More precisely, when $A$ is squarefree, consider
the following well-known implications, which are recalled below in Theorem
7.4.

$\textstyle{A(\mathbb{F}_{q})\cong
A^{\vee}(\mathbb{F}_{q})}$$\textstyle{A\cong\operatorname{Jac}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\text{
has a princ.
pol.}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\cong
A^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\textstyle{\operatorname{End}(A)=\operatorname{End}(A^{\vee})}$

Examples 7.5, 7.6 and 7.8, which are likely unsurprising to experts,
illustrate that none of the reverse implications are true. Additionally,
Example 7.9 exhibits an abelian variety $A$ over $\mathbb{F}_{3}$ such that
$A(\mathbb{F}_{3})\cong A^{\vee}(\mathbb{F}_{3})$ but
$\operatorname{End}(A)\neq\operatorname{End}(A^{\vee})$, hence there is no
downward implication on the right side of the diagram. In each case, there are
many suitable examples. On the other hand, it is unknown whether there are
examples where $A(\mathbb{F}_{q})\not\cong A^{\vee}(\mathbb{F}_{q})$ and
$\operatorname{End}(A)=\operatorname{End}(A^{\vee})$. Observe that, under the
hypotheses of either part of Main Theorem 1,
$\operatorname{End}(A)=\operatorname{End}(A^{\vee})$ implies that
$A(\mathbb{F}_{q})\cong A^{\vee}(\mathbb{F}_{q})$; see also Proposition 7.1.

Main Theorem 4 provides a sufficient condition for $A\not\cong A^{\vee}$ which
only depends on the properties of the orders in the endomorphism algebra. It
is a key ingredient for producing Example 7.8, and may be of independent
interest. It is proved in the text as Proposition 7.7.

######  Main Theorem 4.

Let $A$ be an abelian variety in a squarefree isogeny class $\mathcal{I}_{h}$
over $\mathbb{F}_{q}$, let $S$ be an order in $K=\mathbb{Q}[x]/(h)$ such that
$S=\overline{S}$, and let $\mathfrak{p}$ be a prime of $S$ satisfying
$\mathfrak{p}=\overline{\mathfrak{p}}$ and
$\operatorname{type}_{\mathfrak{p}}(S)=2$. Assume that $A$ is ordinary _(_ Ord
_)_ or that $q$ is prime _(_ CS _)_. If $S\subseteq\operatorname{End}(A)$ and
$S_{\mathfrak{p}}=\operatorname{End}(A)_{\mathfrak{p}}$, then $A\not\cong
A^{\vee}$. In particular, such an $A$ is not principally polarizable and
cannot be a Jacobian.

###  1.5. Related literature

We conclude the introduction by mentioning some additional related results.
There are several papers on the classification of the groups of rational
points of elliptic curves; see for example [Tsf85], [Rüc87], [TVN07] and
[Vol88]. The cases of abelian surfaces and threefolds were studied in [Xin94],
[Xin96], [Ryb12], [Ryb15], [DGS+14], and [Kot19]. Additional results about
cyclic isogeny classes can be found in [GM20], [GM21] and [BGM22].

### Acknowledgements

We thank Jonas Bergström, Valentijn Karemaker, and Bjorn Poonen for providing
comments on a draft version of this paper.

##  2\. Fractional ideals in orders

In this section we recall definitions and properties of orders and their
fractional ideals. These concepts are well-known in the context of number
fields, but we will work in a more general setting. Additional details and
proofs can be found in [Mar22, Sec. 2].

Let $Z$ be a Dedekind domain with field of fractions $Q$. In practice, for the
purpose of this paper, it will be enough to consider $Z=\mathbb{Z}$ and
$Z=\mathbb{Z}_{p}$. Let $K$ be a finite _étale algebra_ over $Q$, that is, a
finite product of finite separable extensions of $Q$. A $Z$-_lattice_ $L$ in
$K$ is a finitely generated free sub-$Z$-module of $K$ such that
$L\otimes_{Z}Q=K$. Given two lattices $L_{1}$ and $L_{2}$ in $K$, we define
the _colon_ as

$(L_{1}:L_{2})=\left\\{x\in K:xL_{2}\subseteq L_{1}\right\\}.$

A $Z$-_order_ $S$ in $K$ is a subring of $K$ which is also a $Z$-lattice.
Observe that $K$ is the total ring of quotients of any $Z$-order $S$ in $K$.
When no confusion can arise, we will drop the base ring $Z$ from the
terminology and simply write lattice and order. When $S\subseteq S^{\prime}$
are orders in $K$, the colon $(S:S^{\prime})$ is called the _conductor_ of $S$
in $S^{\prime}$.

A _fractional $S$-ideal_ $I$ is a finitely generated $S$-submodule of $K$
which is also a lattice. Given a lattice $L$ in $K$ we define its
_multiplicator ring_ as $(L:L)$. Observe that $(L:L)$ is a order in $K$ and
hence $L$ is a fractional $(L:L)$-ideal.

Given two lattices $I$ and $J$ in $K$ (resp. fractional $S$-ideals) then the
sum $I+J$, the intersection $I\cap J$, the product $IJ$, and the colon $(I:J)$
are lattices in $K$ (resp. fractional $S$-ideals).

Let $S$ be an order in $K$. A _prime_ of $S$ is a maximal ideal of $S$. We
denote by $S_{\mathfrak{p}}$ the localization of $S$ at $\mathfrak{p}$, and by
$\widehat{S}_{\mathfrak{p}}$ the completion of $S$ at $\mathfrak{p}$. For an
$S$-module $M$ we put $M_{\mathfrak{p}}=M\otimes_{S}S_{\mathfrak{p}}$ and
$\widehat{M}_{\mathfrak{p}}=M\otimes_{S}\widehat{S}_{\mathfrak{p}}$. Observe
that $\widehat{S}_{\mathfrak{p}}$ is a $\widehat{Z}_{p}$-order, where $p$ is
the contraction of $\mathfrak{p}$ in $Z$. Also, if $I$ is a fractional
$S$-ideal then $\widehat{I}_{\mathfrak{p}}$ is a fractional
$\widehat{S}_{\mathfrak{p}}$-ideal. We will say that $I$ is _principal at
$\mathfrak{p}$_ if $I_{\mathfrak{p}}$ is a principal
$S_{\mathfrak{p}}$-module, or, equivalently, $\widehat{I}_{\mathfrak{p}}$ is a
principal fractional $\widehat{S}_{\mathfrak{p}}$-ideal. A fractional
$S$-ideal $I$ is called _invertible_ if $I(S:I)=S$ or, equivalently, if it is
principal at $\mathfrak{p}$ for every prime $\mathfrak{p}$ of $S$. See for
example [Mar22, Lem. 2.12 and 2.17]. Given orders $S\subseteq S^{\prime}$ in
$K$, we can consider $S^{\prime}$ as a fractional $S$-ideal. Lemma 2.1 below
tells us when $S^{\prime}$ is principal at a prime $\mathfrak{p}$ of $S$.

######  Lemma 2.1.

Let $S\subseteq S^{\prime}$ be orders. Given a prime $\mathfrak{p}$ of $S$,
the following statements are equivalent:

  1. (a)

$(S:S^{\prime})\subseteq\mathfrak{p}$.

  2. (b)

${S^{\prime}}_{\mathfrak{p}}$ is not a principal $S_{\mathfrak{p}}$-module.

  3. (c)

$S_{\mathfrak{p}}\neq{S^{\prime}}_{\mathfrak{p}}$.

###### Proof.

We first prove that ${S^{\prime}}_{\mathfrak{p}}$ is a principal
$S_{\mathfrak{p}}$-module if and only if
$S_{\mathfrak{p}}={S^{\prime}}_{\mathfrak{p}}$. One implication is trivial.
For the other, assume that ${S^{\prime}}_{\mathfrak{p}}=\alpha
S_{\mathfrak{p}}$. Hence $\alpha\in{S^{\prime}}_{\mathfrak{p}}^{\times}$,
which shows that

$S_{\mathfrak{p}}=\frac{1}{\alpha}{S^{\prime}}_{\mathfrak{p}}={S^{\prime}}_{\mathfrak{p}}.$

To conclude, observe that
$S_{\mathfrak{p}}\subsetneq{S^{\prime}}_{\mathfrak{p}}$ if and only if
$(S_{\mathfrak{p}}:{S^{\prime}}_{\mathfrak{p}})\subsetneq S_{\mathfrak{p}}$,
which occurs if and only if $(S:S^{\prime})\subseteq\mathfrak{p}$. ∎

Later in the paper we will study groups of rational points of abelian
varieties over finite fields by describing them in terms of quotient of
fractional ideals. In turn, we describe such quotients locally.

######  Lemma 2.2.

Let $J\subseteq I$ be fractional ideals over an order $S$. Then we have an
isomorphism of $S$-modules

$\frac{I}{J}\simeq\bigoplus_{\mathfrak{p}}\left(\frac{I}{J}\right)_{\mathfrak{p}},$

where the direct sum is over the finitely many primes for which
$I_{\mathfrak{p}}\neq J_{\mathfrak{p}}$.

###### Proof.

The quotient $I/J$ is a finitely generated torsion $Z$-module; see for example
[Coh00, Sec. 1.7]. Hence, $I/J$ is an Artinian and Noetherian $S$-module, so
the result follows from [Eis95, Thm. 2.13.(b)]. ∎

######  Proposition 2.3.

Let $S\subsetneq S^{\prime}$ be orders. If $r\in S$ is not a zero-divisor and
$rS$ is not coprime to the conductor $\mathfrak{f}=(S:S^{\prime})$, then the
quotient $M=S^{\prime}/rS^{\prime}$ is not a cyclic $S$-module.

###### Proof.

By assumption there exists a maximal ideal $\mathfrak{p}$ of $S$ such that

$rS+\mathfrak{f}\subseteq\mathfrak{p},$

which implies

$rS^{\prime}+\mathfrak{f}\subseteq\mathfrak{p}S^{\prime}.$

Since $\mathfrak{p}$ is above the conductor $\mathfrak{f}$, by Lemma 2.1 we
have that ${S^{\prime}}_{\mathfrak{p}}$ is not a principal
$S_{\mathfrak{p}}$-module, or equivalently the $S/\mathfrak{p}$-vector space
$S^{\prime}/\mathfrak{p}S^{\prime}$ has dimension strictly bigger than $1$.
Observe that

$\frac{M}{\mathfrak{p}M}\cong\dfrac{S^{\prime}/rS^{\prime}}{\mathfrak{p}S^{\prime}/rS^{\prime}}\cong\frac{S^{\prime}}{\mathfrak{p}S^{\prime}}.$

Hence the $S_{\mathfrak{p}}$-module $M_{\mathfrak{p}}$ is not cyclic. By Lemma
2.2, we conclude that $M$ is not a cyclic $S$-module. ∎

The isomorphism class of a quotient of fractional ideals can be deduced from
local information at finitely many primes, as explained in the following
lemma.

######  Lemma 2.4.

Let $I$ and $J$ be fractional ideals over an order $S$, and let $r\in S$ be a
non-zero divisor. Let $\mathcal{S}$ be the set of primes of $S$ containing
$r$. Assume that we have an $S_{\mathfrak{p}}$-linear isomorphism
$\varphi_{\mathfrak{p}}:I_{\mathfrak{p}}\overset{\sim}{\to}J_{\mathfrak{p}}$
for every $\mathfrak{p}\in\mathcal{S}$. Then there is an $S$-linear
isomorphism

$\psi:\frac{I}{rI}\overset{\sim}{\longrightarrow}\frac{J}{rJ},$

such that $\psi\otimes S_{\mathfrak{p}}=\varphi_{\mathfrak{p}}\otimes(S/rS)$
for every prime $\mathfrak{p}\in\mathcal{S}$.

###### Proof.

Set $M=I/rI$ and $N=J/rJ$. By assumption, we have isomorphisms

$\varphi_{\mathfrak{p}}\otimes\left(\frac{S}{rS}\right):M_{\mathfrak{p}}\overset{\sim}{\longrightarrow}N_{\mathfrak{p}},$

for each $\mathfrak{p}\in\mathcal{S}$. We now claim that, given a prime
$\mathfrak{p}$ of $S$, we have $M_{\mathfrak{p}}\neq 0$ if and only if
$r\in\mathfrak{p}$. If $r\not\in\mathfrak{p}$ then $(S/rS)_{\mathfrak{p}}=0$
and hence $M_{\mathfrak{p}}=(I\otimes_{S}(S/rS))=0$ as well. Conversely, if
$r\in\mathfrak{p}$ then $rI\subseteq\mathfrak{p}I$ and hence we obtain a
surjective map $M_{\mathfrak{p}}\to I/\mathfrak{p}I$, which shows that
$M_{\mathfrak{p}}\neq 0$, completing the proof of the claim. The same is true
for the $S$-module $N$.

By Lemma 2.2, we have

$M\cong\bigoplus_{\mathfrak{p}\in\mathcal{S}}M_{\mathfrak{p}}\text{ and
}N\cong\bigoplus_{\mathfrak{p}\in\mathcal{S}}N_{\mathfrak{p}}.$

We conclude by setting

$\psi=\bigoplus_{\mathfrak{p}\in\mathcal{S}}\varphi_{\mathfrak{p}}\otimes\left(\frac{S}{rS}\right).$

∎

The étale algebra $K$ comes equipped with a _trace_ map

$\operatorname{Tr}_{K/Q}:K\to Q$

that associates to every element $x\in K$ the trace of the matrix representing
the multiplication-by-$x$ map with respect to any $Q$-basis of $K$. The
existence of such a non-degenerate trace implies that the integral closure
$\mathcal{O}_{K}$ of $Z$ in $K$ is an order, called the _maximal order_ ,
since every other order is contained in $\mathcal{O}_{K}$. Recall that
$\mathcal{O}_{K}$ is characterized by the fact that every localization is a
principal ideal ring.

The following proposition refines Lemma 2.4, in the sense that we only need
local information above the conductor to understand the isomorphism class.

######  Proposition 2.5.

Let $S$ be an order in $K$ and let $r$ be a non-zero divisor of $K$. Assume
that $rS$ is coprime to the conductor $\mathfrak{f}=(S:\mathcal{O}_{K})$. Then
for every fractional $S$-ideal $I$ we have an $S$-linear isomorphism

$\frac{I}{rI}\cong\frac{S}{rS}.$

###### Proof.

Because $rS$ is coprime to the conductor, by Lemma 2.1 we have
$S_{\mathfrak{p}}=\mathcal{O}_{K,\mathfrak{p}}$ for any prime $\mathfrak{p}$
containing $r$. This implies that $I_{\mathfrak{p}}\cong S_{\mathfrak{p}}$ for
every such prime $\mathfrak{p}$. We conclude by Lemma 2.4. ∎

Given a lattice $L$ in $K$ we define its _trace dual_ as

$L^{t}=\left\\{x\in K:\operatorname{Tr}_{K/Q}(xL)\subseteq Z\right\\}.$

In the following lemma, we record some well known properties of trace duals.

######  Lemma 2.6.

Let $L$,$L_{1}$ and $L_{2}$ be lattices and let $S$ be an order in $K$. Then

  1. (a)

$(L^{t})^{t}=L$.

  2. (b)

$(L_{1}:L_{2})=(L_{1}^{t}L_{2})^{t}\leavevmode\nobreak\ $.

  3. (c)

$(L_{1}:L_{2})=(L_{2}^{t}:L_{1}^{t})$.

  4. (d)

$(L:L)=S$ if and only if $LL^{t}=S^{t}$.

###### Proof.

See for example [Voi21, Sec. 15.6]. ∎

Let $S$ be an order, $\mathfrak{p}$ be a prime of $S$ and $p$ its contraction
in $Z$. Denote by $\widehat{Q}_{p}$ the fraction field of $\widehat{Z}_{p}$
and by $\widehat{K}_{\mathfrak{p}}$ the total ring of quotient of
$\widehat{S}_{\mathfrak{p}}$. The trace $\operatorname{Tr}_{K/Q}$ naturally
induces a trace
$\operatorname{Tr}_{\widehat{K}_{\mathfrak{p}}/\widehat{Q}_{p}}$. It follows
that taking trace duals commutes with completion. So, given a fractional
$S$-ideal $I$, the notation $\widehat{I}^{t}_{\mathfrak{p}}$ is not ambiguous.

######  Lemma 2.7.

Let $S$ be an order and $\mathfrak{p}$ be a prime of $S$. For fractional
$S$-ideals $I$ and $J$, we have that $I_{\mathfrak{p}}\cong J_{\mathfrak{p}}$
as $S_{\mathfrak{p}}$-modules if and only if
$(I^{t})_{\mathfrak{p}}\cong(J^{t})_{\mathfrak{p}}$.

###### Proof.

By [Eis95, Ex. 7.5, p. 203], it is enough to prove the equivalent statement
with completion instead of localization. Assume that
$\widehat{I}_{\mathfrak{p}}\cong\widehat{J}_{\mathfrak{p}}$, that is,
$\widehat{I}_{\mathfrak{p}}=\alpha\widehat{J}_{\mathfrak{p}}$ for some
$\alpha\in\widehat{K}^{\times}_{\mathfrak{p}}$. Then
$\widehat{I}^{t}_{\mathfrak{p}}=\alpha^{-1}\widehat{J}_{\mathfrak{p}}^{t}$.
Hence $\widehat{I}^{t}_{\mathfrak{p}}\cong\widehat{J}_{\mathfrak{p}}^{t}$, as
required. We used here that completions and trace duals commute, as noted
above. The converse direction follows from Lemma 2.6.a. ∎

Recall that Lemma 2.4 provides an $S$-linear isomorphism of quotients of
fractional ideals, under certain local conditions. However, when comparing a
fractional ideal and its dual, we can obtain a $Z$-linear isomorphism between
quotients, regardless of the local behavior.

######  Lemma 2.8.

Let $S$ be an order with fractional ideals $I$ and $J$, and let $r\in S$ be a
non-zero divisor. Then we have a $Z$-linear isomorphism

$\frac{I}{rI}\cong\frac{I^{t}}{rI^{t}}.$

###### Proof.

This is a special case of [Mar22, Lem. 2.4.(iv)], which is an application of
Matlis duality; see for example [Ooi76, Thm. 1.7]. ∎

We now recall some properties of orders that were studied in [Mar22, Sec. 3].
The _Cohen-Macaulay type_ of an order $S$ at a prime $\mathfrak{p}$, denoted
by $\operatorname{type}_{\mathfrak{p}}(S)$, is the minimal number of
generators of $(S^{t})_{\mathfrak{p}}$ as an $S_{\mathfrak{p}}$\- This
definition is equivalent to the usual one; see [Mar22, Sec. 3]. We say that an
order $S$ is _Gorenstein_ at a prime $\mathfrak{p}$ if its Cohen-Macaulay
$\operatorname{type}_{\mathfrak{p}}(S)$ is equal to $1$, that is, if
$(S^{t})_{\mathfrak{p}}$ is a principal $S_{\mathfrak{p}}$-module. We say that
$S$ is Gorenstein if it is so at every prime. This definition of Gorenstein is
equivalent to the ones typically used in the literature. For example, a ring
with finite (Krull) dimension is called Gorenstein if it has finite injective
dimension. See [Bas63, Sec. 1] for other equivalent definition, and see
[Bas63, Thm. 6.3] and [BL94, Prop. 2.7] for the proof of the equivalence with
the one used in this paper. In fact, using the latter reference, one can
deduce the following lemma. We give a complete proof for convenience.

######  Proposition 2.9.

Let $S$ be an order and $\mathfrak{p}$ be prime of $S$. Then $S$ is Gorenstein
at $\mathfrak{p}$ if and only if every fractional $S$-ideal $I$ with
$(I:I)_{\mathfrak{p}}=S_{\mathfrak{p}}$ is principal at $\mathfrak{p}$.

###### Proof.

Observe that $S$ is Gorenstein at $\mathfrak{p}$ if and only if
$\widehat{S}^{t}_{\mathfrak{p}}=\alpha\widehat{S}_{\mathfrak{p}}$ for some
$\alpha\in\widehat{K}^{\times}_{\mathfrak{p}}$. Assume now that $S$ is
Gorenstein at $\mathfrak{p}$. Pick a fractional $S$-ideal $I$ with
$(I:I)_{\mathfrak{p}}=S_{\mathfrak{p}}$. By taking the completion we get that
$\widehat{(I:I)}_{\mathfrak{p}}=\widehat{S}_{\mathfrak{p}}$. By Lemma 2.6.d,
we obtain that
$\widehat{I}_{\mathfrak{p}}\widehat{I}^{t}_{\mathfrak{p}}=\widehat{S}^{t}_{\mathfrak{p}}=\alpha\widehat{S}_{\mathfrak{p}}$.
Hence, $\widehat{I}_{\mathfrak{p}}$ is invertible. Because invertible
fractional $\widehat{S}_{\mathfrak{p}}$-ideals are principal, we get that
$I_{\mathfrak{p}}$ is a principal $S_{\mathfrak{p}}$-module, as required. For
the converse, it is enough to observe that $S^{t}$ has multiplicator ring $S$.
∎

Similarly to the Gorenstein case, locally, we have a classification of
fractional ideals with multiplicator ring of type $2$.

######  Proposition 2.10 ([Mar22, Thm. 6.2]).

Let $S$ be an order and $\mathfrak{p}$ a prime of $S$ such that
$\operatorname{type}_{\mathfrak{p}}(S)=2$. Then for every fractional $S$-ideal
$I$ such that $(I:I)_{\mathfrak{p}}=S_{\mathfrak{p}}$ then either
$I_{\mathfrak{p}}\cong S_{\mathfrak{p}}$ or
$I_{\mathfrak{p}}\cong(S^{t})_{\mathfrak{p}}$.

We exploit the classifications of Proposition 2.9 and Proposition 2.10 to
understand quotient of fractional ideals, as we show in the next proposition.
We invite the reader to compare the statement with Proposition 2.5, where the
isomorphism is $S$-linear, while here we only get a $Z$-linear isomorphism.

######  Proposition 2.11.

Let $r\in S$ be a non-zero divisor. If
$\operatorname{type}_{\mathfrak{p}}(S)\leq 2$ for all primes $\mathfrak{p}$ of
$S$ containing $r$ then for all fractional $S$-ideal $I$ with
$(I:I)_{\mathfrak{p}}=S_{\mathfrak{p}}$ we have a $Z$-linear isomorphism

$\frac{I}{rI}\cong\frac{S}{rS}.$

###### Proof.

Fix $\mathfrak{p}$ containing $r$. If $S$ is Gorenstein at $\mathfrak{p}$ then
$I_{\mathfrak{p}}\cong S_{\mathfrak{p}}$ by Proposition 2.9. If
$\operatorname{type}_{\mathfrak{p}}(S)=2$ then $I_{\mathfrak{p}}\cong
S_{\mathfrak{p}}$ or $I_{\mathfrak{p}}\cong(S^{t})_{\mathfrak{p}}$ by
Proposition 2.10. Set $M=I/rI$ and $N=S/rS$. If $I_{\mathfrak{p}}\cong
S_{\mathfrak{p}}$ then we have an induced $S_{\mathfrak{p}}$-linear
isomorphism

$M_{\mathfrak{p}}\cong N_{\mathfrak{p}}.$

If $I_{\mathfrak{p}}\cong(S^{t})_{\mathfrak{p}}$ then first we observe that
$\widehat{I}_{\mathfrak{p}}\cong\widehat{S}^{t}_{\mathfrak{p}}$. So we have an
induced $\widehat{S}_{\mathfrak{p}}$-linear isomorphism

$\widehat{M}_{\mathfrak{p}}\cong\frac{\widehat{S}^{t}_{\mathfrak{p}}}{r\widehat{S}^{t}_{\mathfrak{p}}},$

which combined with Lemma 2.8 gives a $\widehat{Z}_{p}$-linear isomorphism

$\widehat{M}_{\mathfrak{p}}\cong\widehat{N}_{\mathfrak{p}}.$

Since $M_{\mathfrak{p}}$ and $N_{\mathfrak{p}}$ are finitely generated
$Z_{p}$-modules, we obtain a $Z_{p}$-linear isomorphism

$M_{\mathfrak{p}}\cong N_{\mathfrak{p}}$

by [Eis95, Ex. 7.5, p. 203].

By Lemma 2.2, we have

$M\cong\bigoplus_{\mathfrak{p}}M_{\mathfrak{p}}\text{ and
}N\cong\bigoplus_{\mathfrak{p}}N_{\mathfrak{p}},$

where the direct sums run over the primes $\mathfrak{p}$ containing $r$, since
$M_{\mathfrak{p}}=N_{\mathfrak{p}}=0$ for all other primes. We conclude that
$M\cong N$ as $Z$-modules. ∎

As previously anticipated, we will apply the results contained in this section
to orders in commutative endomorphism algebras of abelian varieties over
finite fields. Such algebras have an automorphism that corresponds to the
Rosati involution, and acts as complex conjugation. Putting together
previously stated results with this extra structure, we obtain the following
proposition, which will be used to prove that certain abelian varieties are
not self-dual in Proposition 7.7.

######  Proposition 2.12.

Assume that $K$ has an involution $x\mapsto\overline{x}$ which fixes $Q$
point-wise. Let $S$ be an order in $K$ satisfying $S=\overline{S}$. Let
$\mathfrak{p}$ be a prime of $S$ such that
$\operatorname{type}_{\mathfrak{p}}(S)=2$. Then
$\mathfrak{p}=\overline{\mathfrak{p}}$ if and only if all fractional
$S$-ideals $I$ with $(I:I)_{\mathfrak{p}}=S_{\mathfrak{p}}$ satisfy
$I_{\mathfrak{p}}\not\cong(\overline{I}^{t})_{\mathfrak{p}}$.

###### Proof.

Assume that $\mathfrak{p}=\overline{\mathfrak{p}}$. By Proposition 2.10 we
have that $I_{\mathfrak{p}}\cong S_{\mathfrak{p}}$ or
$I_{\mathfrak{p}}\cong(S^{t})_{\mathfrak{p}}$. Assume the former. By Lemma
2.7, we obtain

$(\overline{I}^{t})_{\mathfrak{p}}=(\overline{I}^{t})_{\overline{\mathfrak{p}}}\cong(\overline{S}^{t})_{\overline{\mathfrak{p}}}=(S^{t})_{\mathfrak{p}}.$

Similarly, if $I_{\mathfrak{p}}\cong(S^{t})_{\mathfrak{p}}$ then

$(\overline{I}^{t})_{\mathfrak{p}}=(\overline{I}^{t})_{\overline{\mathfrak{p}}}\cong\overline{S}_{\overline{\mathfrak{p}}}=S_{\mathfrak{p}}.$

In both cases, if $I_{\mathfrak{p}}\cong(\overline{I}^{t})_{\mathfrak{p}}$
then $S^{t}$ is principal at $\mathfrak{p}$, that is, $S$ is Gorenstein at
$\mathfrak{p}$, which is a contradiction.

Now assume that $\mathfrak{p}\neq\overline{\mathfrak{p}}$. Let $d\in
K^{\times}$ such that $dS^{t}\subseteq S$ and $m>0$ such that
$\overline{\mathfrak{p}}^{m}S_{\overline{\mathfrak{p}}}\subseteq(dS^{t})_{\overline{\mathfrak{p}}}$.
Consider the fractional $S$-ideal $I$ defined as:

$I=dS^{t}+\overline{\mathfrak{p}}^{m}.$

For every $\mathfrak{l}\neq\overline{\mathfrak{p}}$ we have

$I_{\mathfrak{l}}=(dS^{t})_{\mathfrak{l}}+S_{\mathfrak{l}}=S_{\mathfrak{l}},$

and

$I_{\overline{\mathfrak{p}}}=(dS^{t})_{\overline{\mathfrak{p}}}+\overline{\mathfrak{p}}^{m}S_{\overline{\mathfrak{p}}}=(dS^{t})_{\overline{\mathfrak{p}}}.$

It follows by Lemma 2.7 that
$(\overline{I}^{t})_{\mathfrak{p}}\cong\overline{S}_{\mathfrak{p}}=S_{\mathfrak{p}}$,
which gives us $I_{\mathfrak{p}}\cong(\overline{I}^{t})_{\mathfrak{p}}$.
Moreover, we see that $(I:I)=S$ by checking the equality locally at every
prime. ∎

######  Remark 2.13.

Note that in the proof of Proposition 2.12 we showed that if
$\mathfrak{p}\neq\overline{\mathfrak{p}}$ then there exists a fractional ideal
$I$ such that $I_{\mathfrak{p}}\cong(\overline{I}^{t})_{\mathfrak{p}}$ with
multiplicator ring $S=(I:I)$ globally not only locally at $\mathfrak{p}$.

##  3\. Groups of rational points and Gorenstein orders

Our goal is to understand groups of rational points $A(\mathbb{F}_{q})$ for
abelian varieties $A$ defined over $\mathbb{F}_{q}$. To accomplish this goal
in practice, it is productive to view $A(\mathbb{F}_{q})$ as not merely a
group, but as a module over its endomorphism ring
$\operatorname{End}_{\mathbb{F}_{q}}(A)$. Although requesting a description of
the additional structure may appear to make the problem harder _a priori_ ,
the module structure can be exploited and cleanly described in many cases,
which allows one to deduce the group structure immediately.

Given a separable endomorphism $s\colon A\to A$ of an abelian variety $A$ over
$\mathbb{F}_{q}$, we denote by $A[s]$ the $\overline{\mathbb{F}}_{q}$-points
of the kernel of $s$.

######  Proposition 3.1.

If $A$ is a squarefree abelian variety over $\mathbb{F}_{q}$ and $s$ is a
separable endomorphism of $A$, then $\\#A[s]=\deg(s)=N_{K/\mathbb{Q}}(s)$
where $K=\operatorname{End}(A)\otimes_{\mathbb{Z}}\mathbb{Q}$.

###### Proof.

Write $A\sim B_{1}\times\ldots\times B_{r}$ as the product of simple pairwise
non-isogenous varieties. Then
$K=\operatorname{End}(A)\otimes_{\mathbb{Z}}\mathbb{Q}\cong\prod_{i=1}^{r}K_{i}$
is the product of the number fields
$K_{i}=\operatorname{End}(B_{i})\otimes_{\mathbb{Z}}\mathbb{Q}$. Without loss
of generality, by [Wat69, Thm. 3.13], we can choose each $B_{i}$ so that
$\operatorname{End}(B_{i})$ is the maximal order $\mathcal{O}_{K_{i}}$ in
$K_{i}$. Since
$\operatorname{End}(A)\subseteq\prod_{i=1}^{r}\mathcal{O}_{K_{i}}$, we can
write $s=(s_{1},\ldots,s_{r})$ for $s_{i}\in\mathcal{O}_{K_{i}}$. It is
therefore enough to consider the case where $A=B_{1}\times\ldots\times B_{r}$.
When $A$ is simple, i.e., when $r=1$, the statement follows from [Mil86, Prop
12.12]. Therefore, by definition,

$N_{K/\mathbb{Q}}(s)=\prod_{i=1}^{r}N_{K_{i}/\mathbb{Q}}(s_{i})=\prod_{i=1}^{r}\deg(s_{i})=\deg(s),\leavevmode\nobreak\
$

where the final equality follows from the fact that $\deg(s)=\\#A[s]$ and
$\deg(s_{i})=\\#B_{i}[s_{i}]$ by separability. ∎

Theorem 3.2 and Corollary 3.3 below were proven in the case where $A$ is an
elliptic curve by Lenstra [Len96], and generalized to the case of simple
abelian varieties in [Spr21]. To obtain the same result for square-free
varieties, we follow the proof method used in the simple case.

######  Theorem 3.2.

Let $A$ be a squarefree abelian variety over $\mathbb{F}_{q}$ and let $s$ be a
separable endomorphism of $A$. If $\operatorname{End}(A)$ is Gorenstein at the
primes containing $s$, then

$A[s]\cong\frac{\operatorname{End}(A)}{s\cdot\operatorname{End}(A)}$

is an isomorphism of $\operatorname{End}(A)$-modules.

###### Proof.

Write $S=\operatorname{End}(A)$, $S_{0}=S/Ss$ and $M=A[s]$. By the universal
property of quotients, if $rA[s]=0$, then $r=ts$ for some $t\in S$. This
implies that $M$ is a faithful $S_{0}$-module.

Moreover, because $S$ is Gorenstein at every prime ideal containing $s$ and
$s$ is a non-zero-divisor, we deduce that $S_{0}$ is a finite Gorenstein ring
by [Mat86, Ex. 18.1]. Therefore, $M$ contains a free $S_{0}$-submodule $N$ of
rank 1 by [Spr21, Lem. 2.3]. By Proposition 3.1, and the fact that the modules
are finite, we have

$\displaystyle\\#M=\deg s=N_{K/\mathbb{Q}}(s)=\\#(S/Ss)=\\#S_{0}.$

We deduce that $M=N\cong S/Ss$, as desired. ∎

######  Corollary 3.3.

Let $A$ be a squarefree abelian variety over $\mathbb{F}_{q}$ and let $\pi$ be
the Frobenius endomorphism of $A$. If $n\geq 1$ and $\operatorname{End}(A)$ is
Gorenstein at the prime ideals containing $1-\pi^{n}$, then there is an
isomorphism of $\operatorname{End}(A)$-modules

$A(\mathbb{F}_{q^{n}})\cong\frac{\operatorname{End}(A)}{(1-\pi^{n})\operatorname{End}(A)}.\leavevmode\nobreak\
$

In particular, $B(\mathbb{F}_{q^{n}})\cong A(\mathbb{F}_{q^{n}})$ are
isomorphic groups for all abelian varieties $B$ which are isogenous to $A$
with $\operatorname{End}(B)=\operatorname{End}(A)$.

###### Proof.

We may apply Theorem 3.2 because $A(\mathbb{F}_{q^{n}})=A[1-\pi^{n}]$ and
$1-\pi^{n}$ is a separable isogeny. ∎

However, note that these results are not true in general when we remove the
assumption that $\operatorname{End}(A)$ is Gorenstein, see Example 6.7.

##  4\. Cyclic isogeny classes

In this section we study isogeny classes containing only abelian varieties
with cyclic groups of rational points.

######  Definition 4.1.

We say that an isogeny class is _cyclic_ if every variety $A$ in the isogeny
class has a cyclic group $A(\mathbb{F}_{q})$ of rational
$\mathbb{F}_{q}$-points.

Any isogeny class of abelian varietes with a single point is trivially cyclic.
In Section 4.1, we show that, except for such trivial factors, every cyclic
isogeny class is squarefree. Then, in Section 4.2, we provide a
characterization of precisely which squarefree isogeny classes are cyclic in
terms of conductor ideals.

###  4.1. Reducing to the squarefree case

In this subsection, we consider abelian varieties whose endomorphism algebras
are non necessarily commutative. In general, if $B$ is an abelian variety over
$\mathbb{F}_{q}$, then $\operatorname{End}(B)$ is an order in the endomorphism
algebra $K=\operatorname{End}(B)\otimes_{\mathbb{Z}}\mathbb{Q}$, that is a
subring which is finitely generated free module over $\mathbb{Z}$, and whose
$\mathbb{Q}$-span is the whole algebra $K$. Observe that this notion of order
specializes to the one introduced in Section 2 in the commutative case.

######  Proposition 4.2.

If $B$ is a simple abelian variety over $\mathbb{F}_{q}$ such that
$B(\mathbb{F}_{q})$ is a nontrivial cyclic group and $\operatorname{End}(B)$
is a maximal order in $\operatorname{End}(B)\otimes_{\mathbb{Z}}\mathbb{Q}$,
then $\operatorname{End}(B)$ is commutative, that is, $B$ is squarefree.

###### Proof.

Let $L$ be the center of the endomorphism algebra
$\operatorname{End}(B)\otimes_{\mathbb{Z}}\mathbb{Q}$. Because $B$ is simple,
$L=\mathbb{Q}(\pi)$ where $\pi$ is a root in $L$ of the polynomial $h_{B}(x)$;
see [WM71, Thm. 8]. The center of $\operatorname{End}(B)$ is $\mathcal{O}_{L}$
by the maximality of $\operatorname{End}(B)$. By [Spr21, Thm. 1.3.(b)], there
is an isomorphism of $\mathcal{O}_{L}$-modules

$B(\mathbb{F}_{q})\cong\left(\frac{\mathcal{O}_{L}}{(1-\pi)\mathcal{O}_{L}}\right)^{d},\leavevmode\nobreak\
$

where $d=2\dim(B)/[L:\mathbb{Q}]$. In particular, because the group of points
is nontrivial and cyclic by hypothesis, we must have $d=1$, which is
equivalent to $\mathcal{O}_{L}=\operatorname{End}(B)$ by [WM71, Thm. 8]. ∎

######  Theorem 4.3.

If $A$ is an abelian variety over $\mathbb{F}_{q}$ whose isogeny class is
cyclic, then $A\sim A_{1}\times A_{\text{sf}}$ for abelian varieties $A_{1}$
and $A_{\text{sf}}$ over $\mathbb{F}_{q}$, possibly of dimension $0$, such
that $A_{\text{sf}}$ is squarefree and $\\#A_{1}(\mathbb{F}_{q})=1$. If $q\geq
5$, then $A$ itself is squarefree.

###### Proof.

Decompose the isogeny class $A\sim B_{1}^{e_{1}}\times\dots\times
B_{r}^{e_{r}}$ into distinct simple factors. By [Wat69, Theorem 3.13], we may
assume that each $\operatorname{End}(B_{j})$ is a maximal order in its
endomorphism algebra.

After possibly reordering, let $r_{1}$ be such that
$\\#B_{j}(\mathbb{F}_{q})=1$ for all $1\leq j\leq r_{1}$ and
$\\#B_{j}(\mathbb{F}_{q})>1$ for all $r_{1}<j\leq r$. Define

$A_{1}=\prod_{1\leq j\leq r_{1}}B_{j}^{e_{j}}\text{ and
}A_{\text{sf}}=\prod_{r_{1}<j\leq r}B_{j}^{e_{j}}.\leavevmode\nobreak\ $

It suffices to show $A_{\text{sf}}$ is square-free. In this case, $e_{j}=1$
for all $r_{1}<j\leq r$ because the isogeny class is cyclic, and Proposition
4.2 shows that

$\operatorname{End}(A_{\text{sf}})\cong\prod_{r_{1}<j\leq
r}\operatorname{End}(B_{j})$

is commutative, that is, $A_{\text{sf}}$ is squarefree. The theorem then
follows because the Weil bound states
$\\#B(\mathbb{F}_{q})\geq(\sqrt{q}-1)^{2\dim(B)}$ for any abelian variety $B$
over $\mathbb{F}_{q}$; see [Wei48]. Thus $\\#B(\mathbb{F}_{q})=1$ implies
$q\leq 4$. ∎

######  Remark 4.4.

There are infinitely many simple abelian varieties over $\mathbb{F}_{2}$ with
a single rational point by [MP77]; see also [Ked21]. However, there is only
one simple isogeny class over each of $\mathbb{F}_{3}$ and $\mathbb{F}_{4}$
with a single rational point; see [Kad21, Theorem 3.2] and LMFDB isogeny
classes 1.3.ad and 1.4.ae [LMF22].

###  4.2. The cyclicity of squarefree abelian varieties

We will use the same notation as in the previous sections. Let
$\mathcal{I}_{h}$ be a squarefree isogeny class of abelian vareities over
$\mathbb{F}_{q}$. Put $K=\mathbb{Q}[x]/(h)=\mathbb{Q}[\pi]$ and
$R=\mathbb{Z}[\pi,\overline{\pi}]$.

######  Theorem 4.5.

The isogeny class $\mathcal{I}_{h}$ is cyclic if and only if $(1-\pi)R$ is
coprime to the conductor $\mathfrak{f}=(\mathcal{O}_{K}:R)$, that is,
$(1-\pi)R+\mathfrak{f}=R$.

This theorem is a straightforward combination of Proposition 4.7 and Corollary
4.9. We remark that the following is a simple application of Theorem 4.5.

######  Corollary 4.6.

If $h(1)$ is a square-free integer, where $h(t)$ is the characteristic
polynomial of $\pi$, then $(1-\pi)R$ is coprime with
$\mathfrak{f}=(R:\mathcal{O}_{K})$.

###### Proof.

Because $h(1)$ is square-free, any finite abelian group of order $h(1)$ is
cyclic. Thus, the isogeny class is cyclic, and we conclude via Theorem 4.5. ∎

The next proposition proves one direction in Theorem 4.5.

######  Proposition 4.7.

If $(1-\pi)R$ is coprime to the conductor $\mathfrak{f}=(R:\mathcal{O}_{K})$,
then, for every $A$ in $\mathcal{I}_{h}$, we have an $R$-linear isomorphism

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

In particular, the isogeny class $\mathcal{I}_{h}$ is cyclic.

###### Proof.

Let $A$ be an abelian variety in $\mathcal{I}_{h}$. Put
$S=\operatorname{End}(A)$, and note that $R\subseteq S$. Hence $(1-\pi)S$ is
coprime to the conductor $(S:\mathcal{O}_{K})$ of $S$ in $\mathcal{O}_{K}$.
For every prime $\mathfrak{p}$ of $S$ containing $(1-\pi)$ we have
$S_{\mathfrak{p}}=\mathcal{O}_{K,\mathfrak{p}}$ by Lemma 2.1. Therefore $S$ is
Gorenstein at $\mathfrak{p}$ for every $\mathfrak{p}$ containing $(1-\pi)$. By
Theorem 3.2 we have that

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

Since $S$ is a fractional $R$-ideal, Proposition 2.5 gives us an $R$-linear
isomorphism $S/(1-\pi)S\cong R/(1-\pi)R$. We conclude by observing that

$\frac{R}{(1-\pi)R}\cong\frac{\mathbb{Z}[x,y]}{(h(1),x-1,y-q)}\cong\frac{\mathbb{Z}}{(h(1))},$

which is immediate from the method of the proof of [MS22, Prop. 2.7]. ∎

We now prove a strong converse to Proposition 4.7.

######  Proposition 4.8.

Let $A$ be a square-free abelian variety over $\mathbb{F}_{q}$ with Gorenstein
endomorphism ring $S=\operatorname{End}(A)$ and Frobenius endomorphism $\pi$.
If $(1-\pi)R$ is not coprime to the conductor $\mathfrak{f}=(R:S)$, then
$A(\mathbb{F}_{q^{n}})$ is a non-cyclic $R$-module for all $n\geq 1$. In
particular, every abelian variety in the isogeny class with endomorphism ring
$S$ has a non-cyclic group of points.

###### Proof.

We have $A(\mathbb{F}_{q^{n}})\cong S/(1-\pi^{n})S$ as $S$-modules by
Corollary 3.3. Observe that $(1-\pi^{n})R$ is not coprime to $\mathfrak{f}$
because $(1-\pi^{n})=(1-\pi)(1+\pi+\ldots+\pi^{n-1})$ implies
$R\supsetneq(1-\pi)R+\mathfrak{f}\supseteq(1-\pi^{n})R+\mathfrak{f}$ for all
$n\geq 1$. Therefore, $A(\mathbb{F}_{q^{n}})$ is a non-cyclic $R$-module by
Proposition 2.3. ∎

######  Corollary 4.9.

If $(1-\pi)R$ is not coprime to the conductor
$\mathfrak{f}=(R:\mathcal{O}_{K})$, then $A(\mathbb{F}_{q^{n}})$ is a non-
cyclic $R$-module for all $n\geq 1$ for every $A$ in $\mathcal{I}_{h}$ with
maximal endomorphism ring. In particular, the isogeny class $\mathcal{I}_{h}$
is non-cyclic.

###### Proof.

Observe that $\mathcal{O}_{K}$ is the endomorphism ring of an abelian variety
in $\mathcal{I}_{h}$ by [Wat69, Theorem 3.13]. Now, apply Proposition 4.8 with
$S=\mathcal{O}_{K}$, which is Gorenstein. ∎

##  5\. Non-cyclic groups of rational points

In Section 4, we gave a characterization of isogeny classes in which every
abelian variety has cyclic group of points. In this section, we go in the
opposite direction. In Theorem 5.7, we characterize isogeny classes
$\mathcal{I}_{h}$ in which every abelian group of order $h(1)$ occurs as the
group of rational points, and we call such isogeny classes _rich_ ; see
Definiton 5.3.

There are two main tools which we require to study rich isogeny classes. The
first is Lemma 5.4. It generalizes a result by Giangreco-Maidana [GM19, Lem.
2.1] which was originally used to study cyclic isogeny classes. We also use a
theorem of Rybakov, recalled below as Theorem 5.6, which provides a criterion
for the existence of abelian varieties in a given squarefree isogeny class
with a prescribed group of rational points.

To conclude the section, we use Rybakov’s theorem again to prove in
Proposition 5.10 the existence of ordinary abelian varieties over
$\mathbb{F}_{q}$ whose $\ell$-primary part has $2$ generators whenever
$q\equiv 1\bmod\ell$. This allows us to deduce an improved version of [MS22,
Thm. 3.3], which we present below as Theorem 5.13.

###  5.1. Lemmas about characteristic polynomials

We provide some basic lemmas concerning minimal polynomials which we will
apply to the polynomial $h$ for a squarefree isogeny class $\mathcal{I}_{h}$.
For an element $\alpha$ in an étale algebra $K$ over $\mathbb{Q}$, we write,
respectively, $h_{\alpha}(x)$ and $m_{\alpha}(x)$ for the characteristic and
minimal polynomials of the $\mathbb{Q}$-linear map on $K$ defined by
multiplication by $\alpha$. Note that if $K=\prod_{j=1}^{t}K_{j}$ for number
fields $K_{1},\dots,K_{t}$, and $\alpha=(\alpha_{1},\dots,\alpha_{t})$, then
we have $h_{\alpha}(x)=\prod_{j=1}^{t}h_{\alpha_{i}}(x)$. Observe that
$\alpha$ is in the maximal order $\mathcal{O}_{K}$ of $K$ if and only if
$m_{\alpha}(x)$ has integer coefficients.

For a squarefree isogeny class $\mathcal{I}_{h}$, the previous notation
applied to the étale algebra $K=\mathbb{Q}[x]/(h)$ leads to
$h(x)=h_{\pi}(x)=m_{\pi}(x)$ where $\pi\in K$ corresponds to the Frobenius
endomorphism on any abelian variety in $\mathcal{I}_{h}$.

######  Lemma 5.1.

Let $K$ be an étale algebra, let $\alpha\in K^{\times}$, and let
$b,c\in\mathbb{Q}$, with $b\neq 0$. Define $r=\deg h_{\alpha}$. Then

$h_{b\alpha+c}(x)=b^{r}\cdot h_{\alpha}\left(\frac{x}{b}-\frac{c}{b}\right)$

and

$h_{1/\alpha}(x)=\frac{x^{r}}{h_{\alpha}(0)}\cdot h_{\alpha}(1/x).$

In other words, we recognize $h_{1/\alpha}(x)$ as the reverse of the
polynomial $h_{\alpha}(x)/h_{\alpha}(0)$.

###### Proof.

First, we prove the statements when $K=\mathbb{Q}(\alpha)$ is a number field.
Both $b^{r}\cdot
m_{\alpha}\left(\frac{x}{b}-\frac{c}{b}\right)\leavevmode\nobreak\ $ and
$\frac{x^{r}}{m_{\alpha}(0)}\cdot m_{\alpha}(1/x)$ are monic, with
coefficients in $\mathbb{Q}$, of degree $r$, and are $0$ when evaluated at
$b\alpha+c$ and $1/\alpha$ respectively. They are irreducible since we have
isomorphisms
$\mathbb{Q}(\alpha)\cong\mathbb{Q}(b\alpha+c)\cong\mathbb{Q}(1/\alpha).$

In general, write $K=\prod_{j=1}^{t}K_{j}$ for number fields
$K_{1},\dots,K_{t}$ and $\alpha=(\alpha_{1},\dots,\alpha_{t})$. Setting
$d_{j}=[K_{j}:\mathbb{Q}(\alpha_{j})]$, we observe that
$h_{\alpha}(x)=\prod_{j=1}^{t}m_{\alpha_{j}}(x)^{d_{j}}$. The claims follow
from the previous case:

$h_{b\alpha+c}(x)=\prod_{j=1}^{t}m_{b\alpha_{j}+c}(x)^{d_{j}}=\prod_{j=1}^{t}b^{d_{j}\deg
m_{\alpha_{j}}}\cdot
m_{\alpha_{j}}\left(\frac{x}{b}-\frac{c}{b}\right)^{d_{j}}=b^{r}\cdot
h_{\alpha}\left(\frac{x}{b}-\frac{c}{b}\right);\leavevmode\nobreak\ $
$h_{1/\alpha}(x)=\prod_{j=1}^{t}m_{1/\alpha_{j}}(x)^{d_{j}}=\prod_{j=1}^{t}\frac{x^{d_{j}\deg
m_{\alpha_{j}}}}{m_{\alpha_{j}}(0)}\cdot
m_{\alpha_{j}}(1/x)^{d_{j}}=\frac{x^{r}}{h_{\alpha}(0)}\cdot
h_{\alpha}(1/x).\leavevmode\nobreak\ $

∎

We apply the previous lemma to obtain a formula for the coefficients of the
minimal polynomials of two related algebraic numbers.

######  Lemma 5.2.

Let $\alpha$ be an element of an étale algebra $K$ over $\mathbb{Q}$
satisfying $1-\alpha\in K^{\times}$, and let $d\in\mathbb{Q}$. The
coefficients of $h_{\frac{d}{(1-\alpha)}}(x)=\sum_{i=0}^{r}a_{i}x^{i}$ are
given by the formula

$a_{i}=\frac{(-1)^{r+i}d^{r-i}h^{(r-i)}_{\alpha}(1)}{(r-i)!h_{\alpha}(1)},$

where $h^{(i)}_{\alpha}(x)$ is the $i$-th derivative of the polynomial
$h_{\alpha}(x)$.

###### Proof.

We will use Lemma 5.1 several times. Let $\beta=\frac{1}{d}(1-\alpha)$. We
have

(1)
$h_{\frac{d}{(1-\alpha)}}(x)=h_{\frac{1}{\beta}}(x)=\frac{x^{r}}{h_{\beta}(0)}h_{\beta}\left(\frac{1}{x}\right);$
(2) $h_{\beta}(x)=\left(\frac{1}{d}\right)^{r}h_{1-\alpha}\left(dx\right);$
(3) $h_{1-\alpha}(x)=(-1)^{r}h_{\alpha}(1-x).$

Using Equations (2) with (3) we get

(4)
$h_{\beta}(0)=\left(\frac{1}{d}\right)^{r}h_{1-\alpha}\left(0\right)=(-1)^{r}\left(\frac{1}{d}\right)^{r}h_{\alpha}(1).$

Combining Equations (4) and (2) with Equation (1), we obtain

(5)
$h_{\frac{d}{1-\alpha}}(x)=\frac{(-x)^{r}}{h_{\alpha}(1)}h_{1-\alpha}\left(\frac{d}{x}\right).$

Define $h_{1-\alpha}(x)=x^{r}+b_{r-1}x^{r-1}+\ldots+b_{1}x+b_{0}$, and set
$b_{r}=1$. By Equation (3), for every $i=1,\ldots,r$ we get

(6)
$b_{i}=\frac{1}{i!}h^{(i)}_{1-\alpha}(x)\bigg{\rvert}_{x=0}=\frac{1}{i!}(-1)^{r+i}h^{(i)}_{\alpha}(1-x)\bigg{\rvert}_{x=0}=\frac{1}{i!}(-1)^{r+i}h^{(i)}_{\alpha}(1).$

Combining Equations (5) and (6), we obtain

$\displaystyle h_{\frac{d}{(1-\alpha)}}(x)$
$\displaystyle=\frac{(-x)^{r}}{h_{\alpha}(1)}\left(\left(\frac{d}{x}\right)^{r}+b_{r-1}\left(\frac{d}{x}\right)^{r-1}+\ldots+b_{1}\left(\frac{d}{x}\right)+b_{0}\right)$
$\displaystyle=\sum_{i=0}^{r}\frac{(-x)^{r}}{h_{\alpha}(1)}\left(\frac{d}{x}\right)^{r-i}b_{r-i}$
$\displaystyle=\sum_{i=0}^{r}\frac{(-x)^{r}}{h_{\alpha}(1)}\left(\frac{d}{x}\right)^{r-i}\cdot\frac{1}{(r-i)!}(-1)^{i}h^{(r-i)}_{\alpha}(1)$
$\displaystyle=\sum_{i=0}^{r}\frac{(-1)^{r+i}d^{r-i}h^{(r-i)}_{\alpha}(1)}{(r-i)!h_{\alpha}(1)}x^{i}.$

This concludes the proof. ∎

###  5.2. Rich isogeny classes

In Section 4, we studied cyclic isogeny classes. Now we study the opposite
extreme.

######  Definition 5.3.

We call an isogeny class $\mathcal{I}_{h}$ _rich_ if every abelian group of
order $h(1)$ occurs as the group of rational points for some abelian variety
in $\mathcal{I}_{h}$.

The following lemma detects the annihilator of the group of rational points in
terms of the existence of certain endomorphisms. It generalizes [GM19, Lem.
2.1], which was originally presented for the sake of studying cyclic isogeny
classes. We use our generalization to study rich isogeny classes instead.

######  Lemma 5.4.

Let $A$ be an abelian variety over $\mathbb{F}_{q}$ with $N$ rational points.
Denote by $\pi$ the Frobenius of $A$. Let $d$ be a divisor of $N$. Then the
following are equivalent:

  1. (a)

$dA(\mathbb{F}_{q})=0\leavevmode\nobreak\ $.

  2. (b)

$A(\mathbb{F}_{q})\subseteq\ker\left(\left[d\right]\right)(\overline{\mathbb{F}}_{q})$.

  3. (c)

There exists $\varphi\in\operatorname{End}_{\mathbb{F}_{q}}(A)$ such that
$\left[d\right]=\varphi\circ(1-\pi)$. Moreover such $\varphi$ lives in the
center of $\operatorname{End}_{\mathbb{F}_{q}}(A)$.

###### Proof.

The first two statements are clearly equivalent. To show that the second and
the third are equivalent, we first observe that
$A(\mathbb{F}_{q})=\ker(1-\pi)(\overline{\mathbb{F}}_{q})$. In particular, it
is clear that the third statement implies the second. Now, assume that the
second statement holds, that is,
$\ker(1-\pi)(\overline{\mathbb{F}}_{q})\subseteq\ker([d])(\overline{\mathbb{F}}_{q})$.
Then by the separability of $1-\pi$, we have also
$\ker(1-\pi)\subseteq\ker([d])$. Consider the following commutative diagram

$\textstyle{A}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[d]}$$\scriptstyle{1-\pi}$$\textstyle{\dfrac{A}{\ker(1-\pi)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\varphi}}$$\scriptstyle{\iota}$$\textstyle{A}$

where the middle arrow is the canonical projection, $\iota$ is the isomorphism
induced by $1-\pi$ and $\tilde{\varphi}$ is the (unique) map induced by the
inclusion $\ker(1-\pi)\subseteq\ker([d])$ via the universal property of the
quotient. Put $\varphi=\tilde{\varphi}\circ\iota^{-1}$. Therefore

$\varphi\circ(1-\pi)=\left[d\right].$

Since both $1-\pi$ and $[d]$ are in the center and defined over
$\mathbb{F}_{q}$, the same holds for $\varphi$, as required. ∎

Given two isogenous abelian varieties $A$ and $B$ over $\mathbb{F}_{q}$, we
identify the endomorphism algebras
$\operatorname{End}(A)\otimes_{\mathbb{Z}}\mathbb{Q}\cong\operatorname{End}(B)\otimes_{\mathbb{Z}}\mathbb{Q}$
with an isomorphism which maps the Frobenius endomorphism of $A$ to the
Frobenius endomorphism of $B$. We denote this element by $\pi$.

######  Proposition 5.5.

Let $A$ be an abelian variety over $\mathbb{F}_{q}$ whose group of rational
points $A(\mathbb{F}_{q})$ is annihilated by an integer $d$. For every maximal
order $\mathcal{O}$ in the endomorphism algebra
$\operatorname{End}(A)\otimes_{\mathbb{Z}}\mathbb{Q}$, there is an abelian
variety $B$ which is isogenous to $A$ such that
$\operatorname{End}(B)\cong\mathcal{O}$. Furthermore, $B(\mathbb{F}_{q})$ is
also annihilated by $d$ for all $B$ with endomorphism ring
$\operatorname{End}(B)\cong\mathcal{O}$.

###### Proof.

The existence statement is [Wat69, Thm 3.13]. Because $d$ kills
$A(\mathbb{F}_{q})$, we see that $d/(1-\pi)$ is an endomorphism of $A$ which
lies in the center of its endomorphism algebra by Lemma 5.4. Because $B$ has
maximal endomorphism ring, $\operatorname{End}(B)$ contains the maximal order
of the center of its endomorphism algebra. In particular, $d/(1-\pi)$ is an
element of $\operatorname{End}(B)$, which implies that $d$ kills
$B(\mathbb{F}_{q})$ by another application of Lemma 5.4. ∎

We emphasize the fact that Lemma 5.4 and Proposition 5.5 apply to any abelian
variety defined over a finite field. In the rest of the section, we restrict
our attention to the squarefree case.

For a positive integer $N$, we write $\mathrm{rad}(N)$ for its radical, that
is, if $N=\prod_{j=1}^{s}\ell_{j}^{e_{j}}$, then we define
$\mathrm{rad}(N)=\prod_{j=1}^{s}\ell_{j}$. We say that $N$ is the _exponent_
of an abelian group $G$ if $N$ is the smallest positive integer which
annihilates $G$. The minimal possible exponent for a finite abelian group of
order $N$ is $\mathrm{rad}(N)$, which is achieved by the “least cyclic” group
of order $N$. This notion is made precise by the following definition.

Let $g\geq 1$ be an integer, and let $\ell$ be prime. Consider a group $H$
with $\ell$-primary part
$H_{\ell}=\prod_{j=1}^{2g}(\mathbb{Z}/\ell^{e_{j}}\mathbb{Z})$ where $0\leq
e_{1}\leq\dots\leq e_{2g}$. The $\ell$-_Hodge polygon_ of $H$ is the polygon
with vertices $(i,\sum_{j=1}^{2g-i}e_{j})$; see [Ryb10, Def. 1.1] for details
and examples. We will use this notion when $H$ is the group of rational points
of an abelian variety $A$, in which case $g=\dim(A)$. Note that the group
$(\mathbb{Z}/\ell\mathbb{Z})^{f}$ has the highest $\ell$-Hodge polygon among
abelian groups of order $\ell^{f}$, while $(\mathbb{Z}/\ell^{f}\mathbb{Z})$
has the lowest.

Given a polynomial $h(x)=\sum_{i=1}^{2g}a_{i}x^{i}\in\mathbb{Z}[x]$ with
$a_{0}\neq 0$, the $\ell$-_Newton polygon_ of $h$ is the boundary of the lower
convex hull of the points $(i,\operatorname{ord}_{\ell}(a_{i}))$ for $0\leq
i\leq 2g$. The following theorem of Rybakov describes the groups of rational
points occuring in squarefree isogeny classes by comparing Hodge polygons and
Newton polygons.

######  Theorem 5.6 ([Ryb10, Thm. 1.1]).

Given a squarefree isogeny class $\mathcal{I}_{h}$ of abelian varieties over
$\mathbb{F}_{q}$, a finite abelian group $G$ of order $h(1)$ occurs as the
group of rational points of some abelian variety $A$ in $\mathcal{I}_{h}$ if
and only if the $\ell$-Hodge polygon of $G$ lies on or below the $\ell$-Newton
polygon of $h(1-t)$ for all primes $\ell$.

We exploit this theorem to characterize which squarefree isogeny classes are
rich.

######  Theorem 5.7.

Consider a squarefree isogeny class $\mathcal{I}_{h}$ of abelian varieties
over $\mathbb{F}_{q}$ of dimension $g$. Let $K=\mathbb{Q}[x]/(h)$ be the
endomorphism algebra, and let $\pi$ be the class of $x$. Write
$N=h(1)=\prod_{j=1}^{s}\ell_{j}^{e_{j}}$ for the number of rational points on
each abelian variety in $\mathcal{I}_{h}$. The following are equivalent.

  1. (a)

$\mathcal{I}_{h}$ is rich, that is, every abelian group of order $N$ arises as
$A(\mathbb{F}_{q})$ for some $A\in\mathcal{I}_{h}$.

  2. (b)

There is an abelian variety $A\in\mathcal{I}_{h}$ whose group of rational
points has exponent $\mathrm{rad}(N)$, that is,

$A(\mathbb{F}_{q})\cong\prod_{j=1}^{s}\left(\frac{\mathbb{Z}}{\ell_{j}\mathbb{Z}}\right)^{e_{j}}.$

  3. (c)

The coefficients of the characteristic polynomial
$h_{\frac{\mathrm{rad}(N)}{1-\pi}}(x)$ are integers.

  4. (d)

For all $1\leq i\leq 2g$, we have

$\frac{h^{(i)}(1)}{i!}\cdot\ell_{1}^{i-e_{1}}\cdots\ell_{s}^{i-e_{s}}\in\mathbb{Z}.$

If one of the equivalent conditions holds, then $A(\mathbb{F}_{q})$ has
exponent $\mathrm{rad}(N)$ for every $A$ in $\mathcal{I}_{h}$ with maximal
endomorphism ring.

###### Proof.

Trivially, a implies b. The reverse direction is a consequence of Theorem 5.6
because, among abelian groups of order $N$, the group of exponent
$\mathrm{rad}(N)$ has the highest $\ell$-Hodge polygon for every prime $\ell$.
If $A(\mathbb{F}_{q})$ has exponent $\mathrm{rad}(N)$ for some
$A\in\mathcal{I}_{h}$, then $\mathrm{rad}(N)/(1-\pi)$ is an element of
$\operatorname{End}(A)$ by Lemma 5.4. In particular, $\mathrm{rad}(N)/(1-\pi)$
is an integral element, so its minimal polynomial has integer coefficients.
Since $K=\mathbb{Q}[\pi]$, the characteristic and minimal polynomials
$h_{\frac{\mathrm{rad}(N)}{1-\pi}}(x)=m_{\frac{\mathrm{rad}(N)}{1-\pi}}(x)$
coincide. It follows that b implies c.

Conversely, assume that the minimal polynomial of $\mathrm{rad}(N)/(1-\pi)$
has integer coefficients. Then $\mathrm{rad}(N)/(1-\pi)$ is contained in the
maximal order $\mathcal{O}_{K}$ of the endomorphism algebra
$K=\mathbb{Q}[x]/(h)$. By [Wat69, Thm 3.13], there is always at least one
abelian variety $A\in\mathcal{I}_{h}$ whose endomorphism ring
$\operatorname{End}(A)$ is the maximal order $\mathcal{O}_{K}$ in
$K=\operatorname{End}(A)\otimes_{\mathbb{Z}}\mathbb{Q}=\mathbb{Q}(\pi)$.
Therefore, by Lemma 5.4, $A(\mathbb{F}_{q})$ is killed by $\mathrm{rad}(N)$,
so c implies b.

Recall that $h(x)=h_{\pi}(x)$. Applying Lemma 5.2 with $d=\mathrm{rad}(N)$, we
see that the polynomial $h_{\frac{\mathrm{rad}(N)}{1-\pi}}(x)$ is in
$\mathbb{Z}[x]$ if and only if

$\frac{h^{(i)}(1)}{i!}\frac{\mathrm{rad}(N)^{i}}{N}=\frac{h^{(i)}(1)}{i!}\cdot\ell_{1}^{i-e_{1}}\cdots\ell_{s}^{i-e_{s}}\in\mathbb{Z}$

for $i=1,\ldots,2g$. Hence parts c and d are also equivalent.

The final claim about varieties with maximal endomorphism ring follows from
combining Proposition 5.5 and part b. ∎

As an application of Theorem 5.7, we show the following.

######  Corollary 5.8.

A squarefree isogeny class is rich if and only if its simple factors are rich.

###### Proof.

Consider a squarefree isogeny class $\mathcal{I}_{h}$ over $\mathbb{F}_{q}$
whose abelian varieties have $N$ rational points. Let $A$ be an abelian
variety in $\mathcal{I}_{h}$ with maximal endomorphism ring. By Theorem 5.7,
the isogeny class $\mathcal{I}_{h}$ is rich if and only if the exponent of
$A(\mathbb{F}_{q})$ is a squarefree integer, namely $\mathrm{rad}(N)$.

Because $\operatorname{End}(A)$ is maximal,
$\operatorname{End}(A)\cong\prod_{j=1}^{r}\mathcal{O}_{K_{j}}$ where
$K_{j}=\mathbb{Q}[x]/(h_{j})$ according to the factorization $h=h_{1}\dots
h_{r}$ into irreducible polynomials. Moreover, $A\cong A_{1}\times\dots\times
A_{r}$ where each $A_{j}$ is simple and has maximal endomorphism ring
$\operatorname{End}(A_{j})\cong\mathcal{O}_{K_{j}}$. Observe that the exponent
of $A(\mathbb{F}_{q})\cong\prod_{j=1}^{r}A_{j}(\mathbb{F}_{q})$ is the least
common multiple of the exponents of $A_{j}(\mathbb{F}_{q})$ for $1\leq j\leq
r$. Therefore, the exponent of $A(\mathbb{F}_{q})$ is a squarefree integer if
and only if the exponent of $A_{j}(\mathbb{F}_{q})$ is a squarefree integer
for all $1\leq j\leq r$. ∎

It is now straightforward to determine when a squarefree isogeny class is rich
because Theorem 5.7.d provides a criterion which only involves computing the
derivatives of the characteristic polynomial. Although Theorem 4.5 provides a
criterion for cyclcity in terms of conductor ideals, it is faster to use
[GM19, Thm. 2.2], which only requires computing one derivative of the
characteristic polynomial. We exhibit some statistics over small finite fields
in the following example.

######  Example 5.9.

Let $\mathcal{I}^{\text{sf}}(g,q)$ be the set of squarefree isogeny classes of
dimension $g$ over $\mathbb{F}_{q}$, and let $\mathcal{R}(g,q)$ and
$\mathcal{C}(g,q)$ be the subsets containing the rich and cyclic isogeny
classes, respectively. In Table 1, we collect statistics concerning the
cardinalities of these sets for small values of $g$ and $q$.

$\mathbb{F}_{q}$ | $g$ |  $\mathcal{R}\setminus\mathcal{C}$ |  $\mathcal{C}\setminus\mathcal{R}$ |  $\mathcal{R}\cap\mathcal{C}$ |  $\mathcal{I}^{\text{sf}}\setminus(\mathcal{R}\cup\mathcal{C})\leavevmode\nobreak\ $
---|---|---|---|---|---
Only Rich | Only Cyclic | Both | Neither
$\mathbb{F}_{2}$ | 1 | 0% | 20.0% | 80.0% | 0%
2 | 3.45% | 17.2% | 75.9% | 3.45%
3 | 8.66% | 18.4% | 66.0% | 7.02%
4 | 10.5% | 19.8% | 61.0% | 8.67%
5 | 10.7% | 20.5% | 58.6% | 10.2%
| 6 | 10.0% | 21.3% | 58.5% | 10.2%
$\mathbb{F}_{3}$ | 1 | 14.3% | 0% | 85.8% | 0%
2 | 23.6% | 5.45% | 67.2% | 3.64%
3 | 21.6% | 8.38% | 60.9% | 9.17%
4 | 19.4% | 9.31% | 59.8% | 11.5%
| 5 | 18.1% | 9.83% | 60.0% | 12.1%
$\mathbb{F}_{4}$ | 1 | 0% | 28.6% | 71.4% | 0%
2 | 11.9% | 16.4% | 61.2% | 10.5%
3 | 13.1% | 14.8% | 60.1% | 12.0%
| 4 | 12.9% | 15.8% | 60.3% | 11.0%
$\mathbb{F}_{5}$ | 1 | 11.1% | 11.1% | 66.6% | 11.1%
2 | 16.8% | 9.25% | 61.4% | 12.6%
3 | 17.0% | 10.4% | 59.5% | 13.1%
| 4 | 16.7% | 10.4% | 60.0% | 12.9%
Table 1.  For $2\leq q\leq 5$, we count the number of cyclic and rich
squarefree isogeny classes of small dimension $g$ over $\mathbb{F}_{q}$ by
applying [GM19, Thm. 2.2] and Theorem 5.7 to data in the LMFDB [LMF22]. See
also [DKRV21].

An isogeny class is simultaneously rich and cyclic precisely when the
properties are trivially satisfied, namely when the number of rational points
$N=\mathrm{rad}(N)$ is squarefree. As a result, the intersection
$\mathcal{R}(g,q)\cap\mathcal{C}(g,q)$ can be considered the set of trivial
examples. Asymptotically, the proportion of integers which are squarefree is
$6/\pi^{2}\approx 60\%$.

###  5.3. Groups with two generators

To conclude this section, we consider the existence of abelian varieties whose
groups of rational points are not cyclic, but are, locally, the product of
only two cyclic factors. As an application, we improve [MS22, Thm. 3.3] in the
case of ordinary abelian varieties over $\mathbb{F}_{4}$.

######  Proposition 5.10.

Let $\ell$ be a prime and $1\leq s_{1}\leq s_{2}$. If $q\equiv
1\bmod\ell^{s_{1}}$ is a prime power, then every squarefree isogeny class
$\mathcal{I}_{h}$ over $\mathbb{F}_{q}$ with
$\operatorname{ord}_{\ell}(h(1))=s_{1}+s_{2}$ contains an abelian variety $A$
such that the $\ell$-primary part of the group of rational points is

$A(\mathbb{F}_{q})_{\ell}\cong(\mathbb{Z}/\ell^{s_{1}}\mathbb{Z})\times(\mathbb{Z}/\ell^{s_{2}}\mathbb{Z}).\leavevmode\nobreak\
$

###### Proof.

The group
$(\mathbb{Z}/\ell^{s_{1}}\mathbb{Z})\times(\mathbb{Z}/\ell^{s_{2}}\mathbb{Z})$
has $\ell$-Hodge polygon defined by the points
$(0,s_{1}+s_{2}),(1,s_{1}),(2,0)$. Therefore, using the notation
$h(1-x)=\sum_{j=0}^{2g}b_{j}x^{j}$, it is enough show that
$\operatorname{ord}_{\ell}(b_{0})\geq s_{1}+s_{2}$ and
$\operatorname{ord}_{\ell}(b_{1})\geq s_{1}$ by Theorem 5.6. The first holds
by hypothesis because $b_{0}=h(1)$.

By the $q$-symmetry of $h(x)$, we write

$\displaystyle h(x)$
$\displaystyle=\left(\sum_{j=1}^{g-1}a_{2g-j}(x^{2g-j}+q^{g-j}x^{j})\right)+a_{g}x^{g}.$

Because $q\equiv 1\bmod\ell^{s_{1}}$, we deduce that $2g-j+jq^{g-j}\equiv
g(1+q^{g-j})\bmod\ell^{s_{1}}$ for all $0\leq j\leq g-1$. Combined with the
fact that the polynomial $(1-x)^{j}$ has linear coefficient $-j$ for any
$j\geq 0$, we observe

$\displaystyle-b_{1}$
$\displaystyle=\left(\sum_{j=1}^{g-1}a_{2g-j}(2g-j+jq^{g-j})\right)+ga_{g}$
$\displaystyle\equiv\left(\sum_{j=1}^{g-1}ga_{2g-j}(1+q^{g-j})\right)+ga_{g}$
$\displaystyle\equiv gh(1)\bmod\ell^{s_{1}}.$

Thus, $\operatorname{ord}_{\ell}(b_{1})\geq s_{1}$ because
$\operatorname{ord}_{\ell}(h(1))=s_{1}+s_{2}$, and we are done. ∎

######  Corollary 5.11.

Let $q$ be an odd prime power. If $\mathcal{I}$ is a squarefree cyclic isogeny
class over $\mathbb{F}_{q}$, then its point count is not divisible by $4$.

###### Proof.

If the point count is divisible by $4$, then we apply Proposition 5.10 with
$\ell=2$ and $s_{1}=1$ to find a non-cyclic variety. ∎

######  Corollary 5.12.

For every $s\geq 1$, the group
$\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3^{s}\mathbb{Z}$ arises as the group
of rational points of a squarefree ordinary abelian variety over
$\mathbb{F}_{4}$.

###### Proof.

Apply Proposition 5.10 with $\ell=3$, $q=4$, $s_{1}=1$ and $s_{2}=s$. The
existence of ordinary isogeny classes with the desired number of points
follows from Theorem 1.13 and Remark 1.16 in [vCL+21]. ∎

For every $N\geq 1$, there is an abelian variety $A$ over $\mathbb{F}_{4}$
with $A(\mathbb{F}_{4})\cong(\mathbb{Z}/N\mathbb{Z})$, and $A$ can be taken to
be ordinary if and only if $N\neq 3$. In particular, by taking products, this
shows that every finite abelian group arises as the group of rational points
of an abelian variety over $\mathbb{F}_{4}$ which is not necessarily ordinary;
see [MS22, Thm. 3.3]. Corollary 5.12 extends [MS22, Thm. 3.3] by proving that
the abelian variety can be taken to be ordinary in additional non-cyclic
cases. We record the improved theorem here.

######  Theorem 5.13.

Every finite abelian group $G$ arises as the group of rational points $G\cong
A(\mathbb{F}_{q})$ for an abelian variety $A$ over $\mathbb{F}_{4}$. Moreover,
$A$ can be taken to be ordinary, except possibly if
$G=(\mathbb{Z}/3\mathbb{Z})^{n}$ for an odd integer $n$.

###### Proof.

It is already established by [MS22, Thm. 3.3] that every finite abelian group
occurs as the group of rational points of an abelian variety over
$\mathbb{F}_{4}$, and that the abelian variety can be taken to be ordinary
except possibly when $G$ takes the form
$(\mathbb{Z}/3\mathbb{Z})^{n_{1}}\times\prod_{j>1}(\mathbb{Z}/3^{j}\mathbb{Z})^{n_{j}}$,
where $n_{1}$ is odd. Now consider one of the exceptional groups $G$ where
additionally $n_{s}\geq 1$ for some $s>1$. There are ordinary abelian
varieties $A_{1}$ and $A_{2}$ over $\mathbb{F}_{4}$ whose groups of rational
points are

$A_{1}(\mathbb{F}_{4})\cong(\mathbb{Z}/3\mathbb{Z})\times(\mathbb{Z}/3^{s}),$
$A_{2}(\mathbb{F}_{4})\cong(\mathbb{Z}/3\mathbb{Z})^{n_{1}-1}\times(\mathbb{Z}/3^{s}/\mathbb{Z})^{n_{s}-1}\times\prod_{j\not\in\\{1,s\\}}(\mathbb{Z}/3^{j}\mathbb{Z})^{n_{j}}$

by Corollary 5.12 and [MS22, Thm. 3.3], respectively. Therefore,
$G\cong(A_{1}\times A_{2})(\mathbb{F}_{4})$, as desired. ∎

##  6\. Groups of rational points and categorical equivalences

In this section, we first present Theorem 6.2, which describes groups of
rational points by deploying a categorical equivalence between abelian
varieties in certain squarefree isogeny classes and fractional ideals in étale
algebras. In such an isogeny class, we deduce in Proposition 6.5 that every
abelian variety $A$ with endomorphism ring $\operatorname{End}(A)=S$ has the
same group of rational points if $S$ satisfies a certain local condition. In
Remark 6.6, we compare this result to Corollary 3.3.

Throughout this section we use the usual notation. We denote by
$\mathcal{I}_{h}$ a squarefree isogeny class over $\mathbb{F}_{q}$. We set
$K=\mathbb{Q}[x]/(h)$ and $R=\mathbb{Z}[\pi,\overline{\pi}]$ where $\pi$ is
the class of $x$ in $K$.

######  Definition 6.1.

We say that $\mathcal{I}_{h}$ satisfies

  * •

Ord if $\mathcal{I}_{h}$ is ordinary.

  * •

CS if $q$ is prime.

######  Theorem 6.2.

If $\mathcal{I}_{h}$ satisfies Ord (resp. CS) then there exists a covariant
(resp. contravariant) equivalence between $\mathcal{I}_{h}$ and the category
of fractional $R$-ideals (with $R$-linear morphisms). Denote by $\mathcal{F}$
be the functor inducing the equivalence. Let $A$ be an abelian variety in
$\mathcal{I}_{h}$, with dual variety $A^{\vee}$. Put $\mathcal{F}(A)=I$, where
$I$ is a fractional $R$-ideal.

  1. (a)

We have $\mathcal{F}(A^{\vee})=\overline{I}^{t}$ and
$\operatorname{End}(A^{\vee})=\overline{\operatorname{End}(A)}$.

  2. (b)

There is a $\mathbb{Z}$-linear isomorphism

$A(\mathbb{F}_{q^{n}})\cong\frac{I}{(1-\pi^{n})I},$

for all $n\geq 1$.

  3. (c)

There are $\mathbb{Z}$-linear isomorphisms

$A^{\vee}(\mathbb{F}_{q^{n}})\cong\frac{\overline{I}^{t}}{(1-\pi^{n})\overline{I}^{t}}\cong\frac{I^{t}}{(1-\overline{\pi}^{n})I^{t}}\cong\frac{I}{(1-\overline{\pi}^{n})I}\cong\frac{\overline{I}}{(1-\pi^{n})\overline{I}},$

for all $n\geq 1$.

###### Proof.

The existence of the equivalence is given by [Del69] in the Ord case, and by
[CS15] in the CS case. Part a is [Mar21, Thm. 5.2] in the Ord case, and
[BKM21, Cor 3.26] in the CS case. Part b, for $n=1$, is [Mar21, Cor. 4.7] for
both cases, but the proof is identical for $n>1$. In the Ord case, the key
ingredients are [How95, Lem. 4.13 and Prop. 4.14], while the proof in the CS
case uses a local argument.

For Part c, observe that the first $\mathbb{Z}$-linear isomorphism is the
combination of Parts a and b, while the second and fourth are the application
of complex conjugation. For the third isomorphism, we use Lemma 2.8 to deduce

$\frac{I^{t}}{(1-\overline{\pi}^{n})I^{t}}\cong\frac{(1-\overline{\pi}^{n})^{-1}I}{I}\cong\frac{I}{(1-\overline{\pi}^{n})I}.$

∎

######  Remark 6.3.

We emphasize that the hypotheses for Theorem 6.2 only impose conditions on the
isogeny class over the base field $\mathbb{F}_{q}$. Observe that property of
being squarefree is not stable under base extension, and the functor we invoke
in the CS case requires the base field to be prime. Nevertheless, we describe
$A(\mathbb{F}_{q^{n}})$ for all $n\geq 1$, because
$A(\mathbb{F}_{q^{n}})=\ker(1-\pi^{n})$ is the kernel of an isogeny defined
over the base field $\mathbb{F}_{q}$.

######  Remark 6.4.

For $n=1$, the $\mathbb{Z}$-linear isomorphisms in Theorem 6.2 are trivially
$R$-linear. Indeed, for part b since $R$ is generated over $\mathbb{Z}$ by
$\pi$ and $\overline{\pi}=q/\pi$, and $I/(1-\pi)I$ is annihilated by
$(1-\pi)$, $R$-linearity trivially follows from $\mathbb{Z}$-linearity. The
same applies for part c.

Now we show that, in the Ord and CS cases, the group of rational points is
uniquely determined by the endomorphism ring under certain conditions.

######  Proposition 6.5.

Let $A$ be an abelian variety in a squarefree isogeny class $\mathcal{I}_{h}$
over $\mathbb{F}_{q}$ satisfying Ord or CS. Write $S=\operatorname{End}(A)$.
For each $n\geq 1$, if $\operatorname{type}_{\mathfrak{p}}(S)\leq 2$ for every
prime $\mathfrak{p}$ of $S$ above $(1-\pi^{n})$, then the group of
$\mathbb{F}_{q^{n}}$-rational points of $A$ is uniquely determined by $S$.
Specifically,

$A(\mathbb{F}_{q^{n}})\cong\frac{S}{(1-\pi^{n})S}$

are isomorphic as $\mathbb{Z}$-modules.

###### Proof.

The statement follows from Proposition 2.11 with Theorem 6.2. ∎

######  Remark 6.6.

As noted in Section 2, an order $S$ is Gorenstein at a prime $\mathfrak{p}$ if
and only if $\operatorname{type}_{\mathfrak{p}}(S)=1$. Hence Proposition 6.5
is a generalization of Corollary 3.3. As a trade-off, the isomorphism is
$\mathbb{Z}$-linear rather than $S$-linear, and we need further hypotheses on
the isogeny class.

In Proposition 6.5, it is important that the local type of the endomorphism
ring is at most 2 at primes above $1-\pi^{n}$. Indeed, in Example 6.7, we
produce abelian varieties with the same endomorphism ring of (local) type $3$
but non-isomorphic groups of points.

######  Example 6.7.

The polynomial

$h=x^{4}+6x^{2}+25=(x^{2}-2x+5)(x^{2}+2x+5)$

defines an isogeny class $\mathcal{I}_{h}$ of ordinary abelian surfaces over
$\mathbb{F}_{5}$, which has label 2.5.a_g on the LMFDB [LMF22]. Consider the
order $S=\mathbb{Z}+2\mathcal{O}_{K}$. This is the unique overorder of $R$
with $[\mathcal{O}_{K}:S]=[S:R]=8$. Moreover, $S$ is the unique overorder of
$R$ with a prime $\mathfrak{p}$ with
$\operatorname{type}_{\mathfrak{p}}(S)=3$. This prime is
$\mathfrak{p}=2\mathcal{O}_{K}$, which is also the conductor of $S$ in
$\mathcal{O}_{K}$.

Using Theorem 6.2 and algorithms from [Mar20], we compute that there are $5$
isomorphism classes of abelian varieties with endomorphism ring $S$,
represented by fractional ideals $S$,$I$,$I^{t}$,$J$ and $S^{t}$. One has
$J\cong J^{t}$. Let $A_{S},A_{I},A_{I^{t}},A_{J}$ and $A_{S^{t}}$ be the
corresponding abelian varieties, respectively; see Theorem 6.2. We have

$A_{S}(\mathbb{F}_{5})\cong A_{S^{t}}(\mathbb{F}_{5})\cong
A_{J}(\mathbb{F}_{5})\cong\frac{\mathbb{Z}}{2\mathbb{Z}}\times\frac{\mathbb{Z}}{2\mathbb{Z}}\times\frac{\mathbb{Z}}{8\mathbb{Z}},$

and

$A_{I}(\mathbb{F}_{5})\cong
A_{I^{t}}(\mathbb{F}_{5})\cong\frac{\mathbb{Z}}{4\mathbb{Z}}\times\frac{\mathbb{Z}}{8\mathbb{Z}}.$

##  7\. The dual abelian variety

In this section, we study the relationship between an abelian variety and its
dual. We use the categorical equivalences presented in Theorem 6.2 which build
a bridge between abelian varieties and fractional ideals. In the first part,
we will prove that we have an isomorphism $A(\mathbb{F}_{q})\cong
A^{\vee}(\mathbb{F}_{q})$ under certain conditions on the endomorphism ring
$\operatorname{End}(A)$ and the fractional ideal associated to $A$. Clearly,
$A(\mathbb{F}_{q})\cong A^{\vee}(\mathbb{F}_{q})$ whenever $A$ is the Jacobian
of a curve or, more generally, a principally polarizable abelian variety. In
fact, the implication only uses that $A$ is a self-dual abelian variety, that
is, $A\cong A^{\vee}$.

In the second part of the section, we investigate when an abelian variety
fails to be self-dual. In particular, we prove that $A$ is not self-dual if
its endomorphism ring $\operatorname{End}(A)$ satisfies a certain local
condition. To conclude, we provide a sequence of examples comparing various
properties implying and implied by self-duality.

In this section, we use the same notation as Section 6. Specifically, we
denote by $\mathcal{I}_{h}$ a squarefree isogeny class over $\mathbb{F}_{q}$.
We set $K=\mathbb{Q}[x]/(h)$ and $R=\mathbb{Z}[\pi,\overline{\pi}]$ where
$\pi$ is the class of $x$ in $K$.

###  7.1. The group of points of the dual abelian variety

In Proposition 7.1, building on results proven previously, we give a list of
conditions that guarantee the existence of an isomorphism
$A(\mathbb{F}_{q^{n}})\cong A^{\vee}(\mathbb{F}_{q^{n}})$. In Example 7.2, we
exhibit an example of a geometrically simple ordinary abelian variety $A$ with
$A(\mathbb{F}_{q})\not\cong A^{\vee}(\mathbb{F}_{q})$. In Example 7.3, we show
that squarefree ordinary examples over $\mathbb{F}_{q}$ always exist in small
dimensions for small finite fields $\mathbb{F}_{q}$.

######  Proposition 7.1.

Let $A$ be in $\mathcal{I}_{h}$ and put $S=\operatorname{End}(A)$. Fix $n\geq
1$. If one of the following assumptions holds, then
$A(\mathbb{F}_{q^{n}})\cong A^{\vee}(\mathbb{F}_{q^{n}})$.

  1. (a)

$S=\overline{S}$ and $S$ is Gorenstein at $\mathfrak{p}$ for every prime
$\mathfrak{p}$ of $S$ above $(1-\pi^{n})$.

  2. (b)

$\mathcal{I}_{h}$ satisfies Ord or CS, $S=\overline{S}$, and
$\operatorname{type}_{\mathfrak{p}}(S)\leq 2$ for every prime $\mathfrak{p}$
of $S$ above $(1-\pi^{n})$.

  3. (c)

$\mathcal{I}_{h}$ satisfies Ord or CS, and one of the following holds, where
$\mathcal{F}(A)=I$.

     * •

For every prime $\mathfrak{p}$ of $R$ above $(1-\pi^{n})$ we have an
$R$-linear $I_{\mathfrak{p}}\cong(\overline{I})_{\mathfrak{p}}$.

     * •

For every prime $\mathfrak{p}$ of $R$ above $(1-\pi^{n})$ we have an
$R$-linear $I_{\mathfrak{p}}\cong(\overline{I}^{t})_{\mathfrak{p}}$.

###### Proof.

Part a follows from Corollary 3.3, because
$\operatorname{End}(A^{\vee})=\overline{S}=S$. Part b follows similarly from
Proposition 6.5. By Theorem 6.2, we have $\mathbb{Z}$-linear isomorphisms

$A(\mathbb{F}_{q^{n}})\cong
I\otimes_{R}\frac{R}{(1-\pi^{n})R}\quad\text{and}\quad
A^{\vee}(\mathbb{F}_{q^{n}})\cong\overline{I}\otimes_{R}\frac{R}{(1-\pi^{n})R}\cong\overline{I}^{t}\otimes_{R}\frac{R}{(1-\pi^{n})R}.$

Combined with Lemma 2.4, this proves Part c. ∎

######  Example 7.2.

Consider the isogeny class $\mathcal{I}_{h}$ of ordinary abelian surfaces over
$\mathbb{F}_{4}$ determined by the polynomial $h=x^{4}+2x^{3}+x^{2}+8x+16$.
According to the LMFDB [LMF22], this isogeny class, which has label 2.4.c_b,
is geometrically simple and contains a Jacobian. Let $\mathcal{O}_{K}$ be the
maximal order of $K$. We have that
$2\mathcal{O}_{K}=\mathfrak{p}^{2}\overline{\mathfrak{p}}^{2}$ where
$\mathfrak{p}$ is a prime of $\mathcal{O}_{K}$. Consider the order
$S=R+\mathfrak{p}^{2}$. It turns out that $R$ has $3$ overorders, namely,
$S$,$\overline{S}$ and $\mathcal{O}_{K}$, and all of these orders are
Gorenstein. Using Theorem 6.2, there is an abelian variety $A$ with
$\operatorname{End}(A)=S$, hence $\operatorname{End}(A^{\vee})=\overline{S}$.
Using Corollary 3.3, one computes that $A(\mathbb{F}_{4})$ and
$A^{\vee}(\mathbb{F}_{4})$ are not isomorphic. Indeed, they are

$\frac{\mathbb{Z}}{28\mathbb{Z}}\quad\text{and}\quad\frac{\mathbb{Z}}{2\mathbb{Z}}\times\frac{\mathbb{Z}}{14\mathbb{Z}}.$

######  Example 7.3.

Consider the following set of pairs of positive intgers $(g,q)$:

$\displaystyle\\{(2,q):2\leq q\leq 128\text{ is a prime power }\\}$
$\displaystyle\cup\\{(3,q):2\leq q\leq 9\text{ is a prime power
}\\}\cup\\{(3,16),(3,25)\\}$ $\displaystyle\cup\\{(4,q):q\in\\{2,3,4\\}\\}$
$\displaystyle\cup\\{(5,2)\\}.$

For each pair $(g,q)$ in the set above, there is a squarefree ordinary abelian
variety $A$ of dimension $g$ over $\mathbb{F}_{q}$ satisfying

$A(\mathbb{F}_{q})\not\cong A^{\vee}(\mathbb{F}_{q}).$

###  7.2. Self-duality

Recall the following well-known theorem.

######  Theorem 7.4.

If $A$ is an abelian variety over $\mathbb{F}_{q}$, then each statement below
implies the next:

  1. (a)

$A$ is a Jacobian variety;

  2. (b)

$A$ is a principally polarizable abelian variety;

  3. (c)

$A\cong A^{\vee}$ is self-dual;

  4. (d)

$A(\mathbb{F}_{q})\cong A^{\vee}(\mathbb{F}_{q})$ are isomorphic groups.

When $A$ is squarefree, the following item d is also implied by self-duality
c:

  1. (d’)

$\operatorname{End}(A)=\overline{\operatorname{End}(A)}$ is stable under
complex conjugation.

We already observed in Example 7.2 that properties d and d do not always hold.
Now we show that there are counterexamples to all of the reverse implications
in Theorem 7.4. For b$\nRightarrow$a, see Example 7.5, and for
c$\nRightarrow$b, see Example 7.6. In Example 7.8, we show that d and d
combined do not imply c. Moreover, in Example 7.9, we show that
d$\nRightarrow$d, which is another exhibition that d$\nRightarrow$c. Note that
we have the implication d$\Rightarrow$d under certain hypotheses; see
Proposition 7.1.

The following example is well-known, but we record it for completeness.

######  Example 7.5 (Principally polarizable but not Jacobian).

It is easy to find principally polarizable varieties which are not Jacobians.
For example, there are currently 30,079 geometrically simple ordinary isogeny
classes on LMFDB which contain a principally polarizable abelian variety, but
no Jacobian varieties; see [LMF22].

######  Example 7.6 (Self-dual but not principally polarizable).

If $\mathcal{I}_{h}$ is a simple ordinary isogeny class, then the class number
of the field $K=\mathbb{Q}[x]/(h)$ is equal to the number of abelian varieties
in $\mathcal{I}_{h}$ whose endomorphism ring is maximal; see [Wat69, Thm.
6.2]. In particular, if $K$ has class number 1, then any abelian variety $A$
in $\mathcal{I}_{h}$ whose endomorphism ring is
$\operatorname{End}(A)=\mathcal{O}_{K}$ must be self-dual. It is easy to find
isogeny classes satisfying this property which do not contain any principally
polarizable abelian varieties by using [How95, Thm. 1.3], for example. See the
LMFDB [LMF22] isogeny class 2.2.ab_ab for a concrete ordinary example, or
4.2.ad_c_f_an for one which is also geometrically simple.

One way to find examples of abelian varieties which are not self-dual is to
first use the categorical equivalence in Theorem 6.2 to compute all
isomorphism classes, and then use Theorem 6.2.a to determine which classes are
self-dual. Alternatively, one may use the following proposition which only
requires inspecting the local properties of orders in the endomorphism
algebra. The latter technique easily finds Example 7.8.

######  Proposition 7.7.

Let $A$ be any abelian variety in $\mathcal{I}_{h}$ satifying Ord or CS, let
$S$ be an order in $K$ such that $S=\overline{S}$, and let $\mathfrak{p}$ be a
prime of $S$ such that $\operatorname{type}_{\mathfrak{p}}(S)=2$ and
$\mathfrak{p}=\overline{\mathfrak{p}}$ . If $S\subseteq\operatorname{End}(A)$
and $S_{\mathfrak{p}}=\operatorname{End}(A)_{\mathfrak{p}}$, then $A$ is not
self-dual. In particular, such an $A$ is not principally polarizable and
cannot be a Jacobian.

###### Proof.

This follows from Proposition 2.12 and Theorem 6.2.a. ∎

######  Example 7.8 (Same endomorphism ring but not self-dual).

We go back to the isogeny class $\mathcal{I}_{h}$ from Example 6.7. One
computes that $R$ has a unique minimal overorder $T$, and $[T:R]=2$. Such an
order is then necessarily stable under complex conjugation, that is,
$T=\overline{T}$. Also, $T$ has a unique prime $\mathfrak{q}$ above $2$ which
also then satisfies $\mathfrak{q}=\overline{\mathfrak{q}}$. This prime is the
unique non-invertible prime of $T$ and we have
$\operatorname{type}_{\mathfrak{q}}(T)=2$. Recall that abelian varieties in
$\mathcal{I}_{h}$ with endomorphism ring $T$ exist by Theorem 6.2. By
Proposition 6.5, every abelian variety with endomorphism ring $T$ has group of
rational points isomorphic to $T/(1-\pi)T$. On the other hand, by Proposition
7.7, we see that $A\not\cong A^{\vee}$ for every abelian variety $A$ with
endomorphism ring $T$.

We observe that the non-self-dual abelian variety $A$ found in Example 7.8
also satisfies $A(\mathbb{F}_{5})\cong A^{\vee}(\mathbb{F}_{5})$, thereby
exhibiting d$\nRightarrow$c in Theorem 7.4. This is also exhibited in the
following example, which additionally proves d$\nRightarrow$d in the same
theorem.

######  Example 7.9 (Isomorphic groups but different endomorphism ring).

Consider the ordinary isogeny class $\mathcal{I}_{h}$ of abelian surfaces
defined over $\mathbb{F}_{3}$ determined by the polynomial

$h=x^{4}-x^{3}+4x^{2}-3x+9=(x^{2}-2x+3)(x^{2}+x+3).$

The order $R=\mathbb{Z}[\pi,\overline{\pi}]$ has index $[\mathcal{O}_{K}:R]=9$
in the maximal order $\mathcal{O}_{K}$ of $K$. Since $h(1)=10$ is coprime with
the conductor $(R:\mathcal{O}_{K})$, we deduce that $\mathcal{I}_{h}$ is
cyclic, by Theorem 4.5. Moreover, since $10$ is a squarefree integer, we get
that $\mathcal{I}_{h}$ is also trivially rich; see Theorem 5.7.

We observe that $R$ has exactly two primes above the singular rational prime
$3$. These two primes are complex conjugates to each other, and we denote them
by $\mathfrak{p}$ and $\overline{\mathfrak{p}}$. There are only two orders
between $R$ and $\mathcal{O}_{K}$, both with index $3$. These can be realized
as the multiplicator rings $S=(\mathfrak{p}:\mathfrak{p})$ and
$\overline{S}=(\overline{\mathfrak{p}}:\overline{\mathfrak{p}})$. By Theorem
6.2, we conclude that there is an abelian variety $A$ in $\mathcal{I}_{h}$
such that $\operatorname{End}(A)=S$ and
$\operatorname{End}(A^{\vee})=\overline{S}$ are not equal, but
$A(\mathbb{F}_{3})\cong A^{\vee}(\mathbb{F}_{3})\cong\mathbb{Z}/10\mathbb{Z}$.

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