# An exploration of degeneracy in abelian varieties of Fermat type

Heidi Goodson  Department of Mathematics, Brooklyn College; 2900 Bedford
Avenue, Brooklyn, NY 11210 USA<EMAIL_ADDRESS>
###### Abstract.

The term _degenerate_ is used to describe abelian varieties whose Hodge rings
contain exceptional cycles – Hodge cycles that are not generated by divisor
classes. We can see the effect of the exceptional cycles on the structure of
an abelian variety through its Mumford-Tate group, Hodge group, and Sato-Tate
group. In this article we examine degeneracy through these different but
related lenses. We specialize to a family of abelian varieties of Fermat type,
namely Jacobians of hyperelliptic curves of the form $y^{2}=x^{m}-1$. We prove
that the Jacobian of the curve is degenerate whenever $m$ is an odd, composite
integer. We explore the various forms of degeneracy for several examples, each
illustrating different phenomena that can occur.

###### Key words and phrases:

Hodge ring, Mumford-Tate group, Hodge group, Sato-Tate group

###### 2020 Mathematics Subject Classification:

11G10, 14C30, 11F80, 14K22

##  1\. Introduction

We say that a complex abelian variety $A$ is _nondegenerate_ if its
(complexified) Hodge ring is generated by divisor classes. In this article we
are interested in studying abelian varieties that do not have this property.
Abelian varieties whose Hodge rings are not generated by divisor classes are
called _degenerate_ , and the Hodge cycles not coming from the divisor classes
are called _exceptional cycles_.

While the definition of degeneracy is a statement about the Hodge ring, we can
see the effects of degeneracy in groups constructed from the Hodge structure
of the abelian variety: the Mumford-Tate group, the Hodge group, and the Sato-
Tate group. The Mumford-Tate group and Hodge group are related to the Hodge
ring via an action on certain cohomology groups. The algebraic Sato-Tate
conjecture and Mumford-Tate conjecture imply a relationship between the Sato-
Tate group and the Mumford-Tate and Hodge groups. Thus, it is natural that
degeneracies of these groups should be intertwined with degeneracies in the
Hodge ring.

In this paper we focus on nonsimple abelian varieties and study how the Hodge
structures of their simple factors interact with each other. This is
particularly interesting in the case where one of the factors is of type IV in
the Albert’s classification since the product of nondegenerate simple abelian
varieties is also nondegenerate as long as none of the factors are of this
type (see Theorem 0.1 of [16]). Work of Moonen and Zarhin [20] gives a
complete classification of degeneracy in dimensions 4 and 5 that is based on
the decomposition of an abelian variety into its simple factors. There are
some explicit examples of degenerate abelian varieties in the literature where
the focus is on the associated Hodge rings (see, for example, [1, 21, 24]).
Recent work of Lombardo [19] examines degeneracy in the Mumford-Tate group of
a dimension 4 Jacobian variety.

In this paper we build on these results by considering degeneracy in the
Mumford-Tate group, the Hodge group, and the Sato-Tate group within a
particularly interesting family of abelian varieties of Fermat type: Jacobians
of curves of the form $C_{m}\colon y^{2}=x^{m}-1$. This family of curves was
studied by Shioda in [24], where he proved the Hodge Conjecture for their
Jacobian varieties. The Hodge Conjecture claims that the Hodge cycles of an
abelian variety are all algebraic cycles, and one way to verify this is by
showing that all of the Hodge cycles are generated by divisor classes and then
using the fact that the divisor classes are all algebraic. Shioda, on the
other hand, proves the Hodge Conjecture for these Jacobian varieties without
appealing to divisor classes, and in certain cases he proves that the Hodge
rings have exceptional cycles. In this paper we prove the following extension
of his result.

######  Theorem 1.1.

Let $m$ be an odd composite integer. Then the Jacobian
$\operatorname{Jac}(C_{m})$ of the curve $y^{2}=x^{m}-1$ is degenerate in the
sense that the Hodge ring contains exceptional cycles.

We prove this result by first showing that, for any prime divisors $p,q$ of
$m$ (not necessarily distinct), the Jacobian of $y^{2}=x^{pq}-1$ can be viewed
as a factor of $\operatorname{Jac}(C_{m})$. We then apply Proposition 3.3
where we prove that $\operatorname{Jac}(C_{pq})$ has exceptional Hodge cycles.

With the general result of Theorem 1.1 in hand, we aim to understand how
degeneracy in the Hodge ring appears as degeneracy in the Mumford-Tate, Hodge,
and Sato-Tate groups of the Jacobian varieties. The Mumford-Tate and Hodge
groups of degenerate abelian varieties are, in some sense, “smaller” than what
we would see for a nondegenerate abelian variety. The Sato-Tate group, on the
other hand, can exhibit stranger behavior in that the identity component will
be “smaller” but the component group can be “larger” than what we would see
for a nondegenerate abelian variety.

Determining Sato-Tate groups of abelian varieties is the source of ongoing
interest and work. There has been recent progress on computing Sato-Tate
groups of nondegenerate abelian varieties (see, for example, [2, 7, 9, 10, 11,
12, 14, 17]). A nice property of nondegenerate abelian varieties is that the
component group of the Sato-Tate group is isomorphic to
$\operatorname{Gal}(K/F)$, where $K$ is the endomorphism field and $F$ is the
field of definition of the abelian variety – this is due to the fact that the
Sato-Tate group of the abelian variety is connected over this field. However,
the landscape for degenerate abelian varieties is still wide open. In
particular, there are no examples in the literature where the component groups
for these degenerate Sato-Tate groups are computed. One difficulty is that we
may need a larger field than the endomorphism field in order to have a
connected Sato-Tate group, and it’s not clear what field is needed. In this
paper we have several results regarding the identity component of the Sato-
Tate group (see Propositions 5.1 and 5.5) as well as the following conjecture
regarding the full Sato-Tate group.

######  Conjecture 1.2 (Conjecture 5.2).

Let $\operatorname{Jac}(C_{9})$ denote the Jacobian of the curve
$y^{2}=x^{9}-1$. Up to conjugation in $\operatorname{USp}(8)$, the Sato-Tate
group of $\operatorname{Jac}(C_{9})$ is

$\operatorname{ST}(\operatorname{Jac}(C_{9}))\simeq\left\langle\operatorname{diag}(U_{1},U_{2},U_{3},\overline{U}_{1}U_{2}U_{3}),\gamma\right\rangle,$

where $\gamma$ is defined in Equation (5).

In addition to this example, we explore degeneracy for other values of $m$,
carefully examining the Mumford-Tate groups, the Hodge rings and Hodge groups,
and the Sato-Tate groups of the varieties. We see different phenomena
occurring for different values of $m$, which makes this an intriguing problem
to study.

### Organization of the paper

In Section 2 we provide background information for the main objects of study.
In Section 3 we take a deep dive into degeneracy. We begin by generalizing a
result of Lombardo [19] for degeneracy in Mumford-Tate groups. We prove
Proposition 3.3, which states that $\operatorname{Jac}(C_{pq})$ has
exceptional Hodge cycles for any odd primes $p,q$ (not necessarily distinct).
We then explain how the Hodge ring and Hodge group are related to each other.
Finally, we describe how degeneracy can affect the Sato-Tate group of an
abelian variety. In Section 4 we proves results for the decompositions of the
Jacobian varieties into simple factors and we prove Theorem 1.1. In Section 5
we work on explicit examples that demonstrate various phenomena that can occur
for degenerate abelian varieties. For each example, we examine the Mumford-
Tate group, Hodge ring, and Hodge group. For $m=9$ and $m=15$, we also study
the Sato-Tate groups. While the Sato-Tate group of
$\operatorname{Jac}(C_{15})$ is somewhat mysterious, we show that it is less
mysterious for $\operatorname{Jac}(C_{9})$. We make a conjecture for the full
Sato-Tate group of $\operatorname{Jac}(C_{9})$ and provide moment statistics
to support the conjecture.

### Notation and conventions

For any $K$-vector space $W$ and $K$-algebra $R$, let $W_{R}:=W\otimes_{K}R.$

The curve $y^{2}=x^{m}-1$ is denoted by $C_{m}$ and its Jacobian is denoted by
$J_{m}$. We will write $\zeta_{m}$ for a primitive $m^{th}$ root of unity.

Let $I$ denote the $2\times 2$ identity matrix and define the matrix

$\displaystyle J=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}.$

We embed $\operatorname{U}(1)$ in $\operatorname{SU}(2)$ via $u\mapsto
U=\begin{pmatrix}u&\\\ &\overline{u}\end{pmatrix}$ and, for any positive
integer $n$, define the following subgroup of the unitary symplectic group
$\operatorname{USp}(2n)$

$\operatorname{U}(1)^{n}:=\left\langle\operatorname{diag}(U_{1},U_{2},\ldots,U_{n})\right\rangle.$

##  2\. Background

###  2.1. Hodge Structures

In this section, we follow the exposition in Chapters 1 and 17 of [4].

Let $A$ be a nonsingular projective variety over $\mathbb{C}$. We denote the
first homology group of $A$ by $V(A):=H_{1}(A,\mathbb{Q})$ and its dual (the
first cohomology group) by $V^{*}(A):=H^{1}(A,\mathbb{Q})$. The complex vector
space $V(A)_{\mathbb{C}}$ has a weight $-1$ Hodge structure, i.e., a
decomposition $V(A)_{\mathbb{C}}=V(A)^{-1,0}\oplus V(A)^{0,-1}$ where
$\overline{V(A)^{-1,0}}=V(A)^{0,-1}$. This corresponds to the following weight
1 Hodge structure of its dual

$V^{*}_{\mathbb{C}}=H(A)^{1,0}\oplus H(A)^{0,1},$

where $V(A)^{-1,0}=H^{1,0}(A)^{*}$ and $V(A)^{0,-1}=H^{0,1}(A)^{*}$. The
notation in the decomposition of $H^{1}(A,\mathbb{Q})$ is defined by
$H^{a,b}(A)=H^{a}(\Omega^{b}_{A}),$ where $\Omega_{A}^{b}$ is the sheaf of
holomorphic $b$-forms on $A$. In general, we can define $H^{a,b}(A)$ by

$\displaystyle
H^{a,b}(A)\simeq\bigwedge^{a}H^{1,0}(A)\otimes\bigwedge^{b}H^{0,1}(A).$

Hodge structures of weight $n$,
$H^{n}(A,\mathbb{C})\simeq\bigwedge^{n}H^{1}(A,\mathbb{C})$, satisfy
$H^{n}(A,\mathbb{C})\simeq\bigoplus_{a+b=n}H^{a,b}(A).$

###  2.2. Hodge and Mumford Tate Groups

The Hodge structure of the previous section determines a representation
$\mu_{\infty,A}:\mathbb{G}_{m}(\mathbb{C})\rightarrow\operatorname{GL}(V_{\mathbb{C}})$
acting as multiplication by $z$ on $V(A)^{-1,0}$ and trivially on
$V(A)^{0,-1}$. With this setup, we define the Mumford-Tate group of $A$,
denoted $\operatorname{MT}(A)$, to be the smallest $\mathbb{Q}$-algebraic
subgroup of $\operatorname{GL}_{V}$ such that
$\mu_{\infty,A}(\mathbb{G}_{m}(\mathbb{C}))\subseteq\operatorname{MT}(A)_{\mathbb{C}}$.
We define the Hodge group of $A$ to be the connected component of the identity
of $\operatorname{MT}(A)\cap\operatorname{SL}_{V}$.

The Hodge group can also be formed by restricting the representation
$\mu_{\infty,A}$ to the circle group
$\mathbb{S}^{1}:=\\{z\in\mathbb{C}\mid|z|=1\\}$. With this setup, the Hodge
group is the smallest $\mathbb{Q}$-algebraic subgroup of
$\operatorname{GL}_{V}$ such that
$\mu_{\infty,A}(\mathbb{S}^{1})\subseteq\operatorname{Hg}(A)_{\mathbb{C}}$.
The image of this restriction of $\mu_{\infty,A}$ lies in
$\operatorname{SL}(V_{\mathbb{C}})$, and so the Hodge group is a
$\mathbb{Q}$-algebraic subgroup of $\operatorname{SL}_{V}$. In fact, one can
show that the image of the representation $\mu_{\infty,A}$ is contained in the
symplectic group $\operatorname{GSp}({V_{\mathbb{C}}})$. Hence,
$\operatorname{MT}(A)$ and $\operatorname{Hg}(A)$ are $\mathbb{Q}$-algebraic
subgroups of $\operatorname{GSp}_{V}$ and $\operatorname{Sp}_{V}$,
respectively.

In our work we will be interested in nonsimple abelian varieties. For $n\geq
1$, we can identify $\operatorname{Hg}(A^{n})$ with $\operatorname{Hg}(A)$ and
the action is performed diagonally on $V(A^{n})=(V(A))^{n}$. More generally,
for $n_{1},n_{2},\ldots,n_{t}\geq 1$, $\operatorname{Hg}(A_{1}^{n_{1}}\times
A_{2}^{n_{2}}\times\cdots\times A_{t}^{n_{t}})$ is isomorphic to
$\operatorname{Hg}(A_{1}\times A_{2}\times\cdots\times A_{t})$. Even more
generally, if $A$ and $B$ are abelian varieties then
$\operatorname{Hg}(A\times
B)\subseteq\operatorname{Hg}(A)\times\operatorname{Hg}(B)$. In our work we
will be interested in studying cases where the containment is strict.

###  2.3. The Hodge Ring

In this section we use the notation found in [24]. We denote the
(complexified) Hodge ring of $A$ by

$\mathscr{B}^{*}(A):=\displaystyle\sum_{d=0}^{\dim(A)}\mathscr{B}^{d}(A),$

where $\mathscr{B}^{d}(A)=(H^{2d}(A,\mathbb{Q})\cap
H^{d,d}(A))\otimes\mathbb{C}$ is the $\mathbb{C}$-span of Hodge cycles of
codimension $d$ on $A$. The subring of $\mathscr{B}^{*}(A)$ generated by the
divisor classes, i.e. generated by $\mathscr{B}^{1}(A)$ , is

$\mathscr{D}^{*}(A):=\displaystyle\sum_{d=0}^{\dim(A)}\mathscr{D}^{d}(A),$

where $\mathscr{D}^{d}(A)$ is the $\mathbb{C}$-span of classes of intersection
of $d$ divisors.

The relationship between these spaces is relevant to the Hodge conjecture. Let
$\mathscr{C}^{d}(A)$ be the subspace of $\mathscr{B}^{d}(A)$ generated by the
classes of algebraic cycles on $A$ of codimension $d$. Then

$\mathscr{D}^{d}(A)\subseteq\mathscr{C}^{d}(A)\subseteq\mathscr{B}^{d}(A)$

and the Hodge conjecture for $A$ asserts that every Hodge cycle is algebraic:
$\mathscr{C}^{d}(A)=\mathscr{B}^{d}(A)$ for all $d$ (see [1, 24]). One way to
verify the conjecture is to prove the Hodge conjecture in codimension $d$ by
proving the equality $\mathscr{D}^{d}(A)=\mathscr{B}^{d}(A)$. However this
equality does not always hold, even when the Hodge conjecture is known to be
true. In Section 3.2 we will further study the relationship between
$\mathscr{D}^{d}(A)$ and $\mathscr{B}^{d}(A)$.

###  2.4. An $\ell$-adic construction of the Sato-Tate group

We follow the exposition of [7] and [26, Section 3.2]. See also [22, Chapter
8].

For any prime $\ell$, we define the Tate module
$T_{\ell}:=\varprojlim_{n}A[\ell^{n}]$ to be a free $\mathbb{Z}_{\ell}$-module
of rank $2g$, and the rational Tate module
$V_{\ell}:=T_{\ell}\otimes_{\mathbb{Z}}\mathbb{Q}$ to be a
$\mathbb{Q}_{\ell}$-vector space of dimension $2g.$ The Galois action on the
Tate module is given by an $\ell$-adic representation

$\displaystyle\rho_{A,\ell}:\operatorname{Gal}(\overline{F}/F)\rightarrow\operatorname{Aut}(V_{\ell})\cong\operatorname{GL}_{2g}(\mathbb{Q}_{\ell}).$
(1)

The $\ell$-adic monodromy group of $A$, denoted $G_{A,\ell}$, is the Zariski
closure of the image of this map in
$\operatorname{GL}_{2g}(\mathbb{Q}_{\ell})$, and we define
$G^{1}_{A,\ell}:=G_{A,\ell}\cap\operatorname{Sp}_{2g}(\mathbb{Q}_{\ell})$.

######  Conjecture 2.1 (Algebraic Sato-Tate Conjecture).

[3, Conjecture 2.1] There is an algebraic subgroup $\operatorname{AST}(A)$ of
$\operatorname{GSp}_{2g}$ over $\mathbb{Q}$, called the algebraic Sato-Tate
group of $A$, such that the connected component of the identity
$\operatorname{AST}^{0}(A)$ is reductive and, for each prime $\ell$,
$G^{1}_{A,\ell}=\operatorname{AST}(A)\otimes_{\mathbb{Q}}\mathbb{Q}_{\ell}$.

With this set up, we define the Sato-Tate group of $A$, denoted
$\operatorname{ST}(A)$, to be a maximal compact Lie subgroup of
$G^{1}_{A,\ell}\otimes_{\mathbb{Q}_{\ell}}\mathbb{C}=\operatorname{AST}(A)\otimes_{\mathbb{Q}}\mathbb{C}$
contained in $\operatorname{USp}(2g)$. It is conjectured that
$\operatorname{ST}(A)$ is, up to conjugacy in $\operatorname{USp}(2g)$,
independent of the choice of the prime $\ell$ and of the embedding of
$\mathbb{Q}_{\ell}$ in $\mathbb{C}$. While the Sato-Tate group is a compact
Lie group, it may not be connected [13]. We denote the connected component of
the identity (also called the identity component) of $\operatorname{ST}(A)$ by
$\operatorname{ST}^{0}(A)$.

There are many cases where the algebraic Sato-Tate conjecture is known to be
true. Banaszak and Kedlaya prove that the conjecture holds for all abelian
varieties of dimension at most 3 [3, Theorem 6.11], for many examples of
simple abelian varieties [3, Theorem 6.9], and for all abelian varieties of CM
type [3, Theorem 6.6]. There are examples of infinite families of higher
dimensional Jacobian varieties for which the algebraic Sato-Tate conjecture is
known to be true in [7, 13, 14].

###  2.5. The Mumford-Tate conjecture

Deligne showed in [6] that the identity component of $G^{1}_{A,\ell}$ is
always a subgroup of $\operatorname{MT}(A)_{\mathbb{Q}_{\ell}}$, and it is
conjectured that the inclusion is actually equality (see [26]).

######  Conjecture 2.2 (Mumford-Tate Conjecture).

The identity component of $G_{A,\ell}$ is equal to
$\operatorname{MT}(A)_{\mathbb{Q}_{\ell}}$.

Since the identity component of $G^{1}_{A,\ell}$ is a subgroup of
$\operatorname{Hg}(A)_{\mathbb{Q}_{\ell}}$, the Mumford-Tate conjecture
predicts that this inclusion is also an equality.

The Mumford-Tate conjecture is known to be true for many examples of abelian
varieties, including for abelian varieties of CM type (see, for example, [21,
27]) and for all abelian varieties of dimension $g\leq 3$ [3, Theorem 6.11].
More generally, Commelin proved that if the Mumford-Tate conjecture is true
for abelian varieties $A$ and $B$ defined over a finitely generated field then
it is also true for the product $A\times B$ [5].

When the Mumford-Tate conjecture holds, it follows from results of [3, Section
3] that, up to conjugation in $\operatorname{USp}(2g)$, the Mumford-Tate group
and the Hodge group uniquely determine the identity component of the Sato-Tate
group. Unfortunately we will not be able to gain information about the
component group of the Sato-Tate group from these two groups.

###  2.6. Moment Statistics

In Section 5 we will compute moment sequences of the Sato-Tate groups of
Jacobian varieties. These moment statistics can be used to verify the
equidistribution statement of the generalized Sato-Tate conjecture by
comparing them to moment statistics obtained for the traces $a_{i}$ in the
normalized $L$-polynomial. The numerical moment statistics are an
approximation since one can only ever compute them up to some prime.

The following background information has been adapted from [7, 14]. We define
the $n^{th}$ moment (centered at 0) of a probability density function to be
the expected value of the $n$th power of the values, i.e. $M_{n}[X]=E[X^{n}]$.
For independent variables $X$ and $Y$ we have $E[X+Y]=E[X]+E[Y]$ and
$E[XY]=E[X]E[Y]$, which yield the following identity

$\displaystyle
M_{n}[X_{1}+\cdots+X_{m}]=\sum_{\beta_{1}+\cdots+\beta_{m}=n}\binom{n}{\beta_{1},\ldots,\beta_{m}}M_{\beta_{1}}[X_{1}]\cdots
M_{\beta_{m}}[X_{m}].$

In Section 5, we will work with the unitary group $\operatorname{U}(1)$ and
consider the trace map on $U\in\operatorname{U}(1)$ defined by
$z:=\operatorname{tr}(U)=u+\overline{u}=2\cos(\theta)$, where $u=e^{i\theta}$.
From here we see that $dz=2\sin(\theta)d\theta$ and

$\mu_{\operatorname{U}(1)}=\frac{1}{\pi}\frac{dz}{\sqrt{4-z^{2}}}$

gives a uniform measure of $\operatorname{U}(1)$ (see [26, Section 2]). We can
deduce the following pushforward measure

$\mu_{\operatorname{U}(1)^{k}}=\prod_{i=1}^{k}\frac{1}{\pi}\frac{dz_{i}}{\sqrt{4-z_{i}^{2}}}.$

We can now define the moment sequence $M[\mu]$, where $\mu$ is a positive
measure on some interval $I=[-d,d]$. The $n^{th}$ moment $M_{n}[\mu]$ is, by
definition, $\mu(\phi_{n})$, where $\phi_{n}$ is the function $z\mapsto
z^{n}$. This yields $M_{n}[\mu_{\operatorname{U}(1)}]=\binom{n}{n/2}$, where
$\binom{n}{n/2}\colon=0$ if $n$ is odd. Hence,
$M[\mu_{\operatorname{U}(1)}]=(1,0,2,0,6,0,20,0,\ldots).$ From here, we take
binomial convolutions to obtain

$\displaystyle M_{n}[\mu_{\operatorname{U}(1)^{k}}]$
$\displaystyle=\sum_{\beta_{1}+\cdots+\beta_{k}=n}\binom{n}{\beta_{1},\ldots,\beta_{k}}M_{\beta_{1}}[\mu_{\operatorname{U}(1)}]\cdots
M_{\beta_{k}}[\mu_{\operatorname{U}(1)}].$

For a dimension $g$ abelian variety, denote by $\mu_{i}$ the projection of the
Haar measure onto the interval $\left[-\binom{2g}{i},\binom{2g}{i}\right]$,
where $i\in\\{1,2,\ldots,g\\}$. We compute $M_{n}[\mu_{i}]$ by averaging over
the components of the Sato-Tate group.

##  3\. Degenerate Abelian Varieties

We define an abelian variety $A$ to be _degenerate_ if its Hodge ring is not
generated by divisor classes, i.e., the containment
$\mathscr{D}^{*}(A)\subset\mathscr{B}^{*}(A)$ is strict. Results of Hazama
[16] show that $A$ is stably nondegenerate if and only if the rank of the
Hodge group is maximal ($A$ is stably nondegenerate if
$\mathscr{D}^{*}(A^{n})=\mathscr{B}^{*}(A^{n})$ for all $n>0$), which implies
a similar statement for the Mumford-Tate group. Hazama also proves an
interesting result regarding products of abelian varieties: if $A$ and $B$ are
both stably nondegenerate, then the only scenario in which $A\times B$ could
be degenerate is if one of the simple factors is of type IV in the Albert’s
classification.

The main goal of this section is to investigate degeneracy via Mumford-Tate
groups, Hodge rings, and Hodge groups. The results of Sections 3.1 and 3.3 are
general, while the result in Section 3.2 holds for Jacobians of the curves
$y^{2}=x^{m}-1$. In Section 3.4 we will show how degeneracy can affect the
Sato-Tate group of an abelian variety.

###  3.1. Degeneracy via the Mumford-Tate group

In this section we study degeneracy of nonsimple abelian varieties via their
Mumford-Tate groups. Consider a product $A_{1}\times A_{2}\times\cdots\times
A_{n}$ of nonisogenous abelian varieties. The question we wish to answer is:
when is the Mumford-Tate group of this product smaller than expected? In
particular, we wish to know when one of the canonical projections

$\operatorname{MT}(A_{1}\times A_{2}\times\cdots\times
A_{n})\longrightarrow\operatorname{MT}(A_{i})$

is an isomorphism or an isogeny. To answer this, we generalize results of
Lombardo [19]. We first introduce some notation.

####  3.1.1. Notation

Let $L$ be a number field and let $E$ be a subfield. For any number field $E$,
let $T_{E}=Res_{E/\mathbb{Q}}(\mathbb{G}_{m})$ be the corresponding algebraic
torus over $\mathbb{Q}$. A specific case of this is
$T_{E}(\mathbb{Q})=E^{\times}$. See Section 2.3.1 of [19] for more details.

Let $G=\operatorname{Gal}(L/\mathbb{Q})$ and let $H$ be the subgroup of $G$
associated to $E$: $H=\operatorname{Gal}(L/E)$. We can identify a CM type
$\Phi$ of $E$ with a subset of right cosets of $H\setminus G$, and we define
the set $\widetilde{\Phi}=\\{g\in G\mid Hg\in\Phi\\}$. In our work, the reflex
field of $E$ will always satisfy $E^{*}=E$ though its CM-type $\Phi^{*}$ will
usually be distinct from $\Phi$.

There are several important maps that we will carefully define. The first of
these is the usual norm function $N\colon L\to E$ defined by
$N(x)=\prod_{\sigma}\sigma(x)$, where the product is taken over all $\sigma$
in $\operatorname{Gal}(L/E)$. In the special case where $L=E$, this norm map
is simply the identity. This norm map induces a map on the associated tori.

We define two dual maps on the character groups of the tori $T_{E}$, denoted
by $\widehat{T}_{E}$. The first is a map on characters associated to $E$:

$\displaystyle\phi^{*}\colon\widehat{T}_{E}$
$\displaystyle\to\widehat{T}_{E^{*}}$ $\displaystyle[\beta]$
$\displaystyle\mapsto\sum_{\psi\in\Phi^{*}}[\psi\beta].$

Here $[\beta]$ denotes the character of $T_{E}$ corresponding to the embedding
$\beta\colon E\to\overline{\mathbb{Q}}$. The second map sends characters
associated to $E^{*}$ to those associated to $L$:

$\displaystyle N^{*}\colon\widehat{T}_{E^{*}}$
$\displaystyle\to\widehat{T}_{L}$ $\displaystyle[Hg]$
$\displaystyle\mapsto\sum_{h\in H}[hg].$

Here we are defining the map by considering the isomorphism
$\widehat{T}_{E^{*}}\simeq\mathbb{Z}[H\setminus G]$ (see the exposition in
[19]). In our work, we will be interested in the composition of these maps:
$N^{*}\phi^{*}\colon\widehat{T}_{E}\to\widehat{T}_{L}$.

####  3.1.2. The canonical projection map

The work in this section closely follows Section 4 of [19] where the results
are stated for a product of two absolutely simple abelian varieties. Let
$E_{i}$ denote the CM field of the absolutely simple abelian variety $A_{i}$
for $1\leq i\leq n$. We begin with a result for character maps.

######  Lemma 3.1.

The image of the map

$T_{L}\xrightarrow{(N_{1},N_{2},\ldots,N_{n})}T_{E_{1}^{*}}\times
T_{E_{2}^{*}}\times\cdots\times
T_{E_{n}^{*}}\xrightarrow{(\phi_{1},\phi_{2},\ldots,\phi_{n})}T_{E_{1}}\times
T_{E_{2}}\times\cdots\times T_{E_{n}}$

is $\operatorname{MT}(A_{1}\times A_{2}\times\cdots\times A_{n})$.

###### Proof.

See the proof of Lemma 4.1 of [19], which does not depend on the number of
factors in the product. ∎

Now consider the projection $\pi_{i}\colon\operatorname{MT}(A_{1}\times
A_{2}\times\cdots\times A_{n})\to\operatorname{MT}(A_{i})$.

######  Lemma 3.2.

Let
$\widehat{T}_{i}=\\{0\\}\times\cdots\times\\{0\\}\times\widehat{T}_{E_{i}}\times\\{0\\}\times\cdots\times\\{0\\}.$
The canonical projection $\pi_{i}$ is an isomorphism if and only if

$\widehat{T}_{i}+\ker(N_{1}^{*}\phi_{1}^{*}+\cdots+N_{n}^{*}\phi_{n}^{*})=\widehat{T}_{E_{1}}\times\cdots\times\widehat{T}_{E_{n}}.$

Similarly, $\pi_{i}$ is an isogeny if and only if
$\widehat{T}_{i}+\ker(N_{1}^{*}\phi_{1}^{*}+\cdots+N_{n}^{*}\phi_{n}^{*})$ has
finite index in the product
$\widehat{T}_{E_{1}}\times\cdots\times\widehat{T}_{E_{n}}$.

###### Proof.

This result is an easy generalization of Lemma 4.2 of [19], and so we include
just enough detail in the proof to aid in understanding the result.

Using the notation and definitions in Section 3.1.1 and the results of Lemma
3.1, we can build the following commutative diagram

$\textstyle{T_{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\phi_{1}N_{1},\,\ldots,\,\phi_{n}N_{n})\;\;\;\;}$$\scriptstyle{(N_{1},\ldots,N_{n})}$$\textstyle{\operatorname{MT}(A_{1}\times\cdots\times
A_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{i}}$$\textstyle{\operatorname{MT}(A_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{E_{1}^{*}}\times\cdots\times
T_{E_{n}^{*}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\phi_{1},\,\ldots,\,\phi_{n})}$$\textstyle{T_{E_{1}}\times\cdots\times
T_{E_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{i}}$$\textstyle{T_{E_{i}}}$

and its dual

$\textstyle{\widehat{T}_{L}}$$\textstyle{\widehat{\operatorname{MT}}(A_{1}\times\cdots\times
A_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{N_{1}^{*}\phi_{1}^{*}+\cdots+N_{n}^{*}\phi_{n}^{*}\;\;\;\;}$$\textstyle{\widehat{\operatorname{MT}}(A_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{i}^{*}}$$\textstyle{\widehat{T}_{E_{1}^{*}}\times\cdots\times\widehat{T}_{E_{n}^{*}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{N_{1}^{*}+\cdots+N_{n}^{*}}$$\textstyle{\widehat{T}_{E_{1}}\times\cdots\times\widehat{T}_{E_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\phi_{1}^{*},\,\ldots,\,\phi_{n}^{*})}$$\textstyle{\widehat{T}_{E_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{i}^{*}}$

where $\pi_{i},p_{i}$ are the canonical projections on the $i$-th factor, and
we denote by $f^{*}$ the map induced on characters by a morphism $f$ of
algebraic tori.

Proving that the (surjective) projection $\pi_{i}$ is injective is equivalent
to proving that $\pi_{i}^{*}$ is surjective. Similarly, proving that the
projection $\pi_{i}$ is an isogeny is equivalent to proving that $\pi_{i}^{*}$
has finite cokernel. By diagram chasing, in both cases we need only consider
the kernel and cokernel of

$\widehat{T}_{E_{i}}\longrightarrow\widehat{T}_{E_{1}}\times\cdots\times\widehat{T}_{E_{n}}\longrightarrow\widehat{\operatorname{MT}}(A_{1}\times\cdots\times
A_{n}).$

The kernel of the latter map coincides with the kernel of
$\widehat{T}_{E_{1}}\times\cdots\times\widehat{T}_{E_{n}}\xrightarrow{N_{1}^{*}\phi_{1}^{*}+\cdots+N_{n}^{*}\phi_{n}^{*}}\widehat{T}_{L}$
since $\widehat{\operatorname{MT}}(A_{1}\times\cdots\times
A_{n})\to\widehat{T}_{L}$ is injective. Hence,
$\widehat{\operatorname{MT}}(A_{1}\times\cdots\times A_{n})$ can be identified
with
$({\widehat{T}_{E_{1}}\times\cdots\times\widehat{T}_{E_{n}}})/{\ker(N_{1}^{*}\phi_{1}^{*}+\cdots+N_{n}^{*}\phi_{n}^{*})}.$

Under this identification, the map $\pi_{i}^{*}$ is surjective if and only if
$p_{i}^{*}\left(\widehat{T}_{E_{i}}\right)$, which equals $\widehat{T}_{i}$,
surjects onto
${\widehat{T}_{E_{1}}\times\cdots\times\widehat{T}_{E_{n}}}/{\ker(N_{1}^{*}\phi_{1}^{*}+\cdots+N_{n}^{*}\phi_{n}^{*})}$.
Similarly, $\pi_{i}^{*}$ is an isogeny if and only if
$p_{i}^{*}\left(\widehat{T}_{E_{i}}\right)$ maps to
${\widehat{T}_{E_{1}}\times\cdots\times\widehat{T}_{E_{n}}}/{\ker(N_{1}^{*}\phi_{1}^{*}+\cdots+N_{n}^{*}\phi_{n}^{*})}$
with finite cokernel. ∎

As in [19, Lemma 4.3], this can all be rephrased in terms of matrices. In the
notation of Section 3.1.1, let $E_{i}$ denote the CM field of the absolutely
simple abelian variety $A_{i}$ for $1\leq i\leq n$ and let $L$ be a finite
Galois extension of $\mathbb{Q}$ containing the Galois closure of each
$E_{i}$. Furthermore, let $G$ denote the Galois group of $L$ over
$\mathbb{Q}$, and $H_{i}\leq G$ be the subgroup corresponding to $E_{i}$.

We let $M_{i}$ denote the matrix that represents the map
$N_{i}^{*}\phi_{i}^{*}$. The rows of $M_{i}$ are indexed by elements of $G$
whereas the columns of $M_{i}$ are indexed by the quotient group
$H_{i}\setminus G$. We then compute the composition $N_{i}^{*}\phi_{i}^{*}$
for each $[\beta]\in\widehat{T}_{E_{i}}$ in order to determine the entries of
$M_{i}$. In the special case where $E_{i}=L$, the matrix $M_{i}$ is a square
matrix with rows and columns indexed by $G$ (since $H_{i}=\\{1\\}$). Its entry
in position $(g_{j},H_{i}g_{k})$ is given by

$(g_{j},H_{i}g_{k})=\begin{cases}1&\text{if
}g_{j}g_{k}^{-1}\in\widetilde{\Phi_{i}^{*}}\\\ 0&\text{otherwise.}\end{cases}$

The matrix $M$ representing the map
$N_{1}^{*}\phi_{1}^{*}+\cdots+N_{n}^{*}\phi_{n}^{*}$ is obtained by
concatenating the matrices $M_{1},\ldots,M_{n}$ horizontally. This
construction allows to easily examine the kernel of the map when working out
explicit examples.

We will apply these results to Jacobians of curves of the form
$y^{2}=x^{m}-1$. In Section 4 we give the decompositions of the Jacobian
varieties into simple factors, and in Section 5 we use these decompositions to
write the matrix $M$ and study the degeneracy of the Mumford-Tate groups. We
see examples where the canonical projection of the Mumford Tate group is an
isomorphism and ones where it is merely an isogeny.

###  3.2. Degeneracy via Hodge Rings

Recall that the Hodge ring of $A$ is denoted $\mathscr{B}^{*}(A)$ and the
subring generated by divisor classes is denoted $\mathscr{D}^{*}(A)$. As noted
in Section 2.3, we have the containment
$\mathscr{D}^{*}(A)\subseteq\mathscr{B}^{*}(A)$. The question we wish to
answer is: when is the containment strict? In other words, we wish to know
when $\mathscr{B}^{*}(A)$ is not generated by the divisor classes
$\mathscr{B}^{1}(A)$. We can think of this as the Hodge ring being larger than
expected. The additional Hodge cycles that are not generated by divisor
classes are referred to as exceptional cycles, and so we can also ask: in what
codimensions do we have exceptional cycles?

In general, these are difficult questions to answer. However, if we specify to
Jacobian varieties $J_{m}$ of curves $C_{m}\colon y^{2}=x^{m}-1$ then we can
give satisfactory answers.

######  Proposition 3.3.

Let $m=pq$, with $q,p$ odd primes, not necessarily distinct. Suppose $q$ is
relatively prime to $((p+1)/2)!$. Then $\mathcal{B}^{d}(J_{m})$ has
exceptional Hodge cycles when $d=(p+1)/2$.

###### Proof.

Consider the quotient

$\mathcal{B}^{d}(J_{m})\bigg{/}\sum_{r=1}^{d-1}\mathcal{B}^{r}(J_{m})\cdot\mathcal{B}^{d-r}(J_{m}).$

If the dimension of this space is greater than 1, then there will be Hodge
cycles in $\mathcal{B}^{d}(J_{m})$ that do not come from lower codimension. In
particular, there will be exceptional cycles that are not generated by divisor
classes.

Shioda relates the dimension of this space to the the number $N_{m}(d)$ of
certain indecomposable elements of a semigroup (see Section 1C of [24]). For
certain values of $m$, Shioda gives a lower bound for this quantity (see Lemma
5.5 of [24]): letting $m=(2d-1)m^{\prime}$, we have

$N_{m}(d)\geq m^{\prime}-1$ (2)

whenever $\gcd(m^{\prime},d!)=1$.

For $m=pq$ and $d=(p+1)/2$, we let $m^{\prime}=q$ so that
$m=(2d-1)m^{\prime}.$ If $\gcd(q,d!)=1$ then the inequality in (2) becomes

$N_{m}(d)\geq q-1$

which is greater than 1 for any $q>2$. Thus, there are exceptional cycles in
$\mathcal{B}^{d}(J_{m})$, where $d=(p+1)/2$. ∎

This result will always hold if we choose $q\geq p$ since then $q>(p+1)/2$ and
is, therefore, relatively prime to every smaller positive integer. In Section
4.1 we use this result to prove Theorem 1.1.

Proposition 3.3 does not give a value or even a bound for the number of
exceptional cycles – we simply get a lower bound on the number of cycles that
do not come from a lower codimension. However, as we will see in Section 5, it
is possible to compute the generators of the Hodge ring for the Jacobian
varieties $J_{m}$ and so we will be able to see which cycles are exceptional.

###  3.3. Degeneracy via Hodge Groups

As noted in Section 2.2, the Hodge and Mumford-Tate groups are closely related
to each other. The Hodge group is also closely related to the Lefschetz group,
denoted $\operatorname{L}(A)$, which is the connected component of the
identity in the centralizer of the endomorphism ring
$\operatorname{End}(A_{\overline{F}})_{\mathbb{Q}}$ in
$\operatorname{Sp}_{V}$. In general,
$\operatorname{Hg}(A)\subseteq\operatorname{L}(A)$ and while the inclusion can
be strict, there are cases where we have equality. In fact, when $\dim(A)\leq
3$ it is known that we always have equality.

The question we wish to answer in this paper is: when is the Hodge group
smaller than the Lefschetz group? We could investigate an answer to this
through the Hodge group’s relationship to the Mumford Tate group, but we
choose a different approach in this paper: we make use of a connection to
Hodge rings.

Let $A$ be an abelian variety of dimension $g$. For any $1\leq d\leq g$,
denote by

$H^{2d}_{\text{Hodge}}(A):=H^{2d}(A,\mathbb{Q})\cap H^{d,d}(A)$

the $\mathbb{Q}$-vector space of Hodge cycles of codimension $d$ on $A$. Note
that, in the notation of Section 3.2,
$\mathscr{B}^{d}(A)=H^{2d}_{\text{Hodge}}(A)\otimes\mathbb{C}$. The following
result relates the $\mathbb{Q}$-span of Hodge cycles with the Hodge group.

######  Theorem 3.4.

[4, Theorem 17.3.3]

$H^{2d}_{\text{Hodge}}(A)=H^{2d}(A,\mathbb{Q})^{\operatorname{Hg}(A)}.$

There is a balance between the size of the Hodge group and the dimension of
$H^{2d}_{\text{Hodge}}(A)$. The existence of exceptional Hodge cycles, which
we sometimes learn of through Proposition 3.3, often coincides with a smaller
than expected Hodge group. In fact, we see something even more interesting:
the exceptional cycles give us information about the embedding of
$\operatorname{Hg}(A)$ into $\operatorname{USp}(2g)$. We will see examples of
this correspondence between Hodge rings and Hodge group in Section 5.

###  3.4. Effects on the Sato-Tate group

As noted in Section 2.5, when the Mumford-Tate conjecture holds for an abelian
variety $A$ then the Mumford-Tate group and Hodge group can be used to gain
information about the Sato-Tate group of $A$. In particular, the Hodge group
will equal the identity component of the algebraic Sato-Tate group. The
results and exposition in Sections 3.1, 3.2, and 3.3 help us understand the
role degeneracy can play in the Sato-Tate group.

The degeneracy we see in the Mumford-Tate group in Section 3.1 tells us which
factor $A_{i}$ of the abelian variety $A$ gives us the Mumford-Tate group of
$A$. When the Mumford-Tate conjecture holds for $A$, this will imply that
$\operatorname{ST}^{0}(A)\simeq\operatorname{ST}^{0}(A_{i})$. As with the
Mumford-Tate group, in this situation we have a smaller than expected identity
component for the Sato-Tate group.

The degeneracy of the Mumford-Tate group gives us some information about the
Hodge group and the identity component of the algebraic Sato-Tate group of
$A$. However in order to compute moment statistics for the Sato-Tate group, we
need to know the explicit embedding of the Hodge group in
$\operatorname{USp}(2g)$, where $g$ is the dimension of $A$. The exceptional
cycles in the Hodge ring of $A$ give extra relations on the elements of the
Hodge ring (see Theorem 3.4); these extra relations can be used to determine
the embedding of the Hodge group.

Since the Mumford-Tate and Hodge groups only give us information about the
identity component of the Sato-Tate group, we will not gain information about
the component group. The Sato-Tate group of a degenerate abelian variety can
have not only a smaller-than-expected identity component, but can also have a
larger component group than what would be explained by endomorphisms. In
general, for an abelian variety $A/F$ we have a canonical surjection

$\operatorname{ST}(A)/\operatorname{ST}^{0}(A)\rightarrow\operatorname{Gal}(K/F),$

where $K$ is the endomorphism field of $A$, but the surjection is not
necessarily an isomorphism if $A$ is degenerate (see, for example, [11]). In
general, we have an isomorphism
$\operatorname{ST}(A)/\operatorname{ST}^{0}(A)\simeq\operatorname{Gal}(L/F)$,
where $L$ is the minimal Galois extension of $F$ for which
$\operatorname{ST}(A_{L})$ is connected. The field $L$ is the fixed field of
the kernel of a map induced by the $\ell$-adic representation $\rho_{A,\ell}$
in Equation (1) (see [26, Theorem 3.12]), and so we say that $L$ is cut out by
the representation. In some cases, we can use results of Zywina [28] to
determine this field $L$. In Section 5 we will see examples where the field
$L$ is the endomorphism field of $A$ and ones where an extension is needed.

##  4\. Decompositions of Jacobians

Let $C_{m}$ denote the smooth projective curve with hyperelliptic equation
$y^{2}=x^{m}-1$, and let $J_{m}$ denote its Jacobian. In this section we
describe how $J_{m}$ decomposes into simple factors. We begin with two cases:
$m=p^{2}$ and $m=pq$, where $p$ and $q$ are distinct odd primes.

In the first case, there is a map from $C_{m}$ to the degree $p$ hyperelliptic
curve $C_{p}\colon y^{2}=x^{p}-1$ given by $\pi_{p}:(x,y)\mapsto(x^{p},y)$. We
let $X$ denote the identity component of the kernel of $(\pi_{p})_{*}\colon
J_{m}\to J_{p}$. This gives the following isogeny over $\mathbb{Q}$

$\displaystyle J_{p^{2}}\sim X\times J_{p}.$ (3)

We can easily compute the dimension of $X$ to be
$\dim(J_{m})-\dim(J_{p})=p(p-1)/2=\phi(m)/2$.

In the second case, there are two such maps:
$\pi_{p}\colon(x,y)\mapsto(x^{q},y)$ from $C_{m}$ to $C_{p}:y^{2}=x^{p}-1$ and
$\pi_{q}\colon(x,y)\mapsto(x^{p},y)$ from $C$ to $C_{q}:y^{2}=x^{q}-1$. We
obtain the following isogeny over $\mathbb{Q}$

$\displaystyle J_{pq}\sim X\times J_{p}\times J_{q},$ (4)

where the dimension of $X$ is computed to be $\phi(m)/2$.

######  Proposition 4.1.

Let $m=p^{2}$, respectively $pq$. Then the Prym variety $X$ given by the
isogeny in Equation (3), respectively Equation (4), is absolutely simple.

By work of Shimura-Taniyama [23], we can prove that $X$ is absolutely simple
by proving that its CM-type is primitive. Before proving Proposition 4.1, we
provide two lemmas that will be useful.

######  Lemma 4.2.

Let $m=pq$, where $p$ and $q$ are odd primes (not necessarily distinct). Let
$X$ be the Prym variety obtained as a quotient of $J_{m}$. Then the CM-type of
$X$ is $(\mathbb{Q}(\zeta_{m});\Phi)$, where

$\Phi=\\{\sigma_{j}\mid\gcd(j,m)=1\text{ and }j\leq g\\},$

$g=(m-1)/2$, and $\sigma_{j}(\zeta_{m})=\zeta_{m}^{j}$.

###### Proof.

For ease of notation, let $C\colon=C_{m}$. Since $C$ is a genus $g=(m-1)/2$
hyperelliptic curve, we may identify the complex uniformization of $J_{m}$
with $H^{0}(C_{\mathbb{C}},\Omega_{C}^{1})=\mathbb{C}\langle x^{i}dx/y\mid
0\leq i<g\rangle$. We briefly split into two cases:

Case 1: $p=q$. Recall that there is a single map
$\pi_{p}:(x,y)\mapsto(x^{p},y)$ from $C$ to $C_{p}$. By pulling back regular
differentials of $C_{p}$, we see that $\pi_{p}^{*}C_{p}$ corresponds to the
vector subspace

$\mathbb{C}\left\langle{x^{ap-1}}dx/y\mid
a=1,2,\ldots,\frac{p-1}{2}\right\rangle$

of $H^{0}(C_{\mathbb{C}},\Omega_{C}^{1})$. Thus, the quotient $J_{m}/J_{p}$
(which is isogenous to $X$) has natural analytic uniformization given by the
quotient of $\mathbb{C}\langle x^{i}dx/y\mid 0\leq i<g\rangle$ by this vector
space. The dimension of the new vector space is
$\frac{m-1}{2}-\frac{p-1}{2}=\phi(m)/2$.

Case 2: $p\not=q$. In this case, there are two such maps:
$\pi_{p}:(x,y)\mapsto(x^{q},y)$ from $C$ to $C_{p}$ and
$\pi_{q}:(x,y)\mapsto(x^{p},y)$ from $C$ to $C_{q}$. As above, we pull back
regular differentials of $C_{p}$ and $C_{q}$. The quotient of $J_{m}$ by the
Jacobians of the two curves (which is isogenous to $X$) has natural analytic
uniformization given by the quotient of $\mathbb{C}\langle x^{i}dx/y\mid 0\leq
i<g\rangle$ by the disjoint union

$\mathbb{C}\left\langle{x^{aq-1}}dx/y\mid
a=1,2,\ldots,({p-1})/{2}\right\rangle\cup\mathbb{C}\left\langle{x^{ap-1}}dx/y\mid
a=1,2,\ldots,({q-1})/{2}\right\rangle.$

The dimension of the new vector space is
$\frac{m-1}{2}-\frac{p-1}{2}-\frac{q-1}{2}=\phi(m)/2$.

In both cases, the automorphism $\alpha\colon(x,y)\mapsto(\zeta_{m}x,y)$
induces an action of $\mathbb{Q}(\zeta_{m})$ on the $(\phi(m)/2)$-dimensional
vector space given above. Since
$\alpha^{*}x^{j}dx/y=\zeta_{m}^{j+1}x^{j}dx/y$, we find that characters
appearing in this uniformization will be of the form
$\zeta_{m}\mapsto\zeta_{m}^{j}$ where $j$ is coprime to $m$.

Thus, we can identify the CM type $\Phi$ of
$\operatorname{Gal}(\mathbb{Q}(\zeta_{m})/\mathbb{Q})$ from the statement of
the lemma as the CM-type of $X$. ∎

######  Lemma 4.3.

Let $m$ be an odd composite number. Define the set

$S_{m}=\left\\{1,\ldots,\frac{m-1}{2}\right\\}\cap\mathbb{Z}/m\mathbb{Z}^{\times}.$

Then $aS_{m}=S_{m}$ for some $a\in\mathbb{Z}/m\mathbb{Z}^{\times}$ if and only
if $a=1$.

###### Proof.

First note that $aS_{m}=S_{m}$ is clearly true if $a=1$. Also, $aS_{m}$ cannot
equal $S_{m}$ if $a>\frac{m-1}{2}$ since then $a\cdot 1\not\in S_{m}$.

Now suppose $a\in S_{m}$ satisfies $1<a\leq\frac{m-1}{2}$. We will show that
there is a value $b\in S_{m}$ such that $ab\not\in S_{m}$.

Note that there is an integer in the interval
$\left(\frac{m-1}{2a},\frac{m-1}{a}\right)$. Indeed, the only way for this to
not be true is if $(m-1)/a<(m-1)/2a+1$, which would imply $m-1<(m-1)/2+a$.
However this is not possible since $a\in S_{m}$ and, hence, $a<(m-1)/2$.

Now let $b$ be an integer in this interval: $\frac{m-1}{2a}<b<\frac{m-1}{a}$.
This implies that $(m-1)/2<ab<m-1,$ which proves that $ab$ is not in $S_{m}$.
Thus, we have proved the desired result.

∎

###### Proof of Proposition 4.1.

As noted below the statement of Proposition 4.1, it is sufficient to prove
that the CM-type of $X$ is primitive. Proposition 26 of [23] claims the
following: A CM-type $S$ is primitive if and only if $\gamma S=S$ for $\gamma$
in the Galois group only when $\gamma$ is the identity.

We gave the CM-type $\Phi$ of $X$ in Lemma 4.2. Identifying the automorphisms
in $\Phi$ with the set
$S_{m}=\\{1,\ldots,(m-1)/2\\}\cap\mathbb{Z}/m\mathbb{Z}^{\times}$ allows us to
apply the results of Lemma 4.3 and conclude that $\gamma\Phi=\Phi$ only when
$\gamma$ is the identity element. Hence, $\Phi$ is a primitive CM type and $X$
is absolutely simple. ∎

######  Corollary 4.4.

The dual CM type is $(\mathbb{Q}(\zeta_{m});\Phi^{*})$, where

$\Phi^{*}=\\{\sigma_{j}\mid\gcd(j,m)=1\text{ and }j=1\text{ or }j>g\\}.$

###### Proof.

This follows from Lemma 4.2 and the definition of the dual CM type in
Proposition 28 in [23]. ∎

We can see the decompositions in Equations (3) and (4) at the level of
Frobenius polynomials. Let $P_{A,\mathfrak{p}}(x)$ denote the Frobenius
polynomial of the abelian variety $A$ at $\mathfrak{p}$. Then for
$A=J_{p^{2}}$ we have $P_{A,\mathfrak{p}}(x)=P_{X,\mathfrak{p}}(x)\cdot
P_{J_{p},\mathfrak{p}}(x)$ and for $A=J_{pq}$ we have
$P_{A,\mathfrak{p}}(x)=P_{X,\mathfrak{p}}(x)\cdot
P_{J_{p},\mathfrak{p}}(x)\cdot P_{J_{q},\mathfrak{p}}(x)$ for all good primes
$\mathfrak{p}$.

We can compute Frobenius polynomials for $J_{m}$ for more general odd,
composite $m$ in Sage over good primes $\mathfrak{p}$. Upon factoring, we are
able to see how the Jacobian variety $J_{m}$ decomposes into simple factors.
This leads us to the following conjecture.

######  Conjecture 4.5.

Let $m$ be a positive odd integer. Then the decomposition of $J_{m}$ into
simple factors contains exactly one simple factor of dimension $\phi(d)/2$ for
each divisor $d$ of $m$. Furthermore, if $p$ is a prime divisor of $m$ then
the factor of dimension $\phi(p)/2$ is the Jacobian of the curve $C_{p}\colon
y^{2}=x^{p}-1$.

Thus far we have only considered odd values of $m$. In the case where $m$ is
even, a result of [8] can be applied to determine that splitting of the
Jacobian.

######  Proposition 4.6.

[8, Lemma 4.1] Let $g=2k$ be an even integer, and $C_{2g+2}:y^{2}=x^{2g+2}+c$,
where $c\in\mathbb{Q}^{\times}$. Then we have the following isogeny over
$\overline{\mathbb{Q}}$

$\operatorname{Jac}(C_{2g+2})\sim\operatorname{Jac}({C_{g+1}})^{2},$

where $C_{g+1}:y^{2}=x^{g+1}+c$.

This allows us to prove the following statement.

######  Corollary 4.7.

Let $m=p^{2}$ or $pq$. Then the Jacobian of the hyperelliptic curve $C_{2m}$
satisfies the following isogeny over $\mathbb{Q}$

$\displaystyle J_{2m}\simeq\begin{cases}(X\times J_{p})^{2}&\text{if
}m=p^{2},\\\ (X\times J_{p}\times J_{q})^{2}&\text{if }m=pq.\end{cases}$

###### Proof.

In both cases, $J_{2m}\simeq(J_{m})^{2}$ by Lemma 4.1 of [8] since the
dimension of $J_{2m}$ is even: we set $2m=2g+2$ and find that $g=m-1$ is even
since $m$ is odd. The result then follows from Equations (3) and (4). ∎

###  4.1. Proof of Theorem 1.1

We are now in a position to prove that for any odd, composite $m$, the
Jacobian $J_{m}$ of $y^{2}=x^{m}-1$ is degenerate.

###### Proof.

Let $m$ be an odd, composite integer divisible by $pq$, where $p$ and $q$ are
primes (not necessarily distinct). As in the the beginning of Section 4, we
can view the Jacobian of the curve $y^{2}=x^{pq}-1$ as a factor of $J_{m}$ via
a map on the curve: $\pi_{pq}\colon C_{m}\to C_{pq}$ defined by
$(x,y)\mapsto(x^{m/pq},y)$. We proved in Proposition 3.3 that the Jacobian
variety $J_{pq}$ has exceptional Hodge cycles. This proves the desired result
since these exceptional cycles will yield exceptional cycles in the Hodge ring
of $J_{m}$. ∎

##  5\. Examples

In this section we explore the various forms of degeneracy for several
Jacobian varieties $J_{m}$, each illustrating different phenomena that can
occur. The examples are presented in a non-hodgepodge manner: we carefully
examine degeneracy in the Mumford-Tate groups, the Hodge rings and Hodge
groups, and the Sato-Tate groups of the varieties. Most of the computations
were done in Sage (code available at [15]).

###  5.1. $y^{2}=x^{9}-1$

We first consider the genus 4 curve $C_{9}\colon y^{2}=x^{9}-1$. The isogeny
in Equation (3) tells us that its Jacobian factors as $J_{9}\sim X\times
J_{3}$. Here $J_{3}$ denotes the CM elliptic curve $y^{2}=x^{3}-1$ and, by
Proposition 4.1, $X$ is an absolutely simple, 3-dimensional abelian variety.
Lemma 4.2 tells us that the CM field of $X$ is $\mathbb{Q}(\zeta_{9})$.

There are some results in the literature that give us information about the
degeneracy of $J_{9}$. Remark 5.6 of [19] shows that the projection
$\operatorname{MT}(J_{9})\rightarrow\operatorname{MT}(X)$ is an isomorphism;
this is proved using results similar to those in Section 3.1. In [24, Example
6.1], Shioda works out the Hodge ring for $J_{9}$. In particular, Shioda uses
a result that is a special case of Proposition 3.3 to determine that there are
exceptional Hodge cycles in codimension 2. The exceptional cycles in
codimension 2 can be described by

$\mathcal{B}^{2}(J_{9})/\mathcal{D}^{2}(J_{9})=\\{\omega_{1}\wedge\omega_{4}\wedge\overline{\omega}_{3}\wedge\overline{\omega}_{2},\;\;\omega_{2}\wedge\omega_{3}\wedge\overline{\omega}_{4}\wedge\overline{\omega}_{1}\\}.$

There are no exceptional cycles in other codimensions.

We now show how we can determine the Hodge group of $J_{9}$ from this. Recall
that the Hodge group $\operatorname{Hg}(J_{9})$ is an algebraic subgroup of
$\operatorname{SL}_{V}$ and that $\operatorname{Hg}(J_{9})_{\mathbb{C}}$
contains $h(\mathbb{S}^{1})$. Furthermore, Theorem 3.4 tells us
$H^{2d}_{\text{Hodge}}(J_{9})=H^{2d}(J_{9},\mathbb{Q})^{\operatorname{Hg}(J_{9})}.$

We can identify $\operatorname{SL}_{V}$ with the group of $2g\times 2g$
matrices with determinant 1. Since $J_{9}$ is of CM-type, Theorem 17.3.5 of
[4] tells us that its Hodge group is commutative. Thus, we can say that
elements of $\operatorname{Hg}(J_{9})$ are of the form

$U=\operatorname{diag}(U_{1},U_{2},U_{3},U_{4}),$

where $U_{i}=\operatorname{diag}(u_{i},\overline{u}_{i})$ and
$u_{i}\in\mathbb{S}^{1}$, and where there may be some relations among the
$U_{i}$. We compute $U\cdot v$ for each element $v$ of the space of Hodge
cycles in order to determine what relations are required in order to obtain
$U\cdot v=v$. For example,

$\displaystyle
U\cdot(\omega_{1}\wedge\omega_{4}\wedge\overline{\omega}_{3}\wedge\overline{\omega}_{2})$
$\displaystyle=u_{1}u_{4}\overline{u}_{2}\overline{u}_{3}(\omega_{1}\wedge\omega_{4}\wedge\overline{\omega}_{3}\wedge\overline{\omega}_{2}).$

This yields the relation $u_{4}=\overline{u}_{1}u_{2}u_{3}$. The other
exceptional cycle yields the same relation. Thus, we may conclude that the
elements of $\operatorname{Hg}(J_{9})$ are of the form

$U=\operatorname{diag}(U_{1},U_{2},U_{3},\overline{U}_{1}U_{2}U_{3}).$

####  5.1.1. Sato-Tate group

Since the Mumford-Tate conjecture holds for $J_{9}$, the Hodge group equals
the identity component of the algebraic Sato-Tate group. The identity
component of the Sato-Tate group is a maximal compact subgroup of this, and so
we have the following result.

######  Proposition 5.1.

Up to conjugation in $\operatorname{USp}(8)$, the identity component of the
Sato-Tate group of $J_{9}$ is isomorphic to $\operatorname{U}(1)^{3}$. More
specifically,

$\operatorname{ST}^{0}(J_{9})\simeq\langle\operatorname{diag}(U_{1},U_{2},U_{3},\overline{U}_{1}U_{2}U_{3})\rangle.$

This matches the result of Theorem 6.9 of [8], though here we used a different
method to obtain the result and we give the additional relations needed for
the embedding in $\operatorname{USp}(8)$.

In order to determine the full Sato-Tate group of $J_{9}$, we first need to
know the minimal field $L$ for which $\operatorname{ST}((J_{9})_{L})$ is
connected. As noted in Section 3.4, this field can be computed using results
of Zywina. In fact, Zywina determines that this field is equal to the CM field
of $C_{9}$ (see Section 1.7 of [28]). Hence,
$\operatorname{ST}(J_{9})/\operatorname{ST}^{0}(J_{9})\simeq\operatorname{Gal}(\mathbb{Q}(\zeta_{9})/\mathbb{Q}).$

This is enough information to compare moment statistics for the identity
component of the Sato-Tate group to numerical moments since
$\operatorname{ST}^{0}(J_{9})=\operatorname{ST}((J_{9})_{\mathbb{Q}(\zeta_{9})})$.
The numerical moments in Table 1 were computed over the field
$\mathbb{Q}(\zeta_{9})$ and for all primes up to $p<2^{23}$.

| $M_{2}$ | $M_{4}$ | $M_{6}$ | $M_{8}$ | $M_{10}$
---|---|---|---|---|---
$\mu_{1}$ | 8 | 216 | 8000 | 343000 | 16003008
$a_{1}$ | 8.01253 | 216.204 | 7997.25 | 342072 | 15901600
Table 1. Table of $\mu_{1}$\- and (rounded) $a_{1}$-moments for
$\operatorname{ST}(J_{9})$ over $\mathbb{Q}(\zeta_{9})$.

The errors for the even moments are within a margin of $1\%$ and are
consistent with the errors for the odd moments (which should all be zero).
Choosing a larger bound for $p$ will lead to better estimates.

For nondegenerate abelian varieties, we can determine generators of the
component group of the Sato-Tate group through the twisted Lefschetz group
(see [3]). The twisted Lefschetz group, denoted $\operatorname{TL}(A)$, is a
closed algebraic subgroup of $\operatorname{Sp}_{2g}$ defined by

$\displaystyle\operatorname{TL}(A):=\bigcup_{\tau\in\operatorname{Gal}(\overline{F}/F)}\operatorname{L}(A)(\tau),$

where
$\operatorname{L}(A)(\tau):=\\{\gamma\in\operatorname{Sp}_{2g}\mid\gamma\alpha\gamma^{-1}=\tau(\alpha)\text{
for all }\alpha\in\operatorname{End}(A_{\overline{F}})_{\mathbb{Q}}\\}$.

We can compute the twisted Lefschetz group for the Jacobian variety $J_{9}$
even though this is a degenerate abelian variety. The nontrivial Lefschetz
sets $\operatorname{L}(J_{9})(\tau)$ are those coming from elements $\tau$ in
$G=\operatorname{Gal}(\mathbb{Q}(\zeta_{9})/\mathbb{Q})$ since
$\mathbb{Q}(\zeta_{9})$ is the endomorphism field of $J_{9}$. The Galois group
$G$ is generated by the order-6 automorphism
$\tau_{2}\colon\zeta_{9}\to\zeta_{9}^{2}$ and we have $\bigcup_{\tau\in
G}\operatorname{L}(A)(\tau)=\left\langle\gamma\right\rangle,$ where

$\displaystyle\begin{pmatrix}0&I&0&0\\\ 0&0&0&I\\\ 0&0&J&0\\\
J&0&0&0\end{pmatrix}.$ (5)

The matrix $\gamma$ was computed using the techniques of [7] and [14].

We now compute moment statistics of the group
$\langle\operatorname{diag}(U_{1},U_{2},U_{3},\overline{U}_{1}U_{2}U_{3}),\;\gamma\rangle$
and compare them to the numerical moments over $\mathbb{Q}$. The numerical
moments in Table 2 were computed for all primes up to $p<2^{23}$.

| $M_{2}$ | $M_{4}$ | $M_{6}$ | $M_{8}$ | $M_{10}$ | $M_{12}$
---|---|---|---|---|---|---
$\mu_{1}$ | 2 | 38 | 1340 | 57190 | 2667252 | 131481812
$a_{1}$ | 2 | 38 | 1338 | 57010 | 2649180 | 129958000
Table 2. Table of $\mu_{1}$\- and (rounded) $a_{1}$-moments for $J_{9}$ over
$\mathbb{Q}$.

As in Table 1, the errors for the even moments shown in Table 2 are within a
margin of $1\%$ and are consistent with the errors for the odd moments. This
leads us to make the following conjecture.

######  Conjecture 5.2.

Up to conjugation in $\operatorname{USp}(8)$, the Sato-Tate group of $J_{9}$
is

$\operatorname{ST}(J_{9})\simeq\left\langle\operatorname{diag}(U_{1},U_{2},U_{3},\overline{U}_{1}U_{2}U_{3}),\gamma\right\rangle.$

As further evidence supporting this claim, we recall that the Sato-Tate group
of $J_{9}$ is connected over the field $\mathbb{Q}(\zeta_{9})$ and note that
the component group of the above group is isomorphic to the Galois group
$G=\operatorname{Gal}(\mathbb{Q}(\zeta_{9})/\mathbb{Q})$.

####  5.1.2. $y^{2}=x^{18}-1$

We first apply Proposition 4.6 to the Jacobian of the curve $C_{18}\colon
y^{2}=x^{18}-1$ and find that

$\displaystyle J_{18}\simeq(J_{9})^{2}.$

Hence, by the discussion in Section 2.2, the Hodge group
$\operatorname{Hg}(J_{18})$ is isomorphic to $\operatorname{Hg}(J_{9})$, which
acts on $V(J_{18})$ diagonally. Since the Mumford-Tate conjecture holds for
$J_{9}$, it also holds for $J_{18}\simeq(J_{9})^{2}$ (see Section 2.5). Thus,
the Hodge group equals the identity component of the algebraic Sato-Tate
group. The identity component of the Sato-Tate group is a maximal compact
subgroup of this, and so we have the following result.

######  Proposition 5.3.

Up to conjugation in $\operatorname{USp}(16)$, the identity component of the
Sato-Tate group of $J_{18}$ is isomorphic to $(\operatorname{U}(1)^{3})_{2}$.
More specifically,

$\operatorname{ST}^{0}(J_{18})=\langle\operatorname{diag}(U_{1},U_{2},U_{3},\overline{U}_{1}U_{2}U_{3},U_{1},U_{2},U_{3},\overline{U}_{1}U_{2}U_{3})\rangle.$

This matches the result in Section 6.6 of [8], though here we give the
additional relations needed for the embedding in $\operatorname{USp}(16)$.
These additional relations enable us to compute moment statistics for the
identity component (see Table 3).

| $M_{2}$ | $M_{4}$ | $M_{6}$ | $M_{8}$ | $M_{10}$ | $M_{12}$
---|---|---|---|---|---|---
$\mu_{1}$ | 32 | 3456 | 512000 | 87808000 | 16387080192 | 3231289442304
Table 3. Table of $\mu_{1}$-moments for $J_{18}$ over
$\mathbb{Q}(\zeta_{18})$.

###  5.2. $y^{2}=x^{15}-1$

We now consider the curve $C_{15}\colon y^{2}=x^{15}-1$. The isogeny in
Equation (4) tells us that its Jacobian factors as $J_{15}\sim X\times
J_{5}\times J_{3}$, where $X$ is an absolutely simple 4-dimensional abelian
variety. By Lemma 4.2, both $J_{15}$ and $X$ have CM by
$\mathbb{Q}(\zeta_{15})$.

While the Hodge ring and Hodge group degeneracies are similar to those we saw
in Section 5.1, we see interesting new phenomena for the Mumford-Tate group
and Sato-Tate group.

####  5.2.1. Mumford-Tate group

In the notation of Section 3.1.1, let
$L=E_{1}=\mathbb{Q}(\zeta_{15}),E_{2}=\mathbb{Q}(\zeta_{5}),$ and
$E_{3}=\mathbb{Q}(\zeta_{3})$. Let $G$ denote the Galois group of $L$ over
$\mathbb{Q}$, and $H_{i}\leq G$ be the subgroup corresponding to $E_{i}$:
$H_{i}=\operatorname{Gal}(L/E_{i})$.

The exposition after the proof of Lemma 3.2 gives a method for representing
the map $N_{1}^{*}\phi_{1}^{*}+N_{2}^{*}\phi_{2}^{*}+N_{3}^{*}\phi_{3}^{*}$ as
a matrix $M$, which is the concatenation of matrices $M_{1},M_{2}$ and $M_{3}$
corresponding to the factors of $J_{15}$. The rows of each matrix will be
indexed by

$\displaystyle
G=\\{\tau_{1},\tau_{2},\tau_{4},\tau_{7},\tau_{8},\tau_{11},\tau_{13},\tau_{14}\\},$
(6)

where $\tau_{j}(\zeta_{15})=\zeta_{15}^{j}$.

For an explicit example, we show how to compute a column for the matrix
$M_{2}$. The CM-type of $E_{2}=\mathbb{Q}(\zeta_{5})$ is
$\Phi_{2}=\\{\sigma_{1},\sigma_{2}\\}$, where
$\sigma_{j}(\zeta_{5})=\zeta_{5}^{j}$, and the reflex type is
$\Phi_{2}^{*}=\\{\sigma_{1},\sigma_{3}\\}$. The elements of
$H_{2}=\operatorname{Gal}(L/E_{2})$ are $\tau_{1}$ and $\tau_{11}$. Consider
the character $\sigma_{2}\in\widehat{T}_{E_{2}}$. We see that

$\displaystyle N_{2}^{*}\phi_{2}^{*}([\sigma_{2}])$
$\displaystyle=N_{2}^{*}([\sigma_{1}\sigma_{2}]+[\sigma_{3}\sigma_{2}])$
$\displaystyle=N_{2}^{*}([\sigma_{2}]+[\sigma_{1}])$
$\displaystyle=([\tau_{2}]+[\tau_{7}])+([\tau_{1}]+[\tau_{11}]).$

Taking into account the labeling of the rows of $M$ coming from Equation (6),
we can conclude that the second column of $M_{2}$ is $(1,1,0,1,0,1,0,0)$.

We repeat this process for the matrices $M_{1},M_{2},M_{3}$ and obtain

$\displaystyle M$
$\displaystyle=\left(\begin{tabular}[]{cccccccc|cccc|cc}1&1&1&1&0&0&0&0&1&1&0&0&1&0\\\
0&1&1&0&1&0&0&1&0&1&0&1&0&1\\\ 1&0&1&0&1&0&1&0&0&0&1&1&1&0\\\
0&0&1&1&0&0&1&1&0&1&0&1&1&0\\\ 1&1&0&0&1&1&0&0&1&0&1&0&0&1\\\
0&1&0&1&0&1&0&1&1&1&0&0&0&1\\\ 1&0&0&1&0&1&1&0&1&0&1&0&1&0\\\
0&0&0&0&1&1&1&1&0&0&1&1&0&1\\\ \end{tabular}\right),$ (15)

where the vertical lines are included to visually separate the $M_{i}$
matrices.

######  Proposition 5.4.

The canonical projection $\operatorname{MT}(J_{15})\rightarrow X$ is a degree
2 isogeny.

###### Proof.

We use Sage to compute the (right) kernel of the matrix $M$ in Equation (15).
We then determine that the space spanned by kernel of $M$ and
$\mathbb{Z}^{8}\times\\{0\\}\times\\{0\\}$ is an index-2 submodule of
$\mathbb{Z}^{8}\times\mathbb{Z}^{4}\times\mathbb{Z}^{2}$. This corresponds to
$\widehat{T}_{1}\times\\{0\\}\times\\{0\\}+\ker(N_{1}^{*}\phi_{1}^{*}+N_{2}^{*}\phi_{2}^{*}+N_{3}^{*}\phi_{3}^{*})$
having index 2 in the product
$\widehat{T}_{E_{1}}\times\widehat{T}_{E_{2}}\times\widehat{T}_{E_{3}}$. Thus,
by Lemma 3.2, the projection is a degree 2 isogeny. ∎

####  5.2.2. Hodge ring and Hodge group

Proposition 3.3 tells us that there will be exceptional Hodge cycles in
codimension 2. The 12 exceptional cycles in codmension 2 are given by Shioda
in [24, Section 6.2]. Shioda also identified exceptional Hodge cycles in
codimension 3, though in this case we have
$\mathscr{B}^{3}=\mathscr{B}^{1}\cdot\mathscr{B}^{2}$.

The Hodge group of $J_{15}$ is commutative, and so we can identify elements
$\operatorname{Hg}(J_{15})$ with matrices of the form
$U\in\operatorname{U}(1)^{7}$. We use the method of Section 5.1 to identify
the additional relations on the elements of the Hodge group coming from the
exceptional cycles in the Hodge ring. We find that elements of the Hodge group
are of the form

$\displaystyle
U=\operatorname{diag}(U_{1},U_{2},U_{3},U_{4},\overline{U}_{2}U_{3}U_{4},\overline{U}_{1}U_{3}U_{4},\overline{U}_{1}\overline{U}_{2}U_{3}^{2}U_{4}).$

Note that, when identified with a matrix group, the Hodge group is isomorphic
to $\operatorname{U}(1)^{4}$. This is consistent with what we would expect
from our work with the Mumford-Tate group in Section 5.2.1 where we determined
that $\operatorname{MT}(J_{15})$ is isogenous to the Mumford-Tate group of a
4-dimensional simple abelian variety.

####  5.2.3. Sato-Tate group

Based on the above work and the relationship between the Sato-Tate group and
the Hodge group of an abelian variety, we have the following result.

######  Proposition 5.5.

Up to conjugation in $\operatorname{USp}(14)$, the identity component of the
Sato-Tate group of $J_{15}$ is

$\operatorname{ST}^{0}(J_{15})=\langle\operatorname{diag}(U_{1},U_{2},U_{3},U_{4},\overline{U}_{2}U_{3}U_{4},\overline{U}_{1}U_{3}U_{4},\overline{U}_{1}\overline{U}_{2}U_{3}^{2}U_{4})\rangle.$

This is consistent with the result in Table 2 of [8] where it is given that
the identity component is isomorphic to $\operatorname{U}(1)^{4}$. In
Proposition 5.5 we improve upon the earlier result by giving the additional
relations needed for the embedding in $\operatorname{USp}(14)$.

We can easily compute the moment statistics of the Sato-Tate group, but we see
an interesting phenomenon when attempting to compare these to the numerical
moments coming from the curve $C_{15}$. Recall from Section 3.4 that, in
general,
$\operatorname{ST}(A)/\operatorname{ST}^{0}(A)\simeq\operatorname{Gal}(L/F)$,
where $L$ is the minimal Galois extension of of $F$ for which
$\operatorname{ST}(A_{L})$ is connected. We saw in Section 5.1 that for
$J_{9}$ the field $L$ is exactly the endomorphism field of the variety.
However that is not the case for $J_{15}$ – we need a degree 2 extension of
$\mathbb{Q}(\zeta_{15})$.

We get our first hint that we may need a larger field by examining the moment
statistics in Table 4. The numerical moments were computed over the field
$\mathbb{Q}(\zeta_{15})$ for primes $p<2^{32}$ by Sutherland [25]

| $M_{2}$ | $M_{3}$ | $M_{4}$ | $M_{5}$ | $M_{6}$
---|---|---|---|---|---
$\mu_{1}$ | 14 | 0 | 834 | 0 | 78260
$a_{1}$ | 13.992 | 0.037 | 641.326 | 5.495 | 49354.840
Table 4. Table of $\mu_{1}$\- and $a_{1}$-moments for
$\operatorname{ST}(J_{15})$ over $\mathbb{Q}(\zeta_{15})$.

We quickly see that these moment statistics do not match – the errors for
$M_{4}$ and $M_{6}$ are quite large and are not consistent with the errors for
the odd moments.

We can use a technique developed in [28] to gain information about the field
$L$ – we will describe the strategy as it applies to the abelien variety
$J_{15}$. For a prime $p\equiv 1\pmod{15}$, we form the group
$\Phi_{J_{15},p}$ generated by the set of roots in $\overline{\mathbb{Q}}$ of
the Frobenious polynomial of $J_{15}$. If the group $\Phi_{J_{15},p}$ is
torsion-free then $p$ splits completely in $L$ (see Section 1.5 of [28]). By
computing this group for many primes, it appears that roughly half of the
primes congruent to 1 modulo 15 split completely in $L$ (there is Magma code
to assist with these computations available at [29]). This indicates that $L$
is a degree 2 extension of the CM field $\mathbb{Q}(\zeta_{15})$.

We use the L-functions and Modular Forms Database [18] to gain further
information about the field $L$. There is only one field listed that is a
degree 2 extension of $\mathbb{Q}(\zeta_{15})$ with the same ramified primes
(3 and 5):

http://www.lmfdb.org/NumberField/16.0.3243658447265625.1

In order to further confirm that this is the field $L$ for which
$\operatorname{ST}(A_{L})$ is connected, we compute numerical moment
statistics. The numerical moments in Table 5 were computed over the above
field for primes $p<2^{32}$ by Sutherland [25]

| $M_{2}$ | $M_{3}$ | $M_{4}$ | $M_{5}$ | $M_{6}$
---|---|---|---|---|---
$\mu_{1}$ | 14 | 0 | 834 | 0 | 78260
$a_{1}$ | 13.993 | 0.093 | 833.023 | 11.991 | 78067.503
Table 5. Table of $\mu_{1}$\- and $a_{1}$-moments for
$\operatorname{ST}(J_{15})$ over $L$.

We see that the errors for the even moments are within a margin of $1\%$ and
are consistent with the errors for the odd moments.

Further work is needed in order to determine the full Sato-Tate group of
$J_{15}$ (over $\mathbb{Q}$). We know that the component group of the Sato-
Tate group will satisfy
$\operatorname{ST}(J_{15})/\operatorname{ST}^{0}(J_{15})\simeq\operatorname{Gal}(L/\mathbb{Q})$,
however it is not clear how one could find explicit generators for the
component group.

###  5.3. $y^{2}=x^{21}-1$

For our third example we consider the curve $C_{21}\colon y^{2}=x^{21}-1.$ The
isogeny in Equation (4) tells us that its Jacobian factors as $J_{21}\simeq
X\times J_{7}\times J_{3}$, where $X$ is an absolutely simple abelian variety
of dimension 6. By Lemma 4.2, both $J_{21}$ and $X$ have CM by
$\mathbb{Q}(\zeta_{21})$. In this case we see new phenomena in both the Hodge
ring setting, as well as for the Mumford-Tate group.

####  5.3.1. Mumford-Tate Group

In order to study canonical projections of the Mumford-Tate group, we build
the matrix $M$ representing the map
$N_{1}^{*}\phi_{1}^{*}+N_{2}^{*}\phi_{2}^{*}+N_{3}^{*}\phi_{3}^{*}$ described
in Section 3.1.2 (see Section 5.2.1 for more details on how to build this
matrix). The matrix $M$ is obtained by concatenating matrices
$M_{1},M_{2},M_{3}$ corresponding to the factors $X,J_{7},J_{3}$,
respectively:

$\displaystyle
M=\left(\begin{array}[]{rrrrrrrrrrrr|rrrrrr|rr}1&1&1&1&1&1&0&0&0&0&0&0&1&1&1&0&0&0&1&0\\\
0&1&1&0&1&1&0&0&1&0&0&1&0&1&0&1&0&1&0&1\\\
0&0&1&0&1&0&1&0&1&0&1&1&1&0&0&1&1&0&1&0\\\
0&0&1&1&1&1&0&0&0&0&1&1&1&0&1&0&1&0&0&1\\\
1&0&0&0&1&0&1&0&1&1&1&0&1&1&1&0&0&0&0&1\\\
0&0&0&0&1&1&0&0&1&1&1&1&0&1&1&0&0&1&1&0\\\
1&1&1&1&0&0&1&1&0&0&0&0&1&0&0&1&1&0&0&1\\\
0&1&1&1&0&1&0&1&0&0&0&1&0&0&0&1&1&1&1&0\\\
1&1&0&0&0&0&1&1&1&1&0&0&0&1&0&1&0&1&1&0\\\
1&1&0&1&0&1&0&1&0&1&0&0&0&1&1&0&0&1&0&1\\\
1&0&0&1&0&0&1&1&0&1&1&0&1&0&1&0&1&0&1&0\\\
0&0&0&0&0&0&1&1&1&1&1&1&0&0&0&1&1&1&0&1\end{array}\right),$ (28)

where the vertical lines are included to visually separate the $M_{i}$
matrices. We use Sage to compute the (right) kernel of the matrix $M$ in
Equation 28. The space spanned by the kernel of $M$ and
$\mathbb{Z}^{12}\times\\{0\\}\times\\{0\\}$ does not have finite index in
$\mathbb{Z}^{12}\times\mathbb{Z}^{6}\times\mathbb{Z}^{2}$, which tells us that
the projection $\operatorname{MT}(J_{21})\to\operatorname{MT}(X)$ is not an
isomorphism nor an isogeny. However, we do find that the space spanned by
$\ker(M)$ and $\mathbb{Z}^{12}\times\\{0\\}\times\mathbb{Z}^{2}$ has index 1
in $\mathbb{Z}^{12}\times\mathbb{Z}^{6}\times\mathbb{Z}^{2}$. This yields the
following result.

######  Proposition 5.6.

The projection $\operatorname{MT}(J_{21})\to\operatorname{MT}(X\times J_{3})$
is an isomorphism.

####  5.3.2. Hodge Ring and Hodge Group

The exceptional Hodge cycles for $J_{21}$ are given in [24, Section 6].
Applying the technique described in Section 5.1, we find that the elements of
Hodge group are of the form

$\displaystyle\operatorname{diag}(U_{1},\,U_{2},\,U_{3},\,\overline{U}_{1}U_{2}U_{3},\,U_{5},\,U_{6},U_{7},\,\overline{U}_{1}U_{3}U_{6},\,\overline{U}_{2}U_{5}U_{6},\,\overline{U}_{1}U_{5}U_{6}).$

This implies that the Hodge group is isomorphic to $\operatorname{U}(1)^{6}$,
which is surprising since the Mumford-Tate group of $J_{21}$ is isomorphic to
the Mumford-Tate group of a 7-dimensional subvariety. It is possible that this
phenomenon is related to the fact that the abelian variety $X$ is degenerate.
This fact is proven by Shioda in [24, Section 6] by noting that there are
exceptional cycles of $J_{21}$ that come from $X$. This differs from our
examples for $m=9,15$, where in those cases all of the exceptional Hodge
cycles came from the Jacobian factors of $J_{m}$.

###  5.4. $y^{2}=x^{27}-1$

An interesting example to consider next is the genus 13 curve $C_{27}\colon
y^{2}=x^{27}-1$, which is not one of the examples worked out by Shioda. Its
Jacobian factors as $J_{27}\simeq X_{2}\times X_{1}\times J_{3}$, where
$X_{1}$ and $J_{3}$ are the simple abelian varieties appearing in the
decomposition of $J_{9}$ in Section 5.1 and $X_{2}$ is an absolutely simple
abelian variety. To find the dimension of $X_{2}$ we simply compute
$\dim(J_{27})-\dim(J_{9})=9$. We can easily extend Lemma 4.2 and conclude that
the CM field of $X_{2}$ is $\mathbb{Q}(\zeta_{27})$.

####  5.4.1. Mumford-Tate Group

Using the same technique described in detail in Section 5.2.1, we are able to
prove the following result.

######  Proposition 5.7.

The canonical project $\operatorname{MT}(J_{27})\to\operatorname{MT}(X_{2})$
is an isomorphism.

###### Proof.

We use Sage to compute the (right) kernel of the matrix $M$ obtained using the
techniques described in Section 3.1.2. The order of character group
$\widehat{T}_{E}$ associated to $X_{2}$ is $\phi(27)=18$. We determine that
the space spanned by kernel of $M$ and
$\mathbb{Z}^{18}\times\\{0\\}\times\\{0\\}$ spans all of
$\mathbb{Z}^{18}\times\mathbb{Z}^{6}\times\mathbb{Z}^{2}$. Thus, by Lemma 3.2,
the projection is an isomorphism. ∎

####  5.4.2. Hodge Ring and Hodge Group

The Hodge ring of $J_{27}$ is not worked out in Shioda’s paper [24], but we
can use Theorem 5.2 of the paper to determine the exceptional cycles that
appear in the Hodge ring. For example, using code written in Sage, we find
that there are 8 exceptional cycles in codimension 2:

$\displaystyle\omega_{1}\wedge\omega_{10}\wedge\overline{\omega}_{3}\wedge\overline{\omega}_{8}$
$\displaystyle\omega_{3}\wedge\omega_{8}\wedge\overline{\omega}_{1}\wedge\overline{\omega}_{10}$
$\displaystyle\omega_{2}\wedge\omega_{11}\wedge\overline{\omega}_{6}\wedge\overline{\omega}_{7}$
$\displaystyle\omega_{6}\wedge\omega_{7}\wedge\overline{\omega}_{2}\wedge\overline{\omega}_{11}$
$\displaystyle\omega_{3}\wedge\omega_{12}\wedge\overline{\omega}_{6}\wedge\overline{\omega}_{9}$
$\displaystyle\omega_{6}\wedge\omega_{9}\wedge\overline{\omega}_{3}\wedge\overline{\omega}_{12}$
$\displaystyle\omega_{4}\wedge\omega_{13}\wedge\overline{\omega}_{5}\wedge\overline{\omega}_{12}$
$\displaystyle\omega_{5}\wedge\omega_{12}\wedge\overline{\omega}_{4}\wedge\overline{\omega}_{13}$

As in Sections 5.1 and 5.2, we can use the exceptional Hodge cycles to
determine extra relations in the Hodge group of $J_{27}$ from this. We find
that the elements of $\operatorname{Hg}(J_{27})$, identified as $2g\times 2g$
matrices with determinant 1, are of the form

$U=\operatorname{diag}(U_{1},U_{2},U_{3},U_{4},U_{5},U_{6},U_{7},U_{8},U_{9},\overline{U}_{1}U_{3}U_{8},\overline{U}_{2}U_{6}U_{7},\overline{U}_{3}U_{6}U_{9},\overline{U}_{3}\overline{U}_{4}U_{5}U_{6}U_{9}).$

Note that this implies that $\operatorname{Hg}(J_{27})$ is isomorphic to
$\operatorname{U}(1)^{9}$ with embedding in $\operatorname{USp}(26)$ given by
the relations noted above.

###  5.5. Further work for more general $y^{2}=x^{m}-1$

In this section we will investigate further examples and make conjectures
about the degeneracy of Jacobian varieties $J_{m}$. We first consider examples
where $m=p^{2}$, for $p$ prime.

Recall from Section 4 that the Jacobian of $C_{p^{2}}$ satisfies the following

$J_{p^{2}}\simeq X\times J_{p},$

where $X$ is an absolutely simple abelian variety of dimension $p(p-1)/2$.

The following result has been verified by computing the kernel of the matrix
$M$ described in Section 3.1.2.

######  Proposition 5.8.

Let $J_{p^{2}}$ and $X$ be defined as above. Then the projection
$\operatorname{MT}(J_{p^{2}})\rightarrow\operatorname{MT}(X)$ is an
isomorphism for odd primes $p\leq 29$.

It seems likely that this holds more generally, and so we make the following
conjecture.

######  Conjecture 5.9.

Let $J_{p^{2}}$ and $X$ be defined as above. Then the projection
$\operatorname{MT}(J_{p^{2}})\rightarrow\operatorname{MT}(X)$ is an
isomorphism.

The next generalization we consider is $m=p^{3}$. As in Section 5.4 we have
that the Jacobian $J_{p^{3}}$ satisfies the following

$J_{p^{3}}\simeq X_{2}\times X_{1}\times J_{p},$

where $X_{1}$ and $J_{p}$ are the simple abelian varieties appearing in the
decomposition of $J_{p^{2}}$ and $X_{2}$ is an absolutely simple abelian
variety. To find the dimension of $X_{2}$ we simply compute
$\dim(J_{p^{3}})-\dim(J_{p^{2}})$.

The following result is verified using the same techniques as those in Section
5.4.

######  Proposition 5.10.

Let $J_{p^{3}}$ and $X$ be defined as above. Then the projection
$\operatorname{MT}(J_{p^{3}})\rightarrow\operatorname{MT}(X_{2})$ is an
isomorphism for odd primes $p\leq 13$.

As we suspect for $m=p^{2}$, it seems likely that this holds more generally,
and so we make the following conjecture.

######  Conjecture 5.11.

Let $J_{p^{3}}$ and $X_{2}$ be defined as above. Then the projection
$\operatorname{MT}(J_{p^{3}})\rightarrow\operatorname{MT}(X_{2})$ is an
isomorphism.

We believe that these results could be generalized to $m=p^{k}$, for $k>1$.
For values of $m$ with at least two distinct odd prime factors, it is more
difficult to make a general statement regarding the canonical projection of
the Mumford-Tate group onto its factors. It remains an open question of when
the projection onto one or more factors is an isogeny or isomorphism.

In terms of the Hodge rings and Hodge groups, Proposition 3.3 tells us that
there are exceptional Hodge cycles in codimension $(p+1)/2$ for the Jacobian
$J_{m}$, though there may be exceptional cycles in other codimensions as well.
One can compute all of the Hodge cycles using Sage code that implements
Theorem 5.2 of [24] and then determine which of these are exceptional. These
exceptional cycles are used to determine extra relations in the Hodge ring,
which in turn give us the embedding of the identity component of the Sato-Tate
group of $J_{m}$ in $\operatorname{USp}(2g)$. In the examples worked out in
the preparation of this paper, there was not a clear pattern in the number or
the format of the exceptional cycles. It would be interesting to work out more
examples and determine if there is a way to characterize the exceptional
cycles based on the factorization of $m$.

One of the initial goals for this project was to better understand the Sato-
Tate groups of degenerate abelian varieties. By determining the additional
relations in the Hodge group (coming from exceptional cycles in the Hodge
ring), we can describe the identity component of the Sato-Tate group. Beyond
this, there is still much work to be done. The first main question to consider
further is: over what extension $L/\mathbb{Q}$ is the Sato-Tate group
connected? We saw that in the case of $J_{9}$, $L$ is exactly the endomorphism
field, but for $J_{15}$ it is a degree 2 extension of this field. Can we make
a general statement about the field $L$?

The second main question is: how do we find generators of the component group
of the Sato-Tate group? This is particularly unclear in the situation where
the field $L$ strictly contains the endomorphism field, which also means that
any results in this direction will be particularly interesting.

## Acknowledgements

This project began while the author was a visiting research scientist at the
Max-Planck-Institut für Mathematik. She is grateful for the supportive
research environment they provided. The author would like to thank Drew
Sutherland and Davide Lombardo for their helpful conversations while working
on Sections 5.1 and 5.2, and Maria Fox for her feedback on an earlier draft of
the Introduction.

The author was partially supported by NSF grant DMS-2201085 and by a PSC-CUNY
Award, jointly funded by The Professional Staff Congress and The City
University of New York.

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