# Vanishing criteria for Ceresa cycles

Jef Laga  Department of Pure Mathematics and Mathematical Statistics,
Wilberforce Road, Cambridge, CB3 0WB, UK<EMAIL_ADDRESS>and  Ari
Shnidman  Einstein Institute of Mathematics, Hebrew University of Jerusalem,
Israel<EMAIL_ADDRESS>
###### Abstract.

Let $C$ be a smooth projective curve, and let $J$ be its Jacobian. We prove
vanishing criteria for the Ceresa cycle
$\kappa(C)\in\mathrm{CH}_{1}(J)\otimes{\rm\mathbb{Q}}$ in the Chow group of
1-cycles on $J$. Namely,

  1. $(A)$

If $\operatorname{H}^{3}(J)^{{\rm Aut}(C)}=0$, then $\kappa(C)$ vanishes;

  2. $(B)$

If $\operatorname{H}^{0}(J,\Omega_{J}^{3})^{{\rm Aut}(C)}=0$, then $\kappa(C)$
vanishes modulo algebraic equivalence,

with criterion $(B)$ conditional on the Hodge conjecture.

We then study the first interesting case where $(B)$ holds but $(A)$ does not,
namely the case of Picard curves $C\colon y^{3}=x^{4}+ax^{2}+bx+c$. Using work
of Schoen on the Hodge conjecture, we show that the Ceresa cycle of a Picard
curve is torsion in the Griffiths group. Moreover, we determine exactly when
it is torsion in the Chow group. As a byproduct, we show that there are
infinitely many plane quartic curves over ${\rm\mathbb{Q}}$ with torsion
Ceresa cycle (in fact, there is a one parameter family of such curves).
Finally, we determine which automorphism group strata are contained in the
vanishing locus of the universal Ceresa cycle over $\mathcal{M}_{3}$.

##  1\. Introduction

Let $k$ be an algebraically closed field and $C$ a smooth, projective, and
connected curve over $k$ of genus $g\geq 2$ with Jacobian variety $J$. Let $e$
be a degree-$1$ divisor of $C$ and let $\iota_{e}\colon C\hookrightarrow J$ be
the Abel-Jacobi map based at $e$. We study the torsion behaviour of the Ceresa
cycle

(1.1) $\displaystyle\kappa_{C,e}:=[\iota_{e}(C)]-(-1)^{*}[\iota_{e}(C)]\in{\rm
CH}_{1}(J)$

in the Chow group modulo rational equivalence. If $\kappa_{C,e}$ is torsion,
then $(2g-2)e=K_{C}$ in ${\rm CH}_{0}(C)\otimes{\rm\mathbb{Q}}$, where $K_{C}$
is the canonical divisor class. Moreover if $(2g-2)e$ is canonical, then the
image of $\kappa_{C,e}$ in ${\rm CH}_{1}(J)\otimes{{\rm\mathbb{Q}}}$ is
independent of $e$ and we denote it by $\kappa(C)$, see §2.7 for these claims.
The class $\kappa(C)$ vanishes if and only if $\kappa_{C,e}$ is torsion for
some degree-$1$ divisor $e$.

We also consider the image $\bar{\kappa}(C)$ of $\kappa(C)$ in the Griffiths
group ${\rm Gr}_{1}(J)\otimes{\rm\mathbb{Q}}$ of homologically trivial
$1$-cycles modulo algebraic equivalence. When $g=2$, or more generally when
$C$ is hyperelliptic, it is easy to see that $\kappa(C)=0$. On the other hand,
Ceresa famously showed that $\bar{\kappa}(C)\neq 0$ for a very general curve
$C$ over ${\rm\mathbb{C}}$ of genus $g\geq 3$ [14].

The vanishing of the Ceresa cycle is interesting for various reasons. For
example, $\kappa(C)=0$ if and only if the Chow motive $\mathfrak{h}(C)$ has a
multiplicative Chow-Künneth decomposition (by [19, Proposition 3.1] and
Proposition 2.12). Moreover $\bar{\kappa}(C)=0$ if and only if the
tautological subring modulo algebraic equivalence is generated by a theta
divisor (by [6, Corollary 3.4]), in which case Poincaré’s formula
$[C]=\frac{\Theta^{g-1}}{(g-1)!}$ holds modulo algebraic equivalence. More
generally, the Ceresa cycle over $\mathcal{M}_{g}$ serves as a testing ground
for the study of homologically trivial algebraic cycles in codimension greater
than 1.

###  1.1. Vanishing criteria

We prove cohomological vanishing criteria for Ceresa cycles of curves with
nontrivial automorphisms. Let $\operatorname{H}^{*}(-)$ be a Weil cohomology
functor, such as $\ell$-adic cohomology with $\ell\neq{\rm char\hskip
2.84526pt}(k)$ or singular cohomology when $k={\rm\mathbb{C}}$. Note that the
finite group ${\rm Aut}(C)$ acts on $\operatorname{H}^{*}(J)$, by
functoriality. Cupping with the principal polarization gives an injection
$\operatorname{H}^{1}(J)(-1)\hookrightarrow\operatorname{H}^{3}(J)$, allowing
us to define the primitive cohomology
$\operatorname{H}^{3}(J)_{\mathrm{prim}}:=\operatorname{H}^{3}(J)/\operatorname{H}^{1}(J)(-1)$.

######  Theorem A.

If $\operatorname{H}^{3}(J)_{\mathrm{prim}}^{{\rm Aut}(C)}=0$, then
$\kappa(C)=0$.

This improves on a recent result of Qiu and Zhang stating that if
$(\operatorname{H}^{1}(C)^{\otimes 3})^{{\rm Aut}(C)}=0$, then $\kappa(C)=0$
[39].111 This formulation is equivalent to theirs, using [46, Theorem 1.5.5].
By contrast, Theorem A requires only the weaker condition that the
subrepresentation
$\operatorname{H}^{3}(J)_{\mathrm{prim}}\subset\operatorname{H}^{3}(J)\simeq\bigwedge^{3}\operatorname{H}^{1}(C)\subset\operatorname{H}^{1}(C)^{\otimes
3}$ has no nontrivial ${\rm Aut}(C)$-fixed points. If the quotient $C/{\rm
Aut}(C)$ has genus $0$, then our hypothesis is equivalent to
$\operatorname{H}^{3}(J)^{{\rm Aut}(C)}=0$.

Our proof of Theorem A is inspired by Beauville’s proof that for the curve
$y^{3}=x^{4}+x$, the image of $\kappa(C)$ under the complex Abel-Jacobi map
vanishes [7]. To achieve a vanishing result in the Chow group, we work
directly with the rational Chow motive $\mathfrak{h}^{3}(J)$ and make crucial
use of the finite-dimensionality results of Kimura [25].

Our second result is a vanishing criterion for $\overline{\kappa}(C)$, however
our proof works only in characteristic $0$ and is conditional on the Hodge
conjecture:

######  Theorem B.

Suppose $\mathrm{char}(k)=0$ and assume the Hodge conjecture for abelian
varieties. If $\operatorname{H}^{0}(J,\Omega_{J}^{3})^{{\rm Aut}(C)}=0$, then
$\bar{\kappa}(C)=0$.

More precisely, we require the Hodge conjecture for $J\times A$, for a
specific abelian variety $A$ described in Proposition 3.3. Note that
$\operatorname{H}^{0}(J,\Omega_{J}^{3})\simeq\bigwedge^{3}\operatorname{H}^{0}(C,\Omega_{C}^{1})$,
so the conditions of Theorems A (when $\mathrm{char}(k)=0$) and B both depend
only on the abstract representation $(G,V)=({\rm
Aut}(C),\operatorname{H}^{0}(C,\Omega_{C}^{1}))$.

The proof of Theorem B is in the same spirit as that of Theorem A. The Hodge
conjecture is used to show that the motive $\mathfrak{h}^{3}(J)^{{\rm
Aut}(C)}$ is isomorphic to $\mathfrak{h}^{1}(A)(-1)$, from which the algebraic
triviality of $\kappa(C)$ follows.

###  1.2. Picard curves

The condition $\operatorname{H}^{3}(J)_{\mathrm{prim}}^{{\rm Aut}(C)}=0$ in
Theorem A is only rarely satisfied, e.g. it holds for exactly two plane
quartic curves over ${\rm\mathbb{C}}$ (see Theorem D below). However, the
condition $\operatorname{H}^{0}(J,\Omega_{J}^{3})^{{\rm Aut}(C)}=0$ of Theorem
B is satisfied for all Picard curves $y^{3}=x^{4}+ax^{2}+bx+c$. For such
curves it turns out that we require a nontrivial case of the Hodge conjecture
to apply Theorem B, namely that a certain Weil class on an abelian fourfold is
algebraic. Fortunately, Schoen has proved the Hodge conjecture in our specific
situation [41]. (See also recent work of Markman [32].) This leads to an
unconditional proof of the vanishing of $\bar{\kappa}(C)$. By further
analyzing $\kappa(C)$ we prove the following precise characterization of the
vanishing of $\kappa(C)$ in the Picard family:

######  Theorem C.

Suppose $\mathrm{char}(k)=0$ and let $C_{f}$ be a smooth projective Picard
curve with model

(1.2) $\displaystyle C_{f}\colon y^{3}=f(x)=x^{4}+ax^{2}+bx+c$

for some $a,b,c\in k$. Consider the point
$P_{f}=(a^{2}+12c,72ac-2a^{3}-27b^{2})$ on the elliptic curve

$E_{f}\colon y^{2}=4x^{3}-27\cdot{\rm disc}(f).$

Then $\bar{\kappa}(C_{f})=0$, and $\kappa(C_{f})=0$ if and only if $P_{f}\in
E_{f}(k)$ is torsion.

Here, ${\rm disc}(f)$ is the discriminant of $f$ and the coordinates of
$P_{f}$ are the usual $I$\- and $J$-invariants of the binary quartic form
$z^{4}f(x/z)$; see §4.1. The fact that $P_{f}$ defines a point on $E_{f}$
follows from a classical relation between the invariants of a binary quartic.
See also §4.2 for a more precise version of Theorem C, which relates order of
$\kappa_{C_{f},P_{\infty}}$ to the order of $P_{f}$, and in particular shows
that $\kappa_{C_{f},P_{\infty}}$ can have arbitrarily large (finite) order in
this family. Theorem C also gives one-parameter families of plane quartic
curves with vanishing $\kappa(C)$, corresponding to components of the torsion
locus of $P_{f}$. Such families exist even over ${\rm\mathbb{Q}}$, e.g. the
family $y^{3}=x^{4}-12x^{2}+tx-12$.

Theorem C generalizes the results of [27] concerning _bielliptic_ Picard
curves, i.e. those with $b=0$ in (1.2). There we exploited the explicit
geometry of the bielliptic cover to characterize the vanishing locus of
$\kappa(C)$. Here we must proceed more indirectly: using Schoen’s work on the
Hodge conjecture, we first show that the vanishing of $\kappa(C)$ is
equivalent to the vanishing of its Abel–Jacobi image in the intermediate
Jacobian. Using some formal considerations concerning isotrivial families of
elliptic curves, we are able to bootstrap the calculations of [27] to
determine the Abel-Jacobi image up to multiple and obtain the characterization
of Theorem C.

In §3, we give two examples of genus 4 families of curves $\mathcal{C}\to B$
for which Theorem B applies. It would be interesting to completely
characterize the vanishing of $\kappa(C)$ in these families as well, and to
find more such families. The general proof strategy would follow that of
Theorem C: using the Hodge conjecture, one shows that the vanishing of
$\kappa(\mathcal{C}_{b})$ is equivalent to the vanishing of $\sigma(b)$, where
$\sigma$ is a section of a certain abelian scheme $\mathcal{A}\to B$. (The
analytification of $\sigma$ is the normal function associated to the Ceresa
cycle over $B$, which lands in the ${\rm Aut}(C)$-fixed points of the
intermediate Jacobian.) One then tries to identify $\mathcal{A}\to B$ and
$\sigma$ explicitly.

###  1.3. Vanishing loci in genus $3$

Let $\mathcal{M}_{g}$ be the coarse moduli space of genus-$g$ curves in
characteristic zero, seen as a variety over ${\rm\mathbb{Q}}$. Let
$V_{g}^{\mathrm{rat}}\subset\mathcal{M}_{g}$ be the subset consisting of all
(geometric) isomorphism classes of curves with vanishing $\kappa(C)$. Let
$V_{g}^{\mathrm{alg}}\subset\mathcal{M}_{g}$ be the analogous subset for
$\bar{\kappa}(C)$. The sets $V_{g}^{\mathrm{rat}}$ and $V_{g}^{\mathrm{alg}}$
are both countable unions of closed algebraic subvarieties (Lemma 5.1).
Moreover $V_{g}^{\mathrm{rat}}\subset V_{g}^{\mathrm{alg}}$ and Ceresa’s
result [14] shows that $V_{g}^{\mathrm{alg}}\neq\mathcal{M}_{g}$ for all
$g\geq 3$. What can be said about the irreducible components of
$V_{g}^{\mathrm{rat}}$ and $V_{g}^{\mathrm{alg}}$ and their dimensions?
Collino and Pirola have shown that $V_{3}^{\mathrm{rat}}$ does not contain
subvarieties of dimension $\geq 4$ that are not themselves contained in the
hyperelliptic locus [16, Corollary 4.3.4]. Theorem C shows that
$V_{3}^{\mathrm{alg}}$ contains the $2$-dimensional Picard locus and that
$V_{3}^{\mathrm{rat}}$ contains infinitely many $1$-dimensional components.

Recall that there is a stratification of the non-hyperelliptic locus of
$\mathcal{M}_{3}$ by locally closed subvarieties $X_{G}$, indexed by certain
finite groups $G$, with the property that a non-hyperelliptic curve
$[C]\in\mathcal{M}_{3}({\rm\mathbb{C}})$ lies in $X_{G}$ if and only if ${\rm
Aut}(C)\simeq G$. It turns out that the isomorphism class of the ${\rm
Aut}(C)$-representation $\operatorname{H}^{0}(C,\Omega_{C}^{1})$ does not
depend on the choice of curve $[C]$ in $X_{G}({\rm\mathbb{C}})$. Our final
theorem determines exactly which $X_{G}$ are contained in
$V_{3}^{\mathrm{rat}}$ or $V_{3}^{\mathrm{alg}}$.

######  Theorem D.

Let $G$ be a finite group isomorphic to the automorphism group of a plane
quartic. Then $X_{G}\subset V_{3}^{\mathrm{rat}}$ if and only if $G=C_{9}$ or
$G_{48}$, and $X_{G}\subset V_{3}^{\mathrm{alg}}$ if and only if $X_{G}$ is
contained in the Picard locus, in other words if and only if
$G=C_{3},C_{6},C_{9}$ or $G_{48}$.

Here, $G_{48}$ is the group with GAP label $(48,33)$. The strata $X_{C_{9}}$
and $X_{G_{48}}$ are single closed points, represented by the curves
$y^{3}=x^{4}-x$ and $y^{3}=x^{4}+1$ respectively.

Theorem D shows that $X_{G}\subset V_{3}^{\mathrm{rat}}$ if and only if
$\operatorname{H}^{3}(J)^{G}=0$ for every (equivalently, some) curve in
$X_{G}$, and that $X_{G}\subset V_{3}^{\mathrm{alg}}$ if and only if
$\operatorname{H}^{0}(J,\Omega_{J}^{3})^{G}=0$ for every curve in $X_{G}$.
Thus, for $g=3$, the criteria of Theorems A and B exactly single out the
strata $X_{G}$ where $\kappa(C)$ or $\bar{\kappa}(C)$ identically vanishes.

Finally, Shou-wu Zhang has recently announced a proof of a Northcott property
for the Beilinson-Bloch height of the modified diagonal cycle $\Delta_{GKS}$
on $C^{3}$. More precisely, for each $g\geq 3$, there exists an open dense
subset $U_{g}\subset\mathcal{M}_{g}$ such that for any $X\in{\rm\mathbb{R}}$
and $d\in{\rm\mathbb{N}}$, the number of $C\in U_{g}(\bar{\rm\mathbb{Q}})$
defined over a number field of degree at most $d$ and with
$\langle\Delta_{GKS},\Delta_{GKS}\rangle<X$ is finite. In order to better
understand Ceresa vanishing loci in families of curves, it would be of great
interest to try to identify the largest such open dense set $U_{g}$, or
equivalently, its complement $Z_{g}:=\mathcal{M}_{g}\setminus U_{g}$. The
Northcott property implies that any positive dimensional component of
$V_{g}^{\mathrm{rat}}$ is contained in $Z_{g}$, but there may be other
components of $Z_{g}$ as well. Indeed, Theorem C shows that $X_{C_{3}}\subset
Z_{3}$, even though $X_{C_{3}}\not\subset V_{3}^{\mathrm{rat}}$; see Remark
5.4.

###  1.4. Structure of paper

In §2 we collect some standard results on Chow groups, Chow motives and Ceresa
cycles. In §3 we prove the cohomological vanishing criteria (Theorems A and B)
and give some examples. In §4 we study the family of Picard curves in detail
and prove Theorem C. Finally, in §5 we introduce the Ceresa vanishing loci
$V_{g}^{\mathrm{rat}},V_{g}^{\mathrm{alg}}\subset\mathcal{M}_{g}$ and
determine which automorphism strata they contain in genus $3$, proving Theorem
D.

###  1.5. Acknowledgements

We thank Jeffrey Achter, Ben Moonen, Burt Totaro and Congling Qiu for helpful
conversations. This research was carried out while the first author was a
Research Fellow at St John’s College, University of Cambridge, which he thanks
for providing excellent working conditions. The second author was funded by
the European Research Council (ERC, CurveArithmetic, 101078157).

##  2\. Notation and background

###  2.1. Chow groups

Let $k$ be field. A variety is by definition a separated scheme of finite type
over $k$. We say a variety is nice if it is smooth, projective and
geometrically integral. If $X$ is a smooth and geometrically integral variety
and $p\in\\{0,\dots,\dim X\\}$, let ${\rm CH}^{p}(X)$ denote the Chow group
(with ${\rm\mathbb{Z}}$-coefficients) of codimension $p$ cycles modulo
rational equivalence. If $Z\subset X$ is a closed subscheme of codimension
$p$, we denote its class in ${\rm CH}^{p}(X)$ by $[Z]$ (using [44, Tag 02QS]
if $Z$ is not integral).

If $X$ is additionally projective, then ${\rm CH}^{p}(X)$ has a filtration by
subgroups

$\displaystyle{\rm CH}^{p}(X)_{\mathrm{alg}}\subset{\rm
CH}^{p}(X)_{\mathrm{hom}}\subset{\rm CH}^{p}(X),$

where ${\rm CH}^{p}(X)_{\mathrm{alg}}$ is the subgroup of algebraically
trivial cycles (in the sense of [2, §3.1]) and ${\rm
CH}^{p}(X)_{\mathrm{hom}}$ the subgroup of homologically trivial cycles (with
respect to a fixed Weil cohomology theory for nice varieties over $k$). The
Griffiths group is by definition ${\rm Gr}^{p}(X)={\rm
CH}^{p}(X)_{\mathrm{hom}}/{\rm CH}^{p}(X)_{\mathrm{alg}}$. We occasionally
write ${\rm CH}_{p}(X)={\rm CH}^{\dim X-p}(X)$ and ${\rm Gr}_{p}(X)={\rm
Gr}^{\dim X-p}(X)$. If $R$ is a ring, we write ${\rm CH}^{p}(X)_{R}={\rm
CH}^{p}(X)\otimes_{{\rm\mathbb{Z}}}R$ and ${\rm Gr}^{p}(X)_{R}={\rm
Gr}^{p}(X)\otimes_{{\rm\mathbb{Z}}}R$.

###  2.2. Base change, specialization and families

We state three lemmas concerning operations on cycles. These seem to be
standard, but we could not locate proofs in the literature.

######  Lemma 2.1.

Let $X/k$ be a nice variety and $K/k$ a (not necessarily finite) extension of
fields.

  1. (1)

The base change maps ${\rm CH}^{p}(X)_{\rm\mathbb{Q}}\rightarrow{\rm
CH}^{p}(X_{K})_{\rm\mathbb{Q}}$ and ${\rm
Gr}^{p}(X)_{\rm\mathbb{Q}}\rightarrow{\rm Gr}^{p}(X_{K})_{\rm\mathbb{Q}}$ are
injective.

  2. (2)

If in addition $k$ is algebraically closed, the base change maps ${\rm
CH}^{p}(X)\rightarrow{\rm CH}^{p}(X_{K})$ and ${\rm Gr}^{p}(X)\rightarrow{\rm
Gr}^{p}(X_{K})$ are injective.

###### Proof.

The proof is an adaptation of [12, Lemma 1A.3, p. 22]. We first prove $(2)$
for ${\rm CH}^{p}(X)\rightarrow{\rm CH}^{p}(X_{K})$. Suppose $\alpha\in{\rm
CH}^{p}(X)$ has trivial image in ${\rm CH}^{p}(X_{K})$. Then $\alpha$ already
has trivial image in ${\rm CH}^{p}(X_{K^{\prime}})$, where $K^{\prime}\subset
K$ is a subfield that is finitely generated over $k$, since the data
witnessing triviality in ${\rm CH}^{p}(X_{K})$ can be defined over such a
subfield. By spreading out, we can find a smooth integral variety $U/k$ with
function field $K^{\prime}$ such that $\alpha$ has trivial image in ${\rm
CH}^{p}(X\times_{k}U)$. Since $k$ is algebraically closed, there exists a
$k$-point $u\in U(k)$. Pulling back along $u$ defines a left-inverse ${\rm
CH}^{p}(X\times_{k}U)\rightarrow{\rm CH}^{p}(X)$ to the map ${\rm
CH}^{p}(X)\rightarrow{\rm CH}^{p}(X\times_{k}U)$. It follows that $\alpha$ is
trivial in ${\rm CH}^{p}(X)$, as desired. The argument for ${\rm Gr}^{p}(X)$
is identical and omitted.

We now prove $(1)$ for ${\rm CH}^{p}(X)_{{\rm\mathbb{Q}}}\rightarrow{\rm
CH}^{p}(X_{K})_{{\rm\mathbb{Q}}}$. There exists a field $L$ containing both
$K$ and an algebraic closure $\bar{k}$ of $k$. It therefore suffices to prove
the two base change maps ${\rm CH}^{p}(X)_{\rm\mathbb{Q}}\rightarrow{\rm
CH}^{p}(X_{\bar{k}})_{\rm\mathbb{Q}}\rightarrow{\rm
CH}^{p}(X_{L})_{{\rm\mathbb{Q}}}$ are both injective. The first one follows
from the fact that for a finite extension $k^{\prime}/k$ the pushforward map
${\rm CH}^{p}(X_{k^{\prime}})\rightarrow{\rm CH}^{p}(X)$, when precomposed
with the base change map, is multiplication by $[k^{\prime}:k]$. The second
follows from Part $(2)$. The case of ${\rm Gr}^{p}(X)_{{\rm\mathbb{Q}}}$ is
again analogous. ∎

To discuss specialization in the next lemma, let $R$ be a discrete valuation
ring with fraction field $K$ and residue field $k$. Let $X\rightarrow{\rm
Spec}(R)$ be a smooth, projective morphism with geometrically integral fibers,
so the generic and special fibers $X_{K}$ and $X_{k}$ are nice varieties over
$K$ and $k$ respectively. In this setting, Fulton has defined [20, §20.3] a
specialization morphism $\mathrm{sp}\colon{\rm CH}^{p}(X_{K})\rightarrow{\rm
CH}^{p}(X_{k})$ for every $0\leq p\leq\dim(X_{K})$. It has the property that
if $Z\subset X$ is a closed integral subscheme of codimension $p$, flat over
$R$, then $\mathrm{sp}([Z_{K}])=[Z_{k}]$.

######  Lemma 2.2.

In the above notation, $\mathrm{sp}$ sends ${\rm
CH}^{p}(X_{K})_{\mathrm{alg}}\otimes{\rm\mathbb{Q}}$ to ${\rm
CH}^{p}(X_{k})_{\mathrm{alg}}\otimes{\rm\mathbb{Q}}$.

###### Proof.

Let $C/K$ be a nice curve with $K$-points $t_{0},t_{1}\in C(K)$ and let
$Z\subset(X_{K})\times_{K}C$ be an integral closed subscheme of codimension
$p$, flat over $C$. By [2, Theorem 1] and the remarks thereafter, it suffices
to prove that $\mathrm{sp}([Z_{t_{0}}]-[Z_{t_{1}}])\in{\rm
CH}^{p}(X_{k})_{\mathrm{alg}}\otimes{\rm\mathbb{Q}}$ for each such tuple
$(C,t_{0},t_{1},Z)$.

Let $\mathcal{C}\rightarrow{\rm Spec}(R)$ be an integral, projective, flat,
proper and regular model of $C$, which exists by [30, Proposition 10.1.8]. Let
$\mathrm{Hilb}_{X/R}\rightarrow{\rm Spec}(R)$ be the Hilbert scheme of the
projective morphism $X\rightarrow{\rm Spec}(R)$. The closed subscheme $Z$
determines a $K$-morphism $\alpha\colon C\rightarrow\mathrm{Hilb}_{X/R}$.
Since $C$ is connected, the image of $\alpha$ is contained in an open and
closed subscheme $\mathrm{Hilb}_{X/R}^{\phi}$ with fixed Hilbert polynomial.
The surface $\mathcal{C}$ is regular,
$\mathrm{Hilb}^{\phi}_{X/R}\rightarrow{\rm Spec}(R)$ is projective [3,
Corollary (2.8)] and we may view $\alpha$ as a rational map
$\mathcal{C}\dashrightarrow\mathrm{Hilb}_{X/R}^{\phi}$. Therefore by [30,
Theorem 9.2.7] there exists a birational map
$\widetilde{\mathcal{C}}\rightarrow\mathcal{C}$, obtained by successively
blowing up closed points in the special fiber, and an $R$-morphism
$\tilde{\alpha}\colon\widetilde{\mathcal{C}}\rightarrow\mathrm{Hilb}^{\phi}_{X/R}$
with generic fiber $\alpha$. In other words, there exists a closed subscheme
$\mathcal{Z}\subset\widetilde{\mathcal{C}}\times_{R}X$, flat over
$\widetilde{\mathcal{C}}$, whose generic fiber equals $Z$.

The $K$-points $t_{i}\in C(K)$ extend to $R$-points $\tilde{t}_{i}\colon{\rm
Spec}(R)\rightarrow\widetilde{\mathcal{C}}$ whose reductions
$\bar{t}_{i}\in\widetilde{\mathcal{C}}_{k}(k)$ land in the smooth locus of
$\widetilde{\mathcal{C}}_{k}$. The closed subschemes
$\mathcal{Z}_{\tilde{t}_{i}}\subset X$ are flat over $R$. Therefore
$\mathrm{sp}([Z_{t_{0}}]-[Z_{t_{1}}])=[\mathcal{Z}_{\bar{t}_{0}}]-[\mathcal{Z}_{\bar{t}_{1}}]$.

By the Zariski connectedness theorem, the special fiber
$\widetilde{\mathcal{C}}_{k}$ is connected. Consequently, by resolving
irreducible components of $\widetilde{\mathcal{C}}_{k}$, we can connect
$\bar{t}_{0}$ to $\bar{t}_{1}$ by a sequence of nice curves: there exists a
finite extension $k^{\prime}/k$, a collection of nice curves
$D_{1},\dots,D_{n}$ over $k^{\prime}$, and for each $1\leq i\leq n$ a pair of
points $s_{i,1},s_{i,2}\in D_{i}(k^{\prime})$ and a morphism
$\varphi_{i}\colon D_{i}\rightarrow\mathcal{C}_{k^{\prime}}$ such that
$\varphi_{1}(s_{1,1})=\bar{t}_{1}$,
$\varphi_{i}(s_{i,2})=\varphi_{i+1}(s_{i+1,1})$ for all $1\leq i\leq n-1$ and
$\varphi_{n}(s_{n,2})=\bar{t}_{2}$. Letting $W^{(i)}$ be the pullback of
$\mathcal{Z}_{k}$ along $\varphi_{i}$, we have

$\displaystyle\mathrm{sp}([Z_{t_{0}}]-[Z_{t_{1}}])_{k^{\prime}}=([W^{(1)}_{s_{1,1}}]-[W^{(1)}_{s_{1,2}}])+\cdots+([W^{(n)}_{s_{n,1}}]-[W^{(n)}_{s_{n,2}}]).$

Since each term $([W^{(i)}_{s_{i,1}}]-[W^{(i)}_{s_{i,2}}])$ lies in ${\rm
CH}^{p}(X_{k^{\prime}})_{\mathrm{alg}}\otimes{\rm\mathbb{Q}}$ by definition of
algebraic triviality, the same is true for their sum. Taking the pushforward
along $X_{k^{\prime}}\rightarrow X_{k}$, we see that
$[k^{\prime}:k]\cdot\mathrm{sp}([Z_{t_{0}}]-[Z_{t_{1}}])\in{\rm
CH}^{p}(X_{k})_{\mathrm{alg}}$, as desired. ∎

######  Remark 2.3.

The proof shows that if $K$ is perfect and $k$ is algebraically closed, then
$\mathrm{sp}$ even sends ${\rm CH}^{p}(X_{K})_{\mathrm{alg}}$ to ${\rm
CH}^{p}(X_{k})_{\mathrm{alg}}$.

######  Lemma 2.4.

Let $X\rightarrow S$ be a smooth proper morphism of smooth varieties over a
field $k$. Let $\alpha$ be a codimension $p$ cycle on $X$. Then the locus of
points $s\in S$ such that the (Gysin) fiber $\alpha_{s}\in{\rm
CH}^{p}(X_{s})_{{\rm\mathbb{Q}}}$ is zero (resp. lies in ${\rm
CH}^{p}(X_{s})_{\mathrm{alg},{\rm\mathbb{Q}}}$) is a countable union of closed
algebraic subvarieties of $S$.

###### Proof.

There exists a countable subfield $k_{0}\subset k$, a smooth proper morphism
$X_{0}\rightarrow S_{0}$ of varieties over $k_{0}$ and a cycle $\alpha_{0}$ on
$X_{0}$ whose base change to $k$ are $X\rightarrow S$ and $\alpha$
respectively. Since $k_{0}$ is countable, $S_{0}$ has only countably many
closed subschemes. Let $\mathcal{F}$ be the collection of integral closed
subschemes $Z\subset S_{0}$ such that $\alpha_{0}$ is zero in ${\rm
CH}^{p}(X_{0,\eta(Z)})_{{\rm\mathbb{Q}}}$, where $\eta(Z)\in S_{0}$ denotes
the generic point of $Z$. Then we claim that the locus of $s\in S$ for which
$\alpha_{s}\in{\rm CH}^{p}(X_{s})_{{\rm\mathbb{Q}}}$ is zero is exactly the
union $\bigcup_{Z\in\mathcal{F}}Z_{k}\subset S$. This follows from Lemmas 2.1
and properties of the specialization map; we omit the details, and the similar
argument for algebraic triviality. ∎

###  2.3. Chow motives

We recall a few relevant facts about the category $\mathsf{Mot}(k)$ of (pure,
contravariant) Chow motives $M$ over $k$ with ${\rm\mathbb{Q}}$-coefficients;
see [42] or [38, §2] for basic definitions.

Denote the Lefschetz motive by $\mathbb{L}$, its $n$th tensor power by
$\mathbb{L}^{n}$ and write $M(n)=M\otimes\mathbb{L}^{n}$. Following [38,
§2.5], the Chow group in codimension $p$ of $M$ is by definition ${\rm
CH}^{p}(M)={\rm Hom}_{\mathsf{Mot}(k)}(\mathbb{L}^{p},M)$. If
$M=\mathfrak{h}(X)$ where $X$ is smooth projective over $k$, then ${\rm
CH}^{p}(M)={\rm CH}^{p}(X)_{{\rm\mathbb{Q}}}$. (Beware that we need to take
${\rm\mathbb{Q}}$-coefficients on the right hand side.) If $C$ is a nice curve
over $k$ and $\varphi\colon\mathfrak{h}(C)\rightarrow M(p-1)$ a morphism in
$\mathsf{Mot}(k)$, we obtain a homomorphism of abelian groups ${\rm
CH}^{1}(\varphi)\colon{\rm CH}^{1}(C)_{{\rm\mathbb{Q}}}\rightarrow{\rm
CH}^{p}(M)$. We define ${\rm CH}^{p}(M)_{\mathrm{alg}}$ to be the union of the
images of ${\rm CH}^{1}(\varphi)$ ranging over all pairs $(C,\varphi)$ as
above. This coincides with ${\rm
CH}^{p}(X)_{\mathrm{alg}}\otimes{\rm\mathbb{Q}}$ when $M=\mathfrak{h}(X)$, see
[2, Corollary 3.13].

Finally, we define motives of fixed points. If $G$ is a finite group acting on
a motive $M$, write $M^{G}$ for the submotive cut out by the idempotent
$\frac{1}{\\#G}\sum_{g\in G}g_{*}\in{\rm End}(M)$. We have ${\rm
CH}^{p}(M^{G})={\rm CH}^{p}(M)^{G}$. If $G$ acts on a nice variety $X$, then
$G$ acts on $\mathfrak{h}(X)$ and ${\rm CH}^{p}(X)$, and ${\rm
CH}^{p}(\mathfrak{h}(X)^{G})={\rm CH}^{p}(X)^{G}\otimes{{\rm\mathbb{Q}}}$.

###  2.4. Chow–Künneth decomposition for abelian varieties

Let $A/k$ be a $g$-dimensional abelian variety. Deninger and Murre [18, §3]
have constructed a canonical Chow–Künneth decomposition

(2.1) $\displaystyle\mathfrak{h}(A)=\bigoplus_{i=0}^{2g}\mathfrak{h}^{i}(A),$

uniquely characterized by the following property: if $(n)\colon A\rightarrow
A$ denotes the multiplication-by-$n$, then $(n)^{*}$ acts on
$\mathfrak{h}^{i}(A)$ via $n^{i}$ for every integer $n$. On the other hand,
Beauville [5] has shown that there exists a direct sum decomposition ${\rm
CH}^{p}(A)_{\rm\mathbb{Q}}=\bigoplus_{s=p-g}^{p}{\rm CH}^{p}_{(s)}(A)$, where

(2.2) $\displaystyle{\rm CH}^{p}_{(s)}(A)=\\{\alpha\in{\rm
CH}^{p}(A)_{\rm\mathbb{Q}}\colon(n)^{*}\alpha=n^{2p-s}\alpha\quad\forall
n\in{\rm\mathbb{Z}}\\}.$

The two decompositions are linked by the formula ${\rm
CH}^{p}(\mathfrak{h}^{i}(A))={\rm CH}^{p}_{(2p-i)}(A)$. Beauville conjectured
that ${\rm CH}^{p}_{(s)}(A)=0$ when $s<0$, and he proved it when
$p\in\\{0,1,g-2,g-1,g\\}$ [5, Proposition 3(a)].

######  Example 2.5.

For $p=1$, the Beauville decomposition ${\rm CH}^{1}(A)_{\rm\mathbb{Q}}={\rm
CH}^{1}_{(0)}(A)\oplus{\rm CH}^{1}_{(1)}(A)$ is the decomposition of a divisor
class into symmetric and anti-symmetric classes.

######  Lemma 2.6.

If $A/k$ is an abelian variety, then ${\rm
CH}^{p}(\mathfrak{h}^{1}(A))_{\mathrm{alg}}={\rm CH}^{p}(\mathfrak{h}^{1}(A))$
for all $p\geq 0$.

###### Proof.

By Lemma 2.1(1), we may assume $k$ is algebraically closed. The only nonzero
Chow group of $\mathfrak{h}^{1}(A)$ is ${\rm CH}^{1}(\mathfrak{h}^{1}(A))={\rm
CH}^{1}_{(1)}(A)$, the set of anti-symmetric elements of ${\rm
CH}^{1}(A)_{{\rm\mathbb{Q}}}$. The lemma follows from the fact that ${\rm
CH}^{1}_{(1)}(A)={\rm CH}^{1}(A)_{\mathrm{hom}}\otimes{\rm\mathbb{Q}}$ and
that homological and algebraic equivalence coincide for codimension-$1$
cycles. ∎

###  2.5. The Lefschetz decomposition for abelian varieties

Let $A/k$ be an abelian variety with polarization $\lambda\colon A\rightarrow
A^{\vee}$. Let $\ell\in{\rm CH}^{1}_{(0)}(A)$ be the unique class satisfying
$2\ell=(1,\lambda)^{*}\mathcal{P}$, where $\mathcal{P}\in{\rm Pic}(A\times
A^{\vee})$ is the Poincaré bundle. Künnemann has shown [26, Theorem 5.2] that
intersecting with $\ell^{g-i}$ induces an isomorphism
$\mathfrak{h}^{i}(A)\rightarrow\mathfrak{h}^{2g-i}(A)(g-i)$ for all $0\leq
i\leq g$. This induces a Lefschetz decomposition of the Chow–Künneth
components $\mathfrak{h}^{i}(A)$, see [26, Theorem 5.1]. We are chiefly
interested in the components $\mathfrak{h}^{3}(A)$ and
$\mathfrak{h}^{2g-3}(A)$ when $g\geq 2$; in that case the Lefschetz
decomposition has the form
$\mathfrak{h}^{3}(A)=\mathfrak{h}^{3}_{\mathrm{prim}}(A)\oplus\ell\cdot\mathfrak{h}^{1}(A)$
and
$\mathfrak{h}^{2g-3}(A)=\mathfrak{h}^{2g-3}_{\mathrm{prim}}(A)\oplus\ell^{g-2}\cdot\mathfrak{h}^{1}(A)$.
It has the property that the isomorphism
$\ell^{g-3}\colon\mathfrak{h}^{3}(A)\rightarrow\mathfrak{h}^{2g-3}(A)(g-3)$ is
a direct sum of isomorphisms
$\mathfrak{h}^{3}_{\mathrm{prim}}(A)\rightarrow\mathfrak{h}^{2g-3}_{\mathrm{prim}}(A)$
and $\ell\cdot\mathfrak{h}^{1}(A)\rightarrow\ell^{g-2}\mathfrak{h}^{1}(A)$.
Taking Chow groups, we get a decomposition ${\rm
CH}^{g-1}(\mathfrak{h}^{2g-3}(A))={\rm
CH}^{g-1}(\mathfrak{h}_{\mathrm{prim}}^{2g-3}(A))\oplus{\rm
CH}^{g-1}(\ell^{g-2}\cdot\mathfrak{h}^{1}(A))$.

###  2.6. Beauville components of $C$

Consider a nice curve $C$ of genus $g\geq 2$ over $k$ with Jacobian $J$. Let
$e$ be a degree-$1$ divisor on $C$ and embed $C$ in $J$ using the Abel–Jacobi
map based at $e$, sending $x\in C$ to the divisor class of $x-e$. Decompose
$[C]=[C]_{0}+\cdots+[C]_{g-1}$ with $[C]_{s}\in{\rm CH}_{(s)}^{g-1}(J)$. In
this subsection we analyze the component $[C]_{1}$ more closely, whose
vanishing is equivalent to the vanishing of $\kappa(C)$. Proposition 2.7 can
be viewed as a determination of the ‘imprimitive part’ of $[C]_{1}$.

For $\alpha\in{\rm CH}_{p}(J)$ and $\beta\in{\rm CH}_{q}(J)$, the Pontryagin
product $\alpha\star\beta$ is the pushforward of $\alpha\times\beta$ under the
addition map $J\times J\rightarrow J$. If $n$ is a positive integer let
$\alpha^{\star n}$ be the $n$-fold Pontryagin product of $\alpha$ with itself.
Let $K_{C}\in{\rm CH}_{0}(C)$ denote the canonical divisor class and let
$x_{e}\in J(k)$ be the point corresponding to the degree-$0$ divisor class
$[(2g-2)e]-K_{C}$.

######  Proposition 2.7.

We have an equality in ${\rm CH}_{(1)}^{1}(J)$:

(2.3)
$\displaystyle(2g-2)\cdot[C]_{0}^{\star(g-2)}\star[C]_{1}=[C]^{\star(g-1)}\star([0]-[x_{e}]).$

(Here we view $[0]-[x_{e}]$ as an element of ${\rm CH}_{0}(J)$.)

###### Proof.

If $D$ is a degree $g-1$ divisor class on $C$, let $\Theta_{D}$ be the image
of the map ${\rm Sym}^{g-1}(C)\rightarrow J$ defined by $x\mapsto[x]-D$. Then
$[C]^{\star(g-1)}=(g-1)![\Theta_{(g-1)e}]$. By Riemann–Roch,

$(-1)_{*}[\Theta_{(g-1)e}]=[\Theta_{K_{C}-(g-1)e}]=[x_{e}]\star[\Theta_{(g-1)e}].$

Combining the last two sentences shows that
$(-1)_{*}([C]^{\star(g-1)})=[x_{e}]\star[C]^{\star(g-1)}$.

Since Pontryagin product sends ${\rm CH}^{g-p}_{(s)}(J)\times{\rm
CH}^{g-q}_{(t)}(J)$ to ${\rm CH}^{g-p-q}_{(s+t)}(J)$ and since ${\rm
CH}^{1}_{(s)}(J)\neq 0$ only if $s\in\\{0,1\\}$, we calculate that
$[C]^{\star(g-1)}=[C]_{0}^{\star(g-1)}+(g-1)\cdot[C]_{0}^{\star(g-2)}\star[C]_{1}$.
Applying $(-1)_{*}$ and $[x_{e}]\star$ to the previous identity, we obtain

$\displaystyle(-1)_{*}[C]^{\star(g-1)}$
$\displaystyle=[C]_{0}^{\star(g-1)}-(g-1)\cdot[C]_{0}^{\star(g-2)}\star[C]_{1},$
$\displaystyle[x_{e}]\star[C]^{\star(g-1)}$
$\displaystyle=[x_{e}]\star[C]_{0}^{\star(g-1)}+(g-1)\cdot[C]_{0}^{\star(g-2)}\star[C]_{1}.$

Note that $[x_{e}]\star$ acts trivially on ${\rm CH}^{1}_{(1)}(J)$ since
$([x_{e}]-[0])\in\oplus_{s\geq 1}{\rm CH}^{g}_{(s)}(J)$ by the explicit
description of the Beauville decomposition for zero-cycles [5, bottom of p.
649]. Equating the right hand sides of the two centered equations proves that
$(2g-2)\cdot[C]_{0}^{\star(g-2)}\star[C]_{1}=[C]_{0}^{\star(g-1)}\star([0]-[x_{e}])$.
Since $[C]_{s}^{\star(g-1)}\star([0]-[x_{e}])\in\oplus_{t\geq s+1}{\rm
CH}^{1}_{(t)}(J)=\\{0\\}$ for all $s\geq 1$, it follows that
$[C]_{0}^{\star(g-1)}\star([0]-[x_{e}])=[C]^{\star(g-1)}\star([0]-[x_{e}])$,
concluding the proof. ∎

######  Corollary 2.8.

Suppose that $[C]_{1}=0$ in ${\rm CH}_{1}(J)_{{\rm\mathbb{Q}}}$. Then
$(2g-2)e=K_{C}$ in ${\rm CH}_{0}(C)_{{\rm\mathbb{Q}}}$.

###### Proof.

If $[C]_{1}=0$, then $[C]^{\star(g-1)}\star([0]-[x_{e}])=0$ by (2.3). On the
other hand, $[C]^{\star(g-1)}=(g-1)![\Theta_{(g-1)e}]$ is multiple of a theta
divisor, in the notation of the proof of Proposition 2.7. Since
$\Theta_{(g-1)e}$ defines a principal polarization, the map
$x\mapsto[\Theta_{(g-1)e+x}]-[\Theta_{(g-1)e}]=[\Theta_{(g-1)e}]\star([x]-[0])$
induces an isomorphism $\varphi\colon J(k)\rightarrow{\rm CH}^{1}(J)_{\hom}$.
Since $\varphi(x_{e})$ is torsion, it follows that $x_{e}\in J(k)$ is itself
torsion, as desired. ∎

The principal polarization defines an ample class $\ell\in{\rm
CH}_{(1)}^{1}(J)$, which induces a Lefschetz decomposition
$\mathfrak{h}^{2g-3}(J)=\mathfrak{h}^{2g-3}_{\mathrm{prim}}(J)\oplus\ell^{g-2}\cdot\mathfrak{h}^{1}(J)$
as in §2.5.

######  Corollary 2.9.

Suppose that $(2g-2)e=K_{C}$ in ${\rm CH}_{0}(C)_{{\rm\mathbb{Q}}}$. Then
$[C]_{1}\in{\rm CH}^{g-1}(\mathfrak{h}^{2g-3}_{\mathrm{prim}}(J))$.

###### Proof.

By definition and the discussion in §2.4, $[C]_{1}\in{\rm
CH}_{(1)}^{g-1}(J)={\rm CH}^{g-1}(\mathfrak{h}^{2g-3}(J))$. Since $x_{e}$ is
torsion, $(n)_{*}([0]-[x_{e}])=0$ for some integer $n\geq 1$. The
decomposition (2.2) implies that $[0]=[x_{e}]$ in ${\rm
CH}_{0}(J)_{{\rm\mathbb{Q}}}$. Therefore (2.3) shows that
$[C]_{0}^{\star(g-2)}\star[C]_{1}=0$. Using properties of the
$\mathfrak{sl}_{2}$-action on ${\rm CH}^{*}(J)_{{\rm\mathbb{Q}}}$ (in the
sense of [33, §1.3]), this implies that $\ell\cdot[C]_{1}=0$. Since ${\rm
CH}^{g-1}(\mathfrak{h}^{2g-3}_{\mathrm{prim}}(J))$ equals the kernel of
$\ell\cdot(-)\colon{\rm CH}^{g-1}(\mathfrak{h}^{2g-3}(J))\rightarrow{\rm
CH}^{g-1}(\mathfrak{h}^{2g-1}(J)(1))$, the corollary follows. ∎

###  2.7. Ceresa cycles

Let $C/k$ be a nice curve of genus $g\geq 2$ with Jacobian $J$. Let $e$ be a
degree-$1$ divisor on $C$ and let $\iota_{e}\colon C\rightarrow J$ be the
Abel–Jacobi map based at $e$. We define $\kappa_{C,e}\in{\rm CH}_{1}(J)$ using
the formula (1.1) from the introduction. Using the Beauville decomposition
(2.2) to write $[C]=[\iota_{e}(C)]=\sum_{s=0}^{g-1}[C]_{s}$ with
$[C]_{s}\in{\rm CH}^{g-1}_{(s)}(J)$, we calculate that

(2.4)
$\displaystyle\kappa_{C,e}=\kappa(C)=2[C]_{1}+2[C]_{3}+\dots+2[C]_{2\lfloor\frac{g-2}{2}\rfloor+1}$

in ${\rm CH}_{1}(J)_{{\rm\mathbb{Q}}}$.

######  Lemma 2.10.

If $\kappa_{C,e}$ is torsion, then $(2g-2)e-K_{C}$ is torsion.

###### Proof.

If $\kappa_{C,e}$ is torsion, then $[C]_{1}=0$ by (2.4). We conclude using
Corollary 2.8. ∎

######  Lemma 2.11.

If $e,e^{\prime}\in{\rm CH}_{0}(C)$ are degree-$1$ divisors such that
$e-e^{\prime}$ is torsion, then $[\iota_{e}(C)]=[\iota_{e^{\prime}}(C)]$ and
$\kappa_{C,e}=\kappa_{C,e^{\prime}}$ in ${\rm CH}_{1}(J)_{{\rm\mathbb{Q}}}$.

###### Proof.

Suppose $e-e^{\prime}$ has order $n$ in ${\rm CH}_{0}(C)$. Then
$(n)\circ\iota_{e}=(n)\circ\iota_{e^{\prime}}$, hence
$(n)_{*}([\iota_{e}(C)]-[\iota_{e^{\prime}}(C)])=0$ in ${\rm CH}_{1}(J)$. On
the other hand, the decomposition (2.2) shows that $(n)_{*}\colon{\rm
CH}_{1}(J)_{{\rm\mathbb{Q}}}\rightarrow{\rm CH}_{1}(J)_{{\rm\mathbb{Q}}}$ is
an isomorphism. Therefore $[\iota_{e}(C)]-[\iota_{e^{\prime}}(C)]$ is torsion.
Hence $\kappa_{C,e}-\kappa_{C,e^{\prime}}$ is torsion too. ∎

Let $\kappa(C)$ be the image of $\kappa_{C,e}$ in ${\rm
CH}_{1}(J)_{\rm\mathbb{Q}}$ for any choice of degree-$1$ divisor $e$ on $C$
such that $(2g-2)e=K_{C}$ in ${\rm CH}_{0}(C)_{{\rm\mathbb{Q}}}$. Lemma 2.11
shows that this class is independent of the choice of $e$. (If $k$ is not
algebraically closed and no such $e$ exists over $k$, there exists a unique
class $\kappa(C)\in{\rm CH}_{1}(J)_{{\rm\mathbb{Q}}}$ such that
$\kappa(C)_{\bar{k}}=\kappa(C_{\bar{k}})$ in ${\rm
CH}_{1}(J_{\bar{k}})_{{\rm\mathbb{Q}}}$, since Chow groups with
${\rm\mathbb{Q}}$-coefficients satisfy Galois descent.) We let
$\bar{\kappa}(C)$ be the image of $\kappa(C)$ in ${\rm
Gr}_{1}(J)_{{\rm\mathbb{Q}}}$. Since all degree-$1$ divisors $e$ on $C$ are
algebraically equivalent, $\bar{\kappa}(C)$ is also the image of
$\kappa_{e}(C)$ in ${\rm Gr}_{1}(J)_{\rm\mathbb{Q}}$ for _any_ degree-$1$
divisor, not necessarily with the property that $(2g-2)e=K_{C}$ in ${\rm
CH}_{0}(C)_{{\rm\mathbb{Q}}}$.

Suppose now that $(2g-2)e=K_{C}$ in ${\rm CH}_{0}(C)_{{\rm\mathbb{Q}}}$. Since
$[\iota_{e}(C)]\in{\rm CH}_{1}(J)_{{\rm\mathbb{Q}}}$ is independent of the
choice of $e$, the same is true for the classes $[C]_{s}$. In particular, they
are ${\rm Aut}(C)$-invariant.

######  Proposition 2.12.

In the above notation, $\kappa(C)=0$ in ${\rm CH}_{1}(J)_{{\rm\mathbb{Q}}}$ if
and only if $[C]_{1}=0$ in ${\rm CH}_{1}(J)_{{\rm\mathbb{Q}}}$ if and only if
$[C]_{s}=0$ for all $s\geq 1$. Moreover $\bar{\kappa}(C)=0$ in ${\rm
Gr}_{1}(J)_{{\rm\mathbb{Q}}}$ if and only if $[C]_{1}\in{\rm
Gr}_{1}(J)_{{\rm\mathbb{Q}}}$ if and only if $[C]_{s}=0$ in ${\rm
Gr}_{1}(J)_{{\rm\mathbb{Q}}}$ for all $s\geq 1$.

###### Proof.

Using the expression (2.4), it suffices to prove $[C]_{1}=0$ implies
$[C]_{s}=0$ for all $s\geq 1$. This follows from the third centered equation
of [46, Theorem 1.5.5]. The proof for $\bar{\kappa}(C)$ is identical. ∎

Proposition 2.12 shows that the vanishing of $\kappa(C)$ is equivalent to the
vanishing of $[C]_{1}$. The next lemma isolates a summand of the Chow group
that contains $[C]_{1}$. Recall that if $M$ is a direct summand of
$\mathfrak{h}(J)$, then ${\rm CH}^{p}(M)$ is naturally a summand of ${\rm
CH}^{p}(\mathfrak{h}(J))={\rm CH}^{p}(J)_{{\rm\mathbb{Q}}}$.

######  Proposition 2.13.

In the above notation, $[C]_{1}\in{\rm
CH}^{g-1}(\mathfrak{h}^{2g-3}_{\mathrm{prim}}(J)^{{\rm Aut}(C)})\subset{\rm
CH}_{1}(J)_{{\rm\mathbb{Q}}}$.

###### Proof.

Combine Corollary 2.9 and the fact that $[C]_{1}$ is ${\rm Aut}(C)$-invariant.
∎

##  3\. Vanishing criteria in the Chow and Griffiths groups

The proof of Theorem A is quite short, but uses in a crucial way Kimura’s
notion of finite dimensional Chow motives [25].

###### Proof of Theorem A.

We use the definitions and notations of §2.3 and §2.7. Choose a degree-$1$
divisor $e$ such that $(2g-2)e=K_{C}$ in ${\rm CH}_{0}(C)_{{\rm\mathbb{Q}}}$
and decompose $[\iota_{e}(C)]=\sum_{s=-1}^{g-1}[C]_{s}$ with $[C]_{s}\in{\rm
CH}^{g-1}_{(s)}(J)$. Let $G={\rm Aut}(C)$. Proposition 2.13 shows that
$[C]_{1}\in{\rm CH}^{g-1}(\mathfrak{h}^{2g-3}_{\mathrm{prim}}(J)^{G})$.

By the hypothesis and the Lefschetz isomorphism,
$\operatorname{H}^{*}(\mathfrak{h}_{\mathrm{prim}}^{2g-3}(J)^{G})=\operatorname{H}^{2g-3}_{\mathrm{prim}}(J)^{G}=0$.
Since any summand of the motive of an abelian variety is finite-dimensional in
the sense of Kimura [25, Example 9.1], it follows from Kimura’s [25, Corollary
7.3] that $\mathfrak{h}^{2g-3}_{\mathrm{prim}}(J)^{G}=0$ and hence $[C]_{1}=0$
in ${\rm CH}^{g-1}(J)_{{\rm\mathbb{Q}}}$. By Proposition 2.12, we conclude
that $\kappa(C)=0$. ∎

######  Example 3.1.

There is exactly one non-hyperelliptic genus 3 curve over ${\rm\mathbb{C}}$
for which the criterion of [39] applies, namely the curve $y^{3}=x^{4}+1$. The
curve $y^{3}=x^{4}+x$ satisfies the weaker hypothesis of Theorem A (as was
observed in [7]), so we deduce that $\kappa(C)=0$ for this curve as well.
Beauville and Schoen studied the specific geometry of this curve and showed
that $\bar{\kappa}(C)=0$ [8].

######  Example 3.2.

The genus $4$ curve $y^{3}=x^{5}+1$ satisfies $\operatorname{H}^{3}(J)^{{\rm
Aut}(C)}=0$ [29, proof of Theorem 3.3], so $\kappa(C)=0$.

Theorem B will follow from the next proposition. In that proposition and its
proof, if $X$ is a nice variety over ${\rm\mathbb{C}}$ we write
$\operatorname{H}^{*}(X)$ for the singular cohomology of $X({\rm\mathbb{C}})$
with ${\rm\mathbb{Q}}$-coefficients, seen as an object in the category of
Hodge structures.

######  Proposition 3.3.

Let $C$ be a smooth, projective, integral curve over ${\rm\mathbb{C}}$ with
Jacobian $J$ and let $G\subset{\rm Aut}(C)$ be a subgroup with
$\operatorname{H}^{0}(J,\Omega_{J}^{3})^{G}=0$. Then there exists an abelian
variety $A/{\rm\mathbb{C}}$ such that
$\operatorname{H}^{3}(J)^{G}\simeq\operatorname{H}^{1}(A)(-1)$. If the Hodge
conjecture holds for $J\times A$, then
$\mathfrak{h}^{3}(J)^{G}\simeq\mathfrak{h}^{1}(A)(-1)$ and
$\bar{\kappa}(C)=0$.

###### Proof.

Let $N^{1}\operatorname{H}^{3}(J)$ be the largest sub-Hodge structure of
$\operatorname{H}^{3}(J)$ of type $(1,2)+(2,1)$. The polarization on
$\operatorname{H}^{1}(J)$ induces a polarization on
$N^{1}\operatorname{H}^{3}(J)$ and so
$N^{1}\operatorname{H}^{3}(J)\simeq\operatorname{H}^{1}(B)(-1)$ for some
abelian variety $B/{\rm\mathbb{C}}$. The assumptions and the Hodge
decomposition imply that $\operatorname{H}^{3}(J)^{G}$ is a Hodge structure of
type $(1,2)+(2,1)$. It follows that $\operatorname{H}^{3}(J)^{G}\subset
N^{1}\operatorname{H}^{3}(J)$, so there exists an abelian subvariety $A\subset
B$ with $\operatorname{H}^{3}(J)^{G}\simeq\operatorname{H}^{1}(A)(-1)$.

We now show that the Hodge conjecture for $J\times A$ implies the claims of
the final sentence. Fix mutually inverse isomorphisms of Hodge structures
$\phi\colon\operatorname{H}^{3}(J)^{G}\rightarrow\operatorname{H}^{1}(A)(-1)$
and $\psi\colon\operatorname{H}^{1}(A)(-1)\rightarrow\operatorname{H}^{3}(J)$.
By the Hodge conjecture, there exist morphisms of motives
$\Phi\colon\mathfrak{h}^{3}(J)^{G}\rightarrow\mathfrak{h}^{1}(A)(-1)$ and
$\Psi\colon\mathfrak{h}^{1}(A)(-1)\rightarrow\mathfrak{h}^{3}(J)^{G}$ (in
other words, cycles on $J\times A$ with certain properties) with
$\operatorname{H}^{*}(\Phi)=\phi$ and $\operatorname{H}^{*}(\Psi)=\psi$. Since
$\mathfrak{h}^{1}(A)(-1)$ and $\mathfrak{h}^{3}(J)^{G}$ are Kimura finite-
dimensional [25, Example 9.1] and
$\operatorname{H}^{*}(\Phi)\circ\operatorname{H}^{*}(\Psi)$ and
$\operatorname{H}^{*}(\Psi)\circ\operatorname{H}^{*}(\Phi)$ are the identity,
it follows from [25, Proposition 7.2(ii)] (see also [4, Corollaire 3.16]) that
$\Psi\circ\Phi$ and $\Phi\circ\Psi$ are themselves isomorphisms. Hence $\Phi$
and $\Psi$ are isomorphisms too and we conclude that
$\mathfrak{h}^{3}(J)^{G}\simeq\mathfrak{h}^{1}(A)(-1)$.

Using the Lefschetz isomorphism
$\mathfrak{h}^{2g-3}(J)\simeq\mathfrak{h}^{3}(J)(-g+3)$ of [26, Theorem 5.2],
we obtain an isomorphism
$\mathfrak{h}^{2g-3}(J)^{G}\simeq\mathfrak{h}^{1}(A)(-g+2)$. Similarly to the
proof of Theorem A, we decompose $[\iota_{e}(C)]=\sum_{s=0}^{g-1}[C]_{s}$ with
$[C]_{s}\in{\rm CH}^{g-1}_{(s)}(J)$ and observe that the class $[C]_{1}$ lies
in ${\rm CH}_{(1)}^{g-1}(J)^{G}={\rm CH}^{g-1}(\mathfrak{h}^{2g-3}(J)^{G})$.
Lemma 2.6 combined with the isomorphism
$\mathfrak{h}^{2g-3}(J)^{G}\simeq\mathfrak{h}^{1}(A)(-g+2)$ shows that ${\rm
CH}^{g-1}(\mathfrak{h}^{2g-3}(J)^{G})={\rm
CH}^{g-1}(\mathfrak{h}^{2g-3}(J)^{G})_{\mathrm{alg}}$, hence every element of
${\rm CH}^{g-1}_{(1)}(J)^{G}$ lies in ${\rm
CH}^{g-1}(J)_{\mathrm{alg},{\rm\mathbb{Q}}}$. Therefore $[C]_{1}\in{\rm
CH}^{g-1}(J)_{\mathrm{alg},{\rm\mathbb{Q}}}$, hence the image of $[C]_{1}$ in
${\rm Gr}_{1}(J)_{{\rm\mathbb{Q}}}$ vanishes. Proposition 2.12 then implies
that $\bar{\kappa}(C)$ vanishes too. ∎

###### Proof of Theorem B.

Since $C$ can be defined over a countable field and since such a field can be
embedded in ${\rm\mathbb{C}}$, Lemma 2.1 shows that we may assume
$k={\rm\mathbb{C}}$. We conclude by Proposition 3.3 applied to $G={\rm
Aut}(C)$. ∎

######  Example 3.4.

Let $C\colon y^{3}=x^{4}+ax^{2}+bx+c$ be a Picard curve. Then
$\operatorname{H}^{0}(C,\Omega_{C}^{1})\simeq\chi\oplus\chi\oplus\chi^{2}$ as
$C_{3}$-representations, where $\chi$ is a character of order $3$. It follows
that $\operatorname{H}^{0}(J,\Omega_{J}^{3})^{{\rm Aut}(C)}=0$, so that the
condition of Theorem B is satisfied. We consider these curves in detail in the
next section.

We exhibit two families of examples of Theorem B in genus $4$. It would be
interesting to study these families in more detail (along the lines of what we
do in the next section for Picard curves).

######  Example 3.5.

Consider for every $t\in{\rm\mathbb{C}}\setminus\\{0,1,-1\\}$ the
$\mu_{5}$-cover of ${\rm\mathbb{P}}^{1}$ with equation

$C\colon y^{5}=x^{3}(x-1)^{2}(x-t).$

By [34, Lemma 2.7], the $\mu_{5}$-representation
$\operatorname{H}^{0}(C,\Omega_{C})$ is the direct sum of the four nontrivial
characters. It follows that
$\operatorname{H}^{0}(J,\Omega_{J}^{3})^{\mu_{5}}=0$ and
$\operatorname{H}^{3}(J)^{\mu_{5}}$ is isomorphic to
$\operatorname{H}^{1}(A)(-1)$, for some abelian surface $A$ (using Proposition
3.3). We have $D_{5}\subset{\rm Aut}(C)$ by [22, Corollary 1]. By the Kani-
Rosen formula [23, Theorem B], the Jacobian $J$ is isogenous to a product of
abelian surfaces of the form $\operatorname{Jac}(C/\tau)$, for some involution
$\tau$. The Hodge conjecture in any codimension is known for products of
abelian surfaces [40, Theorem 3.15], so $\bar{\kappa}(C)=0$ by Proposition
3.3.

######  Example 3.6.

The general $\mu_{3}$-cover of ${\rm\mathbb{P}}^{1}$ with equation

$y^{3}=x^{2}(x-1)^{2}(x^{3}+ax^{2}+bx+c),$

has genus $4$ and there is an isomorphism of $\mu_{3}$-representations
$\operatorname{H}^{0}(C,\Omega_{C})\simeq\chi\oplus\chi\oplus\chi^{2}\oplus\chi^{2}$.
It follows that $\operatorname{H}^{0}(J,\Omega_{J}^{3})^{\mu_{3}}=0$, and the
abelian variety $A$ from Proposition 3.3 is 4-dimensional. If the Hodge
conjecture holds for the $8$-dimensional $J\times A$, then
$\bar{\kappa}(C)=0$.

##  4\. Picard curves

###  4.1. Generalities

Let $k$ be a field of characteristic zero. A Picard curve over $k$ is by
definition a nice curve with an affine model $y^{3}=f(x)=x^{4}+ax^{2}+bx+c$
for some $a,b,c\in k$. Conversely, given such a polynomial $f(x)\in k[x]$ of
nonzero discriminant, the projective closure of $y^{3}=f(x)$ in
${\rm\mathbb{P}}^{2}_{k}$ is a nice curve denoted by $C_{f}$. It has a unique
point at infinity $P_{\infty}$, which is $k$-rational.

For every third root of unity $\omega\in\bar{k}$, the map
$(x,y)\mapsto(x,\omega y)$ defines an automorphism of $C_{f,\bar{k}}$. We view
$\mu_{3}$ as a subgroup of ${\rm Aut}(C_{f,\bar{k}})$ in this way. Then
$\mu_{3}$ also acts on the Jacobian $J_{f,\bar{k}}$ by taking images of
divisors.

The discriminant of $f$ has the following expression:

(4.1) $\displaystyle{\rm
disc}(f)=-4a^{3}b^{2}-27b^{4}+16a^{4}c+144ab^{2}c-128a^{2}c^{2}+256c^{3}.$

We view $f$ as the dehomogenization $F(x,1)$ of the quartic form
$F(X,Z)=X^{4}+aX^{2}Z^{2}+bXZ^{3}+cZ^{4}$, and we define $I(f)$ and $J(f)$ to
be the usual $I$\- and $J$-invariants attached to $F$, as in [10, §2]. Their
explicit formulae in our case are:

$\displaystyle I(f)$ $\displaystyle=a^{2}+12c,$ $\displaystyle J(f)$
$\displaystyle=72ac-2a^{3}-27b^{2}.$

The 19th century invariant theorists observed the identity
$J(f)^{2}=4I(f)^{3}-27\cdot{\rm disc}(f)$, which can be verified by direct
computation. Therefore $P_{f}:=(I(f),J(f))$ is a $k$-point on the elliptic
curve

$E_{f}\colon y^{2}=4x^{3}-27\cdot{\rm disc}(f).$

###  4.2. Ceresa vanishing criteria

Since $(2g-2)P_{\infty}=4P_{\infty}$ is canonical, we may use $P_{\infty}$ to
embed $C_{f}$ in its Jacobian $J_{f}$ and define the Ceresa cycle
$\kappa_{C_{f},P_{\infty}}\in{\rm CH}_{1}(J_{f})$ as in the introduction; we
denote it by $\kappa_{f}$ for simplicity. Recall that $\kappa(C_{f})$ denotes
the image of $\kappa_{f}$ in ${\rm CH}_{1}(J_{f})_{{\rm\mathbb{Q}}}$ and
$\bar{\kappa}(C_{f})$ its image in ${\rm Gr}_{1}(J_{f})_{{\rm\mathbb{Q}}}$.
Theorem C follows from the following slightly stronger theorems, whose proofs
will take up the rest of this section.

######  Theorem 4.1.

There exists an integer $N\geq 1$ $($depending neither on $f$ nor $k)$ such
that $N\cdot\kappa_{f}\in{\rm CH}^{2}(J_{f})_{\mathrm{alg}}$ for every Picard
curve $C_{f}$ over every algebraically closed field $k$ (of characteristic
zero).

######  Theorem 4.2.

The Ceresa cycle $\kappa_{f}\in{\rm CH}_{1}(J)$ is torsion if and only if
$P_{f}\in E_{f}(k)$ is torsion. Moreover, if $k$ is algebraically closed,
there exists an integer $M\geq 1$ with the following property: if $C_{f}$ is a
Picard curve and $\kappa_{f}$ is torsion, then
$\operatorname*{ord}(\kappa_{f})$ divides $M\cdot\operatorname*{ord}(P_{f})$
and $\operatorname*{ord}(P_{f})$ divides
$M\cdot\operatorname*{ord}(\kappa_{f})$.

Theorem 4.1 will be proven in §4.4, and Theorem 4.2 will be proven in §4.7. A
standard argument using Lemma 2.1 shows that we may assume
$k={\rm\mathbb{C}}$. So in the remainder of §4, all varieties will be over
${\rm\mathbb{C}}$, and cohomology will be singular cohomology.

######  Remark 4.3.

Theorem 4.2 generalizes [27, Theorem 5.16], which considered the special case
where $b=0$. There, we exploited the bielliptic cover to show that the Ceresa
cycle maps via a correspondence to a multiple of the point
$Q_{f}=(a^{2}-4c,a(a^{2}-4c))$ on the elliptic curve $E_{f}^{\prime}\colon
y^{2}=x^{3}+16\cdot{\rm disc}(f)$. This is compatible with the general case
since there is a $3$-isogeny $\phi_{f}\colon E^{\prime}_{f}\to E_{f}$, and one
checks using the explicit formula for $\phi_{f}$ [9, Equation (2)] that
$\phi_{f}(Q_{f})=P_{f}$.

######  Remark 4.4.

Is it always the case that $\kappa_{f}\neq 0$? (Recall that $\kappa_{f}$ lies
in the Chow group with ${\rm\mathbb{Z}}$-coefficients.) We cannot conclude
this from our proof of Theorem 4.2 below since we have worked with
${\rm\mathbb{Q}}$-coefficients, and we make use of various isogenies whose
degrees we do not control.

###  4.3. Multilinear algebra

Our first goal (Proposition 4.7) is to explicitly identify the abelian variety
$A$ of Proposition 3.3 for Picard curves.

Write $\mathcal{O}={\rm\mathbb{Z}}[\omega]$ for the ring of Eisenstein
integers with $\omega^{2}+\omega+1=0$ and let $K={\rm\mathbb{Q}}(\sqrt{-3})$
be its fraction field. Let $C$ be a Picard curve over ${\rm\mathbb{C}}$ with
Jacobian variety $J$. The $\mu_{3}$-action on $C$ extends to an embedding
$\mathcal{O}\subset{\rm End}(J)$. Using this action, the singular cohomology
group $\operatorname{H}^{1}(J;{\rm\mathbb{Z}})$ is a free $\mathcal{O}$-module
of rank $3$, and $\operatorname{H}^{1}(J;{\rm\mathbb{Q}})$ is a
$3$-dimensional $K$-vector space. The next lemma says that the criterion of
Theorem B is always satisfied for Picard curves.

######  Lemma 4.5.

$\operatorname{H}^{0}(J,\Omega_{J}^{3})^{\mu_{3}}=0$ and the Hodge structure
$\operatorname{H}^{3}(J;{\rm\mathbb{Q}})^{\mu_{3}}$ is of type $(1,2)+(2,1)$.

###### Proof.

Since
$\operatorname{H}^{0}(J,\Omega_{J}^{3})\simeq\bigwedge^{3}\operatorname{H}^{0}(C,\Omega_{C}^{1})$,
the first claim follows from a calculation with differentials (Example 3.4).
The second claim follows from the Hodge decomposition for
$\operatorname{H}^{3}(J;{\rm\mathbb{Q}})$. ∎

We may view $\operatorname{H}^{1}(J;{\rm\mathbb{Q}})$ either as a $K$-vector
space or ${\rm\mathbb{Q}}$-vector space; when we perform tensor operations, we
will add the subscript $K$ when we view it as a $K$-vector space, and add no
subscript otherwise. For example, cup product induces an isomorphism
$\bigwedge^{3}\operatorname{H}^{1}(J;{\rm\mathbb{Q}})\simeq\operatorname{H}^{3}(J;{\rm\mathbb{Q}})$,
and we will use this identification without further mention.

The universal property of exterior powers induces a canonical
${\rm\mathbb{Q}}$-linear surjection
$\bigwedge^{3}\operatorname{H}^{1}(J;{\rm\mathbb{Q}})\rightarrow\bigwedge^{3}_{K}\operatorname{H}^{1}(J;{\rm\mathbb{Q}})$.
It is well known (see [35, Lemma 12(i)] or [17, Lemma 4.3]) that this map
admits a canonical splitting, which we use to view
$\bigwedge_{K}^{3}\operatorname{H}^{1}(J;{\rm\mathbb{Q}})$ as a direct summand
of $\operatorname{H}^{3}(J;{\rm\mathbb{Q}})$.

######  Lemma 4.6.

We have
$\operatorname{H}^{3}(J;{\rm\mathbb{Q}})^{\mu_{3}}=\bigwedge_{K}^{3}\operatorname{H}^{1}(J;{\rm\mathbb{Q}})$
inside $\operatorname{H}^{3}(J;{\rm\mathbb{Q}})$. Moreover
$\dim_{{\rm\mathbb{Q}}}\operatorname{H}^{3}(J;{\rm\mathbb{Q}})^{\mu_{3}}=2$.

###### Proof.

It suffices to prove the statements after tensoring with ${\rm\mathbb{C}}$.
Let $g\in\mu_{3}\subset{\rm Aut}(C)$ be a nontrivial element. The action of
$g$ on $\operatorname{H}^{1}(J;{\rm\mathbb{Q}})$ has eigenvalues (with
multiplicity) $\omega,\omega,\omega,\omega^{2},\omega^{2},\omega^{2}$ so we
can write $\operatorname{H}^{1}(J;{\rm\mathbb{C}})=V_{1}\oplus V_{2}$ where
$V_{i}$ is the $\omega^{i}$-eigenspace. Since a three element subset of these
eigenvalues have product $1$ if and only if they are all equal,
$\operatorname{H}^{3}(J;{\rm\mathbb{C}})^{\mu_{3}}=(\bigwedge^{3}V_{1})\oplus(\bigwedge^{3}V_{2})$.
On the other hand, an argument similar to the proof of [17, Proposition 4.4]
shows
$(\bigwedge^{3}_{K}\operatorname{H}^{1}(J;{\rm\mathbb{Q}}))\otimes_{K}{\rm\mathbb{C}}=\bigwedge^{3}_{K\otimes{\rm\mathbb{C}}}\operatorname{H}^{1}(J;{\rm\mathbb{C}})=(\bigwedge^{3}V_{1})\oplus(\bigwedge^{3}V_{2})$,
proving the equality. The explicit description of this subspace shows that it
is $2$-dimensional over ${\rm\mathbb{Q}}$. ∎

Let $E$ be the elliptic curve with Weierstrass equation $y^{2}=x^{3}+1$.

######  Proposition 4.7.

There is an isomorphism of Hodge structures
$\operatorname{H}^{3}(J;{\rm\mathbb{Q}})^{\mu_{3}}\simeq\operatorname{H}^{1}(E,{\rm\mathbb{Q}})(-1)$.

###### Proof.

Since there exists a unique elliptic curve up to isogeny with endomorphism
algebra $K$, there exists a unique Hodge structure of dimension $2$, type
$(0,1)+(1,0)$ and carrying an action of $K$. Since both
$\operatorname{H}^{3}(J;{\rm\mathbb{Q}})^{\mu_{3}}$ and
$\operatorname{H}^{1}(E;{\rm\mathbb{Q}})$ have these properties (the former by
Lemma 4.6), they must be isomorphic, and we conclude using Lemma 4.6. ∎

###  4.4. Weil classes and Schoen’s theorem

Our next goal is to upgrade the isomorphism of Proposition 4.7 to an
isomorphism in the category of Chow motives, and hence deduce (using
Proposition 3.3) the vanishing of $\bar{\kappa}(C)$ in the Griffiths group. We
use the following special case of a result of Schoen, which crucially uses the
assumption that the endomorphism algebra of $J\times E$ contains
$K={\rm\mathbb{Q}}(\omega)$:

######  Theorem 4.8 (Schoen).

The Hodge conjecture holds for $J\times E$.

###### Proof.

Since $J\times E$ is four-dimensional, it suffices to prove Hodge classes in
$\operatorname{H}^{4}(J\times E;{\rm\mathbb{Q}})$ are algebraic. By [36,
Theorem (0.1), Part (i) and (iv)], such Hodge classes are sums of products of
divisor classes (which are algebraic) and Weil classes
$W_{K}:=\bigwedge^{4}_{K}\operatorname{H}^{4}(J\times
E;{\rm\mathbb{Q}})\subset\operatorname{H}^{4}(J\times E;{\rm\mathbb{Q}})$.
Since $K={\rm\mathbb{Q}}(\omega)$ and the embedding $K\subset{\rm End}(J\times
E)\otimes{\rm\mathbb{Q}}$ can be chosen to have signature $(2,2)$, Schoen has
shown in [41] that the classes in $W_{K}$ are algebraic, concluding that all
Hodge classes of $\operatorname{H}^{4}(J\times E;{\rm\mathbb{Q}})$ are
algebraic. ∎

Recall from §2.3 our conventions on motives, the canonical Chow–Künneth
components $\mathfrak{h}^{i}(J)$ and $\mathfrak{h}^{j}(E)$, and the motive of
fixed points of a finite group action.

######  Corollary 4.9.

There is an isomorphism
$\mathfrak{h}^{3}(J)^{\mu_{3}}\simeq\mathfrak{h}^{1}(E)(-1)$.

###### Proof.

This follows from Proposition 3.3, using Proposition 4.7 and Theorem 4.8. ∎

######  Corollary 4.10.

If $C$ is a Picard curve over ${\rm\mathbb{C}}$, then $\bar{\kappa}(C)=0$.

###### Proof.

This follows from Proposition 3.3, using Proposition 4.7 and Theorem 4.8. ∎

###### Proof of Theorem 4.1.

Consider the parameter space $S_{0}=\\{(a,b,c)\mid{\rm disc}(f)\neq
0\\}\subset{\rm\mathbb{A}}^{3}_{{\rm\mathbb{Q}}}$ of Picard curves. Let $\eta$
be the generic point of $S_{0}$ with function field
$k(\eta)={\rm\mathbb{Q}}(a,b,c)$, let $C_{\eta}$ be the generic Picard curve
over $k(\eta)$ with Jacobian $J_{\eta}$, and let $\kappa_{C_{\eta}}\in{\rm
CH}_{1}(J_{\eta})$ be the Ceresa cycle of $C_{\eta}$ based at the point at
infinity. Fix an embedding $j\colon k(\eta)\rightarrow{\rm\mathbb{C}}$. By
Corollary 4.10, the base change of $\kappa_{C{\eta}}$ along $j$ is torsion in
the Griffiths group. By Lemma 2.1, this implies $\kappa_{C_{\eta}}$ is itself
torsion in ${\rm Gr}_{1}(J_{\eta})$. Let $N_{0}$ be the finite order of
$\kappa_{C_{\eta}}$ in ${\rm Gr}_{1}(J_{\eta})$. By spreading out, it follows
that $\kappa_{C_{f}}$ is $N_{0}$-torsion in ${\rm Gr}_{1}(J_{f})$ for all $f$
in an open dense subset $U_{0}\subset S_{0}$. By applying the same argument to
the generic points of the irreducible components of $S_{0}\setminus U_{0}$,
there exists an open dense $U_{1}\subset S_{0}$ containing $U_{0}$ and those
generic points such that $\kappa_{C_{f}}$ is $N_{1}$-torsion in ${\rm
Gr}_{1}(J_{f})$ for all $f$ in $U_{1}$. Repeating this process, we obtain a
sequence of open subsets $U_{0}\subset U_{1}\subset\cdots\subset S_{0}$ whose
complements have strictly decreasing codimension. Therefore this sequence must
terminate after finitely many steps hence there exists an integer $N$ such
that $\kappa_{C_{f}}$ is $N$-torsion in ${\rm Gr}_{1}(J_{f})$ for all $f\in
S_{0}(k)$ over every algebraically closed field of characteristic zero. ∎

The remainder of the section is devoted to proving Theorem 4.2. To this end,
we will analyze the Abel–Jacobi image of $\kappa_{f}$ (the “normal function”
associated to $\kappa_{f}$) in the next two subsections.

###  4.5. The Abel–Jacobi map

If $X$ is a smooth variety over ${\rm\mathbb{C}}$, we will use the notion of
an integral (respectively rational) variation of (pure) Hodge structures over
the complex manifold $X({\rm\mathbb{C}})$, called a ${\rm\mathbb{Z}}$-VHS
(respectively ${\rm\mathbb{Q}}$-VHS) for short; see [45, §5.3.1] for
definitions. If $V$ is a Hodge structure of odd weight $2k-1$, its
intermediate Jacobian $\mathrm{J}(V)$ is the complex torus

$\displaystyle(V\otimes_{{\rm\mathbb{Z}}}{\rm\mathbb{C}})/(F^{k}+V_{\tau}),$

where $F^{k}\subset V\otimes_{{\rm\mathbb{Z}}}{\rm\mathbb{C}}$ is a part of
the descending Hodge filtration and $V_{\tau}$ denotes the quotient of $V$ by
its torsion subgroup. More generally, if $\mathbb{V}$ is a
${\rm\mathbb{Z}}$-VHS of weight $2k-1$ over $X({\rm\mathbb{C}})$, we can
define its intermediate Jacobian $\mathrm{J}(\mathbb{V})\rightarrow
X({\rm\mathbb{C}})$, a relative complex torus whose fibers over points $x\in
X({\rm\mathbb{C}})$ are the classical intermediate Jacobians
$\mathrm{J}(\mathbb{V}_{x})$, see [45, §7.1.1]. If $\mathbb{V},\mathbb{W}$ are
two ${\rm\mathbb{Z}}$-VHS of odd weight, then
$\mathrm{J}(\mathbb{V}(p))=\mathrm{J}(\mathbb{V})$ for all
$p\in{\rm\mathbb{Z}}$ and a morphism $\mathbb{V}\rightarrow\mathbb{W}$ of
${\rm\mathbb{Z}}$-VHS induces a homomorphism of (relative) complex tori
$\mathrm{J}(\mathbb{V})\rightarrow\mathrm{J}(\mathbb{W})$.

If $X/{\rm\mathbb{C}}$ is a nice variety and $0\leq p\leq\dim(X)$, we write
$\mathrm{J}^{p}(X)=\mathrm{J}(\operatorname{H}^{2p-1}(X({\rm\mathbb{C}});{\rm\mathbb{Z}}))$.
In this situation there is an Abel–Jacobi map

(4.2) $\displaystyle\mathrm{AJ}^{p}_{X}\colon{\rm
CH}^{p}(X)_{\hom}\rightarrow\mathrm{J}^{p}(X),$

defined in [45, §7.2.1]. Moreover, if $S/{\rm\mathbb{C}}$ is a smooth variety,
$\pi\colon X\rightarrow S$ a smooth projective morphism with geometrically
integral fibres and $p\in{\rm\mathbb{Z}}_{\geq 0}$, then
$R^{p}\pi_{*}{\rm\mathbb{Z}}$ (pushforward of the constant sheaf in the
analytic topology) has the structure of a ${\rm\mathbb{Z}}$-VHS over
$S({\rm\mathbb{C}})$. If $Z$ is a codimension $p$ cycle on $X$ all of whose
components are flat over $S$, and such that $Z_{s}\in{\rm
CH}^{p}(X_{s})_{\hom}$ for every $s\in S({\rm\mathbb{C}})$, then Griffiths has
shown that there exists a holomorphic section ${\rm AJ}(Z)$ of the relative
complex torus $\mathrm{J}(R^{p}\pi_{*}{\rm\mathbb{Z}})\rightarrow
S({\rm\mathbb{C}})$ with the property that ${\rm AJ}(Z)_{s}={\rm
AJ}_{X}^{p}(Z_{s})$ for all $s\in S({\rm\mathbb{C}})$; this is called the
normal function associated to $Z$.

We record the fact that Abel–Jacobi maps are compatible with correspondences.
Let $X,Y$ be nice varieties over ${\rm\mathbb{C}}$ and let $\gamma\in{\rm
CH}^{r+\dim(X)}(X\times Y)$ be a correspondence of degree $r$. This induces
for every $p\geq 0$ a homomorphism $\gamma_{*}\colon{\rm
CH}^{p}(X)\rightarrow{\rm CH}^{p+r}(Y)$ via the formula
$\alpha\mapsto\pi_{Y,*}(\pi_{X}^{*}(\alpha)\cdot\gamma)$, where $\pi_{X}\colon
X\times Y\rightarrow X$ and $\pi_{Y}\colon X\times Y\rightarrow Y$ denote the
projections. The same formula defines morphism of Hodge structures
$\operatorname{H}^{p}(X;{\rm\mathbb{Z}})\rightarrow\operatorname{H}^{p+2r}(Y)(r)$
for every $p$, hence a homomorphism of complex tori
$\gamma_{*}\colon\mathrm{J}^{p}(X)\rightarrow\mathrm{J}^{p+r}(Y)$.

######  Lemma 4.11.

In the above notation, $\gamma_{*}$ sends ${\rm CH}^{p}(X)_{\hom}$ to ${\rm
CH}^{p+r}(Y)_{\hom}$. Moreover for every $\alpha\in{\rm CH}^{p}(X)_{\hom}$,
$\gamma_{*}({\rm AJ}_{X}^{p}(\alpha))={\rm AJ}_{Y}^{p+r}(\gamma_{*}(\alpha))$.

###### Proof.

The first sentence follows from the compatibility of the cycle class map with
correspondences [45, Proposition 9.21]. To prove the compatibility of the
Abel–Jacobi map with correspondences, it suffices to prove the compatibility
with pullbacks, pushforwards and intersection product. The case of pullback is
elementary, using the definition of ${\rm AJ}^{p}_{X}$ in terms of extensions
of Hodge structures, see [15, §2.2]. The case of pushforward follows from that
of pullback and Poincare duality. Finally, compatibility with intersection
product follows from [45, Proposition 9.23]. ∎

Let $S=\\{(a,b,c)\mid{\rm disc}(f)\neq
0\\}\subset{\rm\mathbb{A}}^{3}_{{\rm\mathbb{C}}}$ be the parameter space of
Picard curves over ${\rm\mathbb{C}}$. We will identify
${\rm\mathbb{C}}$-valued points of $S$ with polynomials
$f=x^{4}+ax^{2}+bx+c\in{\rm\mathbb{C}}[x]$ of nonzero discriminant. Let
$\mathcal{C}\rightarrow S$ be the universal Picard curve, and let
$\pi\colon\mathcal{J}\rightarrow S$ its relative Jacobian variety. The point
at infinity defines a section $P_{\infty}$ of $\mathcal{C}$. Let
$\kappa_{\mathcal{C}}\in{\rm CH}_{1}(\mathcal{J})$ be the universal Ceresa
cycle with respect to this section. (Comparing with our earlier notation, we
have $\mathcal{C}_{f}=C_{f}$ and $\kappa_{\mathcal{C},f}=\kappa_{f}$ for every
$f\in S({\rm\mathbb{C}})$.)

Let $\mathbb{V}=R^{3}\pi_{*}{\rm\mathbb{Z}}$ be the ${\rm\mathbb{Z}}$-VHS on
$S({\rm\mathbb{C}})$ interpolating the cohomology groups
$\operatorname{H}^{3}(\mathcal{J}_{f};{\rm\mathbb{Z}})$ for $f\in
S({\rm\mathbb{C}})$. Then the normal function ${\rm AJ}(\kappa_{\mathcal{C}})$
is a section of $\mathrm{J}(\mathbb{V})\rightarrow S({\rm\mathbb{C}})$
interpolating ${\rm AJ}_{J_{f}}^{2}(\kappa_{f})$.

######  Proposition 4.12.

For $f\in S({\rm\mathbb{C}})$, the class $\kappa_{f}\in{\rm
CH}^{2}(J_{f})_{\mathrm{hom}}$ is torsion if and only if ${\rm
AJ}_{J_{f}}^{2}(\kappa_{f})\in\mathrm{J}^{2}(J_{f})$ is torsion. The torsion
order of $\kappa_{f}$, if finite, equals the torsion order of ${\rm
AJ}_{J_{f}}^{2}(\kappa_{f})$.

###### Proof.

Let $E$ be the elliptic curve $y^{2}=x^{3}+1$. Corollary 4.9 shows that there
exists a correspondence $\gamma\in{\rm CH}^{2}(E\times J)$ such that
$\gamma_{*}$ induces isomorphisms ${\rm
CH}^{1}(E)_{{\rm\mathbb{Q}}}\rightarrow{\rm
CH}^{2}(J)^{\mu_{3}}_{{\rm\mathbb{Q}}}$ and
$\operatorname{H}^{1}(E;{\rm\mathbb{Q}})\rightarrow\operatorname{H}^{3}(J;{\rm\mathbb{Q}})^{\mu_{3}}(1)$.
By Lemma 4.11 these form a commutative diagram:

${{{\rm
CH}^{1}(E)_{\hom,{\rm\mathbb{Q}}}}}$${{\mathrm{J}^{1}(E)\otimes{\rm\mathbb{Q}}}}$${{{\rm
CH}^{2}(J_{f})^{\mu_{3}}_{\hom,{\rm\mathbb{Q}}}}}$${{\mathrm{J}^{2}(J_{f})^{\mu_{3}}\otimes{\rm\mathbb{Q}}}}$$\scriptstyle{{\rm
AJ}}$$\scriptstyle{\gamma_{*}}$$\scriptstyle{\gamma_{*}}$$\scriptstyle{{\rm
AJ}}$

All arrows except the horizontal bottom one are isomorphisms of abelian
groups. The same is therefore true for the bottom one ${\rm AJ}\colon{\rm
CH}^{2}(J_{f})^{\mu_{3}}_{\hom,{\rm\mathbb{Q}}}\rightarrow\mathrm{J}^{2}(J_{f})\otimes{\rm\mathbb{Q}}$.
Since $\kappa_{f}\in{\rm CH}^{2}(J_{f})_{\hom}^{\mu_{3}}$, we conclude that
$\kappa_{f}$ is torsion if and only if ${\rm AJ}_{J_{f}}^{2}(\kappa_{f})$ is.
The last claim follows from the fact that ${\rm AJ}^{2}_{J_{f}}$ is injective
on torsion subgroups by a result of Murre [37, Theorem 10.3]. ∎

###  4.6. Identifying the complex torus $\mathrm{J}(\mathbb{V}^{\mu_{3}})$

The $\mu_{3}$-action on $\mathcal{C}$ induces, via functoriality, a
$\mu_{3}$-action on $\mathcal{J}$, $\mathbb{V}$ and $\mathrm{J}(\mathbb{V})$.
The subsheaf of fixed points $\mathbb{V}^{\mu_{3}}$ has the structure of a
${\rm\mathbb{Z}}$-VHS. The connected component of the identity
$\mathrm{J}(\mathbb{V})^{\mu_{3},\circ}$ of $\mathrm{J}(\mathbb{V})^{\mu_{3}}$
is a relative complex torus over $S({\rm\mathbb{C}})$. Moreover, the natural
homomorphism
$\mathrm{J}(\mathbb{V}^{\mu_{3}})\rightarrow\mathrm{J}(\mathbb{V})$ induces an
isomorphism onto $\mathrm{J}(\mathbb{V})^{\mu_{3},\circ}$.

######  Lemma 4.13.

The multiple $3\cdot\mathrm{AJ}(\kappa_{\mathcal{C}})$ lands in
$\mathrm{J}(\mathbb{V})^{\mu_{3},\circ}$.

###### Proof.

Since the Ceresa cycle $\kappa_{\mathcal{C}}$ is $\mu_{3}$-invariant,
$\mathrm{AJ}(\kappa_{\mathcal{C}})$ lands in
$\mathrm{J}(\mathbb{V})^{\mu_{3}}$. The norm map
$\mathrm{N}\colon\mathrm{J}(\mathbb{V})\rightarrow\mathrm{J}(\mathbb{V})$,
defined by $x\mapsto x+\omega\cdot x+\omega^{2}\cdot x$, lands in
$\mathrm{J}(\mathbb{V})^{\mu_{3},\circ}$, since the image must be connected
and $\mu_{3}$-invariant. We conclude that
$\mathrm{N}(\mathrm{AJ}(\kappa_{\mathcal{C}}))=3\cdot\mathrm{AJ}(\kappa_{\mathcal{C}})$
lands in $\mathrm{J}(\mathbb{V})^{\mu_{3},\circ}$. ∎

Therefore $3\cdot\mathrm{AJ}(\kappa_{\mathcal{C}})$ defines a section of
$\mathrm{J}(\mathbb{V})^{\mu_{3},\circ}$, hence we may view it as a section of
the complex torus $\mathrm{J}(\mathbb{V}^{\mu_{3}})\rightarrow
S({\rm\mathbb{C}})$ in what follows. We study this relative complex torus (up
to isogeny) in the next two propositions.

######  Proposition 4.14.

Let $\mathcal{E}\rightarrow S$ be the relative elliptic curve with Weierstrass
equation

$\mathcal{E}\colon y^{2}=4x^{3}-27\cdot{\rm disc}(f).$

Then the relative complex torus $\mathrm{J}(\mathbb{V}^{\mu_{3}})\rightarrow
S$ is isogenous to the relative complex torus
$\mathcal{E}({\rm\mathbb{C}})\rightarrow S({\rm\mathbb{C}})$.

###### Proof.

Recall from §4.3 that we write $\mathcal{O}={\rm\mathbb{Z}}[\omega]$ and
$K={\rm\mathbb{Q}}(\omega)$. The ${\rm\mathbb{Z}}$-VHS
$R^{1}\pi_{*}{\rm\mathbb{Z}}$ interpolating the cohomology groups
$\operatorname{H}^{1}(\mathcal{J}_{f};{\rm\mathbb{Z}})$ comes equipped with an
action of $\mathcal{O}$, and cup product induces an isomorphism
$\bigwedge^{3}R^{1}\pi_{*}{\rm\mathbb{Z}}\simeq
R^{3}\pi_{*}{\rm\mathbb{Z}}=\mathbb{V}$. Let
$\mathbb{W}=\bigwedge^{3}_{\mathcal{O}}R^{1}\pi_{*}{\rm\mathbb{Z}}$ (the third
exterior product of $R^{1}\pi_{*}{\rm\mathbb{Z}}$, viewed as a sheaf of
$\mathcal{O}$-modules), a ${\rm\mathbb{Z}}$-VHS with an $\mathcal{O}$-action.
Lemma 4.6 shows that
$\mathbb{W}\otimes{\rm\mathbb{Q}}\simeq\mathbb{V}^{\mu_{3}}\otimes{\rm\mathbb{Q}}$,
so $\mathrm{J}(\mathbb{W})$ and $\mathrm{J}(\mathbb{V}^{\mu_{3}})$ are
isogenous.

To analyze $\mathrm{J}(\mathbb{W})$, we analyze the ${\rm\mathbb{Z}}$-VHS
$\mathbb{W}(1)$ more closely. It has an $\mathcal{O}$-action by construction,
Lemma 4.6 shows that it has constant rank $2$, and Lemma 4.5 shows that it has
type $(1,0)+(0,1)$. Since $\mathcal{O}$ has class number $1$, there exists a
unique ${\rm\mathbb{Z}}$-Hodge structure with these properties, hence
$\mathbb{W}_{f}(1)\simeq\operatorname{H}^{1}(E;{\rm\mathbb{Z}})$ for every
$f\in S({\rm\mathbb{C}})$, where $E$ is the elliptic curve with Weierstrass
equation $y^{2}=x^{3}+1$.

Therefore $\mathrm{J}(\mathbb{W})\rightarrow S({\rm\mathbb{C}})$ is an
isotrivial family of elliptic curves: analytically locally on
$S({\rm\mathbb{C}})$, it is isomorphic to $E({\rm\mathbb{C}})\times
S({\rm\mathbb{C}})\rightarrow S({\rm\mathbb{C}})$. It follows that the sheaf
of local isomorphisms between $\mathrm{J}(\mathbb{W})$ and
$E({\rm\mathbb{C}})\times S({\rm\mathbb{C}})$ is an ${\rm Aut}(E)$-torsor in
the analytic topology on $S({\rm\mathbb{C}})$. Since ${\rm
Aut}(E)\simeq\mu_{6}$ is finite, this torsor is the analytification of an
étale $\mu_{6}$-torsor on $S$.

To analyze étale $\mu_{6}$-torsors on $S$, consider the following exact
sequence induced by the Kummer exact sequence in étale cohomology:

$\displaystyle{\rm\mathbb{G}}_{m}(S)\xrightarrow{(-)^{6}}{\rm\mathbb{G}}_{m}(S)\rightarrow\operatorname{H}^{1}_{\mathrm{et}}(S,\mu_{6})\rightarrow{\rm
Pic}(S)[6]\rightarrow 0.$

The Picard group ${\rm Pic}(S)$ vanishes, being a quotient of ${\rm
Pic}({\rm\mathbb{A}}^{3}_{{\rm\mathbb{C}}})$, hence the outer term in the
sequence vanishes. We claim that
${\rm\mathbb{G}}_{m}(S)=\operatorname{H}^{0}(S,\mathcal{O}_{S})^{\times}=\\{c\cdot{\rm
disc}^{n}\mid c\in{\rm\mathbb{C}}^{\times},\,n\in{\rm\mathbb{Z}}\\}$. Indeed,
every $c\cdot{\rm disc}^{n}$ is clearly a unit in
$\operatorname{H}^{0}(S,\mathcal{O}_{S})$. Conversely, given a unit
$f\in\operatorname{H}^{0}(S,\mathcal{O}_{S})$, seen as a rational function on
${\rm\mathbb{A}}^{3}_{{\rm\mathbb{C}}}$, its divisor $\mathrm{div}(f)$ of
zeros and poles must be supported on the zero locus of ${\rm disc}$. Since
${\rm disc}\in{\rm\mathbb{C}}[a,b,c]$ is irreducible,
$\mathrm{div}(f)=n\cdot[\\{{\rm disc}=0\\}]$ for some $n\in{\rm\mathbb{Z}}$.
Then $\mathrm{div}(f/{\rm disc}^{n})=0$ as a rational function on
${\rm\mathbb{A}}^{3}_{{\rm\mathbb{C}}}$, hence $f/{\rm disc}^{n}$ is a unit in
${\rm\mathbb{C}}[a,b,c]$, hence $f/{\rm disc}^{n}\in{\rm\mathbb{C}}^{\times}$,
proving the claim. We conclude that the group
$\operatorname{H}^{1}_{\mathrm{et}}(S,\mu_{6})$ classifying $\mu_{6}$-torsors
is generated by the image of ${\rm disc}$.

Let $\mathcal{E}_{i}\rightarrow S$ be the relative elliptic curve with
equation $y^{2}=x^{3}+{\rm disc}(f)^{i}$. The previous paragraph shows that
$\mathrm{J}(\mathbb{W})$ is isomorphic to $\mathcal{E}_{i}({\rm\mathbb{C}})$
for some $i\in\\{0,1,2,3,4,5\\}$. We show that $i=1$, using our previous
results on _bielliptic_ Picard curves in [27]. Let $T\subset S$ be the closed
subscheme where $b=0$, parametrizing even quartic polynomials
$f=x^{4}+ax^{2}+c$. For $f\in T({\rm\mathbb{C}})$, the $\mu_{3}$-action on
$\mathcal{C}_{f}$ extends to a $\mu_{6}$-action. A calculation shows ${\rm
disc}|_{T}=16c(-a^{2}+4c)^{2}$. Applying the singular cohomology realization
functor to [27, Theorem 5.1], the relative complex torus
$\mathrm{J}((\mathbb{V}|_{T({\rm\mathbb{C}})})^{\mu_{6}})\rightarrow
T({\rm\mathbb{C}})$ is isogenous to
$\mathcal{E}_{1}({\rm\mathbb{C}})|_{T({\rm\mathbb{C}})}\rightarrow
T({\rm\mathbb{C}})$. Since
$\mathrm{J}((\mathbb{V}|_{T({\rm\mathbb{C}})})^{\mu_{6}})$ is a subtorus of
$\mathrm{J}((\mathbb{V}|_{T({\rm\mathbb{C}})})^{\mu_{3}})$ of the same
dimension, they must be equal, hence
$\mathrm{J}(\mathbb{W})|_{T({\rm\mathbb{C}})}$ is isogenous to
$\mathcal{E}_{1}({\rm\mathbb{C}})|_{T({\rm\mathbb{C}})}$ over
$T({\rm\mathbb{C}})$. On the other hand, let $i\in\\{0,1,2,3,4,5\\}$ be such
that $\mathrm{J}(\mathbb{W})\simeq\mathcal{E}_{i}({\rm\mathbb{C}})$. Then
$\mathrm{J}(\mathbb{W})|_{T({\rm\mathbb{C}})}\simeq\mathcal{E}_{i}({\rm\mathbb{C}})|_{T({\rm\mathbb{C}})}$,
hence $\mathcal{E}_{1}({\rm\mathbb{C}})|_{T({\rm\mathbb{C}})}$ is isogenous to
$\mathcal{E}_{i}({\rm\mathbb{C}})|_{T({\rm\mathbb{C}})}$.

We show that the latter can happen only if $i=1$. Indeed, let
$\varphi\colon\mathcal{E}_{1}({\rm\mathbb{C}})|_{T({\rm\mathbb{C}})}\rightarrow\mathcal{E}_{i}({\rm\mathbb{C}})|_{T({\rm\mathbb{C}})}$
be an isogeny. Since the domain and target of $\varphi$ are isotrivial
relative elliptic curves with $\mathcal{O}$-multiplication, $\varphi$ factors
as $\psi\circ\gamma$, where $\gamma$ is an endomorphism of
$\mathcal{E}_{1}({\rm\mathbb{C}})|_{T({\rm\mathbb{C}})}$ and $\psi$ an
isomorphism. Therefore
$\mathcal{E}_{1}({\rm\mathbb{C}})|_{T({\rm\mathbb{C}})}$ and
$\mathcal{E}_{i}({\rm\mathbb{C}})|_{T({\rm\mathbb{C}})}$ are isomorphic. Since
the monodromy representations of
$\mathcal{E}_{1}({\rm\mathbb{C}})|_{T({\rm\mathbb{C}})}$ and
$\mathcal{E}_{i}({\rm\mathbb{C}})|_{T({\rm\mathbb{C}})}$ are non-isomorphic
when $i\neq 1$, we conclude that $i=1$ and that $\mathrm{J}(\mathbb{W})$ is
isogenous to $\mathcal{E}_{1}({\rm\mathbb{C}})$ over $S({\rm\mathbb{C}})$.

In summary, we have shown that $\mathrm{J}(\mathbb{V}^{\mu_{3}})$,
$\mathrm{J}(\mathbb{W})$ and $\mathcal{E}_{1}({\rm\mathbb{C}})$ are isogenous.
Since $\mathcal{E}_{1}({\rm\mathbb{C}})$ is isomorphic to
$\mathcal{E}({\rm\mathbb{C}})$, we conclude the proof. ∎

######  Proposition 4.15.

The group of (algebraic) sections of $\mathcal{E}\rightarrow S$ is free of
rank $1$ over $\mathcal{O}={\rm\mathbb{Z}}[\omega]$, and contains the
$\mathcal{O}$-span of $P:=(a^{2}+12c,72ac-2a^{3}-27b^{2})$ as a finite index
subgroup.

###### Proof.

Given $s=(a,b,c)\in S({\rm\mathbb{C}})$, consider the closed subscheme
$T=\\{(at,bt,ct^{2})\colon t\in{\rm\mathbb{A}}^{1}_{{\rm\mathbb{C}}}\\}\cap S$
of $S$ and the restriction $\pi\colon\mathcal{E}|_{T}\rightarrow T$. The
variety $\mathcal{E}|_{T}$ is an open subscheme of an elliptic surface with
Weierstrass equation $y^{2}=x^{3}+t^{4}g(t)$, where
$g(t)\in{\rm\mathbb{C}}[t]$ has degree $\leq 2$, using the formula (4.1).
There exists a dense open $U\subset S$ such that for all $s\in
U({\rm\mathbb{C}})$, $g(t)$ has two distinct nonzero roots. Let $s\in
U({\rm\mathbb{C}})$. Using Tate’s algorithm, we see that the elliptic surface
has a singular fiber above $t=0$ with Kodaira type $\text{\rm IV}^{*}$,
singular fibers above the roots of $g(t)$ with Kodaira type II, and is smooth
above $t=\infty$. The presence of a fiber of type II implies that
$\mathcal{E}|_{T}\rightarrow T$ has no torsion sections [43, Lemma 7.8].
Moreover the Shioda–Tate formula [43, Theorem 6.3, Proposition 6.6 and §8.8]
shows that the group of sections of $\mathcal{E}|_{T}\rightarrow T$ is free of
rank $2$. Since $\mathcal{E}|_{T}$ receives an $\mathcal{O}$-action, its group
of sections is free of rank $1$ over $\mathcal{O}$. We will now show that
these facts can be used to prove the claims of the proposition by varying $s$
in $U({\rm\mathbb{C}})$.

We first show that $\mathcal{E}(S)$ is torsion-free. Suppose
$Q\in\mathcal{E}(S)$ is a torsion section. Then $N\cdot Q=0$ for some $N\geq
1$. Hence $N\cdot Q|_{T}=0$ for every $s\in U({\rm\mathbb{C}})$. Since the
group of sections of $\mathcal{E}|_{T}\rightarrow T$ is torsion-free,
$Q|_{T}=0$ for every $s\in U({\rm\mathbb{C}})$. Hence $Q|_{U}=0$. Since $U$ is
dense in $S$, we must have $Q=0$, as desired.

The fact that $P$ defines a section of $\mathcal{E}\rightarrow S$ can be
verified by direct computation (see §4.1). Let $L={\rm\mathbb{Z}}\langle
P,\omega\cdot P\rangle$ be the $\mathcal{O}$-span of $P$, which is a subgroup
of the group of sections $\mathcal{E}(S)$ of $\mathcal{E}\rightarrow S$. The
previous paragraph shows that $L$ is free of rank $1$ over $\mathcal{O}$. It
remains to show that it has finite index in $\mathcal{E}(S)$. Suppose for the
sake of contradiction that there exists a third section $Q\in\mathcal{E}(S)$
which is not in $L\otimes{\rm\mathbb{Q}}$. Then the locus $S_{\text{dep}}$ of
$s\in S({\rm\mathbb{C}})$ where $P_{s},\omega\cdot P_{s}$ and $Q_{s}$ are
linearly dependent is a countable union of closed proper algebraic
subvarieties. For every $s\in U({\rm\mathbb{C}})$, the group of sections of
$\mathcal{E}|_{T}\rightarrow T$ is free of rank $1$ over $\mathcal{O}$. There
exists a possibly smaller dense open $V\subset U$ such that if $s\in
V({\rm\mathbb{C}})$ then $P_{s}$ is nonzero, hence $\langle P_{s},\omega\cdot
P_{s}\rangle$ is a finite index subgroup of $\mathcal{E}(T)$. Therefore
$Q_{t},P_{1,t},P_{2,t}$ are linearly dependent for all $s\in
V({\rm\mathbb{C}})$. Since $V({\rm\mathbb{C}})\setminus S_{\text{dep}}$ is
nonempty, we obtain a contradiction. ∎

###  4.7. Proof of the vanishing criterion in the Chow group

###### Proof of Theorem 4.2.

We may assume (using Lemma 2.1) that $k={\rm\mathbb{C}}$. Recall that
$3\cdot{\rm AJ}(\kappa_{\mathcal{C}})$ defines a holomorphic section of
$\mathrm{J}(\mathbb{V}^{\mu_{3}})\rightarrow S({\rm\mathbb{C}})$. Choose an
isogeny of complex tori
$\mathrm{J}(\mathbb{V}^{\mu_{3}})\rightarrow\mathcal{E}({\rm\mathbb{C}})$
using Proposition 4.14 and let $\sigma$ be the image of $3\cdot{\rm
AJ}(\kappa_{\mathcal{C}})$ under this isogeny. This is a holomorphic section
of $\mathcal{E}({\rm\mathbb{C}})\rightarrow S({\rm\mathbb{C}})$.

We claim that $\sigma$ is not a torsion section. If it were torsion, then
${\rm AJ}(\kappa_{\mathcal{C}})$ would be a torsion section of
$\mathrm{J}(\mathbb{V}^{\mu_{3}})$, hence, by Proposition 4.12, $\kappa_{f}$
would be torsion for every $f\in S({\rm\mathbb{C}})$. This is not the case,
since $\kappa_{f}$ is of infinite order if $f=x^{4}+x^{2}+1$ by [27, Corollary
2.9]. We conclude that $\sigma$ is not a torsion section.

Next we claim that $\sigma$ is the analytification of an algebraic section of
$\mathcal{E}\rightarrow S$. Let $N\geq 1$ be an integer such that
$N\cdot\kappa_{f}$ is algebraically trivial for every Picard curve $f\in
S({\rm\mathbb{C}})$ (such an integer exists by Theorem 4.1). By the
algebraicity of the Abel–Jacobi map for algebraically trivial cycles ([1,
Theorem 1]), there exists a relative algebraic subtorus
$\mathrm{J}_{a}(\mathbb{V})\subset\mathrm{J}(\mathbb{V})$ with the following
property: the section $\mathrm{AJ}(3N\cdot\kappa_{\mathcal{C}})$ lands in
$\mathrm{J}_{a}(\mathbb{V})$ and the corresponding holomorphic map
$S({\rm\mathbb{C}})\rightarrow\mathrm{J}_{a}(\mathbb{V})$ is algebraic. On the
other hand, ${\rm AJ}(3N\cdot\kappa_{\mathcal{C}})$ also lands in
$\mathrm{J}(\mathbb{V}^{\mu_{3}})$, is not a torsion section by the previous
paragraph, and $\mathrm{J}(\mathbb{V}^{\mu_{3}})$ has relative dimension $1$
over $S({\rm\mathbb{C}})$. Therefore $\mathrm{J}(\mathbb{V}^{\mu_{3}})$ is the
smallest relative subtorus of $\mathrm{J}(\mathbb{V})$ containing the image of
${\rm AJ}(3N\cdot\kappa_{\mathcal{C}})$. Hence
$\mathrm{J}(\mathbb{V}^{\mu_{3}})\subset\mathrm{J}_{a}(\mathbb{V})$. We
conclude that ${\rm AJ}(3N\cdot\kappa_{\mathcal{C}})\colon
S({\rm\mathbb{C}})\rightarrow\mathrm{J}(\mathbb{V}^{\mu_{3}})$ is algebraic,
so ${\rm AJ}(3\cdot\kappa_{\mathcal{C}})$ is algebraic, so $\sigma$ is
algebraic.

Since $\sigma$ is an algebraic section of $\mathcal{E}\rightarrow S$,
Proposition 4.15 shows that there exists an integer $M\geq 1$ and an element
$\gamma\in\mathcal{O}$ such that $M\cdot\sigma=\gamma\cdot P$. Since $\sigma$
is not a torsion section, $\gamma\neq 0$.

Putting everything together, we have for $f\in S({\rm\mathbb{C}})$:
$\kappa_{f}$ is torsion if and only if ${\rm AJ}_{J_{f}}^{2}(\kappa_{f})$
torsion (by Proposition 4.12), if and only if ${\rm
AJ}(\kappa_{\mathcal{C}})_{f}\in\mathrm{J}(\mathbb{V}^{\mu_{3}})_{f}$ torsion,
if and only if $\sigma_{f}\in\mathcal{E}_{f}({\rm\mathbb{C}})$ torsion, if and
only if $P_{f}$ torsion. Tracing through the equivalences, the quotient of the
torsion orders $\operatorname*{ord}(\kappa_{f})/\operatorname*{ord}(P_{f})$
(if defined) takes only finitely many values as $f$ ranges in
$S({\rm\mathbb{C}})$. ∎

###  4.8. Families of Picard curves with torsion Ceresa cycle of arbitrarily
large order

Theorem 4.2 shows that there are infinitely many plane quartic curves over
${\rm\mathbb{Q}}$ with torsion Ceresa cycle, since we may take $a=c=-12$. In
fact, we can find explicit families of torsion Ceresa cycles of arbitrarily
large order by analyzing the function $f\mapsto(I(f),J(f))$.

Let $C_{f}$ be a Picard curve defined by the polynomial
$f(x)=x^{4}+ax^{2}+bx+c$ over a field $k$ with homogenization
$F(x,z)=x^{4}+ax^{2}z^{2}+bxz^{3}+cz^{4}$. Associated to $C_{f}$ is the genus
$1$ curve $D_{f}$ with equation $y^{2}=F(x,z)$. Viewing $D_{f}$ as an elliptic
curve with origin $(1:1:0)$, it is isomorphic to the elliptic curve
$y^{2}=x^{3}-I(f)x/3-J(f)/27$. The point $(1:-1:0)$ is sent to $(-2a/3,b)$ in
this new model. Conversely, given a nonzero $k$-point $(\alpha,\beta)$ on the
elliptic curve $E_{I,J}\colon y^{2}=x^{3}-Ix/3-J/27$, the quartic

(4.3) $\displaystyle f(x)=x^{4}-3\alpha x^{2}/2+\beta x+(I/12-3\alpha^{2}/16)$

has the property that $(I(f),J(f))=(I,J)$. It follows that if $(I,J)\in k$ is
such that $4J^{3}-J^{2}\neq 0$, then $f\mapsto(-2a/3,b)$ induces a bijection
between the set of polynomials $f=x^{4}+ax^{2}+bx+c\in k[x]$ satisfying
$(I(f),J(f))=(I,J)$, and the set of nonzero elements of $E_{I,J}(k)$. Using
this description, we can cook up explicit 1-parameter families of torsion
Ceresa cycles of arbitrarily large order, as in the following corollary of
Theorem 4.2:

######  Corollary 4.16.

For every integer $N$, there exists a number field $k=k(N)$ such that there
exists infinitely many $\bar{\rm\mathbb{Q}}$-isomorphism classes Picard curves
$C_{f}$ defined over $k$ with Ceresa cycle $\kappa_{f}$ of order at least $N$
in ${\rm CH}^{2}(J_{f,\bar{k}})$.

###### Proof.

Let $M$ be an integer satisfying the conclusion of Theorem 4.2. Fix a nonzero
$D\in{\rm\mathbb{Q}}$ and let $(I,J)\in\bar{{\rm\mathbb{Q}}}^{2}$ be a torsion
point of order $M\cdot N$ on the elliptic curve $y^{2}=4x^{3}+D$. Let $k$ be
the field of definition of $(I,J)$. For $t\in k$, define
$\lambda(t)=t^{3}-It/3-J/27$,
$(\alpha_{t},\beta_{t})=(t\lambda(t),\lambda(t)^{2})$ and the quartic
$f_{t}(x)=x^{4}-3\alpha_{t}x^{2}/2+\beta_{t}x+(I/12-3\alpha_{t}^{2}/16)$ using
Formula (4.3). Since $(\alpha_{t},\beta_{t})$ defines a point on the elliptic
curve $E_{\lambda(t)^{2}I,\lambda(t)^{3}J}$, the quartic $f$ has the property
that $(I({f_{t}}),J({f_{t}}))=(\lambda(t)^{2}I,\lambda(t)^{3}J)$. If $t\in k$
is such that $\lambda(t)\neq 0$, then $(\lambda(t)^{2}I,\lambda(t)^{3}J)$ is
an $N$-torsion point on the elliptic curve $y^{2}=4x^{3}+\lambda_{t}^{6}D$. By
Theorem 4.2, $\kappa_{f_{t}}$ is torsion in ${\rm CH}^{2}(J_{f_{t},\bar{k}})$
and its torsion order $\operatorname*{ord}(\kappa_{f_{t}})$ is divisible by
$N$. When varying over $t\in k$ such that $\lambda(t)\neq 0$, the Picard
curves $C_{f_{t}}$ cover infinitely many $\bar{\rm\mathbb{Q}}$-isomorphism
classes, using [13, Lemma 1.21(b)] and the fact that
$\beta_{t}^{2}/\alpha_{t}^{3}=\lambda(t)/t^{3}$ is not a constant function of
$t$. ∎

Conversely, for any number field $k$, the order of a torsion Ceresa cycle
$\kappa(C_{f})$, with $C_{f}$ defined over $k$, is bounded, with the bound
depending only on the degree of $k$ over ${\rm\mathbb{Q}}$:

######  Corollary 4.17.

For every $d\geq 1$, there exists $N=N(d)\geq 1$ such that for every Picard
curve $C_{f}$ over a number field $k$ of degree $d$, the order of $\kappa_{f}$
in ${\rm CH}_{1}(J_{f,\bar{k}})$ is either infinite or less than $N$.

###### Proof.

This follows from Theorem 4.2 and the uniform bound (depending only on $d$) on
the order of a $k$-rational torsion point on an elliptic curve $y^{2}=x^{3}+D$
over any number field $k$ of degree $d$. ∎

##  5\. Automorphism strata and Ceresa vanishing loci in genus $3$

Fix $g\geq 3$ and let $\mathcal{M}_{g}$ be the (coarse) moduli space of
genus-$g$ curves, seen as a variety over ${\rm\mathbb{Q}}$. Let
$V_{g}^{\mathrm{rat}}\subset\mathcal{M}_{g}$ be the subset of curves $[C]$ for
which $\kappa(C)$ vanishes in ${\rm
CH}_{1}(\operatorname{Jac}(C))_{{\rm\mathbb{Q}}}$, in the notation of §2.7.
Since the vanishing of $\kappa(C)$ only depends on the geometric isomorphism
class of $C$ (by Lemma 2.1), this locus is well defined. Similarly define the
locus $V_{g}^{\mathrm{alg}}\subset\mathcal{M}_{g}$ where
$\bar{\kappa}(C)\in{\rm Gr}_{1}(\operatorname{Jac}(C))_{{\rm\mathbb{Q}}}$
vanishes.

######  Lemma 5.1.

The subsets $V_{g}^{\mathrm{rat}},V_{g}^{\mathrm{alg}}\subset\mathcal{M}_{g}$
are countable unions of proper closed algebraic subvarieties.

###### Proof.

Let $\widetilde{\mathcal{M}}_{g}$ be the fine moduli space parametrizing
genus-$g$ curves $C$ with full symplectic level-$5$ structure and a degree-$1$
divisor class $e\in{\rm CH}_{0}(C)$ such that $(2g-2)e$ is canonical.
Considering the universal curve over it together with its degree-$1$ divisor,
we can define a universal Ceresa cycle on the universal Jacobian over
$\widetilde{\mathcal{M}}_{g}$; Lemma 2.4 then implies that the locus in
$\widetilde{\mathcal{M}}_{g}$ where this Ceresa cycle vanishes (with
${\rm\mathbb{Q}}$-coefficients) is a countable union of closed algebraic
subvarieties. Since the forgetful map
$\widetilde{\mathcal{M}}_{g}\rightarrow\mathcal{M}_{g}$ is proper, the same is
true for the image of this locus, which is exactly $V_{g}^{\mathrm{rat}}$. The
proof for $V_{g}^{\mathrm{alg}}$ is identical. ∎

These vanishing loci have the following basic properties:
$V_{g}^{\mathrm{rat}}\subset V_{g}^{\mathrm{alg}}\subset\mathcal{M}_{g}$; the
hyperelliptic locus is contained in $V_{g}^{\mathrm{rat}}$; and
$V_{g}^{\mathrm{alg}}\neq\mathcal{M}_{g}$ by Ceresa’s famous result [14]. It
would be interesting to obtain further information about the components of
$V_{g}^{\mathrm{rat}}$ and $V_{g}^{\mathrm{alg}}$. We end our paper by
determining the automorphism group strata in $\mathcal{M}_{3}$ that are
contained in $V_{3}^{\mathrm{rat}}$ or $V_{3}^{\mathrm{alg}}$.

So let $g=3$ and consider the open subscheme
$\mathcal{M}_{3}^{\mathrm{nh}}\subset\mathcal{M}_{3}$ of non-hyperelliptic
curves. There is a stratification $\mathcal{M}_{3}^{\mathrm{nh}}=\sqcup X_{G}$
into locally closed subvarieties such that a non-hyperelliptic curve $C$ over
${\rm\mathbb{C}}$ belongs to $X_{G}$ if and only if ${\rm Aut}(C)\simeq G$. It
turns out that $X_{G}$ is irreducible and the closure of $X_{G}$ is a union of
other strata. We refer to [31, §2.2] and references therein for a complete
description of the loci $X_{G}$ and the closure relations between them. We
reproduce here a diagram capturing these closure relations:

$\\{\operatorname{Id}\\}$$C_{2}$$C_{2}^{2}$$C_{3}$$D_{4}$$S_{3}$$C_{6}$$G_{16}$$S_{4}$$C_{9}$$G_{48}$$G_{96}$${\rm
GL}_{3}({\rm\mathbb{F}}_{2})$$\dim=6$$\dim=4$$\dim=3$$\dim=2$$\dim=1$$\dim=0$

For $n\in\\{16,48,96\\}$, the symbol $G_{n}$ the group of order $n$ and GAP
label $(16,13),(48,33),$ and $(96,64)$ respectively [21]. See [31, Table 2]
for models for a generic plane quartic in $X_{G}$. We make explicit the strata
that are relevant for us: $\overline{X_{C_{3}}}$ is the locus of Picard curves
studied in §4; $\overline{X_{C_{6}}}$ is the locus of _bielliptic_ Picard
curves studied in [28]; and the zero-dimensional strata each consist of a
single automorphism-maximal curve with equation

$\displaystyle\begin{cases}y^{3}z=x^{4}+xz^{3}&\text{ if }G=C_{9},\\\
y^{3}z=x^{4}+z^{4}&\text{ if }G=G_{48,}\\\ x^{4}+y^{4}+z^{4}=0&\text{ if
}G=G_{96},\\\ x^{3}y+y^{3}z+z^{3}x&\text{ if }G={\rm
GL}_{3}({\rm\mathbb{F}}_{2}).\end{cases}$

If $C$ is a non-hyperelliptic genus $3$ curve over a field $k$, we say $C$ is
a generic curve for $X_{G}$ if the classifying map ${\rm
Spec}(k)\rightarrow\mathcal{M}_{3}$ maps to the generic point of $X_{G}$. If
$X$ is an integral variety over ${\rm\mathbb{C}}$, we say a property hold for
a very general $x\in X({\rm\mathbb{C}})$ if it holds true outside a countable
union of proper closed subvarieties of $X$.

######  Lemma 5.2.

For a group $G$ in the above diagram, the following are equivalent:

  1. (1)

$\kappa(C)\neq 0$ for some generic curve $C$ for $X_{G}$.

  2. (2)

$\kappa(C)\neq 0$ for a very general $C$ in $X_{G}({\rm\mathbb{C}})$.

  3. (3)

$\kappa(C)\neq 0$ for some $C$ in $X_{G}$.

  4. (4)

$X_{G}\not\subset V_{3}^{\mathrm{rat}}$.

  5. (5)

$\overline{X_{G}}\not\subset V_{3}^{\mathrm{rat}}$.

Moreover, the analogous equivalences hold for $\bar{\kappa}(C)\in{\rm
Gr}_{1}(J)_{\rm\mathbb{Q}}$ and $V_{3}^{\mathrm{alg}}$.

###### Proof.

Follows immediately from Lemma 5.1. ∎

We end with the proof of Theorem D, which we restate for convenience:

######  Proposition 5.3.

Let $G$ be a group in the diagram. Then

  1. (1)

$X_{G}\subset V_{3}^{\mathrm{alg}}$ if and only if
$G=C_{3},C_{6},C_{9},G_{48}$.

  2. (2)

$X_{G}\subset V_{3}^{\mathrm{rat}}$ if and only if $G=C_{9},G_{48}$;

###### Proof.

  1. (1)

The equivalence between $(4)$ and $(5)$ of Lemma 5.2 implies that if
$X_{H}\subset\overline{X_{G}}$ and $X_{G}\subset V_{3}^{\mathrm{rat}}$, then
$X_{H}\subset V_{3}^{\mathrm{rat}}$; we will use this observation in the
remainder of the proof.

Our analysis of Picard curves (Theorem C) shows that $X_{C_{3}}\subset
V_{3}^{\mathrm{rat}}$, so $X_{G}\subset V_{3}^{\mathrm{rat}}$ for
$G=C_{6},C_{9}$ and $G_{48}$ as well. On the other hand, the Ceresa cycle of
the Fermat quartic and Klein quartic are known to be of infinite order in the
Griffiths group; see [11, Theorem (4.1)] for the former and [24, §4] for the
latter. By our observation, this means that $X_{G}\not\subset
V_{3}^{\mathrm{rat}}$ for every stratum whose closure contains one of these
curves. Since every stratum not contained in $\overline{X_{C_{3}}}$ has this
property, we conclude the proof.

  2. (2)

Since $V_{3}^{\mathrm{rat}}\subset V_{3}^{\mathrm{alg}}$, Part $(1)$ implies
that $X_{G}\subset V_{3}^{\mathrm{rat}}$ only if $G=C_{3},C_{6},C_{9}$ or
$G_{48}$. The criterion of Theorem A applies to the curves in $X_{C_{9}}$ and
$X_{G_{48}}$ (see Example 3.1), so $V_{3}^{\mathrm{rat}}$ contains these
strata. On the other hand, there exist curves in $X_{C_{6}}$ with nonvanishing
$\kappa(C)$, by [27, Corollary 1.2]. So $X_{C_{6}}\not\subset
V_{3}^{\mathrm{rat}}$ and $X_{C_{3}}\not\subset V_{3}^{\mathrm{rat}}$.

∎

######  Remark 5.4.

Not every irreducible component of $V_{3}^{\mathrm{rat}}$ is of the form
$\overline{X_{G}}$ for some $G$. The proof of Corollary 4.16 shows that
$V_{3}^{\mathrm{rat}}\cap\overline{X_{C_{3}}}$ is a union of countably many
(open, possibly singular) rational curves.

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