# Ceresa cycles of bielliptic Picard curves

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.

We show that the Ceresa cycle $\kappa(C_{t})$ of the genus $3$ curve
$C_{t}\colon y^{3}=x^{4}+2tx^{2}+1$ is torsion if and only if
$Q_{t}=(\sqrt[3]{t^{2}-1},t)$ is a torsion point on the elliptic curve
$y^{2}=x^{3}+1$. This shows that there are infinitely many smooth plane
quartic curves over ${\rm\mathbb{C}}$ (resp. ${\rm\mathbb{Q}}$) with torsion
(resp. infinite order) Ceresa cycle. Over $\overline{{\rm\mathbb{Q}}}$, we
show that the Beilinson–Bloch height of $\kappa(C_{t})$ is proportional to the
Neron–Tate height of $Q_{t}$. Thus, the height of $\kappa(C_{t})$ is
nondegenerate and satisfies a Northcott property. To prove all this, we show
that the Chow motive that controls $\kappa(C_{t})$ is isomorphic to
$\mathfrak{h}^{1}$ of an appropriate elliptic curve.

###### Contents

  1. 1 Introduction
  2. 2 Ceresa and modified diagonal cycles
  3. 3 Bielliptic Picard curves
  4. 4 Background on Chow motives
  5. 5 Proof of main theorem
  6. 6 Algebraic triviality and the Griffiths group

##  1\. Introduction

Let $C$ be a smooth, projective, geometrically integral curve of genus $g\geq
2$ over an algebraically closed field $k$. We embed $C$ in its Jacobian $J$
using a $(2g-2)$-th root of the canonical bundle. The (canonical) Ceresa cycle
of $C$ is the class

$\kappa(C)=[C]-(-1)^{*}[C]\in{\rm CH}_{1}(J)$

in the Chow group of $1$-cycles on $J$ with
${\rm\mathbb{Z}}$-coefficients.111This depends on our choice of $(2g-2)$-th
root, but only up to a $(2g-2)$-torsion class; see Lemma 2.1. While
$\kappa(C)$ is homologically trivial, Ceresa [12] famously showed that it need
not be trivial in ${\rm CH}_{1}(J)$. Indeed, if $k={\rm\mathbb{C}}$ then the
Ceresa cycle of a very general genus $g\geq 3$ curve has infinite order. On
the other hand, if $C$ is hyperelliptic then $\kappa(C)=0$. Beyond this, very
little is known about the set of genus $g$ curves with vanishing or torsion
Ceresa cycle, though see §1.4 for some recent results and examples.

In this paper, we study the Ceresa cycles of bielliptic Picard curves over
fields of characteristic $\neq 2,3$ [28]. As $k$ is algebraically closed, such
a curve has an affine model of the form

(1.1) $\displaystyle y^{3}=x^{4}+2tx^{2}+1,$

for some $t\in k\setminus\\{\pm 1\\}$, unique up to replacing $t$ with $-t$.
Our main result shows that the Ceresa cycles of these curves are governed by
points on the single elliptic curve $\widehat{E}\colon y^{2}=x^{3}+1$.

######  Theorem 1.1.

Let $C_{t}$ be the smooth projective curve with model (1.1), and let $Q_{t}$
be the point $(\sqrt[3]{t^{2}-1},t)\in\widehat{E}(k)$, for some choice of cube
root. Then $\kappa(C_{t})$ is torsion in ${\rm CH}_{1}(J_{t})$ if and only if
$Q_{t}$ is torsion in $\widehat{E}(k)$.

Theorem 1.1 (which is proved in §5.5) appears to be the first characterization
of torsion Ceresa cycles in a positive-dimensional family of curves, aside
from families where $\kappa(C)$ is identically torsion. We deduce from this
result the following corollary (see §5.6).

######  Corollary 1.2.

There are infinitely many ${\rm\overline{\mathbb{Q}}}$-isomorphism classes of
genus $3$ curves over ${\rm\mathbb{Q}}$ with infinite order Ceresa cycle. In
fact, $\kappa(C_{t})$ has infinite order for all
$t\in{\rm\mathbb{Q}}\setminus\\{0,\pm 1,\pm 3\\}$.

Corollary 1.2 is not surprising given Ceresa’s result, but it seems to have
been open until now and is still open in higher genus, as far as we know.

On the other hand, since $\widehat{E}$ has infinitely many torsion points, we
also deduce from Theorem 1.1 that there are infinitely many plane quartic
curves over ${\rm\mathbb{C}}$ with torsion Ceresa cycle. In fact, our proof
shows that there is an algebraic correspondence from $J_{t}$ to $\widehat{E}$
that sends $\kappa(C_{t})$ to a constant multiple of $Q_{t}$. This allows us
to deduce the following stronger statement:

######  Theorem 1.3.

There exist infinitely many plane quartic curves over ${\rm\mathbb{C}}$ with
torsion Ceresa cycle. Moreover, their orders are unbounded.

Note that it was only recently discovered that there exist any plane quartic
curve with torsion Ceresa cycle: Qiu–Zhang [37, §4.1] have shown that
$C_{0}\colon y^{3}=x^{4}+1$ has this property.

Interestingly, all known examples in Theorem 1.3 can be defined over
${\rm\overline{\mathbb{Q}}}$; for the curves $C_{t}$, this follows immediately
from Theorem 1.1. For $t\in{\rm\overline{\mathbb{Q}}}$, we write
$h(\kappa(C_{t}))$ for the Beilinson–Bloch height
$\langle\kappa(C_{t}),\kappa(C_{t})\rangle$ [6, 27] of $\kappa(C_{t})$. We
show that $h(\kappa(C_{t}))$ is proportional to the Neron–Tate height of
$Q_{t}$ (Theorem 5.19), which implies the following nondegeneracy and
Northcott properties.

######  Theorem 1.4.

We have $h(\kappa(C_{t}))=0$ if and only if $\kappa(C_{t})$ is torsion.
Moreover, for any $X\in{\rm\mathbb{R}}$ and $d\in{\rm\mathbb{N}}$, the set

$\left\\{t\in{\rm\overline{\mathbb{Q}}}\setminus\\{\pm 1\\}\colon
h(\kappa(C_{t}))\leq X\hskip 8.53581pt\mbox{and}\hskip
8.53581pt[{\rm\mathbb{Q}}(t):{\rm\mathbb{Q}}]\leq d\right\\}$

is finite.

Since the field of definition of $C_{t}$ is ${\rm\mathbb{Q}}(t^{2})$ (Lemma
3.2), this implies that for every fixed $d\in\mathbb{N}$, there are finitely
many ${\rm\overline{\mathbb{Q}}}$-isomorphism classes of curves $C_{t}$ with
torsion Ceresa cycle that can be defined over a number field of degree $\leq
d$. In particular, there are finitely many torsion Ceresa cycles
$\kappa(C_{t})$ defined over ${\rm\mathbb{Q}}$. In fact, there are three such
curves up to ${\rm\overline{\mathbb{Q}}}$-isormorphism: $C_{0}$, $C_{\pm 3}$
and $C_{\pm\sqrt{-3}}$ (Proposition 5.18). Note that $C_{\sqrt{-3}}$ has
${\rm\mathbb{Q}}$-model $y^{3}=x^{4}+6x^{2}-3$.

Finally, we show that for all $t$, the image of $\kappa(C_{t})$ in the
Griffiths group of homologically trivial cycles modulo algebraic equivalence
is torsion, in the following strong sense:

######  Theorem 1.5.

There exists an integer $N\geq 1$ such that for every bielliptic Picard curve
$C$ over a field of characteristic not $2$ nor $3$, the cycle
$N\cdot\kappa(C)\in{\rm CH}_{1}(J)$ is algebraically trivial.

Theorems 1.3-1.5 are quick corollaries of our proof of Theorem 1.1, which we
discuss below.

###  1.1. Relation with modified diagonal cycles

All results above hold verbatim for the canonical modified diagonal cycle
$\Delta_{GS}(C_{t})\in{\rm CH}_{1}(C_{t}^{3})$ defined by Gross and Schoen
[21]; see §2 for the definition. Indeed, $\kappa(C)$ is torsion if and only if
$\Delta_{GS}(C)$ is torsion [44, Theorem 1.5.5], and our method of proof
implies a variant of Theorem 1.4 for $\Delta_{GS}(C)$ as well.

S. Zhang has shown a similar Northcott property for the height of
$\Delta_{GS}(C)$ in families of smooth curves over smooth projective bases
[44, Theorem 1.3.5], i.e. compact subvarieties inside the moduli space
$\mathcal{M}_{g}$. As the family $C_{t}$ does not extend to a compact curve
inside $\mathcal{M}_{3}$, Theorem 1.4 suggests that Zhang’s Northcott property
might very well extend to an open dense subset of $\mathcal{M}_{g}$.

###  1.2. Summary of proof

A curve $C$ of the form (1.1) has an order $6$ automorphism (or more
canonically, a $\mu_{6}$-action), and the class $\kappa(C)$ is fixed by the
induced $\mu_{6}$-action on $J$. On the other hand, over ${\rm\mathbb{C}}$,
the $\mu_{6}$-invariant part of the Hodge structure
$\operatorname{H}^{3}(J({\rm\mathbb{C}}),{\rm\mathbb{Q}})(1)$ is two-
dimensional and of type $(1,0)+(0,1)$. This suggests that there should exist
an algebraic correspondence from $J_{t}$ to an elliptic curve that sends
$\kappa(C_{t})$ to a point on this elliptic curve. Instead of attempting to
construct this correspondence explicitly, we prove this by maneuvering within
the flexible category of (pure) Chow motives.

Namely, we show that there is an isomorphism of Chow motives

(1.2)
$\displaystyle\mathfrak{h}^{3}(J)^{\mu_{6}}\simeq\mathfrak{h}^{1}(E^{\Delta})(-1),$

where $J$ is the Jacobian of any curve of the form $C\colon
y^{3}=x^{4}+ax^{2}+b$, $E^{\Delta}$ is a certain twist of $\widehat{E}$
depending on $a$ and $b$, and the remainder of the notation will be explained
later. It is crucial for our proof that we find a universal such isomorphism,
in the category of relative Chow motives over the parameter space
$S={\rm\mathbb{A}}^{2}_{{\rm\mathbb{Z}}[1/6]}\setminus\\{\Delta_{a,b}=0\\}$.
The elliptic scheme $\mathcal{E}^{\Delta}\to S$ can then be thought of as the
universal $j$-invariant $0$ elliptic curve with a nonzero marked point
$\mathcal{Q}$. The main ingredients in the construction of this universal
isomorphism are the Chow–Kunneth decomposition for motives of abelian schemes
(via Beauville’s Fourier transform) [3, 13], Moonen’s refinement of this
decomposition in the presence of extra endomorphisms [32], and our study of
the Galois action on the endomorphism algebra of the universal Prym surface
$\mathcal{P}\to S$ [28]. The latter endomorphism algebra is a nonsplit
quaternion algebra over ${\rm\mathbb{Q}}$.

The Ceresa cycle of $C$ naturally lives in ${\rm
CH}^{2}(\mathfrak{h}^{3}(J)^{\mu_{6}})$ (Lemma 5.12), so the question becomes:
what is the corresponding element of ${\rm
CH}^{2}(\mathfrak{h}^{1}(E^{\Delta}(-1))=E^{\Delta}(k)\otimes{\rm\mathbb{Q}}$,
under the isomorphism (1.2)? We show that it corresponds to a
${\rm\mathbb{Q}}^{\times}$-multiple of the marked point $\mathcal{Q}$ by
showing it universally over $S$, using two soft inputs. First, the elliptic
scheme $\mathcal{E}^{\Delta}\rightarrow S$ (which after
${\rm\mathbb{G}}_{m}$-scaling, is essentially an elliptic surface) has
Mordell–Weil group of rank 1, generated by $\mathcal{Q}$. Second, we show by
explicit computation that the universal Ceresa cycle has at least one infinite
order specialization, using reduction mod $p$ arguments based on the methods
of [8, 11, 15], see §2. Therefore $\kappa(\mathcal{C}/S)$ corresponds to a
nonzero section of $\mathcal{E}^{\Delta}$, and hence a multiple of
$\mathcal{Q}$. Specializing at geometric points, we recover Theorem 1.1. The
construction allows us to quickly deduce the remaining results in the
introduction as well.

The “only if” direction of Theorem 1.1 could be proved without the machinery
of Chow motives, by implementing the above argument at the level of cohomology
(via complex Abel-Jacobi maps and Hodge theory). However, this is not enough
for the “if” direction since the injectivity of higher Abel-Jacobi maps for
varieties over ${\rm\overline{\mathbb{Q}}}$ is very much open. The motivic
approach therefore seems crucial for an unconditional result and is in any
case more direct.

###  1.3. Geometric interpretation via bigonal duality

$C_{t}\colon y^{3}=x^{4}+2tx^{2}+1$ is a double cover of the elliptic curve
$E_{t}\colon y^{3}=x^{2}+2tx+1$. Associated to any such double cover is the
bigonal dual cover $\widehat{C}_{t}\to\widehat{E}_{t}$ constructed by Pantazis
and Donagi [36, 14], whose Prym is dual to the Prym of $C_{t}\to E_{t}$; see
[28, §2.7] for a construction in the situation considered here. In our case,
the curve $\widehat{C}_{t}$ has model $y^{3}=(x^{2}+t)^{2}-1$ so that
$\widehat{E}_{t}$ is isomorphic to $\widehat{E}\colon y^{2}=x^{3}+1$, for all
$t$. The branch locus of $\widehat{C}_{t}\to\widehat{E}$ consists of $\infty$
and the three points $(\sqrt[3]{t^{2}-1},t)$. Theorem 1.1 therefore says that
the Ceresa cycle $\kappa(C_{t})$ is torsion if and only if the bigonal dual
$\widehat{C}_{t}$ is branched along torsion points of $\widehat{E}$. We find
this formulation intriguing, but our proof does not actually use the bigonal
dual construction in a direct way.

###  1.4. Previous results

Several examples of nonhyperelliptic curves with torsion Ceresa cycle were
recently found, conditional on Beilinson–Bloch type conjectures [7, 20, 4, 5,
30]. Qiu and W. Zhang then gave some unconditional examples [37], including
one in genus 3 and one-dimensional families in genus $4$ and $5$. Laterveer
also found a two-dimensional family in genus $5$ [29]. Subsequently, Qiu–Zhang
found examples of plane quartic curves (dominated by Shimura curves) with
trivial automorphism group and torsion Ceresa cycle [38].

The proofs of [37] and [29] rely on the vanishing of the Chow motive
$(\mathfrak{h}^{1}(C)^{\otimes 3})^{{\rm Aut}(C)}$, and hence of
$\mathfrak{h}^{3}(J)^{{\rm Aut}(C)}$, for certain curves $C$, generalizing the
observation that hyperelliptic curves have trivial Ceresa cycle. For our
curves $C=C_{t}$ with $t\neq 0$, the motive $\mathfrak{h}^{3}(J)^{{\rm
Aut}(C)}=\mathfrak{h}^{3}(J)^{\mu_{6}}$ does not vanish, but (1.2) shows that
it is a direct summand of the motive of a curve, hence is still reasonably
accessible. It would be interesting to study other families of curves where
$\mathfrak{h}^{3}(J)^{{\rm Aut}(C)}$ is “small” in a suitable sense. More
generally, one can try to study the nonzero isotypic components of $\kappa(C)$
with respect to the action of ${\rm End}(J)$ on $\mathfrak{h}^{3}(J)$. For
example, even though some of the examples in [38] have trivial automorphism
group, they all satisfy ${\rm End}(J)\neq{\rm\mathbb{Z}}$.

Finally, Eskandari and Murty showed that infinitely many Fermat curves have
infinite order Ceresa cycle [16], proving a weak version of Corollary 1.2
(where the genus is allowed to grow). If we fix $g$ but allow for curves
defined over ${\rm\overline{\mathbb{Q}}}$ (and not necessarily over
${\rm\mathbb{Q}}$), then the analogue of Corollary 1.2 holds for every genus
$g\geq 3$ by [44, Theorem 1.3.5].

###  1.5. Further questions

For an integer $N\geq 1$, let $Z_{g}(N)$ be the subset of the coarse moduli
space of curves $\mathcal{M}_{g}$ over ${\rm\mathbb{Q}}$ consisting of those
curves whose canonical Ceresa cycle (with respect to some choice of $(2g-2)$th
root of the canonical bundle) is killed by $N$ in the Chow group. What is the
geometry of $Z_{g}(N)$? More specifically, what are its positive-dimensional
components? Is the union of $Z_{g}(N)$ over all $N$ dense in
$\mathcal{M}_{g}$?

###  1.6. Acknowledgements

We thank Henri Darmon, David Lilienfeldt, and Padma Srinivasan for helpful
conversations and remarks. We thank Adam Logan for numerically testing Theorem
1.1 (while it was still a conjecture), using the forthcoming [15]. We thank
Dick Gross, whose question helped motivate us to prove Theorem 1.1. Finally,
we thank Tim Dokchitser and Drew Sutherland for helping us compute the central
$L$-values mentioned in §6. This research was carried out while the first
author was a Research Fellow at St John’s College, University of Cambridge.
The second author was funded by the European Research Council (ERC,
CurveArithmetic, 101078157).

###  1.7. Notation and conventions

  * •

All fields considered in this paper will be of characteristic $\neq 2,3$.
Given a field $k$, we denote by $k^{\mathrm{sep}}\subset\bar{k}$ a choice of
separable and algebraic closure respectively, and the absolute Galois group by
${\rm Gal}_{k}={\rm Gal}(k^{\mathrm{sep}}/k)$.

  * •

A variety over a field $k$ is a seperated scheme of finite type over $k$. A
variety is nice if it is smooth, projective and geometrically integral.

  * •

Let $X$ be a smooth and quasi-projective scheme over a Dedekind domain (which
might be a field). Then the Chow groups ${\rm CH}^{p}(X)$ of codimension $p$
cycles on $X$ (modulo rational equivalence and with
${\rm\mathbb{Z}}$-coefficients) are defined and come equipped with flat
pullbacks, proper pushforwards and an intersection product, see [19, §20.2].
If $Z\subset X$ is a codimension $p$ reduced closed subscheme write $[Z]$ for
its class in ${\rm CH}^{p}(X)$.

  * •

If $X$ is a nice variety over a field $k$ we write ${\rm
CH}^{p}(X)_{\mathrm{hom}}\subset{\rm CH}^{p}(X)$ for the classes in the kernel
of the $\ell$-adic cycle class map for every prime $\ell$ invertible in $k$.

  * •

If $T\rightarrow S$ is a morphism of schemes and $X$ is an $S$-scheme we write
$X_{T}$ for the $T$-scheme $X\times_{S}T$ and $X(T)$ for the set of sections
of $X_{T}\rightarrow T$. If $T={\rm Spec}(R)$ is affine we sometimes write
$X_{R}$ and $X(R)$ instead of $X_{T}$ and $X(T)$.

  * •

If $A,B\rightarrow S$ are abelian schemes we denote by ${\rm Hom}(A,B)$ the
set of $S$-homomorphisms between $A$ and $B$ and ${\rm Hom}^{0}(A,B)={\rm
Hom}(A,B)\otimes{\rm\mathbb{Q}}$. We similarly write ${\rm End}(A)={\rm
Hom}(A,A)$ and ${\rm End}^{0}(A)={\rm End}(A)\otimes{\rm\mathbb{Q}}$. We write
$A^{\vee}\rightarrow S$ for the dual abelian scheme (see [18, Chapter 1, §1])
and write ${\rm Hom}^{\operatorname{sym}}(A,A^{\vee})$ for the subset of all
self-dual homomorphisms $A\rightarrow A^{\vee}$.

  * •

If $A/S$ is an abelian scheme we denote by $(n)$ or $(n)_{A}$ the
multiplication-by-$n$ endomorphism $A\rightarrow A$.

  * •

Group actions are always assumed to be _left_ actions.

##  2\. Ceresa and modified diagonal cycles

We recall the definitions of Ceresa and modified diagonal cycles, and show, by
means of an example, a way to verify that $\kappa(C)$ is infinite order, for
genus 3 curves over ${\rm\overline{\mathbb{Q}}}$. The latter is based on the
methods of [11] and [15], which are in turn based on ideas of Bloch [8]. In
this section, and for the rest of the paper, we work over a general (not
necessarily algebraically closed) field $k$ of characteristic $\neq 2,3$.

###  2.1. Ceresa and modified diagonal cycles

Let $C$ be a (nice) curve of genus $g\geq 2$ over $k$. If $e\in C(k)$, let
$\iota_{e}\colon C\hookrightarrow J$ be the Abel-Jacobi map sending $x\mapsto
x-e$, and define the Ceresa cycle based at $e$ to be

$\kappa_{e}(C):=[\iota_{e}(C)]-(-1)^{*}[\iota_{e}(C)]\in{\rm CH}^{g-1}(J).$

######  Lemma 2.1.

The map $e\mapsto\kappa_{e}(C)$ extends to a homomorphism
$\alpha_{*}\colon{\rm CH}^{1}(C)\to{\rm CH}^{g-1}(J)$.

###### Proof.

Let $f\colon C^{2}\to C\times J$ be the morphism $(e,x)\mapsto(e,x-e)$ and let
$\alpha^{\prime}=f_{*}[C^{2}]\in{\rm CH}^{2}(C\times J)$. The correspondence
$\alpha:=\alpha^{\prime}-(\mathrm{id}\times(-1))^{*}\alpha^{\prime}\in{\rm
CH}^{2}(C\times J)$ induces the homomorphism $\alpha_{*}\colon{\rm
CH}^{1}(C)\to{\rm CH}^{g-1}(J)$ sending $Z$ to
$p_{2*}((p_{1}^{*}Z)\cdot\alpha)$, where $p_{1},p_{2}$ denote the projections
from $C\times J$. A computation shows that $\alpha_{*}([e])=\kappa_{e}(C)$ for
all $e\in C(k)$. ∎

For a degree 1 divisor class $e\in{\rm CH}^{1}(C)$, we define
$\kappa_{e}(C):=\alpha_{*}(e)$. When $(2g-2)e$ is canonical, we call
$\kappa_{e}(C)$ a canonical Ceresa cycle and simply write $\kappa(C)$. There
are finitely many such $e\in{\rm CH}^{1}(C)$, and they all give the same
canonical Ceresa class up to $(2g-2)$-torsion, by Lemma 2.1.

The modified diagonal cycle of Gross and Schoen [21] is a close relative of
the Ceresa cycle. For $e\in C(k)$, the modified diagonal cycle
$\Delta_{GS,e}=\Delta_{GS,e}(C)$ based at $e$ is

$\Delta_{GS,e}:=\Delta_{123}-\Delta_{12}-\Delta_{13}-\Delta_{23}+\Delta_{1}+\Delta_{2}+\Delta_{3}\in{\rm
CH}^{2}(C^{3}),$

where $\Delta_{123}=\\{(c,c,c)\colon c\in C\\}$, $\Delta_{12}=\\{(c,c,e)\colon
c\in C\\}$, etc. As before, one can define a cycle $\Delta_{GS,e}$ for any
divisor $e$ of degree $1$ on $C$, and if $(2g-2)e$ is a canonical divisor,
then we call $\Delta_{GS,e}$ a canonical modified diagonal cycle and write
$\Delta_{GS}=\Delta_{GS,e}$. Again this is well-defined up to torsion in the
Chow group; see [21, eq. 3.5].

Some useful general facts about these cycles are:

######  Proposition 2.2.

The cycles $\kappa_{e}(C)$ and $\Delta_{GS,e}(C)$ are homologically trivial.

###### Proof.

For $\kappa_{e}(C)$, this is immediate from the fact that $(-1)^{*}$ acts
trivially on even dimensional cohomology. For $\Delta_{GS,e}(C)$, this is [21,
Proposition 3.1]. ∎

######  Theorem 2.3.

[44, Theorem 1.5.5] $\kappa(C)$ has infinite order if and only if
$\Delta_{GS}(C)$ has infinite order.

The following proposition shows that the canonical modified diagonal cycle
$\Delta_{GS}(C)$ is “minimal up to torsion” in a certain sense.

######  Proposition 2.4.

If $\Delta_{GS,e}(C)$ is torsion, then $(2g-2)e-K_{C}$ is torsion in $J(k)$,
where $K_{C}$ is the canonical divisor class.

###### Proof.

This follows from [37, Proposition 2.3(1)], whose proof is valid over any
field $k$. ∎

###  2.2. A non-vanishing criterion

Assume now that $k$ is a number field. Since $\Delta_{GS,e}$ is homologically
trivial, we may consider for every prime $\ell$ its image under the
$\ell$-adic Abel-Jacobi map

${\rm AJ}_{\ell}\colon{\rm
CH}^{2}(C^{3})_{\mathrm{hom}}\longrightarrow\operatorname{H}^{1}({\rm
Gal}_{k},\operatorname{H}^{3}(C_{k^{\operatorname{sep}}}^{3},{\rm\mathbb{Z}}_{\ell}(2)));$

see [22, §9] for more details concerning this map. The following proposition
is based on the methods of [11] and [15].

######  Proposition 2.5.

Let $e\in C(k)$ and let $v$ be a finite place of $k$ of good reduction for
$C$. Let $\ell>3$ be a prime coprime to the norm of $v$. Let $(C_{v},e_{v})$
be the smooth reduction of $(C,e)$ over the finite residue field
${\rm\mathbb{F}}_{v}$. Define

(2.1) $\displaystyle D:=\sum_{c\in C_{v}({\rm\mathbb{F}}_{v})}(c-e_{v})\in
J({\rm\mathbb{F}}_{v}).$

If the order of $D$ is a multiple of $\ell$, then ${\rm
AJ}_{\ell}(\Delta_{GS,e})\neq 0$.

###### Proof.

Consider the correspondence

$\Gamma=\\{(a,\mathrm{Fr}(a),b,b)\colon a,b\in C_{v}\\}\subset C^{3}_{v}\times
C_{v},$

where $\mathrm{Fr}$ is the Frobenius morphism. This gives a homomorphism

$\Gamma_{*}\colon{\rm CH}^{2}(C^{3}_{v})\to{\rm CH}^{1}(C_{v})\simeq{\rm
Pic}(C_{v}),$

defined by $\Gamma_{*}\beta=p_{2*}([\Gamma]\cdot p_{1}^{*}\beta)$, where
$p_{i}$ are the projections from $C^{3}_{v}\times C_{v}$. Let $\Delta_{GS,v}$
be the modified diagonal cycle on $C_{v}^{3}$ with respect to $e_{v}$. A short
computation yields $\Gamma_{*}(\Delta_{GS,v})=\sum_{c\in
C_{v}({\rm\mathbb{F}}_{v})}(c-e_{v})=D\in J({\rm\mathbb{F}}_{v})$.

If ${\rm AJ}_{\ell}(\Delta_{GS,e})=0$, then a specialization argument using
[11, Proposition 3.1] shows that ${\rm
AJ}_{\ell}(D)\in\operatorname{H}^{1}({\rm
Gal}_{k_{v}},\operatorname{H}^{1}(C_{k_{v}^{\mathrm{sep}}},{\rm\mathbb{Z}}_{\ell}(1)))$
is zero too. By [11, Lemma 1.12] this implies that $D$ is zero in
$J(k_{v})\otimes{\rm\mathbb{Z}}_{\ell}$, so $D$ has order coprime to $\ell$.
We conclude that if $D$ has order divisible by $\ell$ then ${\rm
AJ}_{\ell}(\Delta_{GS,e})\neq 0$. ∎

###  2.3. An infinite order example

We now show by example how Proposition 2.5 can be used to verify that
$\kappa(C)$ has infinite order. For the rest of the section, let
$k={\rm\mathbb{Q}}$, let $C$ be the curve with plane model
$y^{3}=x^{4}+x^{2}+1$, and let $\ell=7$. Consider the modified diagonal cycle
$\Delta_{GS}$ based at the unique point $\infty$ at infinity; note that
$(2g-2)\infty=4\infty$ is canonical.

######  Lemma 2.6.

${\rm AJ}_{7}(\Delta_{GS})\neq 0$.

###### Proof.

We take $v=41$ in Proposition 2.5 and show that $D\in J({\rm\mathbb{F}}_{41})$
of (2.1) has order divisible by $7$ by explicit computation. Let $\pi\colon
C\to E\colon y^{3}=x^{2}+x+1,(x,y)\mapsto(x^{2},y)$ be the double cover and
let $\tau\in{\rm Aut}(C)$ be the corresponding covering involution. Pairing
points $c$ and $\tau(c)$ together, we see that $D$ is a sum of points on
$\pi^{*}{\rm Pic}^{0}(E)$ plus some leftover ramification points. Thus $2D$
lies in $\pi^{*}{\rm Pic}^{0}(E)$, and identifying ${\rm Pic}^{0}(E)$ with
$E$, we have

$2D=\pi^{*}\left((0,1)+2\sum_{q\in
E({\rm\mathbb{F}}_{41})_{\mathrm{lift}}}q\right),$

where $E({\rm\mathbb{F}}_{41})_{\mathrm{lift}}$ is the set of non-branch
points in the image of $\pi\colon C({\rm\mathbb{F}}_{41})\to
E({\rm\mathbb{F}}_{41})$. We compute in Sage that the interior sum equals
$(37,15)\in E({\rm\mathbb{F}}_{41})$ which has order $7$. Therefore
$2D=\pi^{*}((37,15))$ has order $7$ too, hence the order of $D$ is divisible
by $7$. ∎

Let ${\rm Sym}^{3}(C):=C^{3}/S_{3}$ be the symmetric cube of $C$, and let
$V:=\operatorname{H}^{3}({\rm
Sym}^{3}(C_{\overline{{\rm\mathbb{Q}}}}),{\rm\mathbb{Z}}_{7}(2))$.

######  Lemma 2.7.

$V\simeq\operatorname{H}^{3}(J_{\overline{{\rm\mathbb{Q}}}},{\rm\mathbb{Z}}_{7}(2))\oplus\operatorname{H}^{1}(C_{\overline{{\rm\mathbb{Q}}}},{\rm\mathbb{Z}}_{7}(1))$
as ${\rm Gal}_{{\rm\mathbb{Q}}}$-representations.

###### Proof.

Since $C$ is a nonhyperelliptic genus $3$ curve, the Abel–Jacobi map
$\Sigma\colon{\rm Sym}^{3}(C)\to J$ sending a degree $3$ divisor $D$ to
$D-3\infty$ is the blow-up of $J$ along the image of those $D$ with the
property that $\dim\operatorname{H}^{0}(C,\mathcal{O}(D))=2$ [33, Theorem
2.3]. By Riemann-Roch, these are exactly the degree $3$ divisors linearly
equivalent to $4\infty-x$ for some unique $x\in C$. We conclude that $\Sigma$
is the blow-up of $J$ along the closed subvariety $\\{\infty-x\mid x\in
C\\}\simeq C$. The lemma now follows from the description of the cohomology of
a blow-up [1, VII, Theoreme 8.1.1]. ∎

######  Lemma 2.8.

The class $\Delta_{GS}$ has infinite order.

###### Proof.

Suppose, for the sake of contradiction, that $\Delta_{GS}$ has finite order.
Then its Abel-Jacobi image ${\rm
AJ}_{7}(\Delta_{GS})\in\operatorname{H}^{1}({\rm
Gal}_{{\rm\mathbb{Q}}},\operatorname{H}^{3}(C^{3}_{\bar{\rm\mathbb{Q}}},{\rm\mathbb{Z}}_{7}))$
has finite order. By Lemma 2.6 it is nonzero. Let $h\colon C^{3}\to{\rm
Sym}^{3}(C)$ be the quotient map and let
$\tilde{\Delta}_{GS}=h_{*}\Delta_{GS}$. Use the same notation as before for
the Abel-Jacobi map ${\rm AJ}_{\ell}\colon{\rm CH}^{2}({\rm
Sym}^{3}(C))_{\mathrm{hom}}\to\operatorname{H}^{1}({\rm
Gal}_{\rm\mathbb{Q}},V).$ Since $h^{*}\tilde{\Delta}_{GS}=6\Delta_{GS}$, we
conclude that ${\rm AJ}_{\ell}(\tilde{\Delta}_{GS})$ is also nonzero and
torsion. On the other hand, we may use Magma to compute the characteristic
polynomial of a Frobenius element acting on
$\operatorname{H}^{1}(C_{{\rm\overline{\mathbb{Q}}}},{\rm\mathbb{Z}}_{7}(1))$,
and hence on $V$ as well (via Lemma 2.7). We check that the action of the
geometric Frobenius $\mathrm{Fr}_{11}$ at $11$ on $V$ satisfies

$\det(\mathrm{Fr}_{11}-1)=29049104246323668435011663307177984\not\equiv
0\pmod{7}.$

Thus $\operatorname{H}^{1}({\rm Gal}_{\rm\mathbb{Q}},V)$ is torsion-free by
[43, 6.1], which is a contradiction. ∎

Combining Lemma 2.8 and Theorem 2.3, we have proven:

######  Corollary 2.9.

The Ceresa cycle $\kappa(C)$ of the curve $C\colon y^{3}=x^{4}+x^{2}+1$ has
infinite order.

##  3\. Bielliptic Picard curves

We recall some results on bielliptic Picard curves from [28]; we refer to that
paper for more details. In particular, we collect the results we will need on
the endomorphism algebra of the Prym variety and its monodromy in §3.3.

###  3.1. Definitions

A bielliptic Picard curve $C$ over a field $k$ (of characteristic $\neq 2,3$)
is a nice plane quartic curve over $k$ with affine model of the form

(3.1) $\displaystyle y^{3}=x^{4}+ax^{2}+b,$

for some $a,b\in k$. We often write $C=C_{a,b}$. Smoothness of $C$ implies
that

$\displaystyle\Delta_{a,b}:=16b(a^{2}-4b)\neq 0,$

and any plane quartic of the form (3.1) with $\Delta_{a,b}\neq 0$ is smooth.
These curves admit a $\mu_{6}$-action given by
$\zeta\cdot(x,y)=(\zeta^{3}x,\zeta^{4}y)$ for every $\zeta\in\mu_{6}$. The
unique fixed point of this action is the unique point $\infty$ at infinity.
Since $\infty$ intersects the line at infinity with multiplicity $4$, the
divisor $4\infty$ is canonical.

Define the $j$-invariant of $C_{a,b}$ to be $j(C_{a,b}):=(4b-a^{2})/4b\in
k^{\times}$.

######  Lemma 3.1.

[28, §2.2-3] Let $C_{a,b}$ and $C_{a^{\prime},b^{\prime}}$ be bielliptic
Picard curves over $k$.

  1. (1)

$C_{a,b}\simeq C_{a^{\prime},b^{\prime}}$ if and only if there exists
$\lambda\in k^{\times}$ such
$(a^{\prime},b^{\prime})=(\lambda^{6}a,\lambda^{12}b)$.

  2. (2)

$C_{a,b}$ is isomorphic to $C_{a^{\prime},b^{\prime}}$ over $\overline{k}$ if
and only if $j(C_{a,b})=j(C_{a^{\prime},b^{\prime}})$.

######  Lemma 3.2.

Let $C_{a,b}$ be a bielliptic Picard curve over $k^{\mathrm{sep}}$. Then $C$
is defined over $k$ (in other words, there exists a curve $X$ over $k$ such
that $X_{k^{\mathrm{sep}}}\simeq C_{a,b}$) if and only if $j(C_{a,b})\in k$.

###### Proof.

Suppose that $j=j(C_{a,b})\in k$. If $j=1$, then $a=0$ and $C_{a,b}$ is
$k^{\mathrm{sep}}$-isomorphic to $C_{0,1}$ and defined over $k$. If $j\neq 1$,
then a computation using Lemma 3.1(1) shows that $C_{a,b}$ is
$k^{\mathrm{sep}}$-isomorphic to $C_{2,(1-j)^{-1}}$ and hence also defined
over $k$. Conversely, suppose that $C_{a,b}$ is defined over $k$. Then
$C_{a,b}\simeq\sigma^{*}(C_{a,b})=C_{\sigma(a),\sigma(b)}$ for all
$\sigma\in{\rm Gal}_{k}$. By Lemma 3.1(2), $j(C_{a,b})=\sigma(j(C_{a,b}))$ for
all $\sigma\in{\rm Gal}_{k}$. In other words, $j(C_{a,b})\in k$. ∎

###  3.2. The Prym variety

The curve $C$ admits an involution $\tau(x,y)=(-x,y)$, and the quotient by
this involution is a double cover $\pi\colon C\to E$ to an elliptic curve $E$
with origin the image of $\infty$. This elliptic curve has equation
$y^{3}=x^{2}+ax+b$ and (after a change of variables) short Weierstrass model
$E\colon y^{2}=x^{3}+16(a^{2}-4b)$. This double cover decomposes the Jacobian
variety $J$ of $C$. Let

(3.2) $\displaystyle P:=\ker(1+\tau^{*}\colon J\rightarrow J)$

be the Prym variety associated to the double cover $C\rightarrow E$.

######  Lemma 3.3.

[28, §2.5] $P$ is an abelian surface and $J$ is isogenous to $P\times E$.

###  3.3. Galois action on the Neron–Severi group

The main ingredient we need from [28] for our proof of Theorem 1.1 is an
explicit description of the ${\rm Gal}_{k}$-action on the Neron–Severi group
${\rm NS}(P_{k^{\operatorname{sep}}})$, as well as the fact that
$P_{k^{\operatorname{sep}}}$ has quaternionic multiplication.

We will need a universal version of this story over the parameter space of all
bielliptic Picard curves, so let

$S:={\rm\mathbb{A}}^{2}_{{\rm\mathbb{Z}}[1/6]}\setminus\\{\Delta_{a,b}=0\\}={\rm
Spec}({\rm\mathbb{Z}}[1/6,a,b,\Delta_{a,b}^{-1}]).$

Let $\mathcal{C}\rightarrow S$ be the universal bielliptic Picard curve with
equation (3.1). The morphism $\mathcal{C}\rightarrow S$ is smooth, proper and
of relative dimension $1$ and every bielliptic Picard curve occurs as a fiber
of this morphism. Let $\mathcal{J}=\mathbf{Pic}_{\mathcal{C}/S}^{0}$ be its
relative Jacobian [10, §9.4, Proposition 4], an abelian scheme over $S$. The
definition (3.2) of the Prym variety works in families and we obtain an
abelian scheme $\mathcal{P}\rightarrow S$ of relative dimension $2$ whose
fibers are the Prym varieties described in §3.2.

We need to pass to a cover of $S$ to "see" all endomorphisms of $\mathcal{P}$.
Let $\tilde{S}\rightarrow S$ be the finite etale cover given by adjoining a
primitive third root of unity $\omega$ and a sixth root of $\Delta_{a,b}$.
More formally, consider the ring extension

$R={\rm\mathbb{Z}}[a,b,\Delta_{a,b}^{-1}]\rightarrow\tilde{R}={\rm\mathbb{Z}}[\omega][1/6,a,b,\Delta_{a,b}^{-1},\varepsilon]/(\varepsilon^{6}-\Delta_{a,b})$

and define $\tilde{S}\rightarrow S$ to be the induced morphism of schemes.
This morphism is finite, etale, and Galois, with Galois group $G:={\rm
Aut}(\tilde{S}/S)$ a dihedral group of order $12$. The group $G$ naturally
acts on ${\rm End}^{0}(\mathcal{P})$. Let $\mathrm{triv}\colon
G\rightarrow{\rm GL}_{1}({\rm\mathbb{Q}})$ be the trivial representation and
let $\mathrm{std}\colon G\rightarrow{\rm GL}_{2}({\rm\mathbb{Q}})$ be a model
for the reflection representation, i.e. the unique irreducible $2$-dimensional
representation of $G$ defined over ${\rm\mathbb{Q}}$. See §1.7 for our
notations concerning abelian schemes.

######  Theorem 3.4.

The endomorphism algebra $B:={\rm End}^{0}(\mathcal{P}_{\tilde{S}})$ is a
quaternion algebra of discriminant $6$ over ${\rm\mathbb{Q}}$. Moreover, the
$G$-representation ${\rm
Hom}^{\operatorname{sym}}(\mathcal{P}_{\tilde{S}},\mathcal{P}^{\vee}_{\tilde{S}})\otimes{\rm\mathbb{Q}}$
is three-dimensional and isomorphic to $\mathrm{triv}\oplus\mathrm{std}$.

###### Proof.

The schemes $S$ and $\tilde{S}$ are integral and normal; let $K$ and
$\widetilde{K}$ denote their function fields and let $P=\mathcal{P}_{K}$
denote the generic fiber of $\mathcal{P}$. By [17, §2, Lemma 1], the natural
map ${\rm End}(\mathcal{P}_{\tilde{S}})\rightarrow{\rm
End}(P_{\widetilde{K}})$ is an isomorphism. By [41, Tag 0BQM], the natural map
$G={\rm Aut}(\tilde{S}/S)\rightarrow{\rm Gal}(\widetilde{K}/K)$ is also an
isomorphism. It therefore suffices to prove all claims over the generic point
${\rm Spec}(K)$. By [28, Lemma 6.16], $P$ is geometrically simple. If
$\bar{K}$ denotes an algebraic closure of $K$ then [28, Lemma 6.2] implies
that ${\rm End}^{0}(P_{\bar{K}})$ is a discriminant $6$ quaternion algebra,
and [28, Corollary 6.6] shows that ${\rm End}^{0}(P_{\bar{K}})={\rm
End}^{0}(P_{\tilde{K}})$. The claim concerning ${\rm
Hom}^{\operatorname{sym}}(\mathcal{P}_{\tilde{S}},\mathcal{P}^{\vee}_{\tilde{S}})\otimes{\rm\mathbb{Q}}={\rm
Hom}(P_{\tilde{K}},P^{\vee}_{\tilde{K}})\otimes{\rm\mathbb{Q}}$ follows from
[28, Corollary 6.8] and the fact that the latter is isomorphic to ${\rm
NS}(P_{k^{\mathrm{sep}}})\otimes{\rm\mathbb{Q}}$. ∎

The $\mu_{3}$-action on $\mathcal{J}$ restricts to a $\mu_{3}$-action on
$\mathcal{P}$, which by functoriality induces a $\mu_{3}$-action on ${\rm
Hom}^{\operatorname{sym}}(\mathcal{P}_{\tilde{S}},\mathcal{P}_{\tilde{S}}^{\vee})$
via $\omega\cdot\lambda=\omega^{\vee}\circ\lambda\circ\omega^{-1}$ for all
$\lambda\in{\rm
Hom}^{\operatorname{sym}}(\mathcal{P}_{\tilde{S}},\mathcal{P}_{\tilde{S}}^{\vee})\otimes{\rm\mathbb{Q}}$
and $\omega\in\mu_{3}$. This $\mu_{3}$-action commutes with the $G$-action, so
preserves the isotypic components $\mathrm{triv}$ and $\mathrm{std}$.

######  Lemma 3.5.

In the decomposition ${\rm
Hom}^{\operatorname{sym}}(\mathcal{P}_{\tilde{S}},\mathcal{P}^{\vee}_{\tilde{S}})\otimes{\rm\mathbb{Q}}=\mathrm{triv}\oplus\mathrm{std}$
of Theorem 3.4, $\mu_{3}$ acts trivially on $\mathrm{triv}$ and $\mu_{3}$ acts
on $\mathrm{std}\otimes{\rm\overline{\mathbb{Q}}}$ as a direct sum of the two
distinct nontrivial characters.

###### Proof.

The proof of [28, Corollary 6.8] shows that if we write
$B=\text{span}\\{1,i,j,ij\\}$ with $i^{2}=-3$, $j^{2}=2$, $ij=-ji$ and
$\omega=(-1+i)/2$, then ${\rm
Hom}^{\operatorname{sym}}(\mathcal{P}_{\tilde{S}},\mathcal{P}^{\vee}_{\tilde{S}})\otimes{\rm\mathbb{Q}}\simeq\text{span}\\{1,j,ij\\}$,
$\mathrm{triv}$ is spanned by $1$, $\mathrm{std}$ is spanned by $\\{i,ij\\}$
and $\omega\in\mu_{3}$ acts via conjugation by $\omega\in B$. The description
of the $\mu_{3}$-action on $\mathrm{triv}$ and $\mathrm{std}$ now follow from
an explicit calculation. ∎

##  4\. Background on Chow motives

Our proof of Theorem 1.1 uses the language of relative Chow motives, so we
recall the necessary definitions and prove some basic statements on motives
for which we could not find a reference. The most important sections are §4.2
and §4.7 and the others can be referred back to when needed in §5.

We call a scheme $S$ adequate if it is smooth and quasiprojective over a
Dedekind domain. For the remainder of this section fix such a scheme $S$.

###  4.1. Correspondences

We denote by $\mathsf{SmProj}(S)$ the category of smooth projective
$S$-schemes. For $X$ in $\mathsf{SmProj}(S)$ the Chow groups ${\rm CH}^{p}(X)$
are well defined, see §1.7. Write ${\rm CH}^{p}(X;{\rm\mathbb{Q}}):={\rm
CH}^{p}(X)\otimes_{{\rm\mathbb{Z}}}{\rm\mathbb{Q}}$ and if $X\rightarrow S$
has equidimensional fibers write $d(X/S)$ for the relative dimension of
$X\rightarrow S$. If $X$ and $Y$ are in $\mathsf{SmProj}(S)$ we define the
${\rm\mathbb{Q}}$-vector space of correspondences by ${\rm Corr}(X,Y):={\rm
CH}^{*}(X\times_{S}Y;{\rm\mathbb{Q}})$. We say a correspondence has degree $r$
if it lies in the subspace

${\rm Corr}^{r}(X,Y):=\bigoplus_{i}{\rm
CH}^{r+d(X_{i}/S)}(X_{i}\times_{S}Y;{\rm\mathbb{Q}})$

where $X=\sqcup_{i}X_{i}$ is a decomposition such that $X_{i}\rightarrow S$ is
equidimensional. If $p\in{\rm Corr}^{r}(X,Y)$ and $q\in{\rm Corr}^{s}(Y,Z)$
are correspondences, denote by $q\circ
p=\pi_{XZ,*}(\pi_{XY}^{*}p\cdot\pi_{YZ}^{*}q)\in{\rm Corr}^{r+s}(X,Z)$ their
composition. If $f\colon X\rightarrow Y$ is an $S$-morphism write
$\Gamma_{f}\subset X\times_{S}Y$ for its graph and ${}^{t}\Gamma_{f}\subset
Y\times_{S}X$ for its transpose. Write $\Delta_{X/S}$ for the graph of the
identity $X\rightarrow X$.

###  4.2. Relative Chow motives

We denote by $\mathsf{Mot}(S)$ the category of relative Chow motives with
respect to graded correspondences over $S$, see [13, §1] for more details. The
results of [13] are stated under the standing assumption that $S$ is smooth
and quasi-projective over a field, but their results continue to hold in our
more general set-up since intersection theory is developed in this generality,
see [25, §1, Remark 1.1].

We recall that an object of $\mathsf{Mot}(S)$ is a triple $(X/S,p,m)$, where
$X/S$ is a smooth projective $S$-scheme, $p\in{\rm Corr}^{0}(X,X)$ is an
idempotent correspondence and $m$ is an integer. Morphisms are given by

${\rm Hom}((X/S,p,m),(Y/S,q,n)):=q\circ{\rm Corr}^{m-n}(X,Y)\circ p.$

Given a motive $M=(X/S,p,m)$ and $n\in{\rm\mathbb{Z}}$ we write
$M(n):=(X/S,p,m+n)$. There is a contravariant functor
$\mathsf{SmProj}(S)\rightarrow\mathsf{Mot}(S)$ sending $X/S$ to
$\mathfrak{h}(X/S):=(X/S,[\Delta_{X/S}],0)$ and sending morphisms $f\colon
X\rightarrow Y$ to $[^{t}\Gamma_{f}]$. Write
$\mathbf{1}=\mathbf{1}_{S}:=\mathfrak{h}(S/S)$ for the unit motive and
$\mathbb{L}:=\mathbf{1}(-1)$ for the Lefschetz motive. There is a notion of
direct sums and tensor products in $\mathsf{Mot}(S)$.

###  4.3. Base changing motives

Suppose that $T$ is another adequate scheme and $f\colon T\rightarrow S$ a
morphism. There are two situations in which we have a good theory of
pullbacks222Pullbacks can be defined in greater generality, see for example
[19, Example 20.1.2], but we restrict ourselves to the concrete cases we need
in this paper. [19, §20.1]:

  1. (1)

If $f$ is flat, there are pullback maps $f^{*}\colon{\rm
CH}^{p}(X)\rightarrow{\rm CH}^{p}(X_{T})$ for every $X/S$ in
$\mathsf{SmProj}(S)$ and $p\in{\rm\mathbb{Z}}_{\geq 0}$.

  2. (2)

If $f$ is an lci morphism, there are Gysin homomorphisms $f^{*}\colon{\rm
CH}^{p}(X)\rightarrow{\rm CH}^{p}(X_{T})$ for every $X/S$ in
$\mathsf{SmProj}(S)$ compatible with flat pullbacks.

We will say that $f$ admits pullbacks if it is a finite composition
$f_{1}\circ\cdots\circ f_{n}$ of morphisms of the above form, and for such a
morphism we define $f^{*}=f_{n}^{*}\circ\cdots\circ f_{1}^{*}\colon{\rm
CH}^{p}(X)\rightarrow{\rm CH}^{p}(X_{T})$ for every $X/S$ in
$\mathsf{SmProj}(S)$. (This does not depend on the choice of $f_{i}$’s.) For
example, if $T$ is the spectrum of a field then $f$ admits pullbacks. When $f$
admits pullbacks we define a functor

$\displaystyle f^{*}\colon\mathsf{Mot}(S)\rightarrow\mathsf{Mot}(T),$

which on objects sends $(X/S,p,m)$ to $(X_{T}/T,f^{*}(p),m)$ and sends a
correspondence $q$ to $f^{*}(q)$.

###  4.4. Descending direct summands of motives

Let $\pi\colon\tilde{S}\rightarrow S$ be a finite etale morphism of adequate
schemes. Suppose additionally that it is Galois and write $G$ for the group of
Deck transformations, so we have an isomorphism of $S$-schemes
$g\colon\tilde{S}\rightarrow\tilde{S}$ for every $g\in G$. Using the pullback
functor from §4.3, we obtain functors
$g^{*}\colon\mathsf{Mot}(\tilde{S})\rightarrow\mathsf{Mot}(\tilde{S})$. Let
$M,N$ be objects of $\mathsf{Mot}(S)$ and write $\tilde{M}=\pi^{*}M$,
$\tilde{N}=\pi^{*}N$ for the corresponding objects of
$\mathsf{Mot}(\tilde{S})$. Then we have canonical identifications
$\tilde{M}=g^{*}\tilde{M}$ and $\tilde{N}=g^{*}\tilde{N}$ for all $g\in G$. It
follows by functoriality that ${\rm Hom}(\tilde{M},\tilde{N})$ inherits a
$G$-action.

######  Lemma 4.1.

In the above notation, the natural map ${\rm Hom}(M,N)\rightarrow{\rm
Hom}(\tilde{M},\tilde{N})^{G}$ is an isomorphism.

###### Proof.

This is well known when $S={\rm Spec}(k)$ [39, Lemma 1.17]; we sketch the
generalization of the classical proof to the relative setting. Write
$M=(X/S,p,m)$ and $N=(Y/S,q,n)$. Then an element of ${\rm
Hom}(\tilde{M},\tilde{N})^{G}$ is the same as a $G$-invariant element $\alpha$
of $q\circ{\rm CH}^{d(X/S)+m-n}(X\times_{S}Y;{\rm\mathbb{Q}})\circ p$. Choose
a cycle $Z$ on $X_{\tilde{S}}\times_{\tilde{S}}Y_{\tilde{S}}$ representing
$\alpha$. Then $\sum_{g\in G}g^{*}Z$ descends to a unique cycle $W$ of
$X\times_{S}Y$. Moreover the rational equivalence class $[W]$ is independent
of the choice of $Z$ representing $\alpha$. It follows that
$\alpha\mapsto\frac{1}{|G|}[W]$ is a well-defined inverse to the base change
map above. ∎

This implies that Galois-invariant summands also satisfy descent:

######  Corollary 4.2.

Let $M$ be an object of $\mathsf{Mot}(S)$ and $\tilde{U}\oplus\tilde{V}$ a
direct sum decomposition of $\tilde{M}=\pi^{*}M$ such that we have an equality
of direct summands $g^{*}\tilde{U}=\tilde{U}$ and $g^{*}\tilde{V}=\tilde{V}$
in $g^{*}\tilde{M}=\tilde{M}$ for all $g\in G$. Then there exists a unique
direct sum decomposition $M=U\oplus V$ such that $\pi^{*}U=\tilde{U}$ and
$\pi^{*}V=\tilde{V}$.

###### Proof.

Since direct sum decompositions correspond bijectively to a pair of orthogonal
idempotents in the endomorphism ring, this immediately follows from Lemma 4.1.
∎

###  4.5. Group actions on motives

Let $M$ be a motive in $\mathsf{Mot}(S)$ and $G$ a finite group acting on $M$.
Then every idempotent in the group algebra ${\rm\mathbb{Q}}[G]$ gives rise to
an idempotent in $M$. For example, if $\rho$ is an irreducible (but not
necessarily absolutely irreducible) representation defined over
${\rm\mathbb{Q}}$ and $e_{\rho}$ is the corresponding central idempotent, we
get a direct summand $M_{\rho}$ of $M$, called the $\rho$-isotypic component.
Applying this to $\rho=$ the trivial representation, we see that the motive of
$G$-fixed points $M^{G}=M_{\rho}$ is well-defined and a direct summand of $M$.

We may also consider actions by nonconstant finite group schemes, using Lemma
4.1. For example, if $n$ is invertible in $S$, a $\mu_{n}$-action on a motive
$M$ in $\mathsf{Mot}(S)$ is an ${\rm Aut}(\tilde{S}/S)$-equivariant map
$\mu_{n}(\tilde{S})\rightarrow{\rm Aut}(f^{*}M)$, where
$f\colon\tilde{S}=S\times_{{\rm\mathbb{Z}}[1/n]}{\rm\mathbb{Z}}[\zeta_{n}][1/n]\rightarrow
S$. The ${\rm Aut}(\tilde{S}/S)$-equivariance implies that the $\mu_{n}$-fixed
points of $f^{*}M$ descend to a well-defined direct summand $M^{\mu_{n}}$ of
$M$.

###  4.6. Relative Artin motives

Assume that $S$ is connected. An Artin motive in $\mathsf{Mot}(S)$ is a direct
summand of a motive of the form $\mathfrak{h}(X/S)$, where $X\rightarrow S$ in
$\mathsf{SmProj}(S)$ has relative dimension zero or equivalently, is finite
etale. When $S$ is the spectrum of a field, Artin motives are the same as
Galois representations on finite-dimensional ${\rm\mathbb{Q}}$-vector spaces;
we give a similar (presumably well known) description in the relative setting.
Let $f\colon\tilde{S}\rightarrow S$ be a connected finite etale cover with
Galois group $G={\rm Aut}(\tilde{S}/S)$. We say an Artin motive $M$ in
$\mathsf{Mot}(S)$ is trivialized by $f$ if
$f^{*}M\simeq\mathbf{1}_{\tilde{S}}^{\oplus n}$ for some
$n\in{\rm\mathbb{Z}}_{\geq 0}$. In that case the ${\rm\mathbb{Q}}$-vector
space ${\rm Hom}(\mathbf{1}_{\tilde{S}},f^{*}M)$ is $n$-dimensional (since
$\tilde{S}$ is connected) and has a natural $G$-action, by the same reasoning
as §4.4.

######  Lemma 4.3.

Every Artin motive in $\mathsf{Mot}(S)$ is trivialized by some connected
finite etale Galois cover $f\colon\tilde{S}\rightarrow S$. Fixing such an $f$,
the assignment $M\mapsto V_{M}:={\rm Hom}(\mathbf{1}_{\tilde{S}},f^{*}M)$
induces an equivalence between the full subcategory of Artin motives in
$\mathsf{Mot}(S)$ trivialized by $f$ and the category of ${\rm
Aut}(\tilde{S}/S)$-representations on finite-dimensional
${\rm\mathbb{Q}}$-vector spaces.

###### Proof.

(Sketch) If $M$ is a direct summand of $\mathfrak{h}(X/S)$, then $M$ is
trivialized by the compositum of the Galois closures of the connected
components of $X$, justifying the first sentence. Fully faithfulness of
$M\mapsto V_{M}$ follows from Lemma 4.1 and the fact that ${\rm
Hom}(f^{*}M,f^{*}N)\rightarrow{\rm Hom}(V_{M},V_{N})$ is an isomorphism when
$\tilde{S}$ is connected. Moreover, again since $\tilde{S}$ is connected
$V_{\mathfrak{h}(\tilde{S}/S)}$ is isomorphic to the regular representation of
$G$. Therefore essential surjectivity follows from the fact that every
$G$-representation is a direct summand of copies of the regular
representation. ∎

###  4.7. Motives of abelian schemes

Let $A\rightarrow S$ be an abelian scheme of relative dimension $g$.
Deninger–Murre [13, §3] have shown that the motive $\mathfrak{h}(A/S)$ has a
canonical decomposition into Chow–Kunneth components.

######  Theorem 4.4 (Deninger–Murre).

There exists a unique direct sum decomposition

(4.1)
$\displaystyle\mathfrak{h}(A/S)=\bigoplus_{i=0}^{2g}\mathfrak{h}^{i}(A/S)$

in $\mathsf{Mot}(S)$ with the property that $[^{t}\Gamma_{(n)}]$ acts as
$n^{i}$ on $\mathfrak{h}^{i}(A/S)$ for every $0\leq i\leq 2g$ and every
$n\in{\rm\mathbb{Z}}$.

We call the decomposition (4.1) the canonical Chow–Kunneth decomposition of
$\mathfrak{h}(A/S)$. In what follows $\mathfrak{h}^{i}(A/S)$ will always
denote the $i$-th component of this decomposition.

We will use the next proposition frequently when doing explicit calculations
with $\mathfrak{h}^{1}$ of abelian schemes.

######  Proposition 4.5.

[24, Proposition 2.2.1] Let $A,B\rightarrow S$ be abelian schemes of the same
relative dimension. Then the pullback map

(4.2) $\displaystyle{\rm Hom}^{0}(A,B)\rightarrow{\rm
Hom}(\mathfrak{h}^{1}(B/S),\mathfrak{h}^{1}(A/S))$

is an isomorphism of ${\rm\mathbb{Q}}$-vector spaces.

We also include the following result on abelian schemes, which follows from
[17, §2, Lemma 1] and the fact that adequate schemes are normal.

######  Proposition 4.6.

Suppose that $S$ is integral with generic point ${\rm Spec}(K)$. Then the map
$A/S\mapsto A_{K}/K$ identifies the category of abelian schemes over $S$ with
a full subcategory of the category of abelian varieties over $K$.

##  5\. Proof of main theorem

This section is the technical heart of the paper. In §5.1 we state and prove
the isomorphism of motives (1.2) promised in the introduction. Next in §5.2 we
translate this isomorphism into a concrete statement about Chow groups, and in
§5.3 we show that one of these Chow groups contains the Ceresa cycle. In §5.4
we identify the image of the Ceresa cycle under our constructed isomorphism.
In the last two subsections we combine everything and prove the theorems and
corollaries stated in the introduction.

###  5.1. An isomorphism of motives

Let $C=C_{a,b}$ be a bielliptic Picard curve (3.1) over a field $k$ with
Jacobian $J$. The $\mu_{6}$-action on $C$ induces, via pullback, a
$\mu_{6}$-action on $J$ and $\mathfrak{h}(J)$. The uniqueness of the
Chow–Kunneth decomposition (4.1) shows that this $\mu_{6}$-action restricts to
a $\mu_{6}$-action on $\mathfrak{h}^{i}(J)$ for each $0\leq i\leq 6$. Let
$E^{\Delta}\colon y^{2}=x^{3}+4b(a^{2}-4b)^{2}$ be the sextic twist of
$E:y^{2}=x^{3}+16(a^{2}-4b)$ by $\Delta=\Delta_{a,b}=16b(a^{2}-4b)$. Recall
from §4.5 that $\mathfrak{h}^{3}(J)^{\mu_{6}}$ is a direct summand of
$\mathfrak{h}^{3}(J)$.

######  Theorem 5.1.

There is an isomorphism in $\mathsf{Mot}(k)$:
$\mathfrak{h}^{3}(J)^{\mu_{6}}\simeq\mathfrak{h}^{1}(E^{\Delta})(-1)$.

In fact, it will be crucial for our proof of Theorem 1.1 to construct this
isomorphism universally. We will use the notation of §3.3, so
$S={\rm\mathbb{A}}^{2}_{{\rm\mathbb{Z}}[1/6]}\setminus\\{\Delta_{a,b}=0\\}$ is
the base scheme and $\mathcal{C}\rightarrow S$ the universal bielliptic Picard
curve with Jacobian $\mathcal{J}\rightarrow S$ and Prym variety
$\mathcal{P}\rightarrow S$. Let $\mathcal{E}\rightarrow S$ be the relative
elliptic curve with equation $y^{2}=x^{3}+16(a^{2}-4b)$ and let
$\mathcal{E}^{\Delta}\rightarrow S$ be the relative elliptic curve with
equation $y^{2}=x^{3}+4b(a^{2}-4b)^{2}$. There is again a $\mu_{6}$-action on
$\mathcal{C}$, $\mathcal{J}$ and $\mathfrak{h}^{3}(\mathcal{J}/S)$. The goal
of this subsection is to show:

######  Theorem 5.2.

There is an isomorphism in $\mathsf{Mot}(S)$:
$\mathfrak{h}^{3}(\mathcal{J}/S)^{\mu_{6}}\simeq\mathfrak{h}^{1}(\mathcal{E}^{\Delta}/S)(-1)$.

Specializing this isomorphism at $k$-points ${\rm Spec}(k)\rightarrow S$
recovers Theorem 5.1. The proof of Theorem 5.2 will be given at the end of
this subsection after combining a few different ingredients.

The starting point is the decomposition

$\mathfrak{h}^{1}(\mathcal{J}/S)=\mathfrak{h}^{1}(\mathcal{P}/S)\oplus\mathfrak{h}^{1}(\mathcal{E}/S)$

induced by the isogeny $\mathcal{J}\sim\mathcal{P}\times_{S}\mathcal{E}$ of
Lemma 3.3. Via the isomorphism
$\mathfrak{h}^{3}(\mathcal{J}/S)\simeq\bigwedge^{3}\mathfrak{h}^{1}(\mathcal{J}/S)$
[26] and similarly for $\mathcal{P}$ and $\mathcal{E}$, we obtain a
$\mu_{6}$-equivariant decomposition

(5.1)
$\displaystyle\mathfrak{h}^{3}(\mathcal{J}/S)=\mathfrak{h}^{3}(\mathcal{P}/S)\oplus(\mathfrak{h}^{2}(\mathcal{P}/S)\otimes\mathfrak{h}^{1}(\mathcal{E}/S))\oplus(\mathfrak{h}^{1}(\mathcal{P}/S)\otimes\mathfrak{h}^{2}(\mathcal{E}/S)).$

######  Lemma 5.3.

$\mathfrak{h}^{3}(\mathcal{J}/S)^{\mu_{6}}=(\mathfrak{h}^{2}(\mathcal{P}/S)\otimes\mathfrak{h}^{1}(\mathcal{E}/S))^{\mu_{3}}$.

###### Proof.

By definition of $\mathcal{P}$ and $\mathcal{E}$ in §3.2, if
$\mu_{2}(S)=\\{1,\tau\\}$ then $\tau$ acts as $(-1)$ on $\mathcal{P}$ and as
the identity on $\mathcal{E}$. Since $[^{t}\Gamma_{(n)}]$ acts as $n^{i}$ on
$\mathfrak{h}^{i}$ of an abelian scheme, we see that $\tau$ acts on
$\mathfrak{h}^{i}(\mathcal{P}/S)\otimes\mathfrak{h}^{3-i}(\mathcal{E}/S)$ as
$-1$ for $i=1,3$ and as the identity for $i=2$. It follows that
$(\mathfrak{h}^{i}(\mathcal{P}/S)\otimes\mathfrak{h}^{3-i}(\mathcal{E}/S))^{\mu_{6}}=(\mathfrak{h}^{i}(\mathcal{P}/S)\otimes\mathfrak{h}^{3-i}(\mathcal{E}/S))^{\mu_{2}}=0$
if $i=1,3$ and
$(\mathfrak{h}^{2}(\mathcal{P}/S)\otimes\mathfrak{h}^{1}(\mathcal{E}/S))^{\mu_{6}}=(\mathfrak{h}^{2}(\mathcal{P}/S)\otimes\mathfrak{h}^{1}(\mathcal{E}/S))^{\mu_{3}}$,
so we conclude by (5.1). ∎

It remains to determine
$(\mathfrak{h}^{2}(\mathcal{P}/S)\otimes\mathfrak{h}^{1}(\mathcal{E}/S))^{\mu_{3}}$.
To this end, we decompose $\mathfrak{h}^{2}(\mathcal{P}/S)$ further using
results of Moonen [32] and the quaternionic action on $\mathcal{P}$.

Recall the finite etale cover $\tilde{S}\rightarrow S$ from §3.3 given by
adjoining a sixth root of unity and a sixth root of the discriminant $\Delta$.
This cover has Galois group $G={\rm Aut}(\tilde{S}/S)\simeq D_{6}$. Also
recall from Theorem 3.4 the $G$-representation ${\rm
Hom}^{\operatorname{sym}}(\mathcal{P}_{\tilde{S}},\mathcal{P}^{\vee}_{\tilde{S}})\otimes{\rm\mathbb{Q}}$
of self-dual homomorphisms; let ${\rm NS}_{\mathcal{P}}$ be the corresponding
Artin motive in $\mathsf{Mot}(S)$ under the equivalence of Lemma 4.3. The
$\mu_{3}$-action on $\mathcal{P}$ induces via pullback a $\mu_{3}$-action on
${\rm NS}_{\mathcal{P}}$ which is described in Lemma 3.5.

######  Proposition 5.4.

There exists a $\mu_{3}$-stable decomposition

(5.2)
$\displaystyle\mathfrak{h}^{2}(\mathcal{P}/S)=\mathfrak{h}_{\mathrm{alg}}^{2}(\mathcal{P}/S)\oplus\mathfrak{h}^{2}_{\mathrm{tr}}(\mathcal{P}/S)$

in $\mathsf{Mot}(S)$ with the following two properties:

  1. (1)

there is a $\mu_{3}$-equivariant isomorphism
$\mathfrak{h}_{\mathrm{alg}}^{2}(\mathcal{P}/S)\simeq{\rm
NS}_{\mathcal{P}}(-1)$;

  2. (2)

The $\mu_{3}$-action on $\mathfrak{h}_{\mathrm{tr}}^{2}(\mathcal{P})$ is
trivial.

###### Proof.

By Corollary 4.2, it suffices to find a $G$-stable decomposition of
$\mathfrak{h}^{2}(\mathcal{P}_{\tilde{S}}/\tilde{S})$ with properties
analogous to $(1)$ and $(2)$. Theorem 3.4 shows that $B={\rm
End}^{0}(\mathcal{P}_{\tilde{S}})$ is a nonsplit quaternion algebra. By
functoriality we obtain a homomorphism $B^{\mathrm{op}}\rightarrow{\rm
End}(\mathfrak{h}^{1}(\mathcal{P}_{\tilde{S}}/\tilde{S}))$, where
$B^{\mathrm{op}}$ is the opposite algebra of $B$. We identify
$B^{\mathrm{op}}$ with $B$ using the canonical involution on $B$. Moonen [32,
Theorem 7.2] has shown that this $B$-action on
$\mathfrak{h}^{1}(\mathcal{P}_{\tilde{S}}/\tilde{S})$ can be used to refine
the Chow–Kunneth decomposition of
$\mathfrak{h}(\mathcal{P}_{\tilde{S}}/\tilde{S})$. Our specific situation is
explained in detail in [32, §8.2]: let

(5.3)
$\displaystyle\mathfrak{h}^{2}(\mathcal{P}_{\tilde{S}}/\tilde{S})=R^{(2,0)}\oplus
R^{(1,1)}$

be the decomposition described in [32, Bottom of p. 104]. There exist
algebraic representations $\rho_{1},\rho_{2}$ of $B^{\times}$ such that ${\rm
Hom}(M,R^{(2,0)})$ is $\rho_{1}$-isotypic and ${\rm Hom}(M,R^{(1,1)})$ is
$\rho_{2}$-isotypic for every motive $M$ in $\mathsf{Mot}(S)$. The
representation $\rho_{1}$ is a twist of the conjugation action of $B^{\times}$
on trace zero elements in $B$, and the representation $\rho_{2}$ is the norm
map $\text{Nm}\colon B^{\times}\rightarrow{\rm\mathbb{Q}}^{\times}$. The
$\mu_{3}$-action on $\mathcal{P}_{\tilde{S}}$ is induced from restricting the
$B^{\times}$-action to a quadratic field
${\rm\mathbb{Q}}(\omega)^{\times}\subset B^{\times}$ and further restricting
to its subgroup
$\mu_{3}=\langle\omega\rangle\subset{\rm\mathbb{Q}}(\omega)^{\times}$. Since
Nm is trivial on $\mu_{3}$, it follows that $\mu_{3}$ acts trivially on
$R^{(1,1)}$. By construction of $R^{(2,0)}$ [32, §6.4], there is a canonical
isomorphism $\alpha\colon\mathbb{L}\otimes({\rm
Hom}^{\operatorname{sym}}(\mathcal{P}_{\tilde{S}},\mathcal{P}_{\tilde{S}}^{\vee})\otimes{\rm\mathbb{Q}})\xrightarrow{\sim}R^{(2,0)}$.
Since $G$ preserves the isomorphism classes of $\rho_{1}$ and $\rho_{2}$, the
decomposition (5.3) is $G$-stable. Therefore by Corollary 4.2,
$R^{(2,0)}\oplus R^{(1,1)}$ descends to a decomposition
$\mathfrak{h}_{\mathrm{alg}}^{2}(\mathcal{P}/S)\oplus\mathfrak{h}^{2}_{\mathrm{tr}}(\mathcal{P}/S)$.
Since the isomorphism $\alpha$ is $G$-equivariant, it descends to an
isomorphism ${\rm
NS}_{\mathcal{P}}(-1)\simeq\mathfrak{h}^{2}_{\mathrm{alg}}(\mathcal{P}/S)$, as
desired. ∎

For the remainder of the paper we will fix a decomposition (5.4). (It is in
fact unique, but we will not use this.)

######  Remark 5.5.

Let $\bar{K}$ be the algebraic closure of the function field of $S$. Then the
base change of the decomposition (5.2) along ${\rm Spec}(\bar{K})\rightarrow
S$ is the decomposition of $\mathfrak{h}^{2}(\mathcal{P}_{\bar{K}})$ into its
algebraic and transcendental part [23], explaining our choice of notation.

######  Corollary 5.6.

$(\mathfrak{h}^{2}(\mathcal{P}/S)\otimes\mathfrak{h}^{1}(\mathcal{E}/S))^{\mu_{3}}\simeq({\rm
NS}_{\mathcal{P}}\otimes\mathfrak{h}^{1}(\mathcal{E}/S))^{\mu_{3}}(-1)$

###### Proof.

By Proposition 5.4, $\mu_{3}$ acts trivially on
$\mathfrak{h}^{2}_{\mathrm{tr}}(\mathcal{P}/S)$. Since the $\mu_{3}$-action on
$\mathcal{E}$ is faithful, $\mathfrak{h}^{1}(\mathcal{E}/S)^{\mu_{3}}=0$. We
conclude that
$(\mathfrak{h}^{2}_{\mathrm{tr}}(\mathcal{P}/S)\otimes\mathfrak{h}^{1}(\mathcal{E}/S)))^{\mu_{3}}=0$
and so
$(\mathfrak{h}^{2}(\mathcal{P}/S)\otimes\mathfrak{h}^{1}(\mathcal{E}/S))^{\mu_{3}}=(\mathfrak{h}_{\mathrm{alg}}^{2}(\mathcal{P}/S)\otimes\mathfrak{h}^{1}(\mathcal{E}/S))^{\mu_{3}}\simeq({\rm
NS}_{\mathcal{P}}\otimes\mathfrak{h}^{1}(\mathcal{E}/S))^{\mu_{3}}(-1)$, again
by Proposition 5.4. ∎

To analyze the piece ${\rm
NS}_{\mathcal{P}}\otimes\mathfrak{h}^{1}(\mathcal{E}/S)$, we define and study
an auxiliary abelian scheme $\mathcal{E}\otimes\mathrm{std}$. Recall from §3.3
the finite etale cover $\tilde{S}\rightarrow S$ with Galois group $G\simeq
D_{6}$ and its two-dimensional representation $\mathrm{std}\colon
G\rightarrow{\rm GL}_{2}({\rm\mathbb{Q}})$, which in fact lands in ${\rm
GL}_{2}({\rm\mathbb{Z}})$. Let $\mathcal{E}\otimes\mathrm{std}$ be the (up to
isomorphism) unique abelian scheme over $S$ that is isomorphic to
$\mathcal{E}^{2}$ over $\tilde{S}$ with descent data given by
$\mathrm{std}\colon G\rightarrow{\rm GL}_{2}({\rm\mathbb{Z}})\subset{\rm
End}(\mathcal{E}^{2})$. (This is the analogue of the construction of [31] in
the relative setting.)

######  Lemma 5.7.

There is an isogeny
$\mathcal{E}\otimes\mathrm{std}\sim\mathcal{E}^{\Delta}\times\mathcal{E}^{\Delta^{-1}}$.

###### Proof.

Let $K={\rm\mathbb{Q}}(a,b)$ be the function field of $S$ and denote the base
change of $\mathcal{E}$ to $K$ by $E$. Proposition 4.6 shows that it suffices
to prove the lemma over $K$. Let $\tilde{K}=K(\omega,\sqrt[6]{\Delta})$ denote
the function field of $\tilde{S}$. For $d\in K^{\times}$, write
$E^{d}:y^{2}=x^{3}+16(a^{2}-4b)d$ for the sextic twist of $E$ by $d$. Then
$E\otimes\mathrm{std}$ is an isogeny factor of

${\rm Res}^{\tilde{K}}_{K}E\sim{\rm
Res}^{K(\sqrt[6]{\Delta})}_{K}(E^{2})\sim\left({\rm
Res}^{K(\sqrt[6]{\Delta})}_{K}E\right)^{2}\sim\left(\prod_{i=0}^{5}E^{\Delta^{i}}\right)^{2}.$

Indeed: the first isogeny follows from the fact that $E$ is isogenous to its
own $(-3)$-quadratic twist (this being true for any $j$-invariant $0$ elliptic
curve), the second is a formal property of restriction of scalars, and the
third is the isogeny decomposition that corresponds (by base change) to the
six characters of $H={\rm Gal}(\tilde{K}/K(\omega))$. Since
$\mathrm{std}|_{H}$ is the sum of the two characters of order $6$, we must
have

$E\otimes\mathrm{std}\sim E^{\Delta}\times E^{\Delta^{5}}\simeq
E^{\Delta}\times E^{\Delta^{-1}},$

as claimed. ∎

######  Lemma 5.8.

$({\rm
NS}_{\mathcal{P}}\otimes\mathfrak{h}^{1}(\mathcal{E}/S))^{\mu_{3}}\simeq\mathfrak{h}^{1}(\mathcal{E}^{\Delta}/S)$.

###### Proof.

Use Theorem 3.4 to decompose the $G$-representation ${\rm NS}_{\mathcal{P}}$
as $\mathrm{triv}\oplus\mathrm{std}$. By Lemma 3.5, $\mu_{3}$ acts trivially
on $\mathrm{triv}$ and as a direct sum of two nontrivial characters on
$\mathrm{std}\otimes{\rm\overline{\mathbb{Q}}}$. The $\mu_{3}$-actions on
$\mathcal{E}$ and $\mathrm{std}$ combine to a $\mu_{3}$-action on
$\mathcal{E}\otimes\mathrm{std}$. Write
$\mathcal{A}:=(\mathcal{E}\otimes\mathrm{std})^{\mu_{3},\circ}$. We then have

(5.4) $({\rm
NS}_{\mathcal{P}}\otimes\mathfrak{h}^{1}(\mathcal{E}/S))^{\mu_{3}}\simeq(\mathrm{std}\otimes\mathfrak{h}^{1}(\mathcal{E}/S))^{\mu_{3}}\simeq(\mathfrak{h}^{1}((\mathcal{E}\otimes\mathrm{std})/S))^{\mu_{3}}\simeq\mathfrak{h}^{1}(\mathcal{A}/S),$

where the first isomorphism follows from the fact that
$\mathfrak{h}^{1}(\mathcal{E}/S)^{\mu_{3}}=0$, and the last two follow from
Lemma 4.1 and Proposition 4.5. To prove the lemma, it remains to show that the
abelian scheme $\mathcal{A}$ is isogenous to $\mathcal{E}^{\Delta}$.

Let $\mathcal{A}^{\prime}$ be the quotient abelian scheme
$(\mathcal{E}\otimes\mathrm{std})/\mathcal{A}$. A tangent space calculation
(using the $\mu_{3}$-action on $\mathrm{std}$ and $\mathcal{E}$) shows that
$\mathcal{A}\rightarrow S$ has relative dimension $1$, hence so does the
quotient $\mathcal{A}^{\prime}\rightarrow S$. Given
$d\in\mathcal{O}(S)^{\times}$ write $\mathcal{E}^{d}\colon
y^{2}=x^{3}+16(a^{2}-4b)d$ for the sextic twist of $\mathcal{E}$ by $d$. By
complete reducibility and Lemma 5.7, $\mathcal{A}$ is isogenous to either
$\mathcal{E}^{\Delta}\colon y^{2}=x^{3}+4b(a^{2}-4b)^{2}$ or
$\mathcal{E}^{\Delta^{-1}}\colon y^{2}=x^{3}+b^{-1}$. We will exclude the
latter possibility using a rather indirect argument.

Fix a unit $d\in\mathcal{O}(S)^{\times}$, consider the isomorphism
$\lambda_{d}\colon S\rightarrow S$ mapping $(a,b)$ to $(da,d^{2}b)$ and let
$\mathcal{J}^{d}=\lambda_{d}^{*}\mathcal{J}$. Then $\mathcal{J}$ and
$\mathcal{J}^{d}$ are sextic twists in the sense that they become isomorphic
over the $\mu_{6}$-cover $f_{d}\colon S_{d}\rightarrow S$ given by adjoining a
sixth root $\delta$ of $d$. There exists an isomorphism
$f_{d}^{*}\mathcal{J}\xrightarrow{\sim}f_{d}^{*}\mathcal{J}^{d}$ of abelian
schemes over $S_{d}$ induced by the isomorphism
$(x,y)\mapsto(\delta^{3}x,\delta^{4}y)$ between $\mathcal{C}$ and
$\lambda_{d}^{*}\mathcal{C}$. This in turn induces an isomorphism
$\mathfrak{h}^{3}(f_{d}^{*}\mathcal{J}^{d}/S_{d})\xrightarrow{\sim}\mathfrak{h}^{3}(f_{d}^{*}\mathcal{J}/S_{d})$.
Lemma 4.1 shows that the induced morphism on $\mu_{6}$-fixed points descends
to an isomorphism
$\mathfrak{h}^{3}(\mathcal{J}/S)^{\mu_{6}}\simeq\mathfrak{h}^{3}(\mathcal{J}^{d}/S)^{\mu_{6}}$
over $S$.

Now assume, for the sake of contradiction, that
$\mathcal{A}\sim\mathcal{E}^{\Delta^{-1}}$. Then
$\mathfrak{h}^{3}(\mathcal{J}/S)^{\mu_{6}}\simeq\mathfrak{h}^{1}(\mathcal{E}^{\Delta^{-1}}/S)(-1)$
by Lemma 5.3, Corollary 5.6 and (5.4). Pulling back this isomorphism along
$\lambda_{d}$ and using the fact that
$\lambda^{*}_{d}\mathcal{E}^{\Delta^{-1}}\simeq\mathcal{E}^{d^{-2}\Delta^{-1}}$,
we get an isomorphism
$\mathfrak{h}^{3}(\mathcal{J}^{d}/S)^{\mu_{6}}\simeq\mathfrak{h}^{1}(\mathcal{E}^{d^{-2}\Delta^{-1}}/S)(-1)$.
Combining the last three isomorphisms shows that
$\mathfrak{h}^{1}(\mathcal{E}^{\Delta^{-1}}/S)\simeq\mathfrak{h}^{1}(\mathcal{E}^{d^{-2}\cdot\Delta^{-1}}/S)$.
By Proposition 4.5, $\mathcal{E}^{\Delta^{-1}}$ and
$\mathcal{E}^{d^{-2}\cdot\Delta^{-1}}$ are isogenous over $S$ for all such
$d$. This is a contradiction, since setting $s=(a,b)=(1,1)\in
S({\rm\mathbb{Q}})$ and $d=3^{-1}\in\mathcal{O}(S)^{\times}$ produces two
curves $\mathcal{E}^{\Delta^{-1}}_{(a,b)}\colon y^{2}=x^{3}+1$ and
$\mathcal{E}^{9\Delta^{-1}}_{(a,b)}\colon y^{2}=x^{3}+9$ that are not
isogenous over ${\rm\mathbb{Q}}$ (by counting ${\rm\mathbb{F}}_{7}$-points,
for example). ∎

###### Proof of Theorem 5.2.

We have

$\displaystyle\mathfrak{h}^{3}(\mathcal{J}/S)^{\mu_{6}}=(\mathfrak{h}^{2}(\mathcal{P}/S)\otimes\mathfrak{h}^{1}(\mathcal{E}/S))^{\mu_{3}}\simeq({\rm
NS}_{\mathcal{P}}\otimes\mathfrak{h}^{1}(\mathcal{E}/S))^{\mu_{3}}(-1)\simeq\mathfrak{h}^{1}(\mathcal{E}^{\Delta}/S)(-1),$

using Lemma 5.3, Corollary 5.6 and Lemma 5.8 respectively. ∎

###  5.2. Chow groups

Our next goal is to translate the isomorphism of Theorem 5.2 into a concrete
statement about Chow groups. The (covariant) Chow groups of a motive $M$ in
$\mathsf{Mot}(S)$ are by definition the ${\rm\mathbb{Q}}$-vector spaces ${\rm
CH}^{p}(M):={\rm Hom}(\mathbb{L}^{\otimes p},M)$ for $p\in{\rm\mathbb{Z}}$,
where we recall from §4.2 that $\mathbb{L}$ denotes the Lefchetz motive. For
example, ${\rm CH}^{p}(\mathfrak{h}(X/S))={\rm CH}^{p}(X;{\rm\mathbb{Q}})$ for
all $X/S$ in $\mathsf{SmProj}(S)$. We will compute ${\rm CH}^{2}$ of
$\mathfrak{h}^{3}(\mathcal{J}/S)^{\mu_{6}}$ and
$\mathfrak{h}^{1}(\mathcal{E}^{\Delta}/S)(-1)$ below. To describe the former
we will use the Beauville decomposition of the Chow groups of an abelian
variety [3], extended by Deninger–Murre to the relative setting [13, Theorem
2.19].

######  Proposition 5.9 (Beauville decomposition).

Let $A/S$ be an abelian scheme and $p\in{\rm\mathbb{Z}}_{\geq 0}$. For
$i\in{\rm\mathbb{Z}}$ let

${\rm CH}^{p}_{(i)}(A;{\rm\mathbb{Q}}):=\\{\alpha\in{\rm
CH}^{p}(A;{\rm\mathbb{Q}})\mid(n)^{*}\alpha=n^{2p-i}\alpha\text{ for all
}n\in{\rm\mathbb{Z}}\\}.$

Then ${\rm CH}^{p}_{(i)}(A;{\rm\mathbb{Q}})$ is nonzero for only finitely many
$i$ and we have a direct sum decomposition

(5.5) $\displaystyle{\rm
CH}^{p}(A;{\rm\mathbb{Q}})=\bigoplus_{i\in{\rm\mathbb{Z}}}{\rm
CH}^{p}_{(i)}(A;{\rm\mathbb{Q}}).$

######  Proposition 5.10.

Let $T$ be an adequate scheme and $T\rightarrow S$ a morphism. Then there are
isomorphisms:

  1. (1)

${\rm
CH}^{2}(\mathfrak{h}^{1}(\mathcal{E}^{\Delta}_{T}/T)(-1))=\mathcal{E}^{\Delta}(T)\otimes{\rm\mathbb{Q}}$.

  2. (2)

${\rm CH}^{2}(\mathfrak{h}^{3}(\mathcal{J}_{T}/T)^{\mu_{6}})={\rm
CH}^{2}_{(1)}(\mathcal{J}_{T};{\rm\mathbb{Q}})^{\mu_{6}}$.

Moreover if $T^{\prime}\rightarrow S$ is another such morphism and if $f\colon
T^{\prime}\rightarrow T$ is an $S$-morphism that admits pullbacks (see §4.3)
then the above isomorphisms are compatible with base change along $f$.

###### Proof.

  1. (1)

We have ${\rm CH}^{2}(\mathfrak{h}^{1}(\mathcal{E}^{\Delta}_{T}/T)(-1))={\rm
CH}^{1}(\mathfrak{h}^{1}(\mathcal{E}^{\Delta}_{T}/T))$. We will prove the
stronger statement that if $X\rightarrow T$ is an abelian scheme of relative
dimension $1$ then ${\rm
CH}^{1}(\mathfrak{h}^{1}(X/T))=X(T)\otimes{\rm\mathbb{Q}}$. This is the
relative version of a classical curve computation [34, Theorem 2.7.2]; we
briefly sketch the details. Write $\Gamma_{e}\subset X\times_{T}X$ for the
graph of the zero section $e\colon T\rightarrow X$. Then
$\mathfrak{h}^{1}(X/T)=(X/T,[\Delta_{X/T}]-[\Gamma_{e}]-[^{t}\Gamma_{e}],0)$.
Using the equivalence between Cartier divisors and line bundles and unwinding
the definition of ${\rm CH}^{1}(\mathfrak{h}^{1}(X/T))$, we see that it is
exactly the ${\rm\mathbb{Q}}$-tensor product of the set of isomorphism classes
of line bundles on $X$, fiberwise of degree zero and rigidified along the zero
section. In other words, ${\rm
CH}^{1}(\mathfrak{h}^{1}(X/T))=\mathbf{Pic}^{0}_{X/T}(T)\otimes{\rm\mathbb{Q}}$,
where $\mathbf{Pic}^{0}_{X/T}$ denotes the identity component of the Picard
functor [10, §8.1, Proposition 4]. Since $X$ is a relative elliptic curve, the
map $P\mapsto[P]-[e]$ induces an isomorphism $X\simeq\mathbf{Pic}^{0}_{X/T}$.

  2. (2)

By definition of the Chow–Kunneth decomposition of Theorem 4.4 and the
Beauville decomposition (5.5), ${\rm CH}^{p}(\mathfrak{h}^{i}(A/S))={\rm
CH}^{p}_{(2p-i)}(A)$ for every abelian scheme $A/S$ and
$i,p\in{\rm\mathbb{Z}}$. Therefore ${\rm
CH}^{2}(\mathfrak{h}^{3}(\mathcal{J}_{T}/T))={\rm
CH}^{2}_{(1)}(\mathcal{J}_{T};{\rm\mathbb{Q}})$. Since ${\rm
CH}^{p}(M^{G})={\rm CH}^{p}(M)^{G}$ for every finite group $G$ acting on a
motive $M$, the result follows from taking $\mu_{6}$-fixed points.

The statement about compatibility with pullbacks follows from the construction
of the isomorphisms. ∎

######  Corollary 5.11.

Let $T$ be an adequate scheme and $T\rightarrow S$ a morphism that admits
pullbacks (see §4.3). Then there exists an isomorphism of
${\rm\mathbb{Q}}$-vector spaces

$\Phi_{T}\colon{\rm
CH}^{2}_{(1)}(\mathcal{J}_{T};{\rm\mathbb{Q}})^{\mu_{6}}\xrightarrow{\sim}\mathcal{E}^{\Delta}(T)\otimes{\rm\mathbb{Q}}$

with the following property: If $T^{\prime}\rightarrow S$ is another such
morphism and $f\colon T^{\prime}\rightarrow T$ a morphism of $S$-schemes that
admits pullbacks then the following diagram is commutative:

${{\rm
CH}^{2}_{(1)}(\mathcal{J}_{T};{\rm\mathbb{Q}})^{\mu_{6}}}$${\mathcal{E}^{\Delta}(T)\otimes{\rm\mathbb{Q}}}$${{\rm
CH}^{2}_{(1)}(\mathcal{J}_{T^{\prime}};{\rm\mathbb{Q}})^{\mu_{6}}}$${\mathcal{E}^{\Delta}(T^{\prime})\otimes{\rm\mathbb{Q}}}$$\scriptstyle{\Phi_{T}}$$\scriptstyle{f^{*}}$$\scriptstyle{f^{*}}$$\scriptstyle{\Phi_{T^{\prime}}}$

###### Proof.

Choose an isomorphism
$\phi\colon\mathfrak{h}^{3}(\mathcal{J}/S)^{\mu_{6}}\xrightarrow{\sim}\mathfrak{h}^{1}(\mathcal{E}^{\Delta}/S)$
using Theorem 5.2. This defines an isomorphism $\phi_{T}$ in $\mathsf{Mot}(T)$
after base change, see §4.3. Taking ${\rm CH}^{2}$ of this isomorphism and
using Proposition 5.10, we get the desired isomorphism $\Phi_{T}$. ∎

###  5.3. The image of the Ceresa cycle

The defining equation (3.1) of $\mathcal{C}$ shows there exists a unique
section $\infty\colon S\rightarrow\mathcal{C}$ at infinity. We view
$\mathcal{C}$ as a closed subscheme of $\mathcal{J}$ using the Abel–Jacobi map
based at $\infty$. We define the universal Ceresa cycle to be the cycle

$\kappa(\mathcal{C}):=[\mathcal{C}]-(-1)^{*}[\mathcal{C}]\in{\rm
CH}^{2}(\mathcal{J}).$

For every field-valued point $s\colon{\rm Spec}(k)\rightarrow S$, the pullback
$s^{*}(\kappa(\mathcal{C}))\in{\rm CH}^{2}(\mathcal{J}_{s})$ equals the
(canonical) Ceresa cycle $\kappa(\mathcal{C}_{s})$ of the curve
$\mathcal{C}_{s}$ over $k$ with respect to the basepoint at infinity defined
in §2.1. The next lemma shows that ${\rm CH}^{2}$ of
$\mathfrak{h}^{3}(\mathcal{J}/S)^{\mu_{6}}$ naturally contains the Ceresa
cycle.

######  Lemma 5.12.

For every field $k$ and $s\in S(k)$, we have $\kappa(\mathcal{C}_{s})\in{\rm
CH}^{2}_{(1)}(\mathcal{J}_{s};{\rm\mathbb{Q}})^{\mu_{6}}={\rm
CH}^{2}(\mathfrak{h}^{3}(\mathcal{J}_{s})^{\mu_{6}})$.

###### Proof.

Write $C=\mathcal{C}_{s}$ and $J=\mathcal{J}_{s}$. Since $J$ is defined over a
field, Beauville has shown [3, Theoreme and Proposition 3(a)] that ${\rm
CH}^{2}_{(1)}(J;{\rm\mathbb{Q}})$ is zero unless $i\in\\{0,1,2\\}$. So using
(5.5) we may write $[C]=[C]_{0}+[C]_{1}+[C]_{2}$ where $(n)^{*}$ acts on
$[C]_{i}$ as $n^{4-i}$. Therefore
$\kappa(C)=[C]-(-1)^{*}[C]=([C]_{0}+[C]_{1}+[C]_{2})-([C]_{0}-[C]_{1}+[C]_{2})=2[C]_{1}\in{\rm
CH}^{2}_{(1)}(J;{\rm\mathbb{Q}})$. Since $\kappa(C)$ is also
$\mu_{6}$-invariant, we conclude the lemma. ∎

Let $\kappa(\mathcal{C})_{(1)}$ be the component in ${\rm
CH}^{2}_{(1)}(\mathcal{J})$ of $\kappa(\mathcal{C})$ in the Beauville
decomposition (5.5) of ${\rm CH}^{2}(\mathcal{J})$. (It might very well be
that $\kappa(\mathcal{C})=\kappa(\mathcal{C})_{(1)}$, but this is not needed
in our analysis.) Since $\kappa(\mathcal{C})$ is $\mu_{6}$-invariant,
$\kappa(\mathcal{C})_{(1)}\in{\rm CH}^{2}_{(1)}(\mathcal{J})^{\mu_{6}}={\rm
CH}^{2}(\mathfrak{h}^{3}(\mathcal{J}/S)^{\mu_{6}})$. Choose for the remainder
of the paper a collection of isomorphisms $\Phi_{T}$ satisfying the
conclusions of Corollary 5.11. Define

$\sigma_{0}:=\Phi_{S}(\kappa(\mathcal{C})_{(1)})\in\mathcal{E}^{\Delta}(S)\otimes{\rm\mathbb{Q}}.$

Let $\text{den}(\sigma_{0})$ be the smallest positive integer such that
$\text{den}(\sigma_{0})\cdot\sigma_{0}\in\mathcal{E}^{\Delta}(S)$. Let
$\sigma=\text{den}(\sigma_{0})\cdot\sigma_{0}$, a section of the abelian
scheme $\mathcal{E}^{\Delta}\rightarrow S$.

######  Corollary 5.13.

For every field $k$ and $s\in S(k)$, $\sigma(s)\in\mathcal{E}^{\Delta}_{s}(k)$
has infinite order if and only if $\kappa(\mathcal{C}_{s})\in{\rm
CH}^{2}(\mathcal{J}_{s})$ is of infinite order.

###### Proof.

Follows from Corollary 5.11 and Lemma 5.12. ∎

###  5.4. Identifying $\sigma$

Our next goal is to explicitly identify the section $\sigma$ of
$\mathcal{E}^{\Delta}\rightarrow S$, at least up to integer multiples. The
equation $y^{2}=x^{3}+4b(a^{2}-4b)^{2}$ allows us to define
${\rm\mathbb{G}}_{m}$-actions on $\mathcal{E}^{\Delta}$ and
$S\subset{\rm\mathbb{A}}^{2}$ by the formulas
$\lambda\cdot(x,y)=(\lambda^{2}x,\lambda^{3}y)$ and
$\lambda\cdot(a,b)=(\lambda a,\lambda^{2}b)$. With these actions, the morphism
$\mathcal{E}^{\Delta}\rightarrow S$ is ${\rm\mathbb{G}}_{m}$-equivariant.
Write $\mathrm{MW}^{{\rm\mathbb{G}}_{m}}(\mathcal{E}^{\Delta}/S)$ for the set
of ${\rm\mathbb{G}}_{m}$-equivariant sections of
$\mathcal{E}^{\Delta}\rightarrow S$. It is an abelian group, the
${\rm\mathbb{G}}_{m}$-equivariant Mordell–Weil group of
$\mathcal{E}^{\Delta}/S$.

######  Lemma 5.14.

$\mathrm{MW}^{{\rm\mathbb{G}}_{m}}(\mathcal{E}^{\Delta}/S)$ is free of rank
$1$, generated by the section $\mathcal{Q}:=((a^{2}-4b),a(a^{2}-4b))$.

###### Proof.

Let $T$ be the closed subscheme of $S$ given by setting $a=1$. Restricting
sections to $T$ induces an injection
$\mathrm{MW}^{{\rm\mathbb{G}}_{m}}(\mathcal{E}^{\Delta}/S)\hookrightarrow\mathcal{E}^{\Delta}(T)$,
so it suffices to prove that $\mathcal{Q}|_{T}$ generates the Mordell–Weil
group of the elliptic surface $\mathcal{E}^{\Delta}|_{T}\rightarrow T$.

Let $\pi\colon\mathcal{X}\rightarrow{\rm\mathbb{P}}^{1}_{{\rm\mathbb{Q}}}$ be
the minimal regular model of $\mathcal{E}^{\Delta}_{T}\rightarrow
T={\rm\mathbb{P}}^{1}_{{\rm\mathbb{Q}}}\setminus\\{0,1/4,\infty\\}$. Using
Tate’s algorithm, we see that the three singular fibers above $b=0,1/4,\infty$
have Kodaira type $\text{\rm II},\text{\rm IV}$ and I${}_{0}^{*}$
respectively. By [40, Lemma 7.8], there are no nonzero torsion sections of
$\pi$. By the Shioda–Tate formula [40, Theorem 6.3, Proposition 6.6 and §8.8],
the Mordell–Weil group of
$\mathcal{X}_{\bar{{\rm\mathbb{Q}}}}\rightarrow{\rm\mathbb{P}}^{1}_{\bar{{\rm\mathbb{Q}}}}$
is free of rank $2$. The $\mu_{3}$-action on $\mathcal{E}^{\Delta}$ shows that
it is also a ${\rm\mathbb{Z}}[\omega]$-module, necessarily of rank $1$. Taking
Galois invariants, it follows that the Mordell–Weil group of
$\mathcal{X}\rightarrow{\rm\mathbb{P}}^{1}_{{\rm\mathbb{Q}}}$ is free of rank
$1$ over ${\rm\mathbb{Z}}$. To show that it is generated by (the closure of)
$\mathcal{Q}|_{T}$, it suffices to find a single $b\in T({\rm\mathbb{Q}})$ for
which $\mathcal{Q}_{b}$ is primitive in
$\mathcal{E}^{\Delta}_{b}({\rm\mathbb{Q}})$, i.e. not divisible by $n$ for
every $n\geq 2$. One may check that this holds for $b=1$. ∎

######  Theorem 5.15.

There exists an integer $N\geq 1$ such that $\sigma=N\cdot\mathcal{Q}$.

###### Proof.

The section $\sigma\in\mathcal{E}^{\Delta}(S)$ is
${\rm\mathbb{G}}_{m}$-equivariant. Moreover by our concrete computation with
$y^{3}=x^{4}+x^{2}+1$ in Lemma 2.8 (combined with Corollary 5.13), it has a
nonzero specialization, so it is not identically zero. We conclude using Lemma
5.14. ∎

###  5.5. Proof of the main theorem

Putting everything together, we can prove the following theorem which implies
Theorem 1.1 of the introduction.

######  Theorem 5.16.

Let $k$ be a field of characteristic $\neq 2,3$ and let $C/k$ be a smooth
projective curve with equation $y^{3}=x^{4}+ax^{2}+b$ and Jacobian $J$. Then
the Ceresa cycle $\kappa(C)\in{\rm CH}^{2}(J)$ with respect to the point at
infinity is torsion if and only if
$\mathcal{Q}_{(a,b)}=((a^{2}-4b),a(a^{2}-4b))$ is a torsion point of the
elliptic curve $E^{\Delta}\colon y^{2}=x^{3}+4b(a^{2}-4b)^{2}$.

###### Proof of Theorem 5.16.

The curve $C$ is isomorphic to the pullback $\mathcal{C}_{s}$ of
$\mathcal{C}\rightarrow S$ along the $k$-point $s=(a,b)\colon{\rm
Spec}(k)\rightarrow S$. By Corollary 5.13, $\kappa(C)$ is torsion if and only
if $\sigma(s)\in\mathcal{E}^{\Delta}_{s}(k)$ is torsion. By Theorem 5.15,
$\sigma(s)$ is a multiple of $\mathcal{Q}_{s}=\mathcal{Q}_{(a,b)}$, so
$\sigma(s)$ is torsion if and only if $\mathcal{Q}_{s}=(a^{2}-4b,a(a^{2}-4b))$
is torsion. ∎

###### Proof of Theorem 1.1.

We take $(a,b)=(2t,1)$. A simple computation (using that $k=\overline{k}$)
shows that there is an isomorphism from $E_{a,b}^{\Delta}$ to
$\widehat{E}\colon y^{2}=x^{3}+1$, sending
$\mathcal{Q}_{(a,b)}=(4t^{2}-4,2t(4t^{2}-4))$ to
$Q_{t}=(\sqrt[3]{t^{2}-1},t)$, for some choice of cube root. Thus
$\kappa(C_{t})$ is torsion if and only if $Q_{t}$ is torsion. ∎

###  5.6. Corollaries of Theorem 1.1

First we prove the classification of torsion Ceresa cycles over
${\rm\mathbb{Q}}$ claimed in the introduction. We use the following
classification of rational torsion points on the elliptic curves
$\widehat{E}_{d}\colon y^{2}=x^{3}+d$ of $j$-invariant $0$.

######  Lemma 5.17.

Let $d\in{\rm\mathbb{Q}}^{\times}$. Then

  1. (1)

$\widehat{E}_{d}({\rm\mathbb{Q}})_{\mathrm{tors}}$ is a subgroup of
${\rm\mathbb{Z}}/6{\rm\mathbb{Z}}$;

  2. (2)

$\widehat{E}_{d}({\rm\mathbb{Q}})$ contains a point of order $2$ if and only
if $d$ is a cube;

  3. (3)

$\widehat{E}_{d}({\rm\mathbb{Q}})$ contains a point of order $3$ if and only
if $d$ is a square or $d=-2^{4}3^{3}m^{6}$.

###### Proof.

Since $\widehat{E}_{d}$ is a CM elliptic curve, it has potentially good
reduction at every prime $p$. The prime-to-$p$ part of
$\widehat{E}_{d}({\rm\mathbb{Q}})_{\mathrm{tors}}$ therefore injects into
$B({\rm\mathbb{F}}_{p})$ for some $j$-invariant $0$ elliptic curve
$B/{\rm\mathbb{F}}_{p}$. Thus $(1)$ follows from the list of
${\rm\mathbb{F}}_{p}$-points of $j$-invariant $0$ elliptic curves over
${\rm\mathbb{F}}_{3}$ and ${\rm\mathbb{F}}_{5}$. Parts $(2)$ and $(3)$ can be
read off from the $2$\- and $3$\- division polynomials of $X_{d}$, which are
$x^{3}+d$ and $x(x^{3}+4d)$, respectively. ∎

In the following proposition we use the notation of the introduction and
denote the bielliptic Picard curve $C_{2t,1}$ by $C_{t}$.

######  Proposition 5.18.

Suppose $C$ is a nice curve over ${\rm\mathbb{Q}}$ that is
${\rm\overline{\mathbb{Q}}}$-isomorphic to some bielliptic Picard curve
$C_{a,b}$ and has torsion Ceresa cycle. Then $C$ is
${\rm\overline{\mathbb{Q}}}$-isomorphic to either $C_{0}$, $C_{3}$, or
$C_{\sqrt{-3}}$.

###### Proof.

By Lemma 3.2, $C$ is ${\rm\overline{\mathbb{Q}}}$-isomorphic to some $C_{a,b}$
with rational $j$-invariant, and the proof of that lemma shows that $a,b$ can
be chosen to be in ${\rm\mathbb{Q}}$. Since $C_{a,b}$ has torsion Ceresa
cycle, the point $\mathcal{Q}_{a,b}=(a^{2}-4b,a(a^{2}-4b))$ is a
${\rm\mathbb{Q}}$-rational torsion point on the $j$-invariant $0$ elliptic
curve $E^{\Delta}_{a,b}:y^{2}=x^{3}+4b(a^{2}-4b)^{2}$. It follows from Lemma
5.17 that $\mathcal{Q}_{a,b}$ has order $2,3$, or $6$. If $\mathcal{Q}_{a,b}$
has order $2$, then the $y$-coordinate $a(a^{2}-4b)^{2}$ of
$\mathcal{Q}_{a,b}$ is zero, so since $\Delta_{a,b}\neq 0$ we have $a=0$.
Hence $C$ is isomorphic to $C_{0,b}$, which is
${\rm\overline{\mathbb{Q}}}$-isomorphic to $C_{0,1}=C_{0}$. Similarly, if
$\mathcal{Q}_{a,b}$ has order $3$, we compute that $a^{2}+12b=0$ hence $C$ is
${\rm\overline{\mathbb{Q}}}$-isomorphic to $C_{\sqrt{-3}}$. If
$\mathcal{Q}_{a,b}$ has order $6$, then Lemma 5.17 shows that there exists an
isomorphism between $E^{\Delta}_{a,b}$ and $y^{2}=x^{3}+1$ that sends
$\mathcal{Q}_{a,b}$ to $(2,3)$. A calculation shows that $a^{2}-36b=0$, in
other words $C$ is ${\rm\overline{\mathbb{Q}}}$-isomorphic to $C_{3}$. ∎

###### Proof of Corollary 1.2.

Follows from Proposition 5.18. ∎

###### Proof of Theorem 1.3.

Recall from §5.3 that $\Phi_{S}(\kappa(\mathcal{C})_{(1)})=\sigma/d$ in
$\mathcal{E}^{\Delta}(S)\otimes{\rm\mathbb{Q}}$, where $d$ is the denominator
of $\sigma_{0}$. Moreover $\sigma=N\cdot\mathcal{Q}$ in
$\mathcal{E}^{\Delta}(S)$ by Theorem 5.15. So
$d\cdot\Phi_{S}(\kappa(\mathcal{C})_{(1)})-N\cdot\mathcal{Q}$ is torsion in
$\mathcal{E}^{\Delta}(S)$, so

(5.6) $\displaystyle
M\cdot(d\cdot\Phi_{S}(\kappa(\mathcal{C})_{(1)})-N\cdot\mathcal{Q})=0\text{ in
}\mathcal{E}^{\Delta}(S)$

for some $M\geq 1$. The orders of the torsion specializations
$\mathcal{Q}_{s}\in\mathcal{E}^{\Delta}_{s}({\rm\mathbb{C}})$, where $s$
varies in $S({\rm\mathbb{C}})$, are unbounded. The relation (5.6) implies that
the same is true for the orders of the torsion specializations of
$\kappa(\mathcal{C}_{s})_{(1)}$, which equals $\kappa(\mathcal{C}_{s})$ by
Lemma 5.12. ∎

Theorem 1.4 will follow from the following corollary of Theorem 5.15. If $A$
is an abelian variety over ${\rm\overline{\mathbb{Q}}}$ of dimension $g$, then
the Beilinson–Bloch height pairing [6]

$\langle\,,\,\rangle_{\mathrm{BB}}\colon{\rm
CH}^{i}(A)_{\mathrm{hom}}\times{\rm
CH}^{g+1-i}(A)_{\mathrm{hom}}\to{\rm\mathbb{R}}$

has been constructed unconditionally by Künnemann [27, Corollary 1.7].

######  Theorem 5.19.

There is a rational number $N$ such that for all
$a,b\in{\rm\overline{\mathbb{Q}}}$ with $\Delta_{a,b}\neq 0$, we have

$\langle\kappa(C_{a,b}),\kappa(C_{a,b})\rangle_{\mathrm{BB}}=N^{2}\langle\mathcal{Q}_{a,b},\mathcal{Q}_{a,b}\rangle_{\mathrm{NT}},$

where $\langle\,,\,\rangle_{\mathrm{NT}}$ is the Neron–Tate height pairing on
$E^{\Delta}({\rm\overline{\mathbb{Q}}})$.

###### Proof.

This follows from Theorem 5.15 and Lemma 5.12, using the fact that the
Beilinson–Bloch height is compatible with correspondences [6, 4.0.3] and
agrees with the Neron–Tate height pairing for divisors on curves [6, 4.0.8]. ∎

###### Proof of Theorem 1.4.

The theorem follows from Theorem 5.19 and the nondegeneracy and Northcott
properties of the Neron–Tate height of an elliptic curve. ∎

##  6\. Algebraic triviality and the Griffiths group

Let $C$ be a nice curve over a field $k$ and $J$ be its Jacobian. The
Griffiths group of $1$-cycles on $J$ is the group ${\rm Gr}_{1}(J)={\rm
Gr}^{g-1}(J)={\rm CH}_{1}(J)_{0}/A_{1}(J)$, where $A_{1}(J)$ is the subgroup
of algebraically trivial $1$-cycles. The very general Ceresa cycle (of a genus
$g\geq 3$ curve over ${\rm\mathbb{C}}$) is known to be of infinite order in
${\rm Gr}_{1}(J)$ [35]. In particular, the Ceresa cycle of a very general
curve is not algebraically trivial, nor is any multiple of it. In this section
we show that this fails for bielliptic Picard curves. We also discuss the
Beilinson–Bloch conjectures in this context, and show that they imply that
${\rm Gr}^{2}(J)$ is finite for certain bielliptic Picard curves.

###  6.1. Proof of Theorem 1.5

We keep the notation of §5.1. Theorem 1.5 follows from the following
proposition. It essentially follows from the fact that homological and
algebraic equivalence on a curve coincide.

######  Proposition 6.1.

There exists an integer $N$, a smooth projective relative curve
$\mathcal{X}\rightarrow S$ with two sections $\sigma_{1},\sigma_{2}$ and a
cycle $\mathcal{Z}$ on ${\rm CH}^{1}(\mathcal{J}\times_{S}X)$, with the
property that $\mathcal{Z}_{\sigma_{1}}=N\cdot\kappa(\mathcal{C})_{(1)}$ and
$\mathcal{Z}_{\sigma_{2}}=0$.

This proposition implies Theorem 1.5, since for every $k$-point $s\colon{\rm
Spec}(k)\rightarrow S$, corresponding to the bielliptic Picard curve
$\mathcal{C}_{s}=C$ with $X=\mathcal{X}_{s}$, $p_{i}=\sigma_{i}(s)$ and
$Z=\mathcal{Z}_{s}$, the cycle $Z\in{\rm CH}^{1}(J\times_{k}X)$ has the
property that $Z_{p_{1}}=N\cdot\kappa(C)_{(1)}$ and $Z_{p_{2}}=0$. Since
$\kappa(C)_{(1)}=\kappa(C)$ by Lemma 5.12, $N\cdot\kappa(C)$ is algebraically
equivalent to zero.

###### Proof.

We will use the isomorphisms $\Phi$ from Corollary 5.11. Recall from §5.3 that
there exists an $N_{1}$ such that
$\Phi_{S}(N_{1}\cdot\kappa(\mathcal{C})_{(1)})=\sigma\in\mathcal{E}^{\Delta}(S)$.
We can deform the section $\sigma$ of $\mathcal{E}^{\Delta}$ to the identity
section $\infty\colon S\rightarrow\mathcal{E}^{\Delta}$. More precisely, there
exists a smooth projective curve $\mathcal{X}\rightarrow S$, a section
$\mathcal{V}\in\mathcal{E}^{\Delta}(\mathcal{X})$ and sections
$\sigma_{1},\sigma_{2}$ of $\mathcal{X}\rightarrow S$, such that
$\mathcal{V}_{\sigma_{1}}=\sigma$ and $\mathcal{V}_{\sigma_{2}}=\infty$. In
fact, $\mathcal{X}=\mathcal{E}^{\Delta}$ with $\sigma_{1}=\sigma$,
$\sigma_{2}=\infty$ and with
$\mathcal{V}=\Delta_{\mathcal{X}/S}\subset\mathcal{X}\times_{S}\mathcal{X}=\mathcal{E}^{\Delta}\times_{S}\mathcal{X}$
will do. There exists an $N_{2}$ such that
$\Phi_{\mathcal{X}}^{-1}(N_{2}\cdot\mathcal{V})\in{\rm
CH}^{2}_{(1)}(\mathcal{J}\times_{S}\mathcal{X};{\rm\mathbb{Q}})$ is the image
of an integral cycle $\mathcal{W}\in{\rm
CH}^{2}(\mathcal{J}\times_{S}\mathcal{X})$. The commutativity of the diagram
of Corollary 5.11 applied to $\mathcal{X}\rightarrow S$ (which is flat, so
admits pullbacks) shows that
$\mathcal{W}_{\sigma_{1}}=N_{1}N_{2}\cdot\kappa(\mathcal{C})$ and
$\mathcal{W}_{\sigma_{2}}=0$ in ${\rm CH}^{2}(\mathcal{J};{\rm\mathbb{Q}})$.
In other words, the elements
$\mathcal{W}_{\sigma_{1}}-N_{1}N_{2}\kappa(\mathcal{C})$ and
$\mathcal{W}_{\sigma_{2}}$ are torsion in ${\rm CH}^{2}(\mathcal{J})$. There
exists an integer $N_{3}$ such that these elements are killed by $N_{3}$, so
the elements $N=N_{1}N_{2}N_{3}$ and $\mathcal{Z}=N_{3}\cdot\mathcal{W}$
satisfy the conclusion of the proposition. ∎

###  6.2. Beilinson–Bloch conjectures

We briefly describe some consequences of the Beilinson–Bloch conjecture [6, 8]
in our setting. Let $X$ be a nice variety over a number field $k$. Fix a prime
$\ell$ and denote the $\ell$-adic cohomology
$\operatorname{H}^{i}(X_{\bar{k}},{\rm\mathbb{Q}}_{\ell})$ simply by
$\operatorname{H}^{i}(X)$. The conjecture predicts that for each
$i\in{\rm\mathbb{Z}}$, the group of homologically trivial cycles ${\rm
CH}^{i}(X)_{\mathrm{hom}}$ is finitely generated, the $L$-function
$L(\mathfrak{h}^{2i-1}(X),s)$ has analytic continuation and functional
equation, and

$\mathrm{rk}\,({\rm
CH}^{i}(X)_{\mathrm{hom}})=\operatorname*{ord}_{s=i}L(\mathfrak{h}^{2i-1}(X),s).$

Similarly, if $\epsilon\in{\rm CH}^{\dim X}(X\times X)$ is an idempotent
cutting out a direct summand of the Chow motive $\mathfrak{h}(X)$, then

$\mathrm{rk}\,(\epsilon\circ{\rm
CH}^{i}(X)_{\mathrm{hom}})=\operatorname*{ord}_{s=i}L(\epsilon\cdot\mathfrak{h}^{2i-1}(X),s).$

Let $C=C_{a,b}$ be a bielliptic Picard curve over $k$ with Jacobian $J$. By
Theorem 5.1 and Proposition 5.10, the Beilinson–Bloch conjecture for the
motive $\mathfrak{h}^{3}(J)^{\mu_{6}}$ reads:

######  Conjecture 6.2.

$\mathrm{rk}\,{\rm
CH}^{2}_{(1)}(J)^{\mu_{6}}=\operatorname*{ord}_{s=1}L(\mathfrak{h}^{1}(E^{\Delta}),s).$

The (weak) Birch and Swinnerton-Dyer conjecture states that
$\operatorname*{ord}_{s=1}L(\mathfrak{h}^{1}(E^{\Delta}),s)={\rm
rk}\,E^{\Delta}(k)$. Therefore the Beilinson–Bloch conjecture for
$\mathfrak{h}^{3}(J)^{\mu_{6}}$ is equivalent to the weak Birch and
Swinnerton-Dyer conjecture for $E^{\Delta}$ (by Corollary 5.11).

###  6.3. The Griffiths group

As detailed in [9, 1.3], the Beilinson–Bloch conjecture should also be
compatible with the coniveau filtrations on ${\rm CH}^{i}(X)$ and
$\operatorname{H}^{2i-1}(X)(i)$. In our setting, this allows us to predict the
rank of the Griffiths group ${\rm Gr}^{2}(J)$, at least in some cases; we
briefly sketch the details below. To formulate it, we use the decomposition
$\mathfrak{h}^{2}(P)=\mathfrak{h}_{\mathrm{alg}}^{2}(P)+\mathfrak{h}_{\mathrm{tr}}^{2}(P)$
from [23] which has the property that the $\ell$-adic realization of
$\mathfrak{h}^{2}_{{\mathrm{tr}}}(P)$ is

$V:=({\rm
NS}(P_{\bar{k}})(-1)\otimes{\rm\mathbb{Q}}_{\ell})^{\perp}\subset\operatorname{H}^{2}(P),$

the transcendental part of $\operatorname{H}^{2}(P)$.

######  Remark 6.3.

If $P$ does not have CM then $V$ is three dimensional. If $P$ is moreover of
${\rm GL}_{2}$-type, then $V$ is (up to twist) the symmetric square of the
associated 2-dimensional ${\rm Gal}_{k}$-representation. If $P$ has CM by an
imaginary quadratic field $K$, then $V$ is $2$-dimensional and the
$L$-function $L(V,s)$ is a product of Hecke $L$-functions attached to the
field $K$.

Using the isogeny decomposition $J\sim P\times E$ and the Kunneth formula, one
checks that the first graded piece $\mathrm{gr}^{0}\operatorname{H}^{3}(J)(2)$
in the coniveau filtration is a nonzero quotient of
$V\otimes\operatorname{H}^{1}(E)(2)$. Define

$\mathfrak{h}_{\mathrm{Gr}}^{3}(J):=\mathfrak{h}_{{\mathrm{tr}}}^{2}(P)\otimes\mathfrak{h}^{1}(E).$

The following prediction then follows from [9, 1.3].

######  Conjecture 6.4.

Let $C$ be a bielliptic Picard curve over a number field $k$. Then

$\mathrm{rk}\,{\rm
Gr}^{2}(J)\leq\operatorname*{ord}_{s=2}L(\mathfrak{h}_{\mathrm{Gr}}^{3}(J),s),$

with equality if $V\otimes\operatorname{H}^{1}(E)$ is an irreducible ${\rm
Gal}_{k}$-representation.

Assuming this conjecture, we exhibit below some new examples of
nonhyperelliptic Jacobians over a number fields with finite Griffiths group.
First we describe some known examples.

######  Example 6.5.

Following up on [7], Gross found two nonhyperelliptic Jacobians with finite
Griffiths group (assuming the Beilinson–Bloch conjecture): the genus $7$
Fricke-Macbeath curve and its genus $3$ quotient [20], both defined over
${\rm\mathbb{Q}}$.

The other previously known example is a certain “special” bielliptic Picard
curve. To put it in context, note that if $P$ is defined over
${\rm\mathbb{Q}}$ and has CM (over ${\rm\overline{\mathbb{Q}}}$) by an
imaginary quadratic field
${\rm\mathbb{Q}}(\sqrt{d})\neq{\rm\mathbb{Q}}(\omega)$, then
$L(\mathfrak{h}_{\mathrm{Gr}}^{3}(J),s)$ is a Hecke $L$-function for the
biquadratic CM field ${\rm\mathbb{Q}}(\omega,\sqrt{d})$.

######  Example 6.6.

The curve $C_{0,1}\colon y^{3}=x^{4}+1$ has CM by ${\rm\mathbb{Q}}(i)$ [28,
Example 6.15]. In [30, Theorem 1.3], Lilienfeldt and the second author show
that the corresponding Hecke $L$-function has nonvanishing central value, and
hence ${\rm Gr}^{2}(J_{0,1})$ is conjecturally finite.

Using Theorem 3.4 and mod $p$ point counts of $E$ and $J$, one can compute the
good Euler factors for the $L$-function
$L(\mathfrak{h}^{3}_{\mathrm{Gr}}(J),s)$, even for non-CM bielliptic Picard
curves $C_{a,b}$. These $L$-functions are not known to have analytic
continuation in general but we can still compute their analytic rank in Magma
assuming that they do.

######  Example 6.7.

Using [2, 42], we have verified (conditional on the analytic continuation of
the $L$-function) that for both $C_{6,-3}$ and $C_{6,1}$, the central value
$L(\mathfrak{h}^{3}_{\mathrm{Gr}}(J),2)$ is nonzero. Hence ${\rm
Gr}^{2}(J_{6,-3})$ and ${\rm Gr}^{2}(J_{6,1})$ are conjecturally finite
groups.

Theorem 1.5 suggests that the behavior in Examples 6.6 and 6.7 is not so
special, and we expect that many bielliptic Picard Jacobians over
${\rm\mathbb{Q}}$ have finite Griffiths group.

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