2.1 Chow groups

Let k be a 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 $\mathrm{CH}^p(X)$ denote the Chow group (with 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 $\mathrm{CH}^p(X)$ by [Z] (using [Sta18, Tag 02QS] if Z is not integral).

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

\begin{align*} \mathrm{CH}^p(X)_{\mathrm{alg}} \subset \mathrm{CH}^p(X)_{\mathrm{hom}} \subset \mathrm{CH}^p(X),\end{align*}

where $\mathrm{CH}^p(X)_{\mathrm{alg}}$ is the subgroup of algebraically trivial cycles (in the sense of [Reference Achter, Casalaina-Martin and VialACMV19b, § 3.1]) and $\mathrm{CH}^p(X)_{\mathrm{hom}}$ 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 $\mathrm{Gr}^p(X) = \mathrm{CH}^p(X)_{\mathrm{hom}}/\mathrm{CH}^p(X)_{\mathrm{alg}}$ . We occasionally write $\mathrm{CH}_p(X) = \mathrm{CH}^{\dim X -p}(X)$ and $\mathrm{Gr}_p(X) = \mathrm{Gr}^{\dim X -p}(X)$ . If R is a ring, we write $\mathrm{CH}^p(X)_R= \mathrm{CH}^p(X)\otimes_{\mathrm{\mathbb{Z}}} R$ and $\mathrm{Gr}^p(X)_R = \mathrm{Gr}^p(X)\otimes_{\mathrm{\mathbb{Z}}} R$ .

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.

1. (i)The base-change maps $\mathrm{CH}^p(X)_\mathrm{\mathbb{Q}} \rightarrow \mathrm{CH}^p(X_K)_\mathrm{\mathbb{Q}}$ and $\mathrm{Gr}^p(X)_\mathrm{\mathbb{Q}}\rightarrow \mathrm{Gr}^p(X_K)_\mathrm{\mathbb{Q}}$ are injective.

2. (ii)If in addition k is algebraically closed, the base-change maps $\mathrm{CH}^p(X) \rightarrow \mathrm{CH}^p(X_K)$ and $\mathrm{Gr}^p(X)\rightarrow \mathrm{Gr}^p(X_K)$ are injective.

We now prove (i) for $\mathrm{CH}^p(X)_{\mathrm{\mathbb{Q}}}\rightarrow \mathrm{CH}^p(X_K)_{\mathrm{\mathbb{Q}}}$ . 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 $\mathrm{CH}^p(X)_\mathrm{\mathbb{Q}} \rightarrow \mathrm{CH}^p(X_{\bar{k}})_\mathrm{\mathbb{Q}} \rightarrow \mathrm{CH}^p(X_L)_{\mathrm{\mathbb{Q}}}$ are both injective. The first one follows from the fact that for a finite extension $k'/k$ the pushforward map $\mathrm{CH}^p(X_{k'}) \rightarrow \mathrm{CH}^p(X)$ , when precomposed with the base-change map, is multiplication by $[k':k]$ . The second follows from (ii). The case of $\mathrm{Gr}^p(X)_{\mathrm{\mathbb{Q}}}$ 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 $X\rightarrow \mathrm{Spec} (R)$ be a smooth, projective morphism with geometrically integral fibres, so the generic and special fibres $X_K$ and $X_k$ are nice varieties over K and k, respectively. In this setting, Fulton [Reference FultonFul98, § 20.3] has defined a specialization morphism $\mathrm{sp}\colon \mathrm{CH}^p(X_K) \rightarrow \mathrm{CH}^p(X_k)$ for every $0\leqslant p\leqslant \dim(X_K)$ . It has the property that if $Z\subset X$ is a closed integral subscheme of codimension p, flat over R, then $\mathrm{sp}([Z_K]) = [Z_k]$ .

Let $\mathcal{C}\rightarrow \mathrm{Spec}(R)$ be an integral, projective, flat, proper and regular model of C, which exists by [Reference LiuLiu02, Proposition 10.1.8]. Let $\mathrm{Hilb}_{X/R}\rightarrow \mathrm{Spec}(R)$ be the Hilbert scheme of the projective morphism $X\rightarrow \mathrm{Spec}(R)$ . 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, $\mathrm{Hilb}^{\phi}_{X/R}\rightarrow \mathrm{Spec}(R)$ is projective [Reference Altman and KleimanAK80, Corollary (2.8)] and we may view $\alpha$ as a rational map $\mathcal{C}\dashrightarrow \mathrm{Hilb}_{X/R}^{\phi}$ . Therefore, by [Reference LiuLiu02, 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 fibre, and an R-morphism $\tilde{\alpha}\colon \widetilde{\mathcal{C}} \rightarrow \mathrm{Hilb}^{\phi}_{X/R}$ with generic fibre $\alpha$ . In other words, there exists a closed subscheme $\mathcal{Z} \subset \widetilde{\mathcal{C}} \times_R X$ , flat over $\widetilde{\mathcal{C}}$ , whose generic fibre equals Z.

The K-points $t_i\in C(K)$ extend to R-points $\tilde{t}_i\colon \mathrm{Spec}(R) \rightarrow \widetilde{\mathcal{C}}$ 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 fibre $\widetilde{\mathcal{C}}_k$ 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 exist a finite extension $k'/k$ , a collection of nice curves $D_1, \dots,D_n$ over k’, and for each $1\leqslant i \leqslant n$ a pair of points $s_{i,1}, s_{i,2}\in D_i(k')$ and a morphism $\varphi_i\colon D_i \rightarrow \mathcal{C}_{k'}$ such that $\varphi_1(s_{1,1}) = \bar{t}_1$ , $\varphi_i(s_{i,2}) = \varphi_{i+1}(s_{i+1,1})$ for all $1\leqslant i\leqslant n-1$ , and $\varphi_{n}(s_{n,2}) = \bar{t}_2$ . Letting $W^{(i)}$ be the pullback of $\mathcal{Z}_k$ along $\varphi_i$ , we have

\begin{align*} \mathrm{sp}([Z_{t_0}]-[Z_{t_1}])_{k'} = ([W^{(1)}_{s_{1,1}}]- [W^{(1)}_{s_{1,2}}]) + \cdots + ([W^{(n)}_{s_{n,1}}]- [W^{(n)}_{s_{n,2}}]). \end{align*}

Since each term $([W^{(i)}_{s_{i,1}}]- [W^{(i)}_{s_{i,2}}])$ lies in $\mathrm{CH}^p(X_{k'})_{\mathrm{alg}}\otimes \mathrm{\mathbb{Q}}$ by definition of algebraic triviality, the same is true for their sum. Taking the pushforward along $X_{k'} \rightarrow X_k$ , we see that $[k':k] \cdot \mathrm{sp}([Z_{t_0}]-[Z_{t_1}]) \in \mathrm{CH}^p(X_k)_{\mathrm{alg}}$ , as desired.

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 Q-coefficients; see [Reference SchollSch94] or [Reference Murre, Nagel and PetersMNP13, § 2] for basic definitions.

Denote the Lefschetz motive by $\mathbb{L}$ , its nth tensor power by $\mathbb{L}^n$ , and write $M(n) = M\otimes \mathbb{L}^n$ . Following [Reference Murre, Nagel and PetersMNP13, § 2.5], the Chow group in codimension p of M is by definition $\mathrm{CH}^p(M) = \mathrm{Hom}_{\mathsf{Mot}(k)}(\mathbb{L}^{p}, M)$ . If $M = {\frak{h}}(X)$ , where X is smooth projective over k, then $\mathrm{CH}^p(M) = \mathrm{CH}^p(X)_{\mathrm{\mathbb{Q}}}$ . (Beware that we need to take Q-coefficients on the right-hand side.) If C is a nice curve over k, a morphism $\varphi\colon {\frak{h}}(C)\rightarrow M(p-1)$ in $\mathsf{Mot}(k)$ induces a homomorphism of abelian groups $\mathrm{CH}^1(\varphi)\colon \mathrm{CH}^1(C)_{\mathrm{alg},\mathrm{\mathbb{Q}}} \rightarrow \mathrm{CH}^p(M)$ . We define $\mathrm{CH}^p(M)_{\mathrm{alg}}$ to be the union of the images of $\mathrm{CH}^1(\varphi)$ ranging over all pairs $(C, \varphi)$ as above. This coincides with $\mathrm{CH}^p(X)_{\mathrm{alg}} \otimes \mathrm{\mathbb{Q}}$ when $M = {\frak{h}}(X)$ ; see [Reference Achter, Casalaina-Martin and VialACMV19b, Corollary 4.13].

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 $({1}/{\#G})\sum_{g \in G} g_* \in \mathrm{End}(M)$ . We have $\mathrm{CH}^p(M^G) = \mathrm{CH}^p(M)^G$ . If G acts on a nice variety X, then G acts on ${\frak{h}}(X)$ and $\mathrm{CH}^p(X)$ , and $\mathrm{CH}^p({\frak{h}}(X)^G) = \mathrm{CH}^p(X)^G\otimes {\mathrm{\mathbb{Q}}}$ .

2.4 Chow–Künneth decomposition for abelian varieties

Let $A/k$ be a g-dimensional abelian variety. Deninger and Murre [Reference Deninger and MurreDM91, § 3] have constructed a canonical Chow–Künneth decomposition

(2.1) \begin{align}{\frak{h}}(A) = \bigoplus_{i = 0}^{2g}{\frak{h}}^i(A),\end{align}

uniquely characterized by the following property: if $(n)\colon A\rightarrow A$ denotes multiplication by n, then $(n)^*$ acts on ${\frak{h}}^i(A)$ via $n^i$ for every integer n. On the other hand, Beauville [Reference BeauvilleBea86] has shown that there exists a direct sum decomposition $\mathrm{CH}^p(A)_\mathrm{\mathbb{Q}} = \bigoplus_{s = p-g}^p \mathrm{CH}^p_{(s)}(A)$ , where

(2.2) \begin{align}\mathrm{CH}^p_{(s)}(A) = \{\alpha \in\mathrm{CH}^p(A)_\mathrm{\mathbb{Q}} \colon (n)^*\alpha = n^{2p-s} \alpha \ \forall n\in \mathrm{\mathbb{Z}}\}.\end{align}

The two decompositions are linked by the formula $\mathrm{CH}^p({\frak{h}}^i(A)) = \mathrm{CH}^p_{(2p-i)}(A)$ . Beauville conjectured that $\mathrm{CH}^p_{(s)}(A) = 0$ when $s<0$ , and he proved it when $p \in \{0,1,g-2,g-1,g\}$ [Reference BeauvilleBea86, Proposition 3(a)].

2.5 The Lefschetz decomposition for abelian varieties

Let $A/k$ be an abelian variety with polarization $\lambda\colon A \rightarrow A^{\vee}$ . Let $\ell \in \mathrm{CH}^1_{(0)}(A)$ be the unique class satisfying $2\ell = (1, \lambda)^* \mathcal{P}$ , where $\mathcal{P}\in \mathrm{Pic}(A\times A^{\vee})$ is the Poincaré bundle. Künnemann has shown [Kün93, Theorem 5.2] that intersecting with $\ell^{g-i}$ induces an isomorphism ${\frak{h}}^i(A) \rightarrow {\frak{h}}^{2g-i}(A)(g-i)$ for all $0\leqslant i \leqslant g$ . This induces a Lefschetz decomposition of the Chow–Künneth components ${\frak{h}}^i(A)$ ; see [Kün93, Theorem 5.1]. We are chiefly interested in the components ${\frak{h}}^3(A)$ and ${\frak{h}}^{2g-3}(A)$ when $g\geqslant 2$ ; in that case the Lefschetz decomposition has the form ${\frak{h}}^3(A) = {\frak{h}}^3_{\mathrm{prim}}(A) \bigoplus \ell \cdot {\frak{h}}^1(A)$ and ${\frak{h}}^{2g-3}(A) = {\frak{h}}^{2g-3}_{\mathrm{prim}}(A) \bigoplus \ell^{g-2}\cdot {\frak{h}}^1(A)$ . It has the property that the isomorphism $\ell^{g-3}\colon {\frak{h}}^3(A) \rightarrow {\frak{h}}^{2g-3}(A)(g-3)$ is a direct sum of isomorphisms ${\frak{h}}^3_{\mathrm{prim}}(A) \rightarrow {\frak{h}}^{2g-3}_{\mathrm{prim}}(A)$ and $\ell\cdot {\frak{h}}^1(A) \rightarrow \ell^{g-2} {\frak{h}}^1(A)$ . Taking Chow groups, we get a decomposition $\mathrm{CH}^{g-1}({\frak{h}}^{2g-3}(A)) = \mathrm{CH}^{g-1}({\frak{h}}_{\mathrm{prim}}^{2g-3}(A))\bigoplus \mathrm{CH}^{g-1}(\ell^{g-2} \cdot {\frak{h}}^1(A))$ .

2.6 Beauville components of C

Consider a nice curve C of genus $g\geqslant 2$ 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 $[C]_s \in \mathrm{CH}_{(s)}^{g-1}(J)$ . In this subsection we analyse 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 $\alpha \in \mathrm{CH}_p(J)$ and $\beta \in \mathrm{CH}_q(J)$ , 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 $K_C\in \mathrm{CH}_0(C)$ 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 $\mathrm{CH}_{(1)}^1(J)$ :

(2.3) \begin{align}(2g-2) \cdot [C]_0^{\star(g-2)} \star [C]_1 = [C]^{\star(g-1)} \star ([0]-[x_e]). \end{align}

Here we view $[0]-[x_e]$ as an element of $\mathrm{CH}_0(J)$ .

Proof. If D is a degree-( $g-1$ ) divisor class on C, let $\Theta_{D}$ be the image of the map $\mathrm{Sym}^{g-1}(C) \rightarrow J$ 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 the Pontryagin product sends $\mathrm{CH}^{g-p}_{(s)}(J) \times \mathrm{CH}^{g-q}_{(t)}(J)$ to $\mathrm{CH}^{g-p-q}_{(s+t)}(J)$ and since $\mathrm{CH}^1_{(s)}(J)\neq 0$ 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

\begin{align*} (-1)_*[C]^{\star(g-1)} &= [C]_0^{\star(g-1)} - (g-1) \cdot [C]_0^{\star(g-2)} \star [C]_1, \\ [x_e]\star [C]^{\star(g-1)} &= [x_e]\star [C]_0^{\star(g-1)} + (g-1) \cdot [C]_0^{\star(g-2)} \star [C]_1. \end{align*}

Note that $[x_e]\star$ acts trivially on $\mathrm{CH}^1_{(1)}(J)$ since $([x_e]-[0])\in \bigoplus_{s\geqslant 1} \mathrm{CH}^g_{(s)}(J)$ by the explicit description of the Beauville decomposition for 0-cycles [Reference BeauvilleBea86, bottom of p.649]. Equating the right-hand sides of the two centred equations proves that $(2g-2) \cdot [C]_0^{\star(g-2)} \star [C]_1 = [C]_0^{\star(g-1)} \star ([0]-[x_e])$ . Since $[C]_s^{\star(g-1)} \star ([0]-[x_e]) \in \bigoplus_{t\geqslant s+1} \mathrm{CH}^1_{(t)}(J) = \{0\}$ for all $s\geqslant 1$ , it follows that $[C]_0^{\star(g-1)} \star ([0]-[x_e]) = [C]^{\star(g-1)} \star ([0]-[x_e])$ , concluding the proof.

The principal polarization defines an ample class $\ell\in \mathrm{CH}_{(0)}^1(J)$ , which induces a Lefschetz decomposition ${\frak{h}}^{2g-3}(J) = {\frak{h}}^{2g-3}_{\mathrm{prim}}(J) \bigoplus \ell^{g-2}\cdot {\frak{h}}^1(J)$ as in § 2.5.

2.7 Ceresa cycles

Let $C/k$ be a nice curve of genus $g\geqslant 2$ 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 $\kappa_{C,e}\in \mathrm{CH}_1(J)$ using formula (1.1). Using the Beauville decomposition (2.2) to write $[C] = [\iota_e(C)] = \sum_{s=0}^{g-1} [C]_s$ with $[C]_s\in \mathrm{CH}^{g-1}_{(s)}(J)$ , we calculate that

(2.4) \begin{align}\kappa_{C,e}=\kappa(C) = 2[C]_1 + 2[C]_3 + \dots + 2[C]_{2\lfloor ({g-2})/{2}\rfloor+1}\end{align}

in $\mathrm{CH}_1(J)_{\mathrm{\mathbb{Q}}}$ .

Let $\kappa(C)$ be the image of $\kappa_{C,e}$ in $\mathrm{CH}_1(J)_\mathrm{\mathbb{Q}}$ for any choice of degree-1 divisor e on C such that $(2g-2)e = K_C$ in $\mathrm{CH}_0(C)_{\mathrm{\mathbb{Q}}}$ . Lemma 2.11 shows that this class is independent of the choice of e. (If no such e exists over k and if k has characteristic not dividing $2g-2$ , then there exists a unique class $\kappa(C)\in \mathrm{CH}_1(J)_{\mathrm{\mathbb{Q}}}$ such that $\kappa(C)_{\bar{k}} = \kappa(C_{\bar{k}})$ in $\mathrm{CH}_1(J_{\bar{k}})_{\mathrm{\mathbb{Q}}}$ , since Chow groups with Q-coefficients satisfy Galois descent.) We let $\bar{\kappa}(C)$ be the image of $\kappa(C)$ in $\mathrm{Gr}_1(J)_{\mathrm{\mathbb{Q}}}$ . Since all degree-1 divisors e on C are algebraically equivalent, $\bar{\kappa}(C)$ is also the image of $\kappa_{e}(C)$ in $\mathrm{Gr}_1(J)_\mathrm{\mathbb{Q}}$ for any degree-1 divisor, not necessarily with the property that $(2g-2)e = K_C$ in $\mathrm{CH}_0(C)_{\mathrm{\mathbb{Q}}}$ .

Suppose now that $(2g-2)e= K_C$ in $\mathrm{CH}_0(C)_{\mathrm{\mathbb{Q}}}$ . Since $[\iota_e(C)]\in \mathrm{CH}_1(J)_{\mathrm{\mathbb{Q}}}$ is independent of the choice of e, the same is true for the classes $[C]_{s}$ . In particular, they are $\mathrm{Aut}(C)$ -invariant. The following result, due to Zhang [Reference ZhangZha10, Theorem 1.5.5], 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 ${\frak{h}}(J)$ , then $\mathrm{CH}^p(M)$ is naturally a summand of $\mathrm{CH}^p({\frak{h}}(J)) = \mathrm{CH}^p(J)_{\mathrm{\mathbb{Q}}}$ .

4.1 Generalities

Let k be a field of characteristic 0. 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 non-zero discriminant, the projective closure of $y^3= f(x)$ in $\mathrm{\mathbb{P}}^2_k$ 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 $\mathrm{Aut}(C_{f,\bar{k}})$ 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) \begin{align}\mathrm{disc}(f)=-4 a^3 b^2 - 27 b^4 + 16 a^4 c + 144 a b^2 c - 128 a^2 c^2 + 256 c^3.\end{align}

We view f as the dehomogenization F(x,1) of the quartic form $F(X,Z) = X^4+aX^2Z^2+bXZ^3+cZ^4$ , and we define I(f) and J(f) to be the usual I- and J-invariants attached to F, as in [Reference Bhargava and ShankarBS15, § 2]. Their explicit formulae in our case are

\begin{align*} I(f) &= a^2+12c, \\ J(f) &= 72ac-2a^3-27b^2.\end{align*}

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

\[ E_f\colon y^2 = 4x^3-27\cdot \mathrm{disc}(f). \]

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 $\kappa_{C_f, P_{\infty}}\in \mathrm{CH}_1(J_f)$ as in the introduction; we denote it by $\kappa_f$ for simplicity. Recall that $\kappa(C_f)$ denotes the image of $\kappa_f$ in $\mathrm{CH}_1(J_f)_{\mathrm{\mathbb{Q}}}$ and $\bar{\kappa}(C_f)$ its image in $\mathrm{Gr}_1(J_f)_{\mathrm{\mathbb{Q}}}$ . Theorem C follows from the following slightly stronger theorems, whose proofs will take up the rest of this section.

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 $k=\mathrm{\mathbb{C}}$ . So in the remainder of § 4, all varieties will be over C, and cohomology will be singular cohomology.

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 $\mathcal{O} = \mathrm{\mathbb{Z}}[\omega]$ for the ring of Eisenstein integers with $\omega^2+ \omega +1=0$ and let $K=\mathrm{\mathbb{Q}}(\sqrt{-3})$ be its fraction field. Let C be a Picard curve over C with Jacobian variety J. The $\mu_3$ -action on C extends to an embedding $\mathcal{O} \subset \mathrm{End}(J)$ . Using this action, the singular cohomology group $\mathrm{H}^1(J;\mathrm{\mathbb{Z}})$ is a free $\mathcal{O}$ -module of rank 3, and $\mathrm{H}^1(J;\mathrm{\mathbb{Q}})$ is a three-dimensional K-vector space. The next lemma says that the criterion of Theorem B is always satisfied for Picard curves.

We may view $\mathrm{H}^1(J;\mathrm{\mathbb{Q}})$ either as a K-vector space or 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, the cup product induces an isomorphism $\bigwedge^3 \mathrm{H}^1(J;\mathrm{\mathbb{Q}}) \simeq \mathrm{H}^3(J;\mathrm{\mathbb{Q}})$ , and we will use this identification without further mention.

The universal property of exterior powers induces a canonical Q-linear surjection $\bigwedge^3 \mathrm{H}^1(J;\mathrm{\mathbb{Q}}) \rightarrow \bigwedge^3_K \mathrm{H}^1(J;\mathrm{\mathbb{Q}})$ . It is well known (see [Reference Moonen and ZarhinMZ98, Lemma 12(i)] or [Reference DeligneDel82, Lemma 4.3]) that this map admits a canonical splitting, which we use to view $\bigwedge_K^3 \mathrm{H}^1(J;\mathrm{\mathbb{Q}})$ as a direct summand of $\mathrm{H}^3(J;\mathrm{\mathbb{Q}})$ .

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

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 $K=\mathrm{\mathbb{Q}}(\omega)$ .

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

The remainder of the section is devoted to proving Theorem 4.2. To this end, we will analyse 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 C, we will use the notion of an integral (respectively, rational) variation of (pure) Hodge structures over the complex manifold $X(\mathrm{\mathbb{C}})$ , called a Z-VHS (respectively, Q-VHS) for short; see [Reference VoisinVoi07, § 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

\begin{align*} (V\otimes_{\mathrm{\mathbb{Z}}} \mathrm{\mathbb{C}})/(F^k + V_{\tau}),\end{align*}

where $F^k\subset V\otimes_{\mathrm{\mathbb{Z}}} \mathrm{\mathbb{C}}$ 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 Z-VHS of weight $2k-1$ over $X(\mathrm{\mathbb{C}})$ , we can define its intermediate Jacobian $\mathrm{J}(\mathbb{V})\rightarrow X(\mathrm{\mathbb{C}})$ , a relative complex torus whose fibres over points $x\in X(\mathrm{\mathbb{C}})$ are the classical intermediate Jacobians $\mathrm{J}(\mathbb{V}_x)$ ; see [Reference VoisinVoi07, § 7.1.1]. If $\mathbb{V}, \mathbb{W}$ are two Z-VHSs of odd weight, then $\mathrm{J}(\mathbb{V}(p)) = \mathrm{J}(\mathbb{V})$ for all $p \in \mathrm{\mathbb{Z}}$ and a morphism $\mathbb{V} \rightarrow \mathbb{W}$ of Z-VHSs induces a homomorphism of (relative) complex tori $\mathrm{J}(\mathbb{V}) \rightarrow \mathrm{J}(\mathbb{W})$ .

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

(4.2) \begin{align} \mathrm{AJ}^p_{X}\colon \mathrm{CH}^p(X)_{\hom} \rightarrow \mathrm{J}^p(X),\end{align}

defined in [Reference VoisinVoi07, § 7.2.1]. Moreover, if $S/\mathrm{\mathbb{C}}$ is a smooth variety, $\pi\colon X\rightarrow S$ a smooth projective morphism with geometrically integral fibres and $p \in \mathrm{\mathbb{Z}}_{\geqslant 0}$ , then $R^p \pi_*\mathrm{\mathbb{Z}}$ (pushforward of the constant sheaf in the analytic topology) has the structure of a Z-VHS over $S(\mathrm{\mathbb{C}})$ . If Z is a codimension p cycle on X all of whose components are flat over S, and such that $Z_s \in \mathrm{CH}^p(X_s)_{\hom}$ for every $s\in S(\mathrm{\mathbb{C}})$ , then Griffiths has shown that there exists a holomorphic section $\mathrm{AJ}(Z)$ of the relative complex torus $\mathrm{J}(R^p\pi_* \mathrm{\mathbb{Z}}) \rightarrow S(\mathrm{\mathbb{C}})$ with the property that $\mathrm{AJ}(Z)_s = \mathrm{AJ}_X^p(Z_s)$ for all $s\in S(\mathrm{\mathbb{C}})$ ; 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 C and let $\gamma \in\mathrm{CH}^{r+\dim(X)}(X\times Y)$ be a correspondence of degree r. This induces for every $p\geqslant 0$ a homomorphism $\gamma_* \colon \mathrm{CH}^p(X) \rightarrow \mathrm{CH}^{p+r}(Y)$ 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 $\mathrm{H}^p(X;\mathrm{\mathbb{Z}}) \rightarrow \mathrm{H}^{p+2r}(Y)(r)$ for every p, hence a homomorphism of complex tori $\gamma_*\colon \mathrm{J}^p(X) \rightarrow \mathrm{J}^{p+r}(Y)$ .

Let $S = \{(a,b,c) \mid \mathrm{disc}(f) \neq 0\}\subset \mathrm{\mathbb{A}}^3_{\mathrm{\mathbb{C}}}$ be the parameter space of Picard curves over C. We will identify C-valued points of S with polynomials $f = x^4+ax^2+bx+c \in \mathrm{\mathbb{C}}[x]$ of non-zero 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 $\kappa_{\mathcal{C}} \in \mathrm{CH}_1(\mathcal{J})$ 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 $f \in S(\mathrm{\mathbb{C}})$ .)

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

Proof. Let E be the elliptic curve $y^2 = x^3+1$ . Corollary 4.9 shows that there exists a correspondence $\gamma \in \mathrm{CH}^2(E\times J)$ such that $\gamma_*$ induces isomorphisms $\mathrm{CH}^1(E)_{\hom, \mathrm{\mathbb{Q}}} \rightarrow \mathrm{CH}^2_{(1)}(J)^{\mu_3}$ and $\mathrm{H}^1(E;\mathrm{\mathbb{Q}}) \rightarrow \mathrm{H}^3(J;\mathrm{\mathbb{Q}})^{\mu_3}(1)$ . Let $\varphi$ be the restriction of $\mathrm{AJ}_{J_f}^2\otimes \mathrm{\mathbb{Q}}$ to $\mathrm{CH}_{(1)}^2(J_f)^{\mu_3}$ . By Lemma 4.11 these form a commutative diagram.

All arrows except $\varphi$ are isomorphisms of abelian groups. Therefore, $\varphi$ is an isomorphism too. Since the image of $\kappa_f$ in $\mathrm{CH}^2(J_f)_{\mathrm{\mathbb{Q}}}$ lies in $\mathrm{CH}^2_{(1)}(J_f)^{\mu_3}$ by (2.4), we conclude that $\kappa_f$ is torsion if and only if $\mathrm{AJ}_{J_f}^2(\kappa_f)$ is. The claim about torsion orders follows from the fact that $\mathrm{AJ}^2_{J_f}$ is injective on torsion subgroups by a result of Murre [Reference MurreMur85, 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 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 $S(\mathrm{\mathbb{C}})$ . Moreover, the natural homomorphism $\mathrm{J}(\mathbb{V}^{\mu_3}) \rightarrow \mathrm{J}(\mathbb{V})$ induces an isomorphism onto $\mathrm{J}(\mathbb{V})^{\mu_3,\circ}$ .

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 $\mathrm{J}(\mathbb{V}^{\mu_3}) \rightarrow S(\mathrm{\mathbb{C}})$ 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

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

Then the relative complex torus $\mathrm{J}(\mathbb{V}^{\mu_3})\rightarrow S$ is isogenous to the relative complex torus $\mathcal{E}(\mathrm{\mathbb{C}}) \rightarrow S(\mathrm{\mathbb{C}})$ .

To analyse $\mathrm{J}(\mathbb{W})$ , we analyse the 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 Z-Hodge structure with these properties, hence $\mathbb{W}_f(1) \simeq \mathrm{H}^1(E;\mathrm{\mathbb{Z}})$ for every $f\in S(\mathrm{\mathbb{C}})$ , where E is the elliptic curve with Weierstrass equation $y^2 = x^3+1$ .

Therefore, $\mathrm{J}(\mathbb{W}) \rightarrow S(\mathrm{\mathbb{C}})$ is an isotrivial family of elliptic curves: analytically locally on $S(\mathrm{\mathbb{C}})$ , it is isomorphic to $E(\mathrm{\mathbb{C}}) \times S(\mathrm{\mathbb{C}}) \rightarrow S(\mathrm{\mathbb{C}})$ . It follows that the sheaf of local isomorphisms between $\mathrm{J}(\mathbb{W})$ and $E(\mathrm{\mathbb{C}}) \times S(\mathrm{\mathbb{C}})$ is an $\mathrm{Aut}(E)$ -torsor in the analytic topology on $S(\mathrm{\mathbb{C}})$ . Since $\mathrm{Aut}(E) \simeq \mu_6$ is finite, this torsor is the analytification of an étale $\mu_6$ -torsor on S.

To analyse étale $\mu_6$ -torsors on S, consider the following exact sequence induced by the Kummer exact sequence in étale cohomology:

\begin{align*} \mathrm{\mathbb{G}}_m(S)\xrightarrow{(-)^6} \mathrm{\mathbb{G}}_m(S) \rightarrow \mathrm{H}^1_{\mathrm{et}}(S, \mu_6) \rightarrow \mathrm{Pic}(S)[6]\rightarrow 0. \end{align*}

The Picard group $\mathrm{Pic}(S)$ vanishes, being a quotient of $\mathrm{Pic}(\mathrm{\mathbb{A}}^3_{\mathrm{\mathbb{C}}})$ , hence the outer term in the sequence vanishes. We claim that $\mathrm{disc} \in \mathrm{\mathbb{C}}[a,b,c]$ is irreducible. Indeed, let

\[ \mathrm{\mathbb{C}}[a,b,c]\rightarrow R = \mathrm{\mathbb{C}}[\omega_1, \omega_2, \omega_3, \omega_4] /(\omega_1 + \omega_2 + \omega_3 + \omega_4) \]

be the ring extension given by adjoining the roots of the universal quartic $x^4 + ax^2+ bx + c \in \mathrm{\mathbb{C}}[a,b,c][x]$ . The symmetric group $S_4$ acts on $\{\omega_i\}$ hence on R by ring automorphisms. By the fundamental theorem of symmetric functions, the subring of $S_4$ -fixed points of R is $\mathrm{\mathbb{C}}[a,b,c]$ . There is a factorization $\mathrm{disc} = \prod_{i\neq j} (\omega_i - \omega_j)$ in R, and $S_4$ acts transitively on the linear factors in this factorization. So disc is indeed irreducible in $\mathrm{\mathbb{C}}[a,b,c]$ . Second, we claim that $\mathrm{\mathbb{G}}_m(S)=\mathrm{H}^0(S,\mathcal{O}_S)^{\times} = \{ c\cdot \mathrm{disc}^n \mid c\in \mathrm{\mathbb{C}}^{\times},\, n \in \mathrm{\mathbb{Z}}\}$ . Indeed, every $c\cdot \mathrm{disc}^n$ is clearly a unit in $\mathrm{H}^0(S, \mathcal{O}_S)$ . Conversely, given a unit $f\in \mathrm{H}^0(S, \mathcal{O}_S)$ , seen as a rational function on $\mathrm{\mathbb{A}}^3_{\mathrm{\mathbb{C}}}$ , its divisor $\mathrm{div}(f)$ of zeros and poles must be supported on the zero locus of disc. Since $\mathrm{disc}\in \mathrm{\mathbb{C}}[a,b,c]$ is irreducible, $\mathrm{div}(f) = n\cdot [\{\mathrm{disc}=0\}]$ for some $n\in \mathrm{\mathbb{Z}}$ . Then $\mathrm{div}(f/\mathrm{disc}^n) = 0$ as a rational function on $\mathrm{\mathbb{A}}^3_{\mathrm{\mathbb{C}}}$ , hence $f/\mathrm{disc}^n$ is a unit in $\mathrm{\mathbb{C}}[a,b,c]$ , hence $f/\mathrm{disc}^n \in \mathrm{\mathbb{C}}^{\times}$ , proving the claim. We conclude that the group $\mathrm{H}^1_{\mathrm{et}}(S,\mu_6)$ classifying $\mu_6$ -torsors is generated by the image of disc.

Let $\mathcal{E}_i\rightarrow S$ be the relative elliptic curve with equation $y^2 = x^3 + \mathrm{disc}(f)^i$ . The previous paragraph shows that $\mathrm{J}(\mathbb{W})$ is isomorphic to $\mathcal{E}_i(\mathrm{\mathbb{C}})$ for some $i\in \{0,1,2,3,4,5\}$ . We show that $i=1$ , using our previous results on bielliptic Picard curves in [Reference Laga and ShnidmanLS25]. Let $T\subset S$ be the closed subscheme where $b=0$ , parametrizing even quartic polynomials $f = x^4+ ax^2+c$ . For $f\in T(\mathrm{\mathbb{C}})$ , the $\mu_3$ -action on $\mathcal{C}_f$ extends to a $\mu_6$ -action. A calculation shows $\mathrm{disc}|_T = 16 c (-a^2 + 4 c)^2$ . Applying the singular cohomology realization functor to [Reference Laga and ShnidmanLS25, Theorem 5.1], the relative complex torus $\mathrm{J}((\mathbb{V}|_{T(\mathrm{\mathbb{C}})})^{\mu_6})\rightarrow T(\mathrm{\mathbb{C}})$ is isogenous to $\mathcal{E}_1(\mathrm{\mathbb{C}})|_{T(\mathrm{\mathbb{C}})}\rightarrow T(\mathrm{\mathbb{C}})$ . Since $\mathrm{J}((\mathbb{V}|_{T(\mathrm{\mathbb{C}})})^{\mu_6})$ is a subtorus of $\mathrm{J}((\mathbb{V}|_{T(\mathrm{\mathbb{C}})})^{\mu_3})$ of the same dimension, they must be equal, hence $\mathrm{J}(\mathbb{W})|_{T(\mathrm{\mathbb{C}})}$ is isogenous to $\mathcal{E}_1(\mathrm{\mathbb{C}})|_{T(\mathrm{\mathbb{C}})}$ over $T(\mathrm{\mathbb{C}})$ . On the other hand, let $i\in \{0,1,2,3,4,5\}$ be such that $\mathrm{J}(\mathbb{W})\simeq \mathcal{E}_i(\mathrm{\mathbb{C}})$ . Then $\mathrm{J}(\mathbb{W})|_{T(\mathrm{\mathbb{C}})}\simeq \mathcal{E}_i(\mathrm{\mathbb{C}})|_{T(\mathrm{\mathbb{C}})}$ , hence $\mathcal{E}_1(\mathrm{\mathbb{C}})|_{T(\mathrm{\mathbb{C}})}$ is isogenous to $\mathcal{E}_i(\mathrm{\mathbb{C}})|_{T(\mathrm{\mathbb{C}})}$ .

We show that the latter can happen only if $i=1$ . Indeed, let $\varphi\colon \mathcal{E}_1(\mathrm{\mathbb{C}})|_{T(\mathrm{\mathbb{C}})} \rightarrow \mathcal{E}_i(\mathrm{\mathbb{C}})|_{T(\mathrm{\mathbb{C}})}$ 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 $\mathcal{E}_1(\mathrm{\mathbb{C}})|_{T(\mathrm{\mathbb{C}})}$ and $\psi$ an isomorphism. Therefore, $\mathcal{E}_1(\mathrm{\mathbb{C}})|_{T(\mathrm{\mathbb{C}})}$ and $\mathcal{E}_i(\mathrm{\mathbb{C}})|_{T(\mathrm{\mathbb{C}})}$ are isomorphic. Since the monodromy representations of $\mathcal{E}_1(\mathrm{\mathbb{C}})|_{T(\mathrm{\mathbb{C}})}$ and $\mathcal{E}_i(\mathrm{\mathbb{C}})|_{T(\mathrm{\mathbb{C}})}$ are non-isomorphic when $i\neq 1$ , we conclude that $i=1$ and that $\mathrm{J}(\mathbb{W})$ is isogenous to $\mathcal{E}_1(\mathrm{\mathbb{C}})$ over $S(\mathrm{\mathbb{C}})$ .

In summary, we have shown that $\mathrm{J}(\mathbb{V}^{\mu_3})$ , $\mathrm{J}(\mathbb{W})$ and $\mathcal{E}_1(\mathrm{\mathbb{C}})$ are isogenous. Since $\mathcal{E}_1(\mathrm{\mathbb{C}})$ is isomorphic to $\mathcal{E}(\mathrm{\mathbb{C}})$ , we conclude the proof.

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 $N\geqslant 1$ . Hence, $N\cdot Q|_{T}=0$ for every $s\in U(\mathrm{\mathbb{C}})$ . Since the group of sections of $\mathcal{E}|_T\rightarrow T$ is torsion-free, $Q|_T=0$ for every $s\in U(\mathrm{\mathbb{C}})$ . 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 = \mathrm{\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 $L\otimes \mathrm{\mathbb{Q}}$ . Then the locus $S_{\mathrm{dep}}$ of $s\in S(\mathrm{\mathbb{C}})$ where $P_{s}, \omega\cdot P_{s}$ and $Q_s$ are linearly dependent is a countable union of closed proper algebraic subvarieties. For every $s\in U(\mathrm{\mathbb{C}})$ , 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 $s\in V(\mathrm{\mathbb{C}})$ then $P_{s}$ is non-zero, 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 $s\in V(\mathrm{\mathbb{C}})$ . Since $V(\mathrm{\mathbb{C}})\setminus S_{\mathrm{dep}}$ is non-empty, we obtain a contradiction.

4.7 Proof of the vanishing criterion in the Chow group

We claim that $\sigma$ is not a torsion section. If it were torsion, then $\mathrm{AJ}(\kappa_{\mathcal{C}})$ would be a torsion section of $\mathrm{J}(\mathbb{V}^{\mu_3})$ , hence, by Proposition 4.12, $\kappa_f$ would be torsion for every $f\in S(\mathrm{\mathbb{C}})$ . This is not the case, since $\kappa_f$ is of infinite order if $f = x^4+x^2+1$ by [Reference Laga and ShnidmanLS25, 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 $N\geqslant 1$ be an integer such that $N\cdot \kappa_f$ is algebraically trivial for every Picard curve $f\in S(\mathrm{\mathbb{C}})$ (such an integer exists by Theorem 4.1). By the algebraicity of the Abel–Jacobi map for algebraically trivial cycles ([Reference Achter, Casalaina-Martin and VialACMV19a, 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 $S(\mathrm{\mathbb{C}}) \rightarrow \mathrm{J}_a(\mathbb{V})$ is algebraic. On the other hand, $\mathrm{AJ}(3N\cdot \kappa_{\mathcal{C}})$ 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 $S(\mathrm{\mathbb{C}})$ . Therefore, $\mathrm{J}(\mathbb{V}^{\mu_3})$ is the smallest relative subtorus of $\mathrm{J}(\mathbb{V})$ containing the image of $\mathrm{AJ}(3N \cdot \kappa_{\mathcal{C}})$ . Hence, $\mathrm{J}(\mathbb{V}^{\mu_3})\subset \mathrm{J}_a(\mathbb{V})$ . We conclude that $\mathrm{AJ}(3N \cdot\kappa_{\mathcal{C}}) \colon S(\mathrm{\mathbb{C}}) \rightarrow \mathrm{J}(\mathbb{V}^{\mu_3})$ is algebraic, so $\mathrm{AJ}(3\cdot \kappa_{\mathcal{C}})$ is algebraic, so $\sigma$ is algebraic.

Since $\sigma$ is an algebraic section of $\mathcal{E}\rightarrow S$ , Proposition 33 shows that there exist an integer $M\geqslant 1$ 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 $f\in S(\mathrm{\mathbb{C}})$ that $\kappa_f$ is torsion if and only if $\mathrm{AJ}_{J_f}^2(\kappa_f)$ torsion (by Proposition 4.12), if and only if $\mathrm{AJ}(\kappa_{\mathcal{C}})_f \in \mathrm{J}(\mathbb{V}^{\mu_3})_f$ torsion, if and only if $\sigma_f \in \mathcal{E}_f(\mathrm{\mathbb{C}})$ torsion, if and only if $P_f$ torsion. Tracing through the equivalences, the quotient of the torsion orders ${\rm ord}(\kappa_f)/{\rm ord}(P_f)$ (if defined) takes only finitely many values as f ranges in $S(\mathrm{\mathbb{C}})$ .

4.8 Analysing the torsion locus of P

Theorem 4.2 shows that there are infinitely many plane quartic curves over Q with torsion Ceresa cycle, since we may take $a = c = -12$ . In fact, we can find explicit families of torsion Ceresa cycles over $\overline{\mathrm{\mathbb{Q}}}$ of arbitrarily large order by explicitly computing the torsion locus of the section P of $\mathcal{E}\rightarrow S$ .

Recall that $S\subset \mathrm{\mathbb{A}}^3_{\mathrm{\mathbb{C}}}$ is the parameter space of polynomials $f(x) = x^4 + ax^2 + bx+c$ of non-zero discriminant. For $(I,J) \in \mathbb{C}^2$ such that $4I^3- J^2 \neq 0$ , let $X_{I,J}\subset S$ be the closed subscheme of elements f satisfying $(I(f), J(f)) = (I,J)$ , and let $E_{I,J}$ be the elliptic curve $y^2 = x^3- Ix /3 - J /27$ . Let $E_{I,J}^{\circ}\subset E_{I,J}$ be the complement of the origin.

Proof. Indeed, a calculation shows that an inverse is given by

(4.3) \begin{align}(\alpha, \beta)\mapsto f(x) = x^4 - 3\alpha x^2/2 + \beta x + (I/12 - 3\alpha^2/16).\end{align}

We now describe the locus of $f\in S$ where the specialization $P_f = (I(f), J(f))$ of the section P of $\mathcal{E}\rightarrow S$ is torsion. First we consider the restriction of this locus to the closed subscheme $S_1\subset S$ where $\mathrm{disc}(f) =1$ , so that $\mathcal{E}\rightarrow S$ restricts to the trivial family with fibre the elliptic curve $E_0\colon y^2 = 4x^3-27$ . Let $E_0^{\circ}\subset E_0$ be the complement of the origin. If $(I,J) \in E_0^{\circ}(\mathrm{\mathbb{C}})$ , then $X_{I,J}$ is a closed subscheme of $S_1$ . For G a group, write $G_{\mathrm{tors}}\subset G$ for the subset of torsion elements. Then we tautologically have

\begin{align*}\{f\in S_1(k) \colon P_f \in E_0(\mathrm{\mathbb{C}})_{\mathrm{tors}}\}=\bigsqcup_{(I,J) \in E_0(\mathrm{\mathbb{C}})_{\mathrm{tors}}} X_{I,J}(\mathrm{\mathbb{C}}).\end{align*}

Define a $\mathrm{\mathbb{G}}_m$ -action on S via the formula $\lambda \cdot (a,b,c) = (\lambda^2a,\lambda^3 b, \lambda^4 c)$ . The polynomials I(f), J(f) and $\mathrm{disc}(f)$ are then homogeneous of degree 4,6,12. Using the $\mathrm{\mathbb{G}}_m$ -action on S, we can form the (coarse) quotient $\mathcal{M}_{\mathrm{Pic}} = S/\mathrm{\mathbb{G}}_m$ , an open subscheme of weighted projective space $\mathbb{P}(2,3,4)$ .

The inclusion $S_1\subset S$ determines an isomorphism $S_1/\mu_{12}\simeq \mathcal{M}_{\mathrm{Pic}}$ , where $\mu_{12}\subset \mathrm{\mathbb{G}}_m$ acts via restriction on $S_1$ . Write $\pi\colon S\rightarrow \mathcal{M}_{\mathrm{Pic}}$ and $\pi_1\colon S_1\rightarrow \mathcal{M}_{\mathrm{Pic}}$ for the quotient maps.

Let $V\subset \mathcal{M}_{\mathrm{Pic}}$ be the image of the torsion locus in S of the section P of $\mathcal{E}\rightarrow S$ . Since this torsion locus is $\mathrm{\mathbb{G}}_m$ -invariant, V also equals the image of the torsion locus of $P_f \in E_0$ under $\pi_1$ . Therefore, V equals $\cup \pi_1(X_{I,J})$ , where (I,J) ranges over torsion points of $E_{0}^{\circ}$ . If $\zeta\in \mu_{12}$ , then $\zeta\cdot X_{I,J} = X_{\zeta^4 I , \zeta^6 J}$ , so $\mu_2\subset \mu_{12}$ preserves $X_{I,J}$ . Under the bijection $X_{I,J}\simeq E_{I,J}^{\circ}$ , $-1$ acts as $(\alpha, \beta)\mapsto (\alpha, -\beta)$ on $E_{I,J}^{\circ}$ . Therefore, the quotient $X_{I,J}/\mu_2$ is isomorphic, via the a-coordinate, to $\mathrm{\mathbb{A}}^1$ . It follows that for each torsion point (I,J), the restriction $X_{I,J}\hookrightarrow S_1 \rightarrow \mathcal{M}_{\mathrm{Pic}}$ factors through a map $\varphi_{(I,J)}\colon X_{I,J}/\mu_2 =\mathrm{\mathbb{A}}^1 \rightarrow \mathcal{M}_{\mathrm{Pic}}$ , and V is the union of the images of $\varphi_{(I,J)}$ . If $(I,J) = (\zeta^4 I',\zeta^6 J')$ for some $\zeta \in \mu_{12}$ , then the images of $\varphi_{(I,J)}$ are equal. Otherwise, they are disjoint. We conclude the following result.

Proposition 4.18. We have

\begin{align*}V = \bigcup_{(I,J) \in E_0(\mathbb{C})_{\mathrm{tors}}} \mathrm{Image}(\varphi_{(I,J)}\colon \mathrm{\mathbb{A}}^1 \rightarrow \mathcal{M}_{\mathrm{Pic}}).\end{align*}

The images of two maps $\varphi_{(I,J)}$ are either equal or disjoint. The maps $\varphi_{(I,J)}$ are quasi-finite.

We can identify $\varphi_{(I,J)}(t)$ explicitly, at least when $b\neq 0$ . For $t\in \mathrm{\mathbb{C}}$ , let $g_{(I,J)}(t)= t^3 - It/3 -J/27$ , let $(\alpha_t, \beta_t) = (t g_{(I,J)}(t), g_{(I,J)}(t)^2)$ and let $f_{(I,J),t}(x) = x^4 - 3\alpha_t x^2/2 + \beta_t x + (I/12 - 3\alpha_t^2/16)$ .

Proposition 4.19 gives explicit one-parameter families of Picard curves with equation $C_t\colon y^3 = f_{(I,J),t}(x)$ whose Ceresa cycle is torsion. The explicit description shows that when $(I,J)\in k^2$ , where k is a subfield of $\mathbb{C}$ , the family $C_t$ is defined over k. Since all torsion points of $E_0$ are defined over Q, so are the families $C_t$ .

We can now prove Theorem D.

We now turn to (ii). For every $d \geqslant 1$ , there exists $N = N(d) \geqslant 1$ such that for every Picard curve $C_f$ over a number field k of degree d, the order of $\kappa_f$ in $\mathrm{CH}_1(J_{f,\bar{k}})$ is either infinite or less than N. Indeed, 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.