Non-torsion algebraic cycles on the Jacobians of Fermat quotients

Abstract

We study the Abel-Jacobi image of the Ceresa cycle $W_{k, e}-W_{k, e}^-$, where $W_{k, e}$ is the image of the k-th symmetric product of a curve X with a base point e on its Jacobian variety. For certain Fermat quotient curves of genus g, we prove that for any choice of the base point and $k \leq g-2$, the Abel-Jacobi image of the Ceresa cycle is non-torsion. In particular, these cycles are non-torsion modulo rational equivalence.

1 Introduction

Let X be a smooth projective curve of genus g over $\mathbb {C}$ and $\operatorname {Jac}(X)$ be its Jacobian. Let $\operatorname {CH}_k(\operatorname {Jac}(X))_{\hom }$ be the Chow group of homologically trivial algebraic cycles of dimension k on $\operatorname {Jac}(X)$ modulo rational equivalence. To study this group, we consider the Abel-Jacobi map

$$ \begin{align*}\Phi_k \colon \operatorname{CH}_k(\operatorname{Jac}(X))_{\mathrm{hom}} \to J_k(\operatorname{Jac}(X))\quad (k=1, \ldots, g-1).\end{align*} $$

Here, $J_k(\operatorname {Jac}(X))$ is a complex torus, which is called the Griffiths intermediate Jacobian (see Section 3.1). It is well known that $\Phi _{g-1}$ is an isomorphism by the Abel-Jacobi theorem; however, for a general k, $\Phi _k$ is neither injective nor surjective. Fix a base point $e \in X$ and let $\iota _e$ be the embedding defined by

$$ \begin{align*}\iota_e \colon X \to \operatorname{Jac}(X); \quad x \mapsto [x]-[e].\end{align*} $$

Put $X_e=\iota _e(X)$ . We denote $X_e^-$ by the image of $X_e$ under the inversion map. Since the inversion map acts trivially on the cohomology groups of even degree, we have

$$ \begin{align*}X_e-X_e^- \in \operatorname{CH}_1(\operatorname{Jac}(X))_{\mathrm{hom}}.\end{align*} $$

Let $W_{k, e}$ be the image of the k-th symmetric product of X on $\operatorname {Jac}(X)$ . As in the case of $k=1$ , we have

$$ \begin{align*}W_{k,e} - W_{k,e}^- \in \operatorname{CH}_k(\operatorname{Jac}(X))_{\mathrm{hom}}.\end{align*} $$

These cycles are called the Ceresa cycles and for a generic curve X, Ceresa [4] proves that if $1 \leq k \leq g-2$ , then $W_{k,e} - W_{k,e}^-$ is nontrivial modulo algebraic equivalence.

For a positive integer N and integers $a, b \in \{1, \ldots , N-1\}$ , let $C_N^{a, b}$ be the smooth projective curve birational to the affine curve

$$ \begin{align*}y^N=x^a(1-x)^b.\end{align*} $$

Let $F_N$ be the Fermat curve of degree N. Then $C_N^{a, b}$ is a quotient of $F_N$ by a cyclic group $G_N^{a, b}$ (see Section 2). Let g be the genus of $C_N^{a, b}$ . The main theorem of this paper is as follows.

Theorem 1.1 Suppose that N has a prime divisor $p>7$ such that $p \nmid ab$ and $a^2+ab+b^2 \equiv 0 \ \ \pmod {p}$ . Then $\Phi _k(W_{k, e}-W_{k, e}^-) \in J_k(\operatorname {Jac}(C_N^{a, b}))$ is non-torsion for any choice of the base point $e \in C_N^{a, b}$ and $k=1, \ldots , g-2$ .

Remark 1.2 When N does not have a prime divisor $p>7$ , there exist some examples that the Abel-Jacobi image of the Ceresa cycle of $C_N^{a, b}$ is torsion. For example, ${\Phi _1(X_e-X_e^-)}$ is torsion for $X=C_{9}^{1, 2}$ , $C_{12}^{1, 3}$ , $C_{15}^{1, 5}$ and $e=(0, 0)$ ([1, §2 Theorem], [15, Theorem 3.2]).

The algebraical nontriviality of the Ceresa cycles of $F_N$ ( $N \leq 1000$ ) and $C_p^{1, b}$ ( ${p \leq 1000}$ is a prime and $b^2+b+1 \equiv 0 \ \ \pmod {p}$ ) is proved by Harris [10], Bloch [2], Kimura [14], Tadokoro [18, 19, 20], and Otsubo [16]. Moreover, Otsubo [16] and Tadokoro [20] give a sufficient condition for the Ceresa cycles of these to be non-torsion modulo algebraic equivalence; however, it is impossible to confirm numerically these conditions. There are only two explicit examples of non-torsioness modulo algebraic equivalence for $k=1$ : $F_4$ by Bloch [2] and $C_7^{1, 2}$ by Kimura [14]; they prove the non-torsioness of the l-adic Abel-Jacobi image.

Let N be a positive integer divisible by a prime $p> 7$ . Eskandari-Murty [6, 7] prove that $\Phi _1(F_{N, e} -F_{N, e}^-)$ is non-torsion for any $e \in F_N$ ; in particular, $F_{N, e} -F_{N, e}^-$ is non-torsion modulo rational equivalence. Moreover, they conjecture that the same result holds for $C_p^{1, m}$ with $m \in \{1, \ldots , p-2\}$ and $m \neq 1, (p-1)/2, p-2$ [7, Section 4, Remark (2)]. Theorem 1.1 partially but affirmatively answers their conjecture.

We briefly give a sketch of the proof. First, we reduce to the case $k=1$ using a method of Otsubo [16] (see Proposition 3.1). The reduction to the case $N=p$ is easy. The rest of the proof is parallel to the method of Eskandari-Murty [6, 7]. First, the Abel-Jacobi image of the Ceresa cycle is described by an extension of mixed Hodge structures by Harris [10] and Pulte [17] (see Section 3.2). Second, we construct a $1$ -cycle Z on $C_p^{a,b} \times C_p^{a,b}$ and evaluate the extension of mixed Hodge structures at Z. Here, we use the assumptions on a and b so that an automorphism of $F_p$ of order $3$ descends to $C_p^{a, b}$ . Then the extension class is expressed by a rational point $P_Z \in \operatorname {Jac}(C_N^{a, b})$ by formulas of Kaenders [13] and Darmon-Rotger-Sols [5] (see Sections 3.3, 3.4). Finally, since $P_Z$ is non-torsion by a result of Gross-Rohrlich [8] (see Section 2), where we use the assumption $p> 7$ , the theorem follows.

2 Fermat quotient curves

Let $N>3$ be an integer, and for integers $a, b\in \{1, \ldots , N-1\}$ , let $C^{a, b}_N$ be the smooth projective curve birational to

$$ \begin{align*}y^N=x^a(1-x)^b.\end{align*} $$

The map

$$ \begin{align*}C^{a, b}_N \to \mathbb{P}^1; \quad (x, y) \mapsto x\end{align*} $$

is ramified at $x=0, 1$ and $\infty $ . Above $0$ (resp. $1$ , $\infty $ ), there are $\gcd (N, a)$ (resp. $\gcd (N, b)$ , $\gcd (N, a+b)$ ) branches and the ramification index is $N/\gcd (N, a)$ (resp. $N/\gcd (N, b)$ , $N/\gcd (N, a+b)$ ). Therefore, by the Riemann-Hurwitz formula, the genus of $C^{a, b}_N$ is

$$ \begin{align*}\dfrac12(N-(\gcd(N, a)+\gcd(N, b)+\gcd(N, a+b)))+1.\end{align*} $$

We have an isomorphism

$$ \begin{align*} C^{a, b}_N \cong C^{b, a}_N \end{align*} $$

sending x to $1-x$ . If two other integers $a', b' \in \{1, \ldots , N-1\}$ satisfy the relation

$$ \begin{align*}(a', b') = (ha, hb) +(Ni, Nj)\end{align*} $$

for some integers $h, i, j$ with $\gcd (N, h)=1$ , we have

$$ \begin{align*} C^{a, b}_N \cong C^{a', b'}_N; \quad (x, y) \mapsto (x, y^h x^i(1-x)^j). \end{align*} $$

Let $F_N$ be the Fermat curve of degree N defined by

$$ \begin{align*}u^N+v^N=w^N.\end{align*} $$

Then there is a morphism

$$ \begin{align*}\pi_N^{a, b} \colon F_N \to C^{a, b}_N; \quad (u : v : w) \mapsto (x, y)=(u^Nw^{-N}, u^av^bw^{-a-b}).\end{align*} $$

Define a finite group by

$$ \begin{align*}G_N=\mathbb{Z}/N\mathbb{Z} \oplus \mathbb{Z}/N\mathbb{Z}\end{align*} $$

and denote an element $(r, s) \in G_N$ by $g_N^{r, s}$ . Fix a primitive N-th root of unity $\zeta _N$ and let $G_N$ act on $F_N$ by

$$ \begin{align*}g_N^{r, s}(u : v : w)=(\zeta_N^r u : \zeta_N^s v : w).\end{align*} $$

Let $G_N^{a, b}$ be a subgroup of $G_N$ defined by

$$ \begin{align*}G_N^{a, b}= \{g_N^{r, s} \in G_N \mid ar+bs=0\}.\end{align*} $$

If $\gcd (N, a, b)=1$ , $F_N$ is generically Galois over $C^{a, b}_N$ and

$$ \begin{align*}\operatorname{Gal} (F_N/ C^{a, b}_N)=G_N^{a, b} =\langle g_N^{b, -a} \rangle \simeq \mathbb{Z}/N\mathbb{Z}.\end{align*} $$

There is an automorphism $\alpha $ of $F_N$ of order $2$ defined by

$$ \begin{align*} \alpha((u : v : w)) = (v : u : w). \end{align*} $$

When N is odd, there is an automorphism $\beta $ of $F_N$ of order $3$ defined by

$$ \begin{align*} \beta((u : v : w)) = (-v : w : u). \end{align*} $$

Lemma 2.1 (cf. [12, Section 3.1])

Suppose that $\gcd (N, a, b)=1$ . Then,

1. (i) $\alpha $ descends to $C^{a, b}_N$ if and only if $a^2 \equiv b^2 \ \ \pmod {N}$ .

2. (ii) Suppose that N is odd. Then $\beta $ descends to $C^{a, b}_N$ if and only if $a^2+ab+b^2 \equiv 0 \ \ \pmod {N}$ . We denote this automorphism by $\widetilde {\beta }$ .

Proof We only prove (ii) since we use the morphism $\widetilde {\beta }$ to prove Theorem 1.1 and (i) is similarly proved. The automorphism $\beta $ descends to $\widetilde {\beta }$ if and only if

$$ \begin{align*}\pi_N^{a, b}( \beta (g_N^{b, -a} (u : v : w)))=\pi_N^{a, b} (\beta(u : v : w));\end{align*} $$

that is, there exists an integer i such that

$$ \begin{align*}\left(-\zeta_N^{-a} v : w : \zeta_N^{b} u \right)=\left(-\zeta_N^{bi}v : \zeta_N^{-ai}w : u\right)\end{align*} $$

for all $(u : v: w) \in F_N$ . This is equivalent to

(2.1) $$ \begin{align} a+b \equiv -bi \quad \mathrm{and} \quad b \equiv ai \quad\pmod{N}. \end{align} $$

First, (2.1) implies $a^2+ab+b^2 \kern1.3pt{\equiv}\kern1.3pt 0 \,\ \pmod {N}$ . However, if $a^2+ab+b^2 \kern1.3pt{\equiv}\kern1.3pt 0 \,\ \pmod {N}$ , then we have $\gcd (N,a)=\gcd (N, b)=1$ by the assumption $\gcd (N, a, b)=1$ . Therefore, there is an integer i such that $ai \equiv b \ \ \pmod {N}$ , which satisfies (2.1).

Remark 2.2

1. (i) If N is a prime, the condition $a^2+ab+b^2 \equiv 0 \ \ \pmod {N}$ implies that $N \equiv 1 \ \ \pmod {3}$ .

2. (ii) When $N=a^2+ab+b^2$ , the curve $C^{a, b}_N$ is isomorphic to the Hurwitz curve ([12, Lemma 3.8]) which is the smooth projective curve birational to

   $$ \begin{align*}X^{b}Y^{a+b}+Y^{b}Z^{a+b}+Z^{b}X^{a+b}=0.\end{align*} $$

3. (iii) The condition $a^2+ab+b^2 \equiv 0 \ \ \pmod {N}$ (for N prime) appears in Tadokoro [20]. He uses $\widetilde {\beta }$ to construct from a $1$ -form $\omega $ on $C_N^{a, b}$ two other $1$ -forms of the same Hodge type and evaluate the Abel-Jacobi image of the Ceresa cycle for $k=1$ at $\omega \wedge \widetilde {\beta }^* \omega \wedge (\widetilde {\beta }^2)^*\omega $ .

When $\gcd (N, 6)=1$ , the automorphism $\beta $ of $F_N$ has two fixed points

$$ \begin{align*}S=(\zeta_6 : \zeta_6^{-1} :1), \quad \overline{S}=(\zeta_6^{-1} : \zeta_6 : 1),\end{align*} $$

and there is no other fixed point.

Lemma 2.3 Suppose that $\gcd (N, a, b)=\gcd (N, 6)=1$ and $a^2+ab+b^2 \equiv 0 \pmod {N}$ . Then the fixed points of the automorphism $\widetilde {\beta }$ of $C^{a, b}_N$ are $\pi _N^{a, b}(S)$ and $\pi _N^{a, b}(\overline {S})$ , which are distinct.

Proof We regard a, b as elements in $(\mathbb {Z}/N\mathbb {Z})^*$ . Put $\gamma =g_N^{b, -a}$ . Then we have

$$ \begin{align*}\beta \gamma = g_N^{-a-b, -b} \beta =\gamma^{a^{-1}b} \beta\end{align*} $$

since $-a-b=a^{-1}b^2$ by the assumption $a^2+ab+b^2=0$ in $\mathbb {Z}/N\mathbb {Z}$ . For $P \in C_N^{a, b}$ , suppose that $\widetilde {\beta }(P)=P$ and take any $Q \in F_N$ such that $\pi _N^{a, b}(Q)=P$ . Then

$$ \begin{align*}\beta(Q)=\gamma^k Q\end{align*} $$

for some $k \in \mathbb {Z}/N\mathbb {Z}$ . Since $(a-b)^2=3ab \in (\mathbb {Z}/N\mathbb {Z})^*$ , we have $a-b \in (\mathbb {Z}/N\mathbb {Z})^*$ . We take $i=a(a-b)^{-1}k$ . Then we have

$$ \begin{align*}\beta(\gamma^iQ)=\gamma^{a^{-1}b i} \beta(Q)=\gamma^{a^{-1}bi+k} Q= \gamma^i Q,\end{align*} $$

which means that $\gamma ^i Q=S$ or $\overline {S}$ ; hence, $P=\pi _N^{a, b}(S)$ or $\pi _N^{a, b}(\overline {S})$ .

We are to show that $\pi _N^{a, b}(S) \neq \pi _N^{a, b}(\overline {S})$ . Suppose that $\pi _N^{a, b}(S) = \pi _N^{a, b}(\overline {S})$ ; that is, there exists an integer i such that

$$ \begin{align*}\zeta_6 = \zeta_N^{bi}\zeta_6^{-1}, \quad \zeta_6^{-1}= \zeta_N^{-ai} \zeta_6.\end{align*} $$

Then we have $\zeta _6^{2N}=1$ , which contradicts the assumption $\gcd (N, 6)=1$ .

Put $P_0\kern1.3pt{=}\kern1.3pt(0 : 1: 1) \kern1.3pt{\in}\kern1.3pt F_N$ and let $F_N \kern1.3pt{\to}\kern1.3pt \operatorname {Jac}(F_N)$ be the map defined by ${Q \kern1.3pt{\mapsto}\kern1.3pt [Q]-[P_0]}$ . Similarly, we define a map $C^{a, b}_N \to \operatorname {Jac}(C^{a, b}_N)$ by sending $Q'$ to $[Q'] - [\pi _N^{a, b}(P_0)]$ . Then we have a commutative diagram

The following result of Gross and Rohrlich is one of the key ingredients to the proof of Theorem 1.1.

Theorem 2.4 [8, Theorem 2.1]

Let N be an integer such that $\gcd (N, 6)=1$ and N is divisible by a prime $p>7$ . If $a-b, a+2b, 2a+b \not \equiv 0 \ \ \pmod {p}$ , then the point $(\pi _N^{a, b})_*([S]+[\overline {S}] -2[P_0])$ on $\operatorname {Jac}(C^{a, b}_N)$ is non-torsion.

3 Algebraic cycles and Hodge theory of quadratic iterated integrals

3.1 Extension of mixed Hodge structures

Let $R=\mathbb {Z}$ or $\mathbb {Q}$ . An R-mixed Hodge structure H is an R-module $H_R$ of finite rank equipped with an increasing weight filtration $W_{\bullet }$ on $H_{\mathbb {Q}}:=H_R \otimes _{R} \mathbb {Q}$ and a decreasing Hodge filtration $F^{\bullet }$ on $H_{\mathbb {C}}:=H_R \otimes _{R} \mathbb {C}$ such that for each k, $\mathrm {Gr}_k^W(H_{\mathbb {Q}})$ with the induced filtration $F^{\bullet }$ is a pure $\mathbb {Q}$ -Hodge structure of weight k. Let $R(n)$ be the Tate object of pure weight $-2n$ and put $H(n)=H\otimes _{R} R(n)$ . Let $H^{\vee }$ be the dual R-mixed Hodge structure of H.

Let MHS $(R)$ be the category of R-mixed Hodge structures. For R-mixed Hodge structures A, B, let $\mathrm {Ext}_{\mathrm {MHS}(R)}(A, B)$ denote the set of equivalence classes of extensions of R-mixed Hodge structures (i.e., exact sequences

$$ \begin{align*}0 \to B \to E \to A \to 0\end{align*} $$

of R-mixed Hodge structures up to natural equivalence relation). There is a natural operation called the Baer sum which makes $\mathrm {Ext}_{\mathrm {MHS}(R)}(A, B)$ an abelian group. If X is a smooth projective variety over $\mathbb {C}$ , the cohomology group $H^n(X, \mathbb {Z})$ underlies a pure $\mathbb {Z}$ -Hodge structure of weight n, which we denote by $H^n(X)$ .

For a pure $\mathbb {Z}$ -Hodge structure H of weight $-1$ , the intermediate Jacobian is defined by

$$ \begin{align*}JH=H_{\mathbb{C}}/(F^0H_{\mathbb{C}} +H_{\mathbb{Z}}),\end{align*} $$

which is a complex torus. We have Carlson’s isomorphism [3]

$$ \begin{align*}JH \cong \mathrm{Ext}_{\mathrm{MHS}(\mathbb{Z})} (H^{\vee}, \mathbb{Z}(0)).\end{align*} $$

For a smooth projective variety X over $\mathbb {C}$ , $H_{2k+1}(X)(-k)$ is a pure $\mathbb {Z}$ -Hodge structure of weight $-1$ , and

$$ \begin{align*}J_k(X):=JH_{2k+1}(X)(-k) \cong (F^{k+1}H^{2k+1}(X, \mathbb{C}))^{\vee} /H_{2k+1}(X, \mathbb{Z})\end{align*} $$

is the k-th intermediate Jacobian of Griffiths. The Carlson isomorphism is written as

$$ \begin{align*}J_k(X) \cong \mathrm{Ext}_{\mathrm{MHS}(\mathbb{Z})}(H^{2k+1}(X)(k), \mathbb{Z}(0)).\end{align*} $$

Let $\operatorname {CH}_k(X)$ be the Chow group of k-dimensional algebraic cycles on X modulo rational equivalence, and $\operatorname {CH}_k(X)_{\mathrm {hom}}$ be the subgroup of homologically trivial cycles. Then we have the Abel-Jacobi map

$$ \begin{align*}\Phi_k \colon \operatorname{CH}_k(X)_{\mathrm{hom}} \to J_k(X); \quad Z \mapsto \left(\eta \mapsto \int_{\Gamma} \eta \right)\end{align*} $$

for any $\eta \in F^{k+1}H^{2k+1}(X, \mathbb {C})$ , where $\Gamma $ is a topological $(2k+1)$ -chain such that ${\partial \Gamma = Z}$ .

From now on, let X be a smooth projective curve of genus $g \geq 3$ over $\mathbb {C}$ . Let

$$ \begin{align*}\langle \ , \ \rangle \colon H^1(X) \otimes H^1(X) \to H^2(X)= \mathbb{Z}(-1)\end{align*} $$

be the cup product $\varphi \otimes \varphi ' \mapsto \int _X \varphi \wedge \varphi '$ . Choosing a base point $e \in X$ , X is embedded into $\operatorname {Jac}(X)$ sending e to zero. It induces isomorphisms

$$ \begin{align*}H_1(X) \xrightarrow{\simeq} H_1(\operatorname{Jac}(X)), \quad H^1(\operatorname{Jac}(X)) \xrightarrow{\simeq} H^1(X),\end{align*} $$

which do not depend on the choice of e. We identify these and denote them by $H_1$ and $H^1$ , respectively. Recall that the cup product induces an isomorphism

$$ \begin{align*}\wedge^nH^1 \xrightarrow{\simeq} H^n(\operatorname{Jac}(X)).\end{align*} $$

For $e \in X$ , let $\iota _e \colon X \to \operatorname {Jac}(X)$ be the map defined by $P \mapsto [P]-[e]$ . Let $X^k$ (resp. $\operatorname {Jac}(X)^k$ ) be the k-fold product of X (resp. $\operatorname {Jac}(X)$ ) and $\mu \colon \operatorname {Jac}(X)^k \to \operatorname {Jac}(X)$ be the addition. We put

$$ \begin{align*}W_{k, e}=(\mu \circ (\iota_e)^k)(X^k) \quad (1 \leq k \leq g).\end{align*} $$

Then $W_{k, e}$ defines an algebraic k-cycle on $\operatorname {Jac}(X)$ , and $W_{k, e}-W_{k, e}^-$ defines an element of $\operatorname {CH}_k(\operatorname {Jac}(X))_{\mathrm {hom}}$ .

Proposition 3.1 If $\Phi _1(X_e -X_e^-)$ is non-torsion, then $\Phi _k(W_{k, e}-W_{k, e}^-)$ is non-torsion for any $k=2, \ldots , g-2$ .

Proof Let $S=\{e_i, f_i \mid 1 \leq i \leq g\}$ be a symplectic basis of $H^1_{\mathbb {Z}}$ (i.e., $\langle e_i, e_j \rangle = \langle f_i, f_j \rangle =0$ , $\langle e_i, f_j \rangle = \delta _{ij}$ ). Under the identification

$$ \begin{align*}J_k(\operatorname{Jac}(X)) \cong \mathrm{Hom}(\wedge^{2k+1} H^1_{\mathbb{Z}}, \mathbb{R}/\mathbb{Z}),\end{align*} $$

if $\Phi _1(X_e - X_e^-)$ is non-torsion, there exists elements $\varphi _1$ , $\varphi _2$ , $\varphi _3 \in S$ such that

$$ \begin{align*}\Phi_1(X_e - X_e^-)( \varphi_1 \wedge \varphi_2 \wedge \varphi_3)\end{align*} $$

is non-torsion. By renumbering, we may assume that $\varphi _1$ , $\varphi _2$ , $\varphi _3 \in \{e_i, f_i \mid 1 \leq i \leq 3\}$ . For $i=1, \ldots , k-1$ , we put

$$ \begin{align*}\varphi_{2i+2}=e_{i+3}, \quad \varphi_{2i+3}=f_{i+3}.\end{align*} $$

Note that $i+3 \leq g$ by the assumption. Put $\varphi = \varphi _1 \wedge \cdots \wedge \varphi _{2k+1}$ . Then, by [16, Proposition 3.7], we have

$$ \begin{align*} &k! \cdot \Phi_k(W_{k, e}-W_{k, e}^-)(\varphi) \\ &=k! \cdot \sum_{\sigma} \Phi_1(X_e- X_e^-)(\varphi_{\sigma(1)} \wedge \varphi_{\sigma(2)} \wedge \varphi_{\sigma(3)}) \prod_{i=1}^{k-1} \langle \varphi_{\sigma(2i+2)}, \varphi_{\sigma(2i+3)}\rangle \\ & =k! \cdot \Phi_1(X_e- X_e^-)(\varphi_{1} \wedge \varphi_{2} \wedge \varphi_{3}), \end{align*} $$

where $\sigma $ runs through the elements of the symmetric group $S_{2k+1}$ such that ${\sigma (1) < \sigma (2) < \sigma (3)}$ , $\sigma (2i+2) < \sigma (2i+3)$ for $1 \leq i \leq k-1$ , and $\sigma (2i+2) < \sigma (2i+4)$ for $1 \leq i \leq k-2$ . Therefore, $\Phi _k(W_{k, e}-W_{k, e}^-)$ is non-torsion.

Corollary 3.2 Let N be an integer which has a prime divisor $p> 7$ and $X=F_N$ be the Fermat curve of degree N. Then $\Phi _k(W_{k, e}-W_{k, e}^-)$ is non-torsion for any $e \in F_N$ and $k=1, \ldots , g-2$ .

Proof By Proposition 3.1, we are reduced to the case $k=1$ , which is a theorem of Eskandari and Murty [6, Theorem 1.1].

3.2 Harris-Pulte formula

In this subsection, we recall the Harris-Pulte formula, which is a relation between the Abel-Jacobi image of the Ceresa cycle and an extension class of mixed Hodge structures on the space of quadratic iterated integrals on the curve X.

We put

$$ \begin{align*}(H^1 \otimes H^1)' = \operatorname{Ker}(\cup \colon H^1\otimes H^1 \to H^2(\operatorname{Jac}(X))).\end{align*} $$

Then the map

$$ \begin{align*}\phi \colon H^1 \otimes (H^1 \otimes H^1)' \to \wedge^3 H^1,\end{align*} $$

which is obtained by restricting the natural quotient map $(H^1)^{\otimes 3} \to \wedge ^3H^1$ , is surjective ([17, Lemma 4.7]), and induces the injective map

$$ \begin{align*}\phi^* \colon \mathrm{Ext}_{\mathrm{MHS}(\mathbb{Z})}(\wedge^3H^1, \mathbb{Z}(-1)) \to \mathrm{Ext}_{\mathrm{MHS}(\mathbb{Z})}(H^1 \otimes (H^1 \otimes H^1)', \mathbb{Z}(-1)).\end{align*} $$

Let $\pi _1(X, e)$ be the fundamental group. Let I be the augmentation ideal of the group ring $\mathbb {Z}[\pi _1(X, e)]$ – that is, the kernel of the degree map

$$ \begin{align*}\mathbb{Z} [\pi_1(X, e)] \to \mathbb{Z}; \quad \sum n_i \gamma_i \mapsto \sum n_i.\end{align*} $$

By Chen’s $\pi _1$ -de Rham theorem, $\operatorname {Hom}(\mathbb {Z}[\pi _1(X, e)]/I^{s+1}, \mathbb {R})$ is generated by closed iterated integrals of length $\leq s$ . Using this, Hain [9] defines a $\mathbb {Z}$ -mixed Hodge structure on $\mathbb {Z}[\pi _1(X, e)]/I^s$ such that the natural map $\mathbb {Z}[\pi _1(X, e)]/I^s \to \mathbb {Z}[\pi _1(X, e)]/I^t$ for $s \geq t$ is a morphism of mixed Hodge structures. Consider the exact sequence of mixed Hodge structures

(3.1) $$ \begin{align} 0 \to I^2/I^3 \to I/I^3 \to I/I^2 \to 0. \end{align} $$

The map $\pi _1(X, e) \to I/I^2; \gamma \mapsto \gamma -1$ is well-defined and induces an isomorphism

$$ \begin{align*}H_1(X, \mathbb{Z}) \xrightarrow{\simeq} I/I^2\end{align*} $$

of Hodge structures of weight $-1$ . However, the multiplication $I/I^2 \otimes I/I^2 \to I^2/I^3$ induces an isomorphism

$$ \begin{align*}\operatorname{Hom}(I^2/I^3, \mathbb{Z}) \xrightarrow{\simeq} (H^1 \otimes H^1)'\end{align*} $$

of Hodge structures of weight $2$ . Taking the dual of (3.1), we have an exact sequence

$$ \begin{align*}0\to H^1 \to L_2(X, e)\to (H^1 \otimes H^1)' \to 0,\end{align*} $$

where we put $L_2(X, e)=\operatorname {Hom}(I/I^3, \mathbb {Z})$ .

Let $\infty \neq e$ be another point on X. Put $U=X-\{\infty \}$ . We identify $H^1(U)$ and $H^1$ via the map induced by the inclusion $U \subset X$ . Then we can obtain an exact sequence of mixed Hodge structures

$$ \begin{align*}0\to H^1 \to L_2(U, e)\to H^1 \otimes H^1 \to 0\end{align*} $$

similarly as above. We have a commutative diagram

Let $\mathbb {E}_e$ (resp. $\mathbb {E}_e^{\infty }$ ) be an extension class of the top (resp. bottom) row. We regard $\mathbb {E}_e$ as an element of

$$ \begin{align*} \mathrm{Ext}_{\mathrm{MHS}(\mathbb{Z})}( (H^1 \otimes H^1)', H^1) &\cong \mathrm{Ext}_{\mathrm{MHS}(\mathbb{Z})}((H^1)^{\vee} \otimes (H^1 \otimes H^1)', \mathbb{Z}(0)) \\ &\cong \mathrm{Ext}_{\mathrm{MHS}(\mathbb{Z})}(H^1 \otimes (H^1 \otimes H^1)', \mathbb{Z}(-1)), \end{align*} $$

and $\mathbb {E}_e^{\infty }$ as an element of

$$ \begin{align*} \mathrm{Ext}_{\mathrm{MHS}(\mathbb{Z})}( H^1 \otimes H^1, H^1) &\cong \mathrm{Ext}_{\mathrm{MHS}(\mathbb{Z})}((H^1)^{\vee} \otimes H^1 \otimes H^1, \mathbb{Z}(0)) \\ &\cong \mathrm{Ext}_{\mathrm{MHS}(\mathbb{Z})}(H^1 \otimes H^1 \otimes H^1, \mathbb{Z}(-1)). \end{align*} $$

Here, we used the Poincaré duality $H^1(1) \cong (H^1)^{\vee }$ . One sees that $\mathbb {E}_e$ is the restriction of $\mathbb {E}_e^{\infty }$ to $H^1 \otimes (H^1 \otimes H^1)'$ . Then Harris’s formula [10, Section 4], reworked by Pulte [17, Theorem 4.10], is

$$ \begin{align*} \phi^* \circ \Phi_1(X_e-X^-_e) =2 \mathbb{E}_e \end{align*} $$

under the identification $J_1(\operatorname {Jac}(X))=\mathrm {Ext}_{\mathrm {MHS}(\mathbb {Z})}(\wedge ^3 H^1, \mathbb {Z}(-1))$ .

3.3 The decomposition of $(H^1)^{\otimes 3}$

In this subsection, for a $\mathbb {Z}$ -mixed Hodge structure H, we consider the image of H under the forgetful functor $\mathrm {MHS}(\mathbb {Z}) \to \mathrm {MHS(\mathbb {Q})}$ , which we denote by the same letter. The Hodge structure $(H^1)^{\otimes 3}$ can be decomposed in MHS( $\mathbb {Q}$ ) as follows. Let $\xi _{\Delta } \in H^1 \otimes H^1$ be the Künneth component of the Hodge class of the diagonal of X in $H^2(X \times X)$ . Then we have a decomposition

$$ \begin{align*}H^1 \otimes H^1 \otimes H^1 =(H^1 \otimes \langle \xi_{\Delta} \rangle ) \oplus (H^1 \otimes (H^1 \otimes H^1)').\end{align*} $$

Since the Mumford-Tate group of $H^1$ is reductive, the map $\phi $ admits a section $\sigma $ in MHS( $\mathbb {Q}$ ), and we have

$$ \begin{align*}H^1 \otimes (H^1 \otimes H^1)' = \operatorname{ker}(\phi) \oplus \sigma(\wedge^3H^1).\end{align*} $$

Let $\overline {\xi }_{\Delta }$ be the image of $\xi _{\Delta }$ in $\wedge ^2H^1$ . Then we have a decomposition in MHS( $\mathbb {Q}$ )

$$ \begin{align*} H^1 \otimes H^1 \otimes H^1 = (H^1 \otimes \langle \xi_{\Delta} \rangle ) \oplus \operatorname{ker}(\phi) \oplus \sigma(H^1 \wedge \langle \overline{\xi}_{\Delta} \rangle) \oplus \sigma((\wedge^3H^1)_{\mathrm{prim}}), \end{align*} $$

where the last summand (primitive part) is the kernel of the map $\wedge ^3H^1 \to \wedge ^{2g-1}H^1$ given by wedging by $\overline {\xi }_{\Delta }^{g-2}$ (cf. [7, Section 4.2]). We put $\mathbb {E} :=\mathbb {E}_e |_{\sigma ((\wedge ^3H^1)_{\mathrm {prim}})}$ . Then $\mathbb {E}$ is independent of the choice of e ([17, Theorem 3.9] and [10]).

Proposition 3.3

1. (i) Suppose that $-2g[\infty ]+2[e] +K=0$ . Then $\mathbb {E}_{e}^{\infty } =0$ if and only if $\mathbb {E}_{e} =0$ .

2. (ii) Suppose that $(2g-2)[e]-K =0$ . Then $\mathbb {E}_{e}=0$ if and only if $\mathbb {E}=0$ .

Proof (i) The statement follows from that $\mathbb {E}_e^{\infty }|_{H^1 \otimes (H^1 \otimes H^1)'}=\mathbb {E}_e$ and a result of Kaenders [13, Theorem 1.2] that

$$ \begin{align*}\mathbb{E}_e^{\infty}|_{H^1 \otimes \langle \xi_{\Delta} \rangle}=-2g[\infty]+2[e] +K\end{align*} $$

under the identification (cf. [7, Section 4.3.1])

$$ \begin{align*}\mathrm{Ext}_{\mathrm{MHS}(\mathbb{Q})}(H^1 \otimes \langle \xi_{\Delta} \rangle, \mathbb{Q}(-1)) \cong \operatorname{CH}_0(X)_{\mathrm{hom}} \otimes \mathbb{Q}.\end{align*} $$

(ii) The statement follows from that results of Harris [10, Section 3] and Pulte [17, Theorem 4.10]] that $\mathbb {E}_e|_{\operatorname {ker}(\phi )} \in \mathrm { Ext}_{\mathrm {MHS}(\mathbb {Q})}(\ker (\phi ), \mathbb {Q}(-1))$ is zero, and Pulte [17, Corollary 6.7] that

$$ \begin{align*}\mathbb{E}_e|_{\sigma(H^1 \wedge \langle \overline{\xi}_{\Delta} \rangle)}=(2g-2)[e]-K\end{align*} $$

under the identification (cf. [7, Section 4.3.3])

$$ \begin{align*}\mathrm{Ext}_{\mathrm{MHS}(\mathbb{Q})}(\sigma(H^1 \wedge \langle \overline{\xi}_{\Delta} \rangle), \mathbb{Q}(-1)) \cong \operatorname{CH}_0(X)_{\mathrm{hom}} \otimes \mathbb{Q}.\\[-36pt]\end{align*} $$

3.4 Darmon-Rotger-Sols formula

Let $\Delta \in \operatorname {CH}_1(X \times X)$ be the diagonal of X and

$$ \begin{align*}p_i \colon X \times X \to X \quad (i=1, 2)\end{align*} $$

be the projection to the i-th component. For $Z \in \operatorname {CH}_1(X \times X)$ , put

$$ \begin{align*} Z_{12}&=(p_1)_*(Z \cdot \Delta)=(p_2)_*(Z \cdot \Delta), \\ Z_1&=(p_1)_*(Z \cdot (X \times \{e\})), \quad Z_2=(p_2)_*(Z \cdot (\{e\} \times X)) \in \operatorname{CH}_0(X). \end{align*} $$

Put

$$ \begin{align*}P_Z=Z_{12}-Z_1-Z_2-(\deg(Z_{12})-\deg(Z_1)-\deg(Z_2))[e] \in \operatorname{Jac}(X).\end{align*} $$

Then the point $P_Z$ is related to the extension $\mathbb {E}_e^{\infty }$ as follows. Let $\xi _Z$ be the $H^1 \otimes H^1$ -Künneth component of the class of Z in $H^2(X \times X)$ . Consider the map

$$ \begin{align*} \xi_Z^{-1} \colon \mathrm{Ext}_{\mathrm{MHS}(\mathbb{Z})}((H^1)^{\otimes 3}, \mathbb{Z}(-1)) \to \mathrm{Ext}_{\mathrm{MHS}(\mathbb{Z})}(H^1(-1), \mathbb{Z}(-1)) \cong J_0(X) =\operatorname{Jac}(X), \end{align*} $$

where the first arrow is the pullback along the morphism $H^1(-1) \to (H^1)^{\otimes 3}$ defined by $\omega \mapsto \omega \otimes \xi _Z$ . Then we have the following.

Proposition 3.4 [5, Corollary 2.6]

For any $Z \in \operatorname {CH}_1(X \times X)$ , we have

$$ \begin{align*} \xi^{-1}_Z(\mathbb{E}_e^{\infty}) =\left(\int_{\Delta}\xi_Z \right) ([\infty] -[e])-P_Z \end{align*} $$

in $\operatorname {Jac}(X)$ .

4 Proof of Theorem 1.1

There are $3N$ points on $F_N$

$$ \begin{align*}P_i=(0 : \zeta_N^i : 1), \quad Q_i= (\zeta_N^i : 0 : 1), \quad R_i =(\xi_N \zeta_N^i : 1: 0), \quad (i \in \mathbb{Z}/N\mathbb{Z}),\end{align*} $$

where we put $\xi _N=\exp (\pi i /N)$ . Fix $P_0$ as the base point; then the above points are torsion points in $\operatorname {Jac}(F_N)$ [8]. Therefore, for the base point $\pi _N^{a, b}(P_0)$ , the images of these points under $(\pi _N^{a, b})_*$ are also torsion in $\operatorname {Jac}(C_N^{a, b})$ . We shall continue to use the notation as in the previous section, specializing $X=C_N^{a, b}$ , $e=\pi _N^{a, b}(P_0)$ and ${\infty = \pi _N^{a, b}(Q_0)}$ .

Lemma 4.1 Let $K_C$ (resp. g) be the canonical divisor (resp. genus) of $C_N^{a, b}$ . Then ${K_C-(2g-2)[e]}$ , $K_C-2g[\infty ]+2[e] \in \operatorname {Jac}(C_N^{a, b})$ are torsion points.

Proof Since

$$ \begin{align*}K_C-2g[\infty]+2[e]=K_C -(2g-2)[e]-2g([\infty]-[e])\end{align*} $$

and $[\infty ]-[e]$ is a torsion point, it suffices to show that $K_C -(2g-2)[e]$ is a torsion point. Let $K_F$ be the canonical divisor of $F_N$ and $R_{\pi _N^{a, b}}$ be the ramification divisor of $\pi _N^{a, b}$ ; that is,

$$ \begin{align*} &K_F=(N-1) \sum_{i=0}^{N-1}Q_i -2 \sum_{i=0}^{N-1} R_i, \\ &R_{\pi_N^{a, b}}=(\gcd(N, a)-1)\sum_{i=0}^{N-1} P_i + (\gcd(N, b)-1)\sum_{i=0}^{N-1} Q_i +(\gcd(N, a+b)-1)\sum_{i=0}^{N-1} R_i. \end{align*} $$

Then we have

$$ \begin{align*}K_F= (\pi_N^{a, b})^*(K_C) + R_{\pi_N^{a, b}}\end{align*} $$

up to principal divisor (cf. [11, Proposition 2.3, Chap.IV]). Therefore, we have

$$ \begin{align*} N(K_C- (2g-2)[e])= (\pi_N^{a, b})_*\left( K_F -R_{\pi_N^{a, b}}-(2g-2)N[P_0]\right) \end{align*} $$

in $\operatorname {Jac}(C_N^{a, b})$ . Since $P_i$ , $Q_i$ , and $R_i$ are torsion in $\operatorname {Jac}(F_N)$ , $K_F -R_{\pi _N^{a, b}}-(2g-2)N[P_0]$ is torsion, which finishes the proof.

Proof of Theorem 1.1

First, by Proposition 3.1, it suffices to show the case when $k=1$ . Secondly, consider the map

$$ \begin{align*}f \colon F_N \to F_p; \quad (x_0 : y_0 : z_0) \mapsto (x_0^{N/p} : y_0^{N/p} : z_0^{N/p}).\end{align*} $$

Let $\langle a \rangle \in \{0, \ldots , p-1\}$ be the representative of a. Then f descends to a map $\overline {f} \colon C^{a, b}_N \to C^{\langle a \rangle , \langle b \rangle }_p$ . Since

$$ \begin{align*}f_{*}(\Phi_1(C_{N, e}^{a, b}-(C^{a, b}_{N,e})^{-}))=\deg \overline{f} \cdot \Phi_1 \left(C_{p, \overline{f}(e)}^{a, b}-(C^{a, b}_{p,\overline{f}(e)})^{-} \right),\end{align*} $$

we are reduced to the case when $N=p$ .

By Lemma 4.1 and Proposition 3.3, it suffices to show that, for the specific choices of e and $\infty $ as above, the element $\mathbb {E}^{\infty }_{e} \in \mathrm {Ext}_{\mathrm {MHS}(\mathbb {Q})}(H^1 \otimes H^1 \otimes H^1, \mathbb {Q}(-1))$ is nonzero. By Lemma 2.1, the automorphism $\beta $ of $F_p$ descends to an automorphism $\widetilde {\beta }$ of $C_p^{a, b}$ ; let Z be the graph of $\widetilde {\beta }$ . Since $[\infty ] - [e]$ is torsion, it suffices to show that $P_Z$ is non-torsion by Proposition 3.4. Since $\widetilde {\beta } \circ \pi _p^{a, b}=\pi _p^{a, b}\circ \beta $ and $\widetilde {\beta }$ has two fixed points by Lemma 2.3, we have

$$ \begin{align*} P_Z&=([\pi_p^{a, b}(S)]+[\pi_p^{a, b}(\overline{S})]-2[e])-([\widetilde{\beta}(e)]+[\widetilde{\beta}^{-1}(e)]-2[e]) \\ &=(\pi_p^{a, b})_*(([S]+[\overline{S}]-2[P_0])-([\beta(P_0)]+[\beta^{-1}(P_0)]-2[P_0])). \end{align*} $$

The point $[\beta (P_0)]+[\beta ^{-1}(P_0)]-2[P_0]$ is a torsion point on $\operatorname {Jac}(F_p)$ ; hence, $(\pi _p^{a, b})_*([\beta (P_0)]+[\beta ^{-1}(P_0)]-2[P_0])$ is a torsion point on $\operatorname {Jac}(C_p^{a, b})$ . However, since $a-b, a+2b, 2a+b \not \equiv 0 \ \ \pmod {p}$ by the assumption $a^2+ab+b^2 \equiv 0 \ \ \pmod {p}$ , the point

$$ \begin{align*}(\pi_p^{a, b})_*([S]+[\overline{S}]-2[P_0]) \in \operatorname{Jac}(C_p^{a, b})\end{align*} $$

is non-torsion by Theorem 2.4. Therefore, the point $P_Z$ is non-torsion, which finishes the proof.