2.1 Higher Chow cycles of type (2,1)

For a smooth variety X over ${\mathbb {C}}$ , let ${\mathrm {CH}}^p(X,q)$ be the higher Chow group defined by Bloch. In this paper, we treat the case $(p,q)=(2,1)$ . In this case, the following fact is well-known (see, e.g., [Reference Müller-StachMS98, Corollary 5.3]).

Proposition 2.1 The higher Chow group $\mathrm {CH}^2(X,1)$ is isomorphic to the middle homology group of the following complex:

$$ \begin{align*} K_2^{\mathrm{M}} ({\mathbb{C}}(X)) \xrightarrow{T} \displaystyle\bigoplus_{C\in X^{(1)}}{\mathbb{C}}(C)^\times \xrightarrow{\mathrm{div}}\displaystyle\bigoplus_{p\in X^{(2)}}{\mathbb{Z}}\cdot p. \end{align*} $$

Here, $X^{(r)}$ denotes the set of closed subvarieties of X of codimension r. The map T denotes the tame symbol map from the Milnor $K_2$ -group of the function field ${\mathbb {C}}(X)$ .

Therefore, a higher Chow cycle in $\mathrm {CH}^2(X,1)$ is represented by a formal sum

(3) $$ \begin{align} \sum_j (C_j, f_j)\in \displaystyle\bigoplus_{C\in X^{(1)}}{\mathbb{C}}(C)^\times, \end{align} $$

where $C_j$ are prime divisors on X and $f_j\in {\mathbb {C}}(C_j)^\times $ are nonzero rational functions on them such that $\sum _j {\mathrm {div}}_{C_j}(f_j) = 0$ as codimension two cycles on X.

A decomposable cycle in $\mathrm {CH}^2(X,1)$ is an element of the image of the group homomorphism

(4) $$ \begin{align} \mathrm{Pic}(X)\otimes_{\mathbb{Z}} \Gamma(X,\mathcal{O}_X^\times)={\mathrm{CH}}^1(X)\otimes_{\mathbb{Z}} {\mathrm{CH}}^1(X,1) \longrightarrow \mathrm{CH}^2(X,1) \end{align} $$

induced by the intersection product. Let C be a prime divisor on X, and let $[C]\in \mathrm {Pic}(X)$ be the class corresponding to C. The image of $[C]\otimes \alpha $ under (4) is represented by $(C,\alpha |_C)$ in the presentation (3). The cokernel of (4) is denoted by $\mathrm {CH}^2(X,1)_{\mathrm {ind}}$ . For $\xi \in {\mathrm {CH}}^2(X,1)$ , $\xi _{{\mathrm {ind}}}$ denotes its image in $\mathrm {CH}^2(X,1)_{\mathrm {ind}}$ . A cycle $\xi $ is called indecomposable if $\xi _{{\mathrm {ind}} }\neq 0$ .

2.2 The regulator map

By the canonical identification of $\mathrm {CH}^2(X,1)$ with the motivic cohomology $H^3_M(X,{\mathbb {Z}}(2))$ , there exists the map

(5) $$ \begin{align} \mathrm{reg}: \mathrm{CH}^2(X,1) \longrightarrow H^3_{\mathcal D}(X, {\mathbb{Z}}(2)) = \frac{H^2(X,{\mathbb{C}})}{F^2H^2(X,{\mathbb{C}})+H^2(X,{\mathbb{Z}}(2))} \end{align} $$

called the regulator map. The target $H^3_{\mathcal D}(X, {\mathbb {Z}}(2))$ denotes the Deligne cohomology of X. This map can be regarded as the Abel–Jacobi map for ${\mathrm {CH}}^2(X,1)$ . We recall an explicit formula for (5) following [Reference LevineLev88, pp. 458–459].

Let X be a $K3$ surface over ${\mathbb {C}}$ . By the Poincaré duality, the Deligne cohomology of X is isomorphic to the generalized complex torus

(6) $$ \begin{align} H^3_{\mathcal{D}}(X, {\mathbb{Z}}(2)) \simeq \frac{(F^1H^2(X,{\mathbb{C}}))^\vee}{H_2(X,{\mathbb{Z}})}, \end{align} $$

where $(F^1H^2(X,{\mathbb {C}}))^\vee $ is the dual ${\mathbb {C}}$ -vector space of $F^1H^2(X,{\mathbb {C}})$ and we regard $H_2(X,{\mathbb {Z}})$ as a subgroup of $(F^1H^2(X,{\mathbb {C}}))^\vee $ by the integration. Under the isomorphism (6), the image of the cycle $\xi $ represented by a formal sum $\sum _j (C_j, f_j)$ under the regulator map is described as follows.

Let $D_j$ be the normalization of the closed curve $C_j$ on X. Let $\mu _j: D_j\rightarrow X$ denote the composition of $D_j\rightarrow C_j$ and $C_j\rightarrow X$ . We will define a topological 1-chain $\gamma _j$ on $D_j$ . If $f_j$ is constant, we define $\gamma _j = 0$ . If $f_j$ is not constant, we regard $f_j$ as a finite morphism from $D_j$ to ${\mathbb {P}}^1$ . Then we define $\gamma _j = f_j^{-1}([\infty , 0]),$ where $[\infty , 0]$ is a path on ${\mathbb {P}}^1$ from $\infty $ to $0$ along the positive real axis. By the condition, $\sum _j {\mathrm {div}}_{C_j}(f_j) = 0$ , $\gamma = \sum _j (\mu _j)_*\gamma _j$ is a topological 1-cycle on X. Since $H_1(X, {\mathbb {Z}}) = 0$ , there exists a 2-chain $\Gamma $ on X such that $\partial \Gamma = \gamma $ . In this paper, $\gamma $ and $\Gamma $ are called the 1-cycle associated with $\xi $ and a 2-chain associated with $\xi $ , respectively. Then the image of $\xi $ under the regulator map is represented by the pairing

$$ \begin{align*} \langle \mathrm{reg}(\xi), [\omega]\rangle = \displaystyle\int_\Gamma\omega + \sum_j\dfrac{1}{2\pi\sqrt{-1}}\displaystyle\int_{D_j-\gamma_j}\log (f_j)\mu_j^*\omega\quad ([\omega]\in F^1H^2(X,{\mathbb{C}})). \end{align*} $$

Here, $\log (f_j)$ is the pull-back of the logarithmic function on ${\mathbb {P}}^1-[\infty ,0]$ by $f_j$ .

In this paper, we use the following variant of the regulator map.

Definition 2.2 The transcendental regulator map is the composite of the regulator map 5 and the projection induced by $H^{2,0}(X) \hookrightarrow F^1H^2(X,{\mathbb {C}})$ :

We denote this map by r.

By taking the pairing with a nonzero holomorphic 2-form $\omega $ on X, we have an isomorphism $H^{2,0}(X)^\vee /H_2(X,{\mathbb {Z}}) \simeq {\mathbb {C}}/{\mathcal {P}}(\omega ),$ where ${\mathcal {P}}(\omega )$ is the subgroup of ${\mathbb {C}}$ defined by

$$ \begin{align*} {\mathcal{P}}(\omega) = \left\{\displaystyle \int_{\Gamma}\omega \in {\mathbb{C}} : \Gamma \text{ is a topological 2-cycles on }X\right\}, \end{align*} $$

i.e., ${\mathcal {P}}(\omega )$ is the set of periods of X with respect to $\omega $ . By the above formula, the image of $\xi \in {\mathrm {CH}}^2(X,1)$ under the transcendental regulator map is

(7) $$ \begin{align} \langle r(\xi), [\omega]\rangle \equiv \int_{\Gamma}\omega \quad\mod {\mathcal{P}}(\omega), \end{align} $$

where $\Gamma $ is a 2-chain associated with $\xi $ . If $\xi $ is decomposable, $r(\xi ) = 0$ . This implies the following.

2.3 A relative setting

Since it is difficult to prove non-vanishingness of an element of $H^{2,0}(X)^\vee /H_2(X,{\mathbb {Z}})$ , we use its relative version. Let $\pi :{\mathcal {X}}\rightarrow S$ be an algebraic family of $K3$ surfaces over a variety S. We define sheaves ${\mathcal {P}}$ and ${\mathcal {Q}}$ of abelian groups on S by

$$ \begin{align*} \begin{aligned} {\mathcal{P}} &= \mathrm{Im}(R^2\pi_*\underline{{\mathbb{Z}}}_ {{\mathcal{X}}}\rightarrow \mathcal{H}om_{{\mathcal{O}}^{{\mathrm{an}}}_{S}}(\pi_*\Omega^2_{{\mathcal{X}}/S}, {\mathcal{O}}^{\mathrm{an}}_{S})), \\ {\mathcal{Q}} &= \mathrm{Coker}(R^2\pi_*\underline{{\mathbb{Z}}}_ {{\mathcal{X}}}\rightarrow \mathcal{H}om_{{\mathcal{O}}^{{\mathrm{an}}}_{S}}(\pi_*\Omega^2_{{\mathcal{X}}/S}, {\mathcal{O}}^{\mathrm{an}}_{S})), \end{aligned} \end{align*} $$

where $\underline {{\mathbb {Z}}}_{{\mathcal {X}}}$ is the constant sheaf on ${\mathcal {X}}$ and ${\mathcal {O}}^{\mathrm {an}}_{S}$ is the sheaf of holomorphic functions on S. Note that ${\mathcal {P}}$ is a local system on S.

By considering the pairing with a nonzero relative 2-form $\omega \in \Gamma ({\mathcal {X}},\Omega ^2_{{\mathcal {X}}/S})$ , we have an isomorphism $\mathcal {H}om_{{\mathcal {O}}^{{\mathrm {an}}}_{S}}(\pi _*\Omega ^2_{{\mathcal {X}}/S}, {\mathcal {O}}^{\mathrm {an}}_{S})\simeq {\mathcal {O}}^{\mathrm {an}}_{S}$ . Hence, ${\mathcal {P}}$ is isomorphic to the subsheaf ${\mathcal {P}}_\omega $ of ${\mathcal {O}}^{\mathrm {an}}_{S}$ generated by period functions with respect to $\omega $ and ${\mathcal {Q}}$ is isomorphic to ${\mathcal {Q}}_\omega = {\mathcal {O}}^{\mathrm {an}}_{S}/{\mathcal {P}}_\omega $ . For a local section $\varphi $ of ${\mathcal {O}}^{\mathrm {an}}_{S}$ , its image in ${\mathcal {Q}}_\omega $ is denoted by $[\varphi ]$ .

For each $s\in S$ , there exists the evaluation map

$$ \begin{align*} {\mathrm{ev}}_s: \Gamma(S,{\mathcal{Q}}) \rightarrow H^{2,0}({\mathcal{X}}_s)^\vee/H_2({\mathcal{X}}_s,{\mathbb{Z}}). \end{align*} $$

Under the isomorphisms ${\mathcal {Q}}\simeq {\mathcal {Q}}_{\omega }$ and $H^{2,0}({\mathcal {X}}_s)^\vee /H_2({\mathcal {X}}_s,{\mathbb {Z}})\simeq {\mathbb {C}}/{\mathcal {P}}(\omega _s)$ induced by $\omega $ and $\omega _s$ , the evaluation map coincides with the map

$$ \begin{align*} {\mathcal{Q}}_\omega = {\mathcal{O}}^{\mathrm{an}}_{S}/{\mathcal{P}}_\omega \ni [f] \longmapsto f(s)\quad\mod {\mathcal{P}}(\omega_s)\quad \in {\mathbb{C}}/{\mathcal{P}}(\omega_s), \end{align*} $$

where $f(s)\in {\mathbb {C}}$ denotes the value of the holomorphic function f at s. The following elementary lemma is crucial for the result.

Proof Since the question is local, we can shrink S in the sense of the classical topology. By fixing a relative 2-form $\omega \in \Gamma ({\mathcal {X}},\Omega _{{\mathcal {X}}/S}^2)$ , we have the isomorphism ${{\mathcal {Q}} \simeq {\mathcal {O}}_S^{\mathrm {an}}/{\mathcal {P}}_\omega} $ . We may assume that there exist a holomorphic function $\varphi \in {\mathcal {O}}_S^{{\mathrm {an}}}(S)$ such that $\nu = [\varphi ]$ and a free basis $f_1,f_2,\dots ,f_r$ of ${\mathcal {P}}_\omega (S)$ .

For each $\underline {c} = (c_i) \in {\mathbb {Z}}^r$ , we define the holomorphic function $F_{\underline {c}}$ by

$$ \begin{align*} F_{\underline{c}} = \varphi - \sum_{i=1}^r c_if_i. \end{align*} $$

Since $\nu = [\varphi ]$ is nonzero in ${\mathcal {Q}}_\omega (S)$ , $F_{\underline {c}}$ is a nonzero holomorphic function for each $\underline {c} \in {\mathbb {Z}}^r$ . Consider the countable family $\{F_{\underline {c}}\}_{\underline {c}\in {\mathbb {Z}}^r}$ of the holomorphic functions. Then outside of the zeros of the functions in this family, $\varphi (s)\not \in {\mathcal {P}}(\omega _s) = \langle f_1(s),f_2(s),\dots , f_r(s) \rangle _{\mathbb {Z}}$ . Hence ${\mathrm {ev}}_s(\nu )\neq 0$ .

The following corollary is also used in technical propositions.

Finally, we consider the regulator map in the relative setting. Suppose that we have irreducible divisors ${\mathcal {C}}_j$ on ${\mathcal {X}}$ which are smooth over S and nonzero rational functions $f_j$ on ${\mathcal {C}}_j$ whose zeros and poles are also smooth over S. Assume that they satisfy the condition $\sum _j{\mathrm {div}}_{({\mathcal {C}}_j)_s}((f_j)_s) = 0$ for each $s\in S$ . Then we have a family of higher Chow cycles $\xi = \{\xi _s\}_{s\in S}$ such that $\xi _s\in {\mathrm {CH}}^2({\mathcal {X}}_s,1)$ is represented by the formal sum $\sum _{j}(({\mathcal {C}}_j)_s,(f_j)_s)$ . A family of higher Chow cycles constructed in this way is called an algebraic family of higher Chow cycles in this paper.

If we shrink S in the sense of the classical topology, there exists a $C^\infty $ -family of topological 2-chains $\{\Gamma _s\}_{s\in S}$ such that $\Gamma _s$ is a 2-chain associated with $\xi _s$ . If we fix a relative 2-form $\omega $ , the function

$$ \begin{align*} S\ni s \mapsto \int_{\Gamma_s}\omega_s \in {\mathbb{C}} \end{align*} $$

is holomorphic (cf. [Reference Chen and LewisCL05, Proposition 4.1]). Hence, we can define the element ${\nu _{\mathrm {tr}}(\xi ) \in \Gamma (S,{\mathcal {Q}})}$ such that

$$ \begin{align*} {\mathrm{ev}}_s(\nu_{\mathrm{tr}}(\xi)) = r(\xi_s) \end{align*} $$

for every $s\in S$ . This $\nu _{\mathrm {tr}}(\xi )$ can be regarded as a part of the normal function associated with $\xi $ .

By combining Proposition 2.3 and Lemma 2.4, we have the following.

4.1 Construction of higher Chow cycles at fibers

First, we explain the construction at each fiber over $t=(a,b)\in T_0$ . Let ${\mathcal {D}} \subset {\mathbb {P}}^1\times {\mathbb {P}}^1=({\mathcal {Y}}_0)_t$ be the diagonal curve. Since $a\neq b$ , ${\mathcal {D}} $ intersects with the branching locus of $({\mathcal {X}}_0)_t\rightarrow ({\mathcal {Y}}_0)_t$ as Figure 2.

Figure 2:

The diagonal curve ${\mathcal {D}}$ and the branching locus.

Let ${\mathcal {C}} \subset ({\mathcal {X}}_0)_t$ be the pull-back of ${\mathcal {D}}$ by $({\mathcal {X}}_0)_t\rightarrow ({\mathcal {Y}}_0)_t$ , and let $\widetilde {{\mathcal {C}}}$ be the strict transform of ${\mathcal {C}}$ by the blowing-up $(\widetilde {{\mathcal {X}}}_0)_t\rightarrow ({\mathcal {X}}_0)_t$ . Then we see that $\widetilde {{\mathcal {C}}}$ is smooth and $\widetilde {{\mathcal {C}}} \rightarrow {\mathcal {D}} $ is a double covering ramified at two points. Hence, $\widetilde {{\mathcal {C}}}$ is isomorphic to ${\mathbb {P}}^1$ . Furthermore, for each $\bullet \in \{0,1,\infty \}$ , $\widetilde {{\mathcal {C}}} $ intersects with $Q_{(\bullet ,\bullet )}$ at two points $p_{\bullet }^+$ and $p_{\bullet }^-$ (cf. Figure 3).

Figure 3:

The intersections of $\widetilde {{\mathcal {C}}}$ , $Q_{(0,0)}$ , $Q_{(1,1)}$ , and $Q_{(\infty ,\infty )}$ .

Hence, we can find rational functions $\psi _0, \psi _1,\psi _\infty \in {\mathbb {C}}(\widetilde {{\mathcal {C}}})$ and $\varphi _{\bullet } \in {\mathbb {C}}(Q_{(\bullet , \bullet )}) (\bullet = 0,1,\infty \})$ which satisfy the following relations:

(8) $$ \begin{align} \begin{aligned} {\mathrm{div}}_{\widetilde{{\mathcal{C}}}}(\psi_0) &= p_{0}^- - p_{0}^+ = - {\mathrm{div}}_{Q_{(0,0)}}(\varphi_0), \\{\mathrm{div}}_{\widetilde{{\mathcal{C}}}}(\psi_1) &= p_{1}^- - p_{1}^+ = - {\mathrm{div}}_{Q_{(1,1)}}(\varphi_1), \\{\mathrm{div}}_{\widetilde{{\mathcal{C}}}}(\psi_\infty) &= p_\infty^- - p_\infty^+ = - {\mathrm{div}}_{Q_{(\infty,\infty)}}(\varphi_\infty). \end{aligned} \end{align} $$

We define $(\xi _0)_t,(\xi _1)_t,(\xi _\infty )_t\in {\mathrm {CH}}^2((\widetilde {{\mathcal {X}}}_0)_t,1)$ by the formal sums

(9) $$ \begin{align} \begin{aligned} (\xi_0)_t &= (\widetilde{{\mathcal{C}}}, \psi_0) + (Q_{(0,0)},\varphi_0), \\(\xi_1)_t &= (\widetilde{{\mathcal{C}}}, \psi_1) + (Q_{(1,1)},\varphi_1), \\(\xi_\infty)_t &= (\widetilde{{\mathcal{C}}}, \psi_\infty) + (Q_{(\infty,\infty)},\varphi_\infty). \end{aligned} \end{align} $$

4.2 Construction of families of higher Chow cycles

To get families of higher Chow cycles, it is enough to construct rational functions $\varphi _\bullet $ and $\psi _\bullet $ on the family. However, the intersection points $p_{1}^+$ and $ p_{1}^-$ (resp. $p_{\infty }^+$ and $ p_{\infty }^-$ ) interchange by the monodromy of $T_0$ . Hence, it is impossible to construct such rational functions for $\bullet = 1,\infty $ . Thus, it is necessary to take a finite étale base change of $T_0$ to define the families of cycles.

Let $B = B_0[\sqrt {a},\sqrt {b},\sqrt {1-a},\sqrt {1-b}]$ and $T\rightarrow T_0$ be the finite étale cover corresponding to $B_0\rightarrow B$ . The base changes of ${\mathcal {Y}}_0,{\mathcal {X}}_0,{\mathcal {A}}_0, \widetilde {{\mathcal {Y}}}_0, \widetilde {{\mathcal {X}}}_0$ , and $\widetilde {{\mathcal {A}}}_0$ by $T\rightarrow T_0$ are denoted by ${\mathcal {Y}},{\mathcal {X}},{\mathcal {A}}, \widetilde {{\mathcal {Y}}}, \widetilde {{\mathcal {X}}}$ , and $\widetilde {{\mathcal {A}}}$ , respectively.

Let ${\mathcal {D}}\subset {\mathcal {Y}}$ be the closed subscheme defined by the local equation $x=y$ , let ${\mathcal {C}}$ be its pull-back by ${\mathcal {X}}\rightarrow {\mathcal {Y}}$ , and let $\widetilde {{\mathcal {C}}}$ be its strict transform by the blowing up $\widetilde {{\mathcal {X}}}\rightarrow {\mathcal {X}}$ . On the local chart V, $\widetilde {{\mathcal {C}}}\hookrightarrow \widetilde {{\mathcal {X}}}$ is described by the following ring homomorphism:

$$ \begin{align*} B[x,y,v]/(v^2f(x)-g(y)) \:{\longrightarrow}\: B[z,v]/(v^2(1-az)-(1-bz)); x,y,v\mapsto z,z,v. \end{align*} $$

By this description, we see that $Q_{(0,0)}$ and $\widetilde {{\mathcal {C}}}$ intersect at $(x,y,v) = (0,0,\pm 1)$ , and $Q_{(1,1)}$ and $\widetilde {{\mathcal {C}}}$ intersect at $(x,y,v) = \left (1,1,\pm \sqrt {1-b}/\sqrt {1-a}\right )$ . By the local computation on another local chart containing $(\infty ,\infty )\in \Sigma ^2$ , $Q_{(\infty ,\infty )}$ and $\widetilde {{\mathcal {C}}}$ intersect at $(\xi ,\eta ,v')=\left (0,0,\pm \sqrt {b}/\sqrt {a}\right )$ , where local coordinates $\xi ,\eta $ , and $v'$ are defined by $\xi =1/x$ , $\eta =1/y,$ and $v' = x^2v/y^2$ . Hence, we can define rational functions $\psi _\bullet \in {\mathbb {C}}(\widetilde {{\mathcal {C}}})$ and $\varphi _{\bullet }\in {\mathbb {C}}(Q_{(\bullet ,\bullet )})\: (\bullet \in \{0,1,\infty \})$ by the following equations:

$$ \begin{align*} \begin{aligned} & \psi_0 = (v+1)\cdot(v-1)^{-1}, && \varphi_0 = (v-1)\cdot(v + 1)^{-1}, \\ & \psi_1 = \left(v + \frac{\sqrt{1-b}}{\sqrt{1-a}}\right)\cdot\left(v - \frac{\sqrt{1-b}}{\sqrt{1-a}}\right)^{-1}, && \varphi_1 = \left(v-\frac{\sqrt{1-b}}{\sqrt{1-a}}\right)\cdot\left(v+\frac{\sqrt{1-b}}{\sqrt{1-a}}\right)^{-1}, \\ & \psi_\infty = \left(v' + \frac{\sqrt{b}}{\sqrt{a}}\right)\cdot\left(v' - \frac{\sqrt{b}}{\sqrt{a}}\right)^{-1}, && \varphi_\infty =\left(v'- \frac{\sqrt{b}}{\sqrt{a}}\right)\cdot\left(v'+ \frac{\sqrt{b}}{\sqrt{a}}\right)^{-1}. \end{aligned} \end{align*} $$

They satisfy the relations in (8) and we define algebraic families of higher Chow cycles $\xi _0 = \left \{(\xi _0)_t\right \}_{t\in T}, \xi _1 = \left \{(\xi _1)_t\right \}_{t\in T}$ and $\xi _{\infty } =\left \{(\xi _\infty )_t\right \}_{t\in T}$ by the equations (9).

5.1 The Picard–Fuchs differential operator

For $c\in {\mathbb {C}}-{\mathbb {R}}_{\ge 0}$ , we consider the following improper integrals:

$$ \begin{align*} P_1(c)=\int_0^1\frac{dx}{\sqrt{x(1-x)(1-c x)}},\quad P_2(c)=\int_1^\infty\frac{dx}{\sqrt{x(1-x)(1-c x)}}. \end{align*} $$

As is well-known (see, e.g., [Reference Whittaker and WatsonWW62, p. 253]), these integrals converge and give linearly independent solutions of the hypergeometric differential equation

(13) $$ \begin{align} c(1-c)\frac{d^2P}{dc^2} +(1-2c)\frac{dP}{dc} -\frac{1}{4}P=0. \end{align} $$

For any $c\in S_0$ , they can be analytically continued to a neighborhood of c and we use the same notation for the resulting multivalued functions.

Let $\theta $ be the relative 1-form $\frac {dx}{y}$ on the Legendre family of elliptic curves ${\mathcal {E}}\rightarrow S_0$ . For $c\in {\mathbb {C}}-{\mathbb {R}}_{\ge 0}$ , let $\gamma _+,\gamma _-$ (resp. $\delta _+,\delta _-$ ) be lifts of paths $[0,1]$ and $[1,\infty ]$ on ${\mathbb {P}}^1$ by ${\mathcal {E}}_c \rightarrow {\mathbb {P}}^1$ . Let $\theta _c$ be the pull-back of $\theta $ at ${\mathcal {E}}_c$ . Then we have

$$ \begin{align*} \int_{\gamma_+}\theta_c = -\int_{\gamma_-}\theta_c,\quad \int_{\delta_+}\theta_c = -\int_{\delta_-}\theta_c, \end{align*} $$

and they coincide with $\pm P_1(c)$ and $\pm P_2(c)$ , respectively. Since $[\gamma _+]-[\gamma _-]$ and $[\delta _+]-[\delta _-]$ are generators of $H_1({\mathcal {E}}_c,{\mathbb {Z}})$ (see, e.g., [Reference Carlson, Müller-Stach and PetersCMP02, p. 10]), $2P_1$ and $2P_2$ are local basis of the period functions of ${\mathcal {E}}\rightarrow S_0$ with respect to $\theta $ . In particular, hypergeometric differential equation (13) is a Picard–Fuchs differential equation of ${\mathcal {E}}\rightarrow S_0$ .

Next, we will find a Picard–Fuchs differential operator of $\widetilde {{\mathcal {X}}}\rightarrow T$ . Recall that we have T-morphisms

We name the morphisms p and $\pi $ as above. We have the relative 2-form $pr_1^*(\theta )\wedge pr_2^*(\theta )$ on ${\mathcal {A}}\rightarrow T$ , where $pr_i$ is the morphism ${\mathcal {A}}\rightarrow {\mathcal {A}}_0 \hookrightarrow {\mathcal {E}}\times {\mathcal {E}} \xrightarrow {pr_i} {\mathcal {E}}$ . Then its pull-back by p is stable under the covering transformation of $\pi $ , so it descends to $\widetilde {{\mathcal {X}}}$ . We denote the resulting 2-form on $\widetilde {{\mathcal {X}}}$ by $\omega $ . The relative 2-form $\omega $ is described as $\frac {dx\wedge dy}{vf(x)}$ and $\frac {dx\wedge dy}{wg(y)}$ on the local charts V and W. Then we have the following.

Proof By the Künneth formula, we see that $4P_i(a)P_j(b) \:\: (i,j\in \{1,2\})$ is a local basis for the local system generated by period functions of ${\mathcal {A}}\rightarrow T$ with respect to $pr_1^*(\theta )\wedge pr_2^*(\theta )$ . For $t\in T$ , let $\phi $ be the morphism of Hodge structures defined by

where $\pi _!$ is the Gysin morphism induced by $\pi $ . Since the mapping degree of $\pi $ is 2, $\pi _!\circ \pi ^*: H^2(\widetilde {{\mathcal {X}}}_t)\rightarrow H^2(\widetilde {{\mathcal {X}}}_t)$ equals multiplication by 2 (cf. [Reference Voisin and SchnepsVoi02, Remark 7.29]). Since $\pi ^*(\omega ) = p^*\left (pr_1^*(\theta )\wedge pr_2^*(\theta )\right )$ , we have the relation

(14) $$ \begin{align} \phi([\left(pr_1^*(\theta)\wedge pr_2^*(\theta)\right)_t] = 2[\omega_t]. \end{align} $$

Let $\phi ^\vee : H_2(\widetilde {{\mathcal {X}}}_t) \rightarrow H_2({\mathcal {A}}_t)$ be the dual of $\phi $ . For any $[\Gamma ]\in H_2(\widetilde {{\mathcal {X}}}_t,{\mathbb {Z}})$ and $[\Gamma ']\in H_2({\mathcal {A}}_t,{\mathbb {Z}})$ such that $\phi ^\vee ([\Gamma ]) = [\Gamma ']$ , we have

$$ \begin{align*} \int_{\Gamma}\omega_t = \frac{1}{2}\int_{\Gamma'}\left(pr_1^*(\theta)\wedge pr_2^*(\theta)\right)_t \end{align*} $$

by (14). Since the right-hand side is a linear combination of $2P_i(a)P_j(b)$ , we see that any period function of $\widetilde {{\mathcal {X}}}\rightarrow T$ with respect to $\omega $ is a linear combination of $2P_i(a)P_j(b)$ . Furthermore, since $\phi $ is injective and its cokernel has no torsion (cf. [Reference Barth, Hulek, Peters and Van de VenBHPV04], Chapter VIII, Proposition 5.1, and Corollary 5.6), $\phi ^\vee $ is surjective. Thus, $2P_i(a)P_j(b)$ itself is a period function for $i,j\in \{1,2\}$ and we have the result.

Now we can prove Proposition 5.2.

5.2 Calculation of the regulator

For a while, we fix $t_0\in T$ such thatFootnote ² $\sqrt {1-a}$ , $\sqrt {1-b}\in {\mathbb {R}}_{>0}$ . Our first goal is to compute the image of $(\xi _1)_t-(\xi _0)_t$ under the transcendental regulator map locally around $t_0$ . For the computation, we construct a 2-chain and use (7). In the calculation, we replace a 2-chain associated with $(\xi _1)_t-(\xi _0)_t$ with another 2-chain to make the computation easier.

Let $\xi $ be a higher Chow cycle on a $K3$ surface X, and let $\gamma $ be the topological 1-cycle associated with $\xi $ . A topological 2-chain $\Gamma '$ is called a 2-chain associated with $\xi $ in a weak sense if there exists a finite family of curves $\{E_k\}_{k}$ on X and a topological 2-chain $\Gamma _k$ on each $E_k$ such that

(15) $$ \begin{align} \partial \Gamma' = \gamma + \sum_{k}\partial\Gamma_k. \end{align} $$

Let $\Gamma $ be a 2-chain associated with $\xi $ . If $\Gamma '$ is a 2-chain associated with $\xi $ in a weak sense, $\Gamma $ and $\Gamma '+\sum _{k}\Gamma _k$ coincide up to topological 2-cycles. Since the pull-back of a holomorphic 2-form $\omega $ on $E_k$ vanishes, we have

$$ \begin{align*} \int_{\Gamma'}\omega \equiv \int_{\Gamma}\omega \underset{(7)}{\equiv} \langle r(\xi), [\omega]\rangle \quad\mod {\mathcal{P}}(\omega). \end{align*} $$

Hence, we can also use $\Gamma '$ for the computation of the transcendental regulator map.

We will construct a desired 2-chain. For t which is sufficiently close to $t_0$ , let $\triangle _+$ and $\triangle _-$ be the images of the following maps:

(16)

Note that since t is sufficiently close to $t_0$ , we may assume the function $\sqrt {1-ax}$ (resp. $\sqrt {1-by}$ ) do not ramify on $\triangle $ and we can fix the branch of it so that it takes a value $1$ and $\sqrt {1-a}$ at $x=0,1$ (resp. $1$ and $\sqrt {1-b}$ at $y=0,1$ ).

We define $K_+$ and $K_-$ as the closures of $\triangle _+$ and $\triangle _-$ , respectively. Let $\gamma $ be a path $[0,1]$ on ${\mathbb {P}}^1$ reparameterized so that $\gamma (s) = s^2$ (resp. $1-\gamma (s) = (1-s)^2$ ) on a neighborhood of $0$ (resp. $1$ ). Then if we replace x and y in the target in (16) by $\gamma (x)$ and $\gamma (y)$ , respectively, (16) can be extended to a map from a compact-oriented manifold with corners, so $K_+$ and $K_-$ are $C^\infty $ -chains.

Proposition 5.5 For any t which is sufficiently close to $t_0$ , we have

$$ \begin{align*} \langle r((\xi_1)_t-(\xi_0)_t), [\omega_t]\rangle \equiv \int_{K_+}\omega_t -\int_{K_-}\omega_t \quad\mod {\mathcal{P}}(\omega_t). \end{align*} $$

To prove this, we should examine the boundaries of $K_+$ and $K_-$ . Recall that we have the following morphisms.

We regard $\triangle =\{(x,y)\in {\mathbb {R}}^2:0<y<x<1\}$ as a subset of ${\mathcal {Y}}_t={\mathbb {P}}^1\times {\mathbb {P}}^1$ . Let K be the closure of the inverse image of $\triangle $ by $\widetilde {{\mathcal {Y}}}_t\rightarrow {\mathcal {Y}}_t$ (see Figure 4). We define paths $\gamma _c,\gamma _{11},\gamma _{y},\gamma _{10}, \gamma _x,$ and $\gamma _{00}$ on the boundary $\partial K$ as in Figure 4.

Figure 4:

The 2-chain K and its boundary.

Figure 5:

The 2-chains $K_+$ and $K_-$ and their boundaries.

Then $\widetilde {{\mathcal {X}}}_t\rightarrow \widetilde {{\mathcal {Y}}}_t$ induces the homeomorphism $K_+\xrightarrow {\sim } K$ and $K_-\xrightarrow {\sim } K$ (see Figure 5). Here, the bold line in the figure denotes the branching locus of $\widetilde {{\mathcal {X}}}_t\rightarrow \widetilde {{\mathcal {Y}}}_t$ . We define paths $\gamma _{c,\pm }, \gamma _{11,\pm }, \gamma _y,\gamma _{10,\pm },\gamma _x,$ and $\gamma _{00,\pm }$ on $\partial K_+$ and $\partial K_-$ as in Figure 5. They satisfy the following properties:

1. (1) The paths $\gamma _{c,+}$ and $\gamma _{c,-}$ are the lifts of $\gamma _c$ . Since $\gamma _c$ is on the strict transformation of ${\mathcal {D}}$ , they are on the curve $\widetilde {{\mathcal {C}}}$ .

2. (2) Furthermore, since t is close to $t_0$ , $\gamma _{c,+}$ (resp. $\gamma _{c,-}$ ) is a path from $(1,1,\sqrt {1-b}/\sqrt {1-a})$ to $(0,0,1)$ (resp. $(1,1,-\sqrt {1-b}/\sqrt {1-a})$ to $(0,0,-1)$ ).

3. (3) The paths $\gamma _{00,+},\gamma _{10,+}$ , and $\gamma _{11,+}$ (resp. $\gamma _{00,-},\gamma _{10,-}$ , and $\gamma _{11,-}$ ) are the lifts of $\gamma _{00},\gamma _{10}$ , and $\gamma _{11}$ and they are on the exceptional curves $Q_{(0,0)}, Q_{(1,0)}$ , and $Q_{(1,1)}$ , respectively.

4. (4) Since $\gamma _x$ and $\gamma _y$ on $\partial K$ is contained in the branching locus of $\widetilde {{\mathcal {X}}}_t\rightarrow \widetilde {{\mathcal {Y}}}_t$ , there exist the unique lifts of them and their lifts are contained in $\partial K_+\cap \partial K_-$ .

Then we can prove Proposition 5.5.

Proof of Proposition 5.5

It is enough to show that $K_+-K_-$ is a 2-chain associated with $(\xi _1)_t-(\xi _0)_t$ in a weak sense. Note that $(\xi _1)_t-(\xi _0)_t$ is represented by the formal sum

$$ \begin{align*} \left(\widetilde{{\mathcal{C}}}, \psi_0^{-1}\psi_1\right) + \left(Q_{(0,0)}, \varphi_0^{-1}\right) + \left(Q_{(1,1)},\varphi_1\right). \end{align*} $$

Let $\widetilde {\gamma }_c$ (resp. $\widetilde {\gamma }_{00}$ , $\widetilde {\gamma }_{11}$ ) be the 1-chain on $\widetilde {{\mathcal {C}}}$ (resp. $Q_{(0,0)}$ , $Q_{(1,1)}$ ) defined by the pull-back of $[\infty ,0]$ on ${\mathbb {P}}^1$ by the rational function $\psi _0^{-1}\psi _1$ (resp. $\varphi _0^{-1}$ , $\varphi _1$ ). Then $\widetilde {\gamma }_c+\widetilde {\gamma }_{00}+\widetilde {\gamma }_{11}$ is the 1-cycle associated with $\xi $ . By Figure 5, we have

$$ \begin{align*} \partial (K_+ -K_-) = (\gamma_{c,+}-\gamma_{c,-}) + (\gamma_{00,+}-\gamma_{00,-}) + (\gamma_{10,+} -\gamma_{10,-}) + (\gamma_{11,+}-\gamma_{11,-}). \end{align*} $$

Hence, we should show that the 1-chains $(\gamma _{c,+}-\gamma _{c,-}-\widetilde {\gamma }_c)$ , $(\gamma _{00,+}-\gamma _{00,-}-\widetilde {\gamma }_{00})$ , $(\gamma _{10,+} -\gamma _{10,-}),$ and $(\gamma _{11,+}-\gamma _{11,-}-\widetilde {\gamma }_{11})$ are 1-boundaries on the curves $\widetilde {{\mathcal {C}}},Q_{(0,0)}, Q_{(1,0)}$ , and $Q_{(1,1)}$ , respectively. Since $\widetilde {{\mathcal {C}}},Q_{(0,0)},Q_{(1,0)}$ and $Q_{(1,1)}$ are isomorphic to ${\mathbb {P}}^1$ and $H_1({\mathbb {P}}^1)=0$ , it is enough to show that they are 1-cycles. By Figure 5, we have the following relations:

$$ \begin{align*} \begin{aligned} &\partial(\gamma_{c,+}-\gamma_{c,-}) ={\mathrm{div}}_{\widetilde{{\mathcal{C}}}}(\psi_0^{-1}\psi_1) = \partial\widetilde{\gamma}_c,&&\partial(\gamma_{00,+}-\gamma_{00,-}) = {\mathrm{div}}_{Q_{(0,0)}}(\varphi_0^{-1}) = \partial\widetilde{\gamma}_{00,}\\ &\partial(\gamma_{11,+}-\gamma_{11,-}) = {\mathrm{div}}_{Q_{(1,1)}}(\varphi_1) = \partial\widetilde{\gamma}_{11},&& \partial(\gamma_{10,+} -\gamma_{10,-})=0. \end{aligned} \end{align*} $$

Hence, they are 1-cycles and we confirm that $K_+-K_-$ is a 2-chain associated with $(\xi _1)_t-(\xi _0)_t$ in a weak sense.

Let ${\mathcal {L}}$ be the local holomorphic function

(17) $$ \begin{align} {\mathcal{L}}(t) = \int_{K_+}\omega_t-\int_{K_-}\omega_t, \end{align} $$

which is defined around $t_0$ . Using the local description of $\omega _t$ , we see that

$$ \begin{align*} \int_{K_+}\omega_t=-\int_{K_-}\omega_t = \int_{\triangle}\frac{dxdy}{\sqrt{x(1-x)(1-ax)}\sqrt{y(1-y)(1-by)}}. \end{align*} $$

Hence, ${\mathcal {L}}(t)$ is a lift of ${\mathcal {L}}(a,b)$ in (10). Then we can prove Proposition 5.3.

Proof of Proposition 5.3

We will prove (1) and (2) simultaneously. Let $\{U_i\}_i$ be a good open cover of T such that for each i, there exists a local holomorphic function $\varphi _i$ on $U_i$ which represents $\nu _{{\mathrm {tr}}}(\xi _1-\xi _0)|_{U_i}$ . To prove (1) and (2), it is enough to show that ${\mathcal {L}}$ have an analytic continuation on each $U_i$ and the resulting function coincides with $\varphi _i$ up to an elements in ${\mathcal {P}}_\omega (U_i)$ .

By Proposition 5.5, the equation (11) holds for any point t of an open neighborhood $U_0$ of $t_0$ . Since the left-hand side of (11) corresponds to $\langle {\mathrm {ev}}_t(\nu _{{\mathrm {tr}}}(\xi _1-\xi _0)),[\omega _t]\rangle $ , we see that $[{\mathcal {L}}]$ corresponds to $\nu _{{\mathrm {tr}}}(\xi _1-\xi _0)|_{U_0}$ by Corollary 2.5. Thus if $U_i\cap U_0\neq \emptyset $ , we have $[\varphi _i]|_{U_i\cap U_0} = [{\mathcal {L}}]|_{U_i\cap U_0}$ , so ${\mathcal {L}}-\varphi _i \in {\mathcal {P}}_\omega (U_i\cap U_0)$ . Let $p_i$ be the restriction of ${{\mathcal {L}}-\varphi _i}$ on $U_i\cap U_0$ . Then we can extend $p_i$ on $U_i$ . The function $\varphi _i + p_i$ is a desired analytic continuation of ${\mathcal {L}}$ on $U_i$ . For a general $U_i$ , take a finite family $U_{i_1},\dots , U_{i_n}=U_i$ such that $U_0\cap U_{i_1}\neq \emptyset $ and $U_{i_{k-1}}\cap U_{i_{k}}\neq \emptyset $ for $k = 2,\dots ,n$ and repeat the above process.

Next, we will prove (3). We will compute ${\mathscr {D}}_1({\mathcal {L}})$ . It is enough to calculate it on a neighborhood of $t_0$ . Hence, we may assume ${\mathcal {L}}$ is given by (17).

Let $H(a,x)$ be a local holomorphic function defined by $H(a,x) =-\frac {\sqrt {x(1-x)}}{2\sqrt {1-ax}^3}$ . We can check that

$$ \begin{align*} {\mathscr{D}}_1 \left(\frac{1}{\sqrt{x(1-x)(1-ax)}}\right) = \frac{\partial H(a,x)}{\partial x}. \end{align*} $$

Then by Stokes’ theorem,Footnote ³ we have

$$ \begin{align*} \begin{aligned} &{\mathscr{D}}_1\left(\frac{1}{2}{\mathcal{L}}\right) = {\mathscr{D}}_1\left( \int_{K_+}\frac{dxdy}{\sqrt{x(1-x)(1-ax)}\sqrt{y(1-y)(1-by)}}\right) \\ &= \int_{K_+} d\left(\frac{H(a,x)dy}{\sqrt{y(1-y)(1-by)}}\right) = \int_{\partial K_+} \frac{H(a,x)dy}{\sqrt{y(1-y)(1-by)}}, \end{aligned} \end{align*} $$

and since the 1-form $\frac {H(a,x)dy}{\sqrt {y(1-y)(1-by)}}$ vanishes on $\partial K_+$ except $\gamma _{c,+}$ , we have

$$ \begin{align*} =\frac{1}{2}\int_0^1 \frac{dz}{(1-bz)^{\frac{1}{2}}(1-az)^{\frac{3}{2}}} =\frac{1}{a-b} \int_1^{\frac{\sqrt{1-b}}{\sqrt{1-a}}} du = \frac{1}{a-b}\cdot\left(\frac{\sqrt{1-b}}{\sqrt{1-a}}-1\right). \end{align*} $$

Here, we use the coordinate transform $u= \frac {\sqrt {1-bz}}{\sqrt {1-az}}$ . We can compute ${\mathscr {D}}_2({\mathcal {L}})$ similarly.

6.1 Generalities

A scheme with a group action $(S,H,\varphi )$ is a triplet consisting of a ${\mathbb {C}}$ -scheme S, a group H and a group homomorphism $\varphi : H\rightarrow {\mathrm {Aut}}(S)$ . We usually omit $\varphi $ from the notation and write $(S,H)$ . A morphism $(X,G)\rightarrow (S,H)$ between two such objects is a pair of a morphism $X\rightarrow S$ of varieties and a group homomorphism $G\rightarrow H$ satisfying the usual compatibility condition.

In Section 6.2, we use the following fiber product construction. Suppose we have two morphisms $(\pi ,\psi ): (X,G)\rightarrow (S,H)$ and $(f,\varphi ):(S',H')\rightarrow (S,H)$ . Recall that the fiber product of G and $H'$ over H is defined by

$$ \begin{align*} G\times_{H} H' = \{(g,h)\in G\times H': \psi(g) = \varphi(h)\}. \end{align*} $$

Then we define the fiber product of $(X,G)$ and $(S',H')$ over $(S,H)$ by

(18) $$ \begin{align} (X\times_{S} S', G\times_{H}H'). \end{align} $$

This is indeed the fiber product in the category of schemes with group actions.

Let $(X,G)$ be a scheme with a group action. Then we have the natural right G-action on $\Gamma (X,{\mathcal {O}}_X^\times )$ . A 1-cocycle on $\Gamma (X,{\mathcal {O}}_X^\times )$ is a map $\chi : G\rightarrow \Gamma (X,{\mathcal {O}}_X^\times )$ satisfying

$$ \begin{align*} \chi(gh) = h^\sharp(\chi(g))\cdot \chi(h) \end{align*} $$

for any $g,h\in G$ . Here, $h^\sharp : {\mathcal {O}}_X\rightarrow h_*{\mathcal {O}}_{X}$ is the natural morphism induced by ${h:X\rightarrow X}$ .

Let ${\mathscr {L}}$ be a G-linearized line bundle on X. The G-linearization is the same as a collection $(\Phi _{g}: g^*{\mathscr {L}}\xrightarrow {\sim } {\mathscr {L}})_{g\in G}$ of isomorphisms satisfying the relation $\Phi _{h}\circ h^*(\Phi _g) = \Phi _{gh}$ for $g,h\in G$ . We can construct a 1-cocycle as follows. Let s be a global section of ${\mathscr {L}}$ such that ${\mathrm {div}}(s)$ is stable under the G-action. Then there exists the unique 1-cocycle $\chi :G\rightarrow \Gamma (X,{\mathcal {O}}_X^\times )$ such that

$$ \begin{align*} \Phi_g(g^*(s)) = \chi(g)^{-1}\cdot s\quad (g\in G). \end{align*} $$

Finally, we prove liftability of a group action by a cyclic covers. For a line bundle ${\mathscr {L}}$ and a section $s\in \Gamma (X,{\mathscr {L}}^{\otimes (-m)})$ , the cyclic covering $Y\rightarrow X$ associated with $({\mathscr {L}},s)$ is defined as the relative spectrum of the ${\mathcal {O}}_X$ -algebra $\bigoplus _{i=0}^{m-1}{\mathscr {L}}^{\otimes i}$ (cf. [Reference Barth, Hulek, Peters and Van de VenBHPV04, p. 54]). The branching locus of $Y\rightarrow X$ coincides with ${\mathrm {div}}(s)$ .

Proposition 6.3 Let $\pi :Y\rightarrow X$ be the degree m cyclic covering associated with $({\mathscr {L}},s)$ . Suppose a group G acts on X and a G-linearization $(\Phi _g)_{g\in G}$ of ${\mathscr {L}}$ are given. Assume that there exists a 1-cocycle $\chi :G\rightarrow \Gamma (X,{\mathcal {O}}_X^\times )$ such that

$$ \begin{align*} \Phi_g^{\otimes (-m)}(g^*(s)) = \chi(g)^{-m}\cdot s \end{align*} $$

for any $g\in G$ . Then there exists a G-action on Y such that $\pi $ is G-equivariant.

Proof For $g\in G$ , we define an automorphism $\tilde {g}: Y\rightarrow Y$ as follows.

1. 1. Let $Y_1$ be the cyclic covering associated with $(g^*{\mathscr {L}}, g^*(s))$ . Then $Y_1$ is the fiber product of $Y\rightarrow X$ and $X\xrightarrow {\:g\:} X$ . Since g is an isomorphism, $Y_1\rightarrow Y$ is an isomorphism as well.

2. 2. Let $Y_2$ be the cyclic covering associated with $({\mathscr {L}},\chi (g)^{-m}\cdot s)$ . Then we have the isomorphism $Y_2\simeq Y_1$ over X induced by $\Phi _g$ .

3. 3. We have the ${\mathcal {O}}_X$ -module automorphism on ${\mathscr {L}}$ defined by the multiplication of $\chi (g)^{-1}$ . This automorphism induces an isomorphism $Y\simeq Y_2$ over X.

Composing these isomorphisms, we get an automorphism $\tilde {g}\in {\mathrm {Aut}}(Y)$ .

(19)

We can show that $G\rightarrow {\mathrm {Aut}}(Y);g\mapsto \tilde {g}$ is a group homomorphism by the condition on $(\Phi _{g})_{g\in G}$ and the property of the 1-cocycles. Hence, we can construct a G-action on Y and $\pi $ is G-equivariant by the construction.

6.2 Construction of the groups and their actions

In this section, we will construct a group action on $\widetilde {{\mathcal {X}}}\rightarrow T$ by four steps.

1. 1. We construct a $\mathfrak {S}(\Sigma )$ -action on ${\mathbb {P}}^1\times S_0\rightarrow S_0$ . By taking its direct product, we have a $G_{{\mathcal {Y}}_0}(=\mathfrak {S}(\Sigma )^2)$ -action on ${\mathcal {Y}}_0\rightarrow T_0$ which stabilizes $\Sigma ^2$ .

2. 2. By taking the fiber product of the $G_{{\mathcal {Y}}_0}$ -action on ${\mathcal {Y}}_0\rightarrow T_0$ and a $G_{T}(=\mathfrak {S}_4^2)$ -action on $T\rightarrow T_0$ , we have a $G_{{\mathcal {Y}}}$ -action on ${\mathcal {Y}}\rightarrow T$ .

3. 3. By considering a $\mu _2$ -extension $G_{{\mathcal {X}}}$ of $G_{{\mathcal {Y}}}$ , we can construct a $G_{{\mathcal {X}}}$ -action on ${\mathcal {X}} \rightarrow T$ by applying Proposition 6.3.

4. 4. We lift the $G_{{\mathcal {X}}}$ -action on ${\mathcal {X}}\rightarrow T$ to $\widetilde {{\mathcal {X}}}\rightarrow T$ .

Throughout in this section, we identify ${\mathrm {Aut}}(S_0) = \mathfrak {S}(\{0,1,\infty \}) (\simeq \mathfrak {S}_3)$ .

Proof Consider the following contravariant functor from the category of varieties to the category of sets.

For each $(Y\rightarrow X;p_1,p_2,p_3,p_4)$ , by considering the cross-ratio of $p_1(x)$ , $p_2(x)$ , $p_3(x)$ , $p_4(x)\in {\mathbb {P}}^1$ at $x\in X$ , we have the morphism $X\rightarrow S_0$ . Then we can check that $S_0$ represents this functor and $({\mathbb {P}}^1\times S_0\rightarrow S_0;0,1,\infty ,1/c)$ is the universal element. For each $\rho \in \mathfrak {S}_4$ , $\rho $ induces a natural automorphism on this functor. Combining with the fact that an automorphism on ${\mathbb {P}}^1$ which stabilizes four points is identity, this induces a $\mathfrak {S}_4$ -action on ${\mathbb {P}}^1\times S_0\rightarrow S_0$ . The property (2) is clear from this construction. The property (1) follows from the property of the cross-ratio.

We need an explicit description of the $\mathfrak {S}(\Sigma )$ -action on ${\mathbb {P}}^1\times S_0$ for Step 3. We summarize them on Table 1. In the table, for each $\rho \in \mathfrak {S}(\Sigma )$ , the image of c under $\underline {\rho }^\sharp :{\mathcal {O}}_{S_0}\rightarrow \underline {\rho }_*{\mathcal {O}}_{S_0}$ and the image of the local coordinate z on ${\mathbb {P}}^1$ under $\rho ^\sharp : {\mathcal {O}}_{{\mathbb {P}}^1\times S_0}\rightarrow \rho _*{\mathcal {O}}_{{\mathbb {P}}^1\times S_0}$ are given.

Table 1:

The $\mathfrak {S}(\Sigma )$ -action on ${\mathbb {P}}^1\times S_0.$

From Proposition 6.4, we have a morphism $({\mathbb {P}}^1\times S_0,\mathfrak {S}(\Sigma ))\rightarrow (S_0,\mathfrak {S}(\{0,1,\infty \}))$ . By taking the direct product, we have the morphism

(21)

We denote the group $\mathfrak {S}(\Sigma )\times \mathfrak {S}(\Sigma )$ (resp. $\mathfrak {S}(\{0,1,\infty \}) \times \mathfrak {S}(\{0,1,\infty \})$ ) by $G_{{\mathcal {Y}}_0}$ (resp. $G_{T_0}$ ). Since $T_0\subset S_0\times S_0$ is stable under the $G_{T_0}$ -action, (21) induces a morphism $({\mathcal {Y}}_0,G_{{\mathcal {Y}}_0})\rightarrow (T_0,G_{T_0})$ . Furthermore, by the property (2) of Proposition 6.4, we see that $\Sigma ^2\subset {\mathcal {Y}}_0$ is stable under the $G_{{\mathcal {Y}}_0}$ -action on ${\mathcal {Y}}_0$ . We finish Step 1.

Next, we will construct a group action on T. Recall that T is a finite étale cover of $T_0$ corresponding to $B_0 \rightarrow B_0[\sqrt {a},\sqrt {b},\sqrt {1-a},\sqrt {1-b}]$ .

Proof We have the following ring isomorphism:

(22)

Using the ring isomorphism (22), we can check directly that for each $\rho \in {\mathrm {Aut}}(S_0)$ , there exists a lift of $\rho $ in ${\mathrm {Aut}}(S')$ . Moreover, the $S_0$ -automorphism group of $S'$ is isomorphic to $\mu _2\times \mu _2$ . Thus we have the following exact sequence of groups:

$$ \begin{align*} 1 \:{\longrightarrow}\: \mu_2\times \mu_2 \:{\longrightarrow}\: {\mathrm{Aut}}(S'\rightarrow S_0) \:{\longrightarrow}\: {\mathrm{Aut}}(S_0) \:{\longrightarrow}\: 1. \end{align*} $$

To prove the proposition, we should show that ${\mathrm {Aut}}(S'\rightarrow S_0)$ is isomorphic to $\mathfrak {S}_4$ . We will construct an ${\mathrm {Aut}}(S'\rightarrow S_0)$ -action on a set of cardinality $4$ . By the ring isomorphism (22), we have $S' \simeq {\mathbb {P}}^1-\{0,\infty ,\pm 1, \pm \sqrt {-1}\}$ . Thus, ${\mathrm {Aut}}(S'\rightarrow S_0)$ acts on the set $\{0,\infty ,\pm 1, \pm \sqrt {-1}\}$ . Let F be the set

$$ \begin{align*} F = \left\{ \begin{aligned} &\{\{1,\sqrt{-1},\infty\},\{-1,-\sqrt{-1},0\}\}, && \{\{-1,\sqrt{-1},\infty\},\{1,-\sqrt{-1},0\}\} \\ &\{\{1,-\sqrt{-1},\infty\},\{-1,\sqrt{-1},0\}\}, &&\{\{-1,-\sqrt{-1},\infty\},\{1,\sqrt{-1},0\}\} \end{aligned} \right\}. \end{align*} $$

If we plot $\pm 1, \pm \sqrt {-1},0,\infty $ on the Riemann sphere, they are the vertexes of the regular octahedron. Then F is the set of pairs of opposite faces of the octahedron. We can check that ${\mathrm {Aut}}(S'\rightarrow S_0)$ acts on F and the group homomorphism ${\mathrm {Aut}}(S'\rightarrow S_0)\rightarrow \mathfrak {S}(F)$ is surjective. By the exact sequence above, the orders of ${\mathrm {Aut}}(S'\rightarrow S_0)$ and $\mathfrak {S}(F)$ coincide. Thus ${\mathrm {Aut}}(S'\rightarrow S_0)\simeq \mathfrak {S}(F)$ .

From Proposition 6.5, we have a morphism $(S',\mathfrak {S}_4)\rightarrow (S_0,\mathfrak {S}(\{0,1,\infty \}))$ . By taking its direct product, we have

(23)

We denote the group $\mathfrak {S}_4\times \mathfrak {S}_4$ by $G_{T}$ . Since T is a fiber product of $T_0$ and $S'\times S'$ over $S_0\times S_0$ , this induces a morphism $(T,G_{T})\rightarrow (T_0,G_{T_0})$ .

We have already constructed the morphisms $({\mathcal {Y}}_0,G_{{\mathcal {Y}}_0})\rightarrow (T_0,G_{T_0})$ and $(T,G_{T})\rightarrow (T_0,G_{T_0})$ . Then the group

$$ \begin{align*} G_{{\mathcal{Y}}_0}\times_{G_{T_0}} G_{T} = (\mathfrak{S}(\Sigma)^2)\times_{\mathfrak{S}(\{0,1,\infty\})^2} (\mathfrak{S}_4^2)\simeq (\mathfrak{S}_4\times_{\mathfrak{S}_3}\mathfrak{S}_4)^2 \end{align*} $$

acts on ${\mathcal {Y}} = {\mathcal {Y}}_0\times _{T_0} T$ by (18). We denote this group by $G_{\mathcal {Y}}$ . Thus, we have the morphism $({\mathcal {Y}},G_{\mathcal {Y}})\rightarrow (T,G_{T})$ . We often denote an element of $G_{\mathcal {Y}}$ by a 3-tuple

$$ \begin{align*} (\rho_1,\rho_2,\tau) \in \mathfrak{S}(\Sigma)\times \mathfrak{S}(\Sigma)\times G_{T} \end{align*} $$

such that $\tau \in G_{T}$ and $(\rho _1,\rho _2)\in G_{{\mathcal {Y}}_0}$ have the same image in $G_{T_0}=\mathfrak {S}(\{0,1,\infty \})^2$ . We finish Step 2.

We will lift this group action to the double cover ${\mathcal {X}}\rightarrow {\mathcal {Y}}$ . Let ${\mathscr {L}}$ be the line bundle on ${\mathcal {Y}}$ defined by

$$ \begin{align*} {\mathscr{L}} = pr_1^*\Omega^1_{{\mathbb{P}}^1\times S_0/S_0} \otimes pr_2^*\Omega^1_{{\mathbb{P}}^1\times S_0/S_0}, \end{align*} $$

where $pr_i$ is a morphism ${\mathcal {Y}}\rightarrow {\mathcal {Y}}_0 \hookrightarrow ({\mathbb {P}}^1\times S_0)\times ({\mathbb {P}}^1\times S_0) \xrightarrow {pr_i} {\mathbb {P}}^1\times S_0$ . Let s be the section of ${\mathscr {L}}^{\otimes (-2)}$ defined by

$$ \begin{align*} s = f(x)g(y)(dx\otimes dy)^{\otimes(-2)}. \end{align*} $$

We have the natural $G_{{\mathcal {Y}}}$ -linearization $(\Phi _\rho )_{\rho \in G_{{\mathcal {Y}}}}$ of ${\mathscr {L}}$ since it is constructed from sheaves of differentials.

The morphism ${\mathcal {X}}\rightarrow {\mathcal {Y}}$ is the double covering associated with $({\mathscr {L}},s)$ . Since ${\mathrm {div}}(s)$ is equal to the branching locus of ${\mathcal {X}}\rightarrow {\mathcal {Y}}$ , ${\mathrm {div}}(s)$ is stable under the $G_{{\mathcal {Y}}}$ -action. Hence, we have the 1-cocycle $\eta : G_{{\mathcal {Y}}}\rightarrow \Gamma ({\mathcal {Y}},{\mathcal {O}}_{{\mathcal {Y}}}^\times ) = B^\times $ such that

$$ \begin{align*} \Phi_\rho^{\otimes(-2)}(\rho^*(s)) = \eta(\rho)^{-1}\cdot s \end{align*} $$

for any $\rho \in G_{{\mathcal {Y}}}$ . We can compute $\eta $ as follows.

Proposition 6.6 For $\rho = (\rho _1,\rho _2,\tau )\in G_{{\mathcal {Y}}}$ , $\eta (\rho )$ is determined by $\underline {\rho }_1, \underline {\rho }_2 \in \mathfrak {S}(\{0,1,\infty \})$ and calculated by the following formula:

$$ \begin{align*} \eta(\rho) = \eta_1(\underline{\rho}_1)\cdot \eta_2(\underline{\rho}_2), \end{align*} $$

where $\eta _1(\underline {\rho }_1), \eta _2(\underline {\rho }_2)\in B^\times $ are given by the following table.

Proof By the definition of the $G_{{\mathcal {Y}}}$ -linearization on ${\mathscr {L}}$ , we have

$$ \begin{align*} \Phi_{\rho}(\rho^*(dx\otimes dy)) = \left(\frac{\partial }{\partial x}\rho_1^\sharp(x)\right)\left(\frac{\partial }{\partial y}\rho_2^\sharp(y)\right)\cdot (dx\otimes dy). \end{align*} $$

Using Table 1, we can compute $\eta $ for each $\rho _1,\rho _2\in \mathfrak {S}(\Sigma )$ and we get the result.

To apply Proposition 6.3, we should construct a “square root” of the 1-cocycle $\eta $ . We define coboundary 1-cocycles $\phi _1,\phi _2:G_{T}\rightarrow B^\times $ by

$$ \begin{align*} \phi_1(\tau) = \tau^\sharp\left(\frac{\sqrt{a}\sqrt{1-a}}{a^2-a+1}\right)\cdot \left(\frac{\sqrt{a}\sqrt{1-a}}{a^2-a+1}\right)^{-1},\:\phi_2(\tau) = \tau^\sharp\left(\frac{\sqrt{b}\sqrt{1-b}}{b^2-b+1}\right)\cdot \left(\frac{\sqrt{b}\sqrt{1-b}}{b^2-b+1}\right)^{-1} \end{align*} $$

for $\tau \in G_{T}$ . Then we can check that for any $\rho =(\rho _1,\rho _2,\tau )\in G_{{\mathcal {Y}}}$ ,

(24) $$ \begin{align} \eta_i(\underline{\rho}_i) = {\mathrm{sgn}}(\underline{\rho}_i)\phi_i(\tau)^2 \end{align} $$

holds for $i=1,2$ . Hence, $\phi _1\cdot \phi _2$ coincides with a square root of $\eta $ up to sign. To get a square root of sign, we enlarge the group $G_{{\mathcal {Y}}}$ as follows.

We define the group $G_{{\mathcal {X}}}$ as the fiber product $G_{{\mathcal {Y}}}\times _{\mu _2} \mu _4$ , where the group homomorphisms $G_{{\mathcal {Y}}}\rightarrow \mu _2$ and $\mu _4\rightarrow \mu _2$ is given by

$$ \begin{align*} \begin{aligned} &G_{{\mathcal{Y}}} \rightarrow \mu_2; && (\rho_1,\rho_2,\tau) \mapsto {\mathrm{sgn}}(\underline{\rho}_1){\mathrm{sgn}}(\underline{\rho}_2), \\ &\mu_4 \rightarrow \mu_2; && \zeta \mapsto \zeta^2. \end{aligned} \end{align*} $$

We denote an element of $G_{{\mathcal {X}}}$ by a 4-tuple

$$ \begin{align*} (\rho_1,\rho_2,\tau,\zeta) \in \mathfrak{S}(\Sigma)\times \mathfrak{S}(\Sigma)\times G_{T}\times \mu_4. \end{align*} $$

The group $G_{{\mathcal {X}}}$ acts on ${\mathcal {Y}}$ through the surjection $G_{{\mathcal {X}}}\twoheadrightarrow G_{{\mathcal {Y}}}$ . For $\rho = (\rho _1,\rho _2,\tau ,\zeta )\in G_{{\mathcal {X}}}$ , we define

$$ \begin{align*} \chi(\rho) = \zeta\cdot \phi_1(\tau)\cdot \phi_2(\tau) \quad \in B^\times=\Gamma({\mathcal{Y}},{\mathcal{O}}_{\mathcal{Y}}^\times). \end{align*} $$

Then by (24), $\chi :G_{{\mathcal {X}}}\rightarrow \Gamma ({\mathcal {Y}},{\mathcal {O}}_{{\mathcal {Y}}}^\times )$ is a 1-cocycle satisfying the relation

$$ \begin{align*} \chi(\rho)^2 = \eta(\rho). \end{align*} $$

Hence, we have the following result.

Proof The $G_{{\mathcal {Y}}}$ -linearization $(\Phi _\rho )_{\rho \in G_{{\mathcal {Y}}}}$ of ${\mathscr {L}}$ induces the natural $G_{{\mathcal {X}}}$ -linearization of ${\mathscr {L}}$ . This linearization satisfies the equation

$$ \begin{align*} \Phi_{\rho}^{\otimes(-2)}(\rho^*(s)) = \eta(\rho)^{-1}\cdot s = \chi(\rho)^{-2}\cdot s. \end{align*} $$

Hence, by applying Proposition 6.3, we have the result.

Thus, we have the morphism $({\mathcal {X}},G_{{\mathcal {X}}})\rightarrow (T,G_{T})$ . We finish Step 3.

Recall that the set $\Sigma ^2$ on ${\mathcal {Y}}_0$ is stable under the $G_{{\mathcal {Y}}_0}$ -action. Then its base change $\Sigma ^2$ on ${\mathcal {Y}}$ is also stable under the $G_{{\mathcal {Y}}}$ -action. Since $\Sigma ^2\subset {\mathcal {Y}}$ is contained in the branching locus of ${\mathcal {X}}\rightarrow {\mathcal {Y}}$ and ${\mathcal {X}}\rightarrow {\mathcal {Y}}$ is $G_{{\mathcal {X}}}$ -equivariant, $\Sigma ^2$ on ${\mathcal {X}}$ is also $G_{{\mathcal {X}}}$ -stable. Then by the universality of the blowing-up, for any $\rho \in G_{{\mathcal {X}}}$ , there exists the unique lift $\widetilde {\rho }\in {\mathrm {Aut}}(\widetilde {{\mathcal {X}}})$ of $\rho $ . By the uniqueness, $G_{{\mathcal {X}}}\rightarrow {\mathrm {Aut}}(\widetilde {{\mathcal {X}}});\rho \mapsto \widetilde {\rho }$ defines a $G_{{\mathcal {X}}}$ -action on $\widetilde {{\mathcal {X}}}$ . We finish Step 4.

Finally, we can prove the main result of this section.

Proof of Proposition 6.2

We have the morphisms

$$ \begin{align*} (\widetilde{{\mathcal{X}}},G_{{\mathcal{X}}})\:{\longrightarrow}\: ({\mathcal{X}},G_{{\mathcal{X}}}) \:{\longrightarrow}\: ({\mathcal{Y}},G_{{\mathcal{Y}}}) \:{\longrightarrow}\: (T,G_{T}). \end{align*} $$

By composing these morphisms, we have the $G_{{\mathcal {X}}}$ -action on the family $\widetilde {{\mathcal {X}}}\rightarrow T$ .

At the end of this section, we see an explicit description of the $G_{{\mathcal {X}}}$ -action on $\widetilde {{\mathcal {X}}}$ using the local coordinates $x,y,v$ on V. Let $\rho =(\rho _1,\rho _2,\tau ,\zeta )\in G_{{\mathcal {X}}}$ . Since $\widetilde {{\mathcal {X}}}\rightarrow {\mathcal {Y}}$ is $G_{{\mathcal {X}}}$ -equivariant, we see that

$$ \begin{align*} x\mapsto \rho_1^\sharp(x),\quad y\mapsto \rho_2^\sharp(y). \end{align*} $$

Locally on $U\subset {\mathcal {Y}}$ , the morphism ${\mathcal {X}}\rightarrow {\mathcal {Y}}$ is described by the ring homomorphism

where u corresponds to the local section $(dx\otimes dy)$ of ${\mathscr {L}}$ . By the construction of the $G_{{\mathcal {X}}}$ -action, the local coordinate u transforms

$$ \begin{align*} u \mapsto \chi(\rho)^{-1}\frac{\partial}{\partial x}(\rho_1^\sharp(x))\frac{\partial}{\partial y}(\rho_2^\sharp(y)) u. \end{align*} $$

Since $u=vf(x)$ , by the formula $\rho _1^\sharp (f(x))=\eta _1(\underline {\rho }_1)^{-1}\left (\frac {\partial }{\partial x}\rho _1^\sharp (x)\right )^2f(x)$ , v transforms as follows:

$$ \begin{align*} v\mapsto \frac{\eta_1(\underline{\rho}_1)}{\chi(\rho)}\frac{\frac{\partial}{\partial y}(\rho_2^\sharp(y))}{\frac{\partial}{\partial x}(\rho_1^\sharp(x))}v. \end{align*} $$

7.1 The group action on the higher Chow cycles

First, we produce more families of higher Chow cycles from $\xi _0,\xi _1$ and $\xi _\infty $ by using the group action. For $t\in T$ , $\rho =(\rho _1,\rho _2,\tau ,\zeta )\in G_{{\mathcal {X}}}$ induces the isomorphism $\rho _t:\widetilde {{\mathcal {X}}}_t\rightarrow \widetilde {{\mathcal {X}}}_{\tau (t)}$ such that

(25)

commutes. The morphism $\rho _t$ induces a map $\rho _t^*:{\mathrm {CH}}^2(\widetilde {{\mathcal {X}}}_{\tau (t)},1)\rightarrow {\mathrm {CH}}^2(\widetilde {{\mathcal {X}}}_t,1)$ . For an algebraic family of higher Chow cycles $\xi =\{\xi _t\}_{t\in T}$ on $\widetilde {{\mathcal {X}}}\rightarrow T$ , we define the algebraic family $\rho ^*\xi $ of higher Chow cycles by $\rho ^*\xi = \{\rho _t^*(\xi _{\tau (t)})\}_{t\in T}$ .

In particular, we have defined families of higher Chow cycles $\rho ^*\xi _0, \rho ^*\xi _1$ and $\rho ^*\xi _\infty $ for $\rho \in G_{{\mathcal {X}}}$ . For each $t\in T$ , we define the subgroup $\Xi _t$ of ${\mathrm {CH}}^2(\widetilde {{\mathcal {X}}}_t,1)_{{\mathrm {ind}}}$ by

$$ \begin{align*} \Xi_t = \left\langle ((\rho^*\xi{}_{\bullet})_{t})_{\mathrm{ind}} : \bullet \in\{ 0,1,\infty\}, \rho\in G_{{\mathcal{X}}}\right\rangle_{{\mathbb{Z}}}. \end{align*} $$

Then we can state the main theorem.

To be more specific, 18 linearly independent cycles in $\Xi _t$ are given as follows. Let $\rho ^1,\rho ^2,\dots ,\rho ^6\in G_{{\mathcal {X}}}$ be lifts of $({\mathrm {id}},{\mathrm {id}})$ , $({\mathrm {id}},(0\:\:1))$ , $({\mathrm {id}},(1\:\:\infty ))$ , $({\mathrm {id}},(0\:\:1\:\:\infty ))$ , $({\mathrm {id}}, (0\:\:\infty ))$ and $({\mathrm {id}}, (0\:\:\infty \:\:1))\in G_{{\mathcal {Y}}_0}$ by $G_{{\mathcal {X}}}\twoheadrightarrow G_{{\mathcal {Y}}_0}=\mathfrak {S}(\Sigma )^2$ , respectively. Then we will show that $(((\rho ^i)^*\xi _{\bullet })_t)_{\mathrm {ind}} \:\:(i=1,2,\dots ,6,\bullet = 0,1,\infty )$ are linearly independent in $\Xi _t$ for very general $t\in T$ . The key of the proof is to compute ${\mathscr {D}}(\nu _{{\mathrm {tr}}}((\rho ^i)^*\xi _{\bullet }))$ for each $i=1,2,\dots ,6$ and $\bullet = 0,1,\infty $ .

Before proceeding to the proof, we explain geometric constructions of cycles in $\Xi _t$ . For $\rho = (\rho _1,\rho _2,\tau ,\zeta )\in G_{{\mathcal {X}}}$ , let ${\mathcal {D}}_{\rho }$ be the $(1,1)$ -curve on ${\mathbb {P}}^1\times {\mathbb {P}}^1$ which passes through $(\rho _1^{-1}(0),\rho _2^{-1}(0))$ , $(\rho _1^{-1}(1),\rho _2^{-1}(1))$ and $(\rho _1^{-1}(\infty ),\rho _2^{-1}(\infty )) \in \Sigma ^2$ . The local equation of ${\mathcal {D}}_{\rho }$ is given by $\rho _1^\sharp (x) = \rho _2^\sharp (y)$ . Let ${\mathcal {C}}_{\rho }\subset {\mathcal {X}}_t$ be its pull-back by ${\mathcal {X}}_t\rightarrow {\mathcal {Y}}_t ={\mathbb {P}}^1\times {\mathbb {P}}^1$ and $\widetilde {{\mathcal {C}}}_{\rho }\subset \widetilde {{\mathcal {X}}}_t$ be the strict transform of ${\mathcal {C}}_{\rho }$ . By definition, ${\mathcal {D}}_{\rho }$ coincides with the inverse image of ${\mathcal {D}}$ by $\rho $ , so we have $\widetilde {{\mathcal {C}}}_{\rho } =\rho ^{-1}_t(\widetilde {{\mathcal {C}}})$ . Moreover, by considering the $G_{{\mathcal {X}}}$ -action on $\Sigma ^2$ , we have $Q_{(\rho _1^{-1}(\bullet ), \rho _2^{-1}(\bullet ))}=\rho _t^{-1}(Q_{(\bullet ,\bullet )})$ for $\bullet = 0,1,\infty $ . Thus we see that $(\rho ^*\xi _\bullet )_t=\rho _t^*((\xi _{\bullet })_{\tau (t)})\:\:(\bullet \in \{0,1,\infty \})$ is represented by the formal sum

(26) $$ \begin{align} (\rho^*\xi_{}\bullet)_{}{t} = \left(\widetilde{{\mathcal{C}}}_{\rho}, \psi_\bullet\circ \rho_t\right) + \left(Q_{(\rho_1^{-1}(\bullet), \rho_2^{-1}(\bullet))}, \varphi_\bullet \circ \rho_t\right). \end{align} $$

For example, for $\rho ^1,\rho ^2,\dots ,\rho ^6\in G_{{\mathcal {X}}}$ defined above, ${\mathcal {D}}_{\rho ^1},{\mathcal {D}}_{\rho ^2},\dots ,{\mathcal {D}}_{\rho ^6}$ are graphs of rational functions $z,1-z,z/(1-z),1/(1-z),1/z,$ and $(z-1)/z$ on ${\mathbb {P}}^1$ , respectively. Hence, $((\rho ^i)^*\xi _{\bullet })_t$ are higher Chow cycles made from such graphs.

7.2 Proof of the main theorem

We define subgroups $N^{\mathrm {can}}$ and N of $\Gamma (T,{\mathcal {Q}})$ as follows:

$$ \begin{align*} \begin{aligned} &N^{\mathrm{can}} = \langle \nu_{\mathrm{tr}}(\xi_0),\nu_{\mathrm{tr}}(\xi_1),\nu_{{\mathrm{tr}}}(\xi_\infty)\rangle_{\mathbb{Z,}} \\ &N = \langle \nu_{{\mathrm{tr}}}(\rho^*\xi_\bullet): \rho\in G_{{\mathcal{X}}}, \bullet \in \{0,1,\infty\}\rangle_{\mathbb{Z}}. \end{aligned} \end{align*} $$

By Proposition 2.3, the transcendental regulator map induces a surjective map $\Xi _t\twoheadrightarrow {\mathrm {ev}}_t(N)$ for each $t\in T$ . Hence, to prove Theorem 7.1, we have to examine the rank of N. To get a rank estimate for N, we use the following proposition, whose proof will be given in the next section.

Proposition 7.3 For $\rho =(\rho _1,\rho _2,\tau ,\zeta )\in G_{{\mathcal {X}}}$ , let $\Theta _{\rho }: \tau ^*({\mathcal {O}}_T^{\mathrm {an}})^{\oplus 2}\rightarrow ({\mathcal {O}}_T^{\mathrm {an}})^{\oplus 2}$ be the morphism defined by

where $\tau ^\sharp :\tau ^*{\mathcal {O}}_T^{\mathrm {an}}\rightarrow {\mathcal {O}}_T^{\mathrm {an}}; \varphi \mapsto \varphi \circ \tau $ . Let $\xi $ be an algebraic family of higher Chow cycles on $\widetilde {{\mathcal {X}}}\rightarrow T$ . Then we have

$$ \begin{align*} {\mathscr{D}}(\nu_{\mathrm{tr}}(\rho^*\xi)) = \Theta_{\rho}({\mathscr{D}}(\nu_{\mathrm{tr}}(\xi))), \end{align*} $$

where ${\mathscr {D}}:{\mathcal {Q}}\simeq {\mathcal {Q}}_\omega \rightarrow ({\mathcal {O}}_T^{{\mathrm {an}}})^{\oplus 2}$ is the Picard–Fuchs differential operator defined in Proposition 5.2.

Thanks to this proposition, we can compute ${\mathscr {D}}(N)$ explicitly and we get the desired rank estimate for N. First, we compute ${\mathscr {D}}(N^{\mathrm {can}})$ .

Proposition 7.4 The images of $\nu _{\mathrm {tr}}(\xi _0),\nu _{\mathrm {tr}}(\xi _1),\nu _{\mathrm {tr}}(\xi _\infty )$ under the Picard–Fuchs differential operator ${\mathscr {D}}$ are as follows:

$$ \begin{align*} {\mathscr{D}}(\nu_{\mathrm{tr}}(\xi_0)) = \frac{2}{a-b} \begin{pmatrix} 1 \\[1.5ex] -1 \end{pmatrix} ,\:\: {\mathscr{D}}(\nu_{\mathrm{tr}}(\xi_1))= \frac{2}{a-b} \begin{pmatrix} \frac{\sqrt{1-b}}{\sqrt{1-a}}\\[1.5ex] -\frac{\sqrt{1-a}}{\sqrt{1-b}} \end{pmatrix} ,\:\: {\mathscr{D}}(\nu_{\mathrm{tr}}(\xi_\infty)) = \frac{2}{a-b} \begin{pmatrix} \frac{\sqrt{b}}{\sqrt{a}} \\[1.5ex] -\frac{\sqrt{a}}{\sqrt{b}} \end{pmatrix}. \end{align*} $$

In particular, ${\mathrm {rank}\:} N^{\mathrm {can}} = 3$ .

Proof The key for the proof is to find $\rho ^a,\rho ^b\in G_{{\mathcal {X}}}$ such that

(27) $$ \begin{align} (\rho^a)^*(\xi_1-\xi_0) = \xi_1+\xi_0,\quad (\rho^b)^*(\xi_1-\xi_0) = \xi_0-\xi_\infty. \end{align} $$

We consider the following elements.

1. (a) $\rho ^a = ({\mathrm {id}},{\mathrm {id}},\tau ^a,1),$ where $\tau ^a\in G_T$ satisfies $(\tau ^a)^\sharp \left (\sqrt {a}\right )=\sqrt {a}$ , $(\tau ^a)^\sharp \left (\sqrt {1-a}\right )=-\sqrt {1-a}$ , $(\tau ^a)^\sharp \left (\sqrt {b}\right )=\sqrt {b}$ , and $(\tau ^a)^\sharp \left (\sqrt {1-b}\right )=\sqrt {1-b}$ . Then we have $\phi _1(\tau ^a) = -1$ and $\phi _2(\tau ^b)=1$ .

2. (b) $\rho ^b = ((1\:\:\:\infty ),(1\:\:\:\infty ),\tau ^b,1),$ where $\tau ^b\in G_T$ satisfies $(\tau ^b)^\sharp \left (\sqrt {a}\right )=\sqrt {1-a}$ , $(\tau ^b)^\sharp \left (\sqrt {1-a}\right )=\sqrt {a}$ , $(\tau ^b)^\sharp \left (\sqrt {b}\right )=\sqrt {1-b}$ , and $(\tau ^b)^\sharp \left (\sqrt {1-b}\right )=\sqrt {b}$ . Then we have $\phi _1(\tau ^b)=1$ and $\phi _2(\tau ^b)=1$ .

These elements stabilize the curve $\widetilde {{\mathcal {C}}}$ . By the local description of the $G_{{\mathcal {X}}}$ -action in the end of Section 6, we see that these elements satisfy (27).

By Proposition 7.3 and (27), we can compute ${\mathscr {D}}(\nu _{\mathrm {tr}}(\xi _0+\xi _1))$ and ${\mathscr {D}}(\nu _{\mathrm {tr}}(\xi _0-\xi _\infty ))$ from Proposition 5.3 as follows:

$$ \begin{align*} \begin{aligned} &{\mathscr{D}}(\nu_{\mathrm{tr}}(\xi_0+\xi_1)) =\Theta_{\rho^a}\left({\mathscr{D}}(\nu_{tr}(\xi_1-\xi_0))\right) = \frac{2}{a-b} \begin{pmatrix} 1+ \frac{\sqrt{1-b}}{\sqrt{1-a}} \\[1.5ex] -1-\frac{\sqrt{1-a}}{\sqrt{1-b}} \end{pmatrix}, \\ &{\mathscr{D}}(\nu_{\mathrm{tr}}(\xi_0-\xi_\infty))= \Theta_{\rho^b}\left({\mathscr{D}}(\nu_{tr}(\xi_1-\xi_0))\right) = \frac{2}{(1-a)-(1-b)} \begin{pmatrix} \frac{\sqrt{b}}{\sqrt{a}}-1 \\[1.5ex] 1-\frac{\sqrt{a}}{\sqrt{b}} \end{pmatrix}. \end{aligned} \end{align*} $$

Hence, the images of $\nu _{\mathrm {tr}}(\xi _0),\nu _{\mathrm {tr}}(\xi _1),$ and $\nu _{\mathrm {tr}}(\xi _\infty )$ under the Picard–Fuchs differential operator are as in the statement. Since they are linearly independent over ${\mathbb {Q}}$ , the latter part follows from these expressions.

Next, we give an estimate for the rank of N. The proof below was simplified by advice from Terasoma. Furthermore, Sreekantan pointed out an error in the Table 2 in the previous version of this paper.

Table 2:

The images $\nu _{\mathrm {tr}}((\rho ^i)^*\xi _\bullet )$ under the Picard–Fuchs differential operator ${\mathscr {D.}}$

Proof Let $\rho ^1,\rho ^2,\dots ,\rho ^6 \in G_{{\mathcal {X}}}$ be elements defined after Theorem 7.1. Using Propositions 7.3 and 7.4, we can compute ${\mathscr {D}}(\nu _{\mathrm {tr}}((\rho ^i)^*\xi _\bullet ))$ for $i = 1,\dots , 6$ and $\bullet = 0,1,\infty $ as Table 2. Note that this table contains the ambiguity of signs because we do not fix the $G_{T}$ -components of $\rho ^i$ .

We will show that the vectors in Table 2 are linearly independent over ${\mathbb {Q}}$ . It is enough to show that the first components of these vectors are linearly independent over ${\mathbb {C}}$ . Note that the first components of these vectors are expressed by

$$ \begin{align*} 2\zeta \cdot F_1\cdot F_2, \end{align*} $$

where $\zeta \in \mu _4$ , $F_1$ is either

(28) $$ \begin{align} \dfrac{1}{a-b},\:\:\dfrac{1}{a+b-1},\:\:\dfrac{1}{ab-a-b},\:\:\dfrac{1}{ab-b+1},\:\:\dfrac{1}{ab-1}\text{ or }\dfrac{1}{a-ab-1} \in {\mathrm{Frac}}(B_0) \end{align} $$

and $F_2$ is either

(29) $$ \begin{align} 1,\:\dfrac{\sqrt{b}}{\sqrt{a}},\:\dfrac{\sqrt{1-b}}{\sqrt{1-a}},\:\dfrac{\sqrt{1-b}}{\sqrt{a}},\:\dfrac{\sqrt{b}}{\sqrt{1-a}},\:\dfrac{1}{\sqrt{1-a}},\: \sqrt{1-b},\:\dfrac{1}{\sqrt{a}}\text{ or }\sqrt{b} \in {\mathrm{Frac}}(B). \end{align} $$

Since elements in (28) are linearly independent over ${\mathbb {C}}$ and elements in (29) are linearly independent over ${\mathrm {Frac}}(B_0)$ , their products are linearly independent over ${\mathbb {C}}$ . Hence, we have the result.

Then we can prove the main theorem as follows.

7.3 Precise rank of N

To prove this, we use the following proposition which we prove in Section 8.

The following proof of Proposition 7.6 was simplified by the advice from T. Saito.

Proof of Proposition 7.6

It is enough to prove ${\mathrm {rank}\:} N \le 18$ . By the $G_{{\mathcal {X}}}$ -linearization $(\Psi _{\rho })_{\rho \in G_{{\mathcal {X}}}}$ of ${\mathcal {Q}}$ , N becomes a right $G_{{\mathcal {X}}}$ -module.

We consider the following two subgroups H and I of $G_{{\mathcal {X}}}$ . We define the normal subgroup H by ${\mathrm {Ker}}(G_{{\mathcal {X}}}\rightarrow G_{T})$ . By the definition of $\Psi _{\rho }$ , H acts on N as $\pm 1$ . Next, we define the subgroup I of $G_{{\mathcal {X}}}$ by

$$ \begin{align*} I =\{(\rho_1,\rho_2,\tau,\zeta)\in G_{{\mathcal{X}}}: \rho_1=\rho_2\in \mathfrak{S}(\{0,1,\infty\})\text{ and }\zeta=1\}, \end{align*} $$

where we regard $\mathfrak {S}(\{0,1,\infty \})$ as a subgroup of $\mathfrak {S}(\Sigma )$ by $\{0,1,\infty \}\subset \Sigma $ . By definition, I stabilizes the subset $\{(0,0),(1,1),(\infty ,\infty )\}$ of $\Sigma ^2$ on ${\mathcal {Y}}$ . Thus $\widetilde {{\mathcal {C}}}$ is stable under the I-action. Hence, for $\bullet = 0,1,\infty $ and $\rho \in I$ , $(\rho ^*\xi _\bullet )_t$ coincides with either $\pm (\xi _0)_t,\pm (\xi _1)_t$ , or $\pm (\xi _\infty )_t$ up to decomposable cycles (see (26)). By Proposition 2.3, $\nu _{\mathrm {tr}}(\xi )$ is determined by $\xi _{\mathrm {ind}}$ , therefore $\nu _{\mathrm {tr}}(\rho ^*\xi _\bullet )$ coincides with either $\pm \nu _{\mathrm {tr}}(\xi _0),\pm \nu _{\mathrm {tr}}(\xi _1)$ or $\pm \nu _{{\mathrm {tr}}}(\xi _\infty )$ . This implies that $N^{\mathrm {can}}$ is an I-submodule of N.

Therefore $N^{\mathrm {can}}$ is a $H I$ -submodule of N. Then $N^{\mathrm {can}} \hookrightarrow N$ induces the following $G_{{\mathcal {X}}}$ -module homomorphism:

(30) $$ \begin{align} \mathrm{Ind}_{H I}^{G_{{\mathcal{X}}}} N^{\mathrm{can}} \:{\longrightarrow}\: N, \end{align} $$

where $\mathrm {Ind}_{H I }^{G_{{\mathcal {X}}}} N^{\mathrm {can}}$ is the induced representation. Since the $G_{{\mathcal {X}}}$ -module N is generated by $N^{\mathrm {can}}$ , the map (30) is surjective. Then we have

$$ \begin{align*} {\mathrm{rank}\:} N \le {\mathrm{rank}\:} \mathrm{Ind}_{H I}^{G_{{\mathcal{X}}}} N^{\mathrm{can}} = |G_{{\mathcal{X}}}/H I|\cdot {\mathrm{rank}\:} N^{\mathrm{can}} = 3\cdot |G_{{\mathcal{X}}}/H I|. \end{align*} $$

Hence it is enough to show $ |G_{{\mathcal {X}}}/H I| = 6$ . First, we show $H \cap I = \{{\mathrm {id}}\}$ . Let $\rho =(\rho _1,\rho _2,\tau ,\zeta )\in H\cap I$ . Then we have $\rho _1=\rho _2\in \mathfrak {S}(\{0,1,\infty \})$ , $\tau = {\mathrm {id}}$ and $\zeta = 1$ . Furthermore, the image of $(\rho _1,\rho _2)\in G_{{\mathcal {Y}}_0}$ under $G_{{\mathcal {Y}}_0}\rightarrow G_{T_0}$ is $({\mathrm {id}},{\mathrm {id}})$ by the condition of the fiber product. Since $\mathfrak {S}(\{0,1,\infty \})\hookrightarrow \mathfrak {S}(\Sigma )\twoheadrightarrow {\mathrm {Aut}}(S_0)$ is the isomorphism, we see that $(\rho _1,\rho _2)=({\mathrm {id}},{\mathrm {id}})$ . Hence $\rho ={\mathrm {id}}$ . Second, we calculate $|G_{{\mathcal {X}}}|$ and $|H|$ . We have that $|G_{{\mathcal {X}}}| = 2\cdot |G_{{\mathcal {Y}}}| = 2\cdot |\mathfrak {S}_4\times _{\mathfrak {S}_3}\mathfrak {S}_4|^2 = 2^{11}\cdot 3^2$ . By $|G_{T}| = |\mathfrak {S}_4^2| = 2^6\cdot 3^2$ and the fact that $G_{{\mathcal {X}}}\rightarrow G_{T}$ is surjective, $|H| =|G_{{\mathcal {X}}}|/|G_T| =2^5$ . Third, we calculate $|I|$ . For each $\rho \in \mathfrak {S}(\{0,1,\infty \})$ , the number of elements in $G_{{\mathcal {X}}}$ of the form $(\rho ,\rho ,*,1)$ equals to the cardinality of ${\mathrm {Ker}}(G_{T}\rightarrow G_{T_0})$ , i.e., $2^4$ . Hence $|I|=2^4\cdot |\mathfrak {S}(\{0,1,\infty \})| = 2^5\cdot 3$ . To sum up these results, we have

$$ \begin{align*} |G_{{\mathcal{X}}}/H I| = \frac{|G_{{\mathcal{X}}}|}{|H I|} = \frac{|G_{{\mathcal{X}}}|}{|H|\cdot |I|} = 6.\\[-41pt] \end{align*} $$

8.1 The group action on periods

First, we prove Proposition 7.7. By the property of 1-cocycles, we see that $(\Psi _{\rho })_{\rho \in G_{{\mathcal {X}}}}$ defines a $G_{{\mathcal {X}}}$ -linearization on ${\mathcal {O}}_T^{\mathrm {an}}$ . To prove that this induces the $G_{{\mathcal {X}}}$ -linearization on ${\mathcal {Q}}_\omega $ , we should prove that $\Psi _{\rho }$ preserves the subsheaf ${\mathcal {P}}_\omega $ which is the local system consisting of period function with respect to $\omega $ .

Let $\rho =(\rho _1,\rho _2,\tau ,\zeta )\in G_{{\mathcal {X}}}$ . By the local description of the $G_{{\mathcal {X}}}$ -action on $\widetilde {{\mathcal {X}}}$ , we can check that $\rho ^*\omega = \chi (\rho )\cdot \omega $ . Pulling-back this relation at $\tau (t)\in T$ , we have

(31) $$ \begin{align} (\rho_t^{-1})^*(\omega_{t}) = \chi(\rho^{-1})(\tau(t))\cdot \omega_{\tau(t)}\quad \in \Gamma(\widetilde{{\mathcal{X}}}_{\tau(t)},\Omega^2_{\widetilde{{\mathcal{X}}}_\tau(t)/{\mathbb{C}}}). \end{align} $$

Here, $\rho _t:\widetilde {{\mathcal {X}}}_t\rightarrow \widetilde {{\mathcal {X}}}_{\tau (t)}$ is the isomorphism in the diagram (25) and $\chi (\rho ^{-1})(\tau (t))\in {\mathbb {C}}$ is the value of $\chi (\rho ^{-1})\in B$ at $\tau (t)$ . From the equation (31), for any 2-chain $\Gamma _{\tau (t)}$ on $\widetilde {{\mathcal {X}}}_{\tau (t)}$ , we have

(32) $$ \begin{align} \begin{aligned} \int_{\rho_t^{-1}(\Gamma_{\tau(t)})}\omega_t &= \int_{\Gamma_{\tau(t)}}(\rho^{-1}_t)^*(\omega_t) = \chi(\rho^{-1})(\tau(t))\cdot \int_{\Gamma_{\tau(t)}}\omega_{\tau(t)} \\ &= \tau^\sharp\left(\chi(\rho^{-1})\right)(t)\cdot \int_{\Gamma_{\tau(t)}}\omega_{\tau(t)} = \chi(\rho)(t)^{-1}\cdot \int_{\Gamma_{\tau(t)}}\omega_{\tau(t),} \end{aligned} \end{align} $$

where the last equality follows from the property of 1-cocycles. This is the key for the proof of Proposition 7.7.

Proof of Proposition 7.7

We prove (1). We prove that $\Psi _{\rho }(\tau ^*{\mathcal {P}}_\omega ) = {\mathcal {P}}_{\omega }$ . Since $\rho ^*(\Psi _{\rho ^{-1}})=\Psi _{\rho }^{-1}$ , it is enough to show only $(\subset )$ . Let $U'$ be an open subset of T in the classical topology and $\varphi \in \tau ^*{\mathcal {P}}_\omega (U')={\mathcal {P}}_\omega (\tau (U'))$ . Then there exists a $C^\infty $ -family of 2-cycles $\{\Gamma _t\}_{t\in \tau (U)}$ such that $\varphi (t) = \int _{\Gamma _t}\omega _t$ . For any $t\in U'$ , we have

$$ \begin{align*} (\Psi_{\rho}(\varphi))(t) = \chi(\rho)(t)^{-1}\cdot \varphi(\tau(t)) = \chi(\rho)(t)^{-1}\cdot \int_{\Gamma_{\tau(t)}}\omega_{\tau(t)} \underset{(32)}{=} \int_{\rho_t^{-1}(\Gamma_{\tau(t)})}\omega_t\in {\mathcal{P}}(\omega_t). \end{align*} $$

Hence, we have $\Psi _{\rho }(\varphi )\in {\mathcal {P}}_\omega (U')$ . Thus, $(\Psi _{\rho })_{\rho \in G_{{\mathcal {X}}}}$ induces the $G_{{\mathcal {X}}}$ -linearization on ${\mathcal {Q}}_\omega = {\mathcal {O}}_T^{\mathrm {an}}/{\mathcal {P}}_\omega $ .

Next, we prove (2). By Corollary 2.5, it is enough to show that

(33) $$ \begin{align} \langle {\mathrm{ev}}_t\left(\nu_{{\mathrm{tr}}}(\rho^*\xi)\right), [\omega_t]\rangle = \langle {\mathrm{ev}}_t\left(\Psi_{\rho}(\nu_{{\mathrm{tr}}}(\xi))\right), [\omega_t]\rangle \end{align} $$

for any $t\in T$ . We take a neighborhood $U'$ of $\tau (t)$ and a $C^\infty $ -family of 2-chains $\{\Gamma _{t'}\}_{t'\in U'}$ such that $\Gamma _{t'}$ is a 2-chain associated with $\xi _{t'}$ . We will compute the right-hand side of (33). Under the isomorphism ${\mathcal {Q}}\xrightarrow {\sim }{\mathcal {Q}}_\omega $ , $\Psi _{\rho }(\nu _{\mathrm {tr}}(\xi ))$ is represented by the local holomorphic function

$$ \begin{align*} \tau^{-1}(U')\ni t' \longmapsto \chi(\rho)(t')^{-1}\int_{\Gamma_{\tau(t')}}\omega_{\tau(t')} \underset{({eqn32}{32})}{=} \int_{\rho_{t'}^{-1}\left(\Gamma_{\tau(t')}\right)}\omega_{t'}\in {\mathbb{C}}. \end{align*} $$

The right-hand side of (33) is represented by the value of this function at t. That is,

(34) $$ \begin{align} \langle {\mathrm{ev}}_t\left(\Psi_{\rho}(\nu_{{\mathrm{tr}}}(\xi))\right), [\omega_t]\rangle=\displaystyle \int_{\rho_t^{-1}(\Gamma_{\tau(t)})}\omega_{t} \quad\mod {\mathcal{P}}(\omega_{t}). \end{align} $$

On the other hand, the left-hand side of (33) is $\langle r((\rho ^*\xi )_t), [\omega _t]\rangle = \langle r(\rho _t^*(\xi _{\tau (t)})), [\omega _t]\rangle $ . Since $\rho _t^{-1}(\Gamma _{\tau (t)})$ is a 2-chain associated with $\rho _t^*(\xi _{\tau (t)})$ , the left-hand side of (33) also coincides with (34). Hence we have the result.

8.2 The group action on the Picard–Fuchs operator

Next, we will investigate the $G_{{\mathcal {X}}}$ -action on the differential operator ${\mathscr {D}}$ . For $\tau \in G_{T}$ , we define differential operators ${\mathscr {D}}^{\tau }_1,{\mathscr {D}}^{\tau }_2:{\mathcal {O}}_T^{\mathrm {an}} \rightarrow {\mathcal {O}}_T^{\mathrm {an}}$ as follows:

$$ \begin{align*} \begin{aligned} &{\mathscr{D}}^{\tau}_1 = a'(1-a') \frac{\partial^2}{(\partial a')^2} +(1-2a')\frac{\partial}{\partial a'} -\frac{1}{4}, \\ &{\mathscr{D}}^{\tau}_2 = b'(1-b') \frac{\partial^2}{(\partial b')^2} +(1-2b')\frac{\partial}{\partial b'} -\frac{1}{4}, \end{aligned} \end{align*} $$

where $a'=\tau ^\sharp (a)$ and $b'=\tau ^\sharp (b)$ . By definition,

(35) $$ \begin{align} {\mathscr{D}}^\tau_i(\tau^\sharp(\varphi)) = \tau^\sharp({\mathscr{D}}_i(\varphi))\:\:(i=1,2) \end{align} $$

holds for any local section $\varphi $ of ${\mathcal {O}}_T^{\mathrm {an}}$ . We have the following transformation formulas.

Proposition 8.1 For $\rho =(\rho _1,\rho _2,\tau ,\zeta )\in G_{{\mathcal {X}}}$ , we have the following relations in the ring of differential operators on ${\mathcal {O}}_T^{\mathrm {an}}$ :

$$ \begin{align*} \begin{aligned} &{\mathscr{D}}_1\cdot \chi(\rho)^{-1} = \chi(\rho)^{-1}\cdot \phi_1(\tau)^{-2}\cdot {\mathscr{D}}^\tau_1, \\ &{\mathscr{D}}_2\cdot \chi(\rho)^{-1} = \chi(\rho)^{-1}\cdot \phi_2(\tau)^{-2}\cdot {\mathscr{D}}^\tau_2, \end{aligned} \end{align*} $$

where we regard $\chi (\rho ),\phi _1(\tau )$ and $\phi _2(\tau )$ as differential operators by multiplication.

Proof We prove the first equation. The second equation is proved similarly. Since $\chi (\rho ) = \zeta \phi _1(\tau )\phi _2(\tau )$ and $\zeta \phi _2(\tau )$ commutes with ${\mathscr {D}}_1$ , it is enough to show that

(36) $$ \begin{align} \phi_1(\tau)^3\cdot {\mathscr{D}}_1\cdot \phi_1(\tau)^{-1} = {\mathscr{D}}_1^\tau. \end{align} $$

Let $\tau = (\tau _1,\tau _2)\in G_{T}=\mathfrak {S}_4^2$ . We denote the image of $\tau _1$ under $\mathfrak {S}_4\rightarrow {\mathrm {Aut}}(S_0)=\mathfrak {S}(\{0,1,\infty \})$ by $\underline {\tau }_1$ . By (24) and the table in Proposition 6.6, $\phi _1(\tau )$ is determined by $\underline {\tau }_1$ up to sign and coincides with either $\pm 1, \pm \sqrt {-1}\sqrt {1-a}$ or $\pm \sqrt {-1}\sqrt {a}$ . Using $\frac {\partial }{\partial a}\cdot a^\lambda = \lambda a^{\lambda -1}+a^{\lambda }\frac {\partial }{\partial a}$ and $\frac {\partial }{\partial a}\cdot (1-a)^\lambda = -\lambda (1-a)^{\lambda -1}+(1-a)^{\lambda }\frac {\partial }{\partial a}$ , we can compute the left-hand side of (36) as follows:

(37) $$ \begin{align} \begin{aligned} &a(1-a)\dfrac{\partial^2}{\partial a^2} + (1-2a)\dfrac{\partial}{\partial a} -\dfrac{1}{4} && (\underline{\tau}_1 = {\mathrm{id}}, (0\:\:1)), \\ &-a(1-a)^2\frac{\partial^2}{\partial a^2}-(1-a)^2\frac{\partial}{\partial a} -\frac{1}{4} && (\underline{\tau}_1 = (1\:\;\infty), (0\:\:1\:\:\infty)), \\ &-a^2(1-a)\frac{\partial^2}{\partial a^2}+a^2\frac{\partial}{\partial a}-\frac{1}{4} && (\underline{\tau}_1 = (0\:\:\infty), (0\:\:\infty\:\;1)). \end{aligned} \end{align} $$

On the other hand, ${\mathscr {D}}_1^\tau $ is also determined by $\underline {\tau }_1\in {\mathrm {Aut}}(S_0)=\mathfrak {S}(\{0,1,\infty \})$ by definition. We can check (36) holds for each $\underline {\tau }_1$ . For example, when $\underline {\tau }_1 = (1\:\:\infty )$ , the differential operator ${\mathscr {D}}_1^\tau $ is computed as follows: since $a' = \frac {a}{a-1}$ , we have

$$ \begin{align*} \begin{aligned} &\frac{\partial}{\partial a'} = \frac{\partial a}{\partial a'}\cdot \frac{\partial }{\partial a} = -\frac{1}{(a'-1)^2}\cdot \frac{\partial}{\partial a} = -(a-1)^2\cdot \frac{\partial }{\partial a} \\ &\frac{\partial^2}{(\partial a')^2}= \left(-(a-1)^2\cdot \frac{\partial}{\partial a}\right)^2 = (a-1)^4\frac{\partial^2}{\partial a^2}+2(a-1)^3\frac{\partial }{\partial a}. \end{aligned} \end{align*} $$

By substituting $a',\frac {\partial }{\partial a'}, \frac {\partial ^2}{(\partial a')^2}$ in ${\mathscr {D}}_1^{\tau }$ by them, we can confirm that ${\mathscr {D}}_1^\tau $ coincides with the second differential operator in (37).

Finally, we prove Proposition 7.3.

Proof of Proposition 7.3

By Proposition 8.1, for a local section $\varphi $ of ${\mathcal {O}}_T^{\mathrm {an}}$ , we have

$$ \begin{align*} \begin{aligned} {\mathscr{D}}_1\left(\Psi_{\rho}(\varphi)\right) &= {\mathscr{D}}_1\left(\chi(\rho)^{-1}\cdot \tau^\sharp(\varphi)\right) = \chi(\rho)^{-1}\cdot \phi_1(\tau)^{-2}\cdot {\mathscr{D}}_1^\tau(\tau^\sharp(\varphi)) \\ &\underset{({eqn35}{35})}{=} \chi(\rho)^{-1}\cdot \phi_1(\tau)^{-2}\cdot \tau^\sharp({\mathscr{D}}_1(\varphi)), \\ {\mathscr{D}}_2\left(\Psi_{\rho}(\varphi)\right) &= \chi(\rho)^{-1}\cdot \phi_2(\tau)^{-2}\cdot \tau^\sharp({\mathscr{D}}_2(\varphi)). \end{aligned} \end{align*} $$

In particular, we have ${\mathscr {D}}(\Psi _{\rho }(\varphi )) = \Theta _{\rho }({\mathscr {D}}(\varphi ))$ . Then by Proposition 7.7, we have

$$ \begin{align*} {\mathscr{D}}(\nu_{{\mathrm{tr}}}(\rho^*\xi)) = {\mathscr{D}}(\Psi_{\rho}(\nu_{\mathrm{tr}}(\xi))) = \Theta_{\rho}({\mathscr{D}}(\nu_{{\mathrm{tr}}}(\xi))).\\[-33pt] \end{align*} $$