The $\mathrm {al}$ function of a cyclic trigonal curve of genus three

Abstract

A cyclic trigonal curve of genus three is a $\mathbb {Z}_3$ Galois cover of $\mathbb {P}^1$, therefore can be written as a smooth plane curve with equation $y^3 = f(x) =(x - b_1) (x - b_2) (x - b_3) (x - b_4)$. Following Weierstrass for the hyperelliptic case, we define an “$\mathrm {al}$” function for this curve and $\mathrm {al}^{(c)}_r$, $c=0,1,2$, for each one of three particular covers of the Jacobian of the curve, and $r=1,2,3,4$ for a finite branchpoint $(b_r,0)$. This generalization of the Jacobi $\mathrm {sn}$, $\mathrm {cn}$, $\mathrm {dn}$ functions satisfies the relation:

$$\begin{aligned} \sum _{r=1}^4 \frac{\prod _{c=0}^2\mathrm {al}_r^{(c)}(u)}{f'(b_r)} = 1 \end{aligned}$$

which generalizes $\mathrm {sn}^2u + \mathrm {cn}^2u = 1$. We also show that this can be viewed as a special case of the Frobenius theta identity.

Appendix: Hyperelliptic \(\mathrm {al}\) Functions

In this appendix, we review the hyperelliptic $\mathrm {al}$-function mainly following [3, 4].

Hyperelliptic Curve: We let a (hyper)elliptic curve $C_g$ of genus $g$ $(g>0)$ be defined by the affine equation,

$$\begin{aligned} y^2&= (x-b_0)(x-b_1)(x-b_2)\cdots (x-b_{2g})\nonumber \\&= P(x) Q(x), \end{aligned}$$

(11.1)

where $b_j$’s are distinct complex numbers, $P(x) = (x-b_1)(x-b_3) \cdots (x-b_{2g-1})$ and $Q(x) := y^2 / P(x)$. Let $(b_j,0) = B_j \in C_g$.

For a point $(x, y)\in C_g$, differentials of the first kind (not normalized in the standard way which gives the identity as the matrix of $A$-periods) are defined by,

$$\begin{aligned} {\nu ^{I}}_i := \frac{x^{i-1} d x}{2y}. \end{aligned}$$

The extended Abel map from the $g$-th symmetric product of the universal cover $\Gamma _\infty C_g$ of the curve $C_g$ to ${\mathbb C}^g$ is defined by,

$$\begin{aligned} w :{S}^g \Gamma _\infty C_g \longrightarrow \mathbb C^g, \quad \left( w\left( \Gamma _{\left( x_1,y_1\right) ,\infty },\ldots , \Gamma _{\left( x_g,y_g\right) ,\infty }\right) := \sum _{i=1}^g \int _{ \Gamma _{\left( x_g,y_g\right) ,\infty }} {\nu ^{I}}\right) , \end{aligned}$$

where $\Gamma _{(x_g,y_g),\infty }$ is a path in the path space $\Gamma _\infty C_g$.

Consider $\mathrm {H}_1(C_g, \mathbb Z) =\bigoplus _{j=1}^g\mathbb Z\alpha _{j} \oplus \bigoplus _{j=1}^g\mathbb Z\beta _{j},$ the homology group of the hyperelliptic curve $C_g$, where the intersections are given by $[\alpha _i, \alpha _j]=0$, $[\beta _i, \beta _j]=0$ and $[\alpha _i, \beta _j]=\delta _{i,j}$. Here we employ the choice illustrated in Fig. 5.

Fig. 5

a $C_g$ and b $\hat{C}_{2g-1}$

The (half-period) hyperelliptic integrals of the first kind are defined by,

$$\begin{aligned} {\omega }':=\frac{1}{2}\left[ \left( \int _{\alpha _{j}}{\nu ^{I}}^{}_{i}\right) _{ij}\right] , \quad {\omega }'':=\frac{1}{2}\left[ \left( \int _{\beta _{j}}{\nu ^{I}}^{}_{i}\right) _{ij}\right] , \quad {\omega }:=\left[ \begin{matrix} {\omega }' \\ {\omega }'' \end{matrix}\right] . \end{aligned}$$

If we let:

$$\begin{aligned} \omega _a:=\int ^{B_a}_\infty {\nu ^{I}}, \quad (a=0, 1, 2, \ldots , 2g-1, 2g), \end{aligned}$$

Figure 5 shows:

$$\begin{aligned} \omega '_a = \omega _{2a - 1}, \quad \omega ''_a = \omega _{2a} - \omega _{2a - 1}, \quad a>1. \end{aligned}$$

The Jacobian ${\mathcal J}_g$ is defined as the complex torus,

$$\begin{aligned} {\mathcal J}_g := \mathbb C^g /{{\Lambda }}_g. \end{aligned}$$

Here ${\Lambda }_g$ is a $2g$-dimensional lattice generated by the period matrix given by $2{\omega }$. We also use the same letter $u$ for a vector in $\mathbb C^g$ and a point of the Jacobian ${\mathcal J}_g$.

Using the (unnormalized) differentials of the second kind,

$$\begin{aligned} {\nu ^{II}}_j =\dfrac{1}{2 y}\sum _{k=j}^{2g-j}\left( k+1-j\right) \lambda _{k+1+j} x^k d x, \quad \quad \left( j=1, \ldots , g\right) , \end{aligned}$$

the half-period hyperelliptic matrices of the second kind are defined by,

$$\begin{aligned} {\eta }':=\frac{1}{2}\left[ \left( \int _{\alpha _{j}}{\nu ^{II}}_{i}\right) _{ij}\right] , \quad {\eta }'':=\frac{1}{2}\left[ \left( \int _{\beta _{j}}{\nu ^{II}}_{i}\right) _{ij}\right] . \end{aligned}$$

The hyperelliptic $\sigma$ function, which is a holomorphic function over $u\in \mathbb C^g$, is defined by [4,  p. 336, p. 350], [7, 15],

$$\begin{aligned} \sigma (u):=\sigma (u;C_g): =\ \gamma \mathrm {exp}\left( -\dfrac{1}{2}\ ^t\ u {\eta }'{{\omega }'}^{-1}u\right) \vartheta \!\left[ \begin{matrix} \delta '' \\ \delta ' \end{matrix}\right] \left( \frac{1}{2}{{\omega }'}^{-1}u ; \tau \right) , \end{aligned}$$

(11.2)

where $\gamma$ is a certain constant factor, $\vartheta \left[ \!\right]$ is the Riemann $\theta$ function with characteristics,

$$\begin{aligned} \vartheta \!\left[ \begin{matrix} a \\ b \end{matrix}\right] (z; \tau ) :=\sum _{n \in \mathbb Z^g} \exp \left[ 2\pi \sqrt{-1}\left\{ \dfrac{1}{2} \ ^t\!(n+a) \tau (n+a) + \ ^t\!(n+a)(z+b)\right\} \right] , \end{aligned}$$

with $\tau :={{\omega }'}^{-1}{\omega }''$ for $g$-dimensional vectors $a$ and $b$, and

$$\begin{aligned} \delta ' :=\ ^t\left[ \begin{array}{llll} \dfrac{g}{2}&\quad \dfrac{g-1}{2}&\quad \cdots&\quad \dfrac{1}{2} \end{array}\right] , \quad \delta '':=\ ^t\left[ \begin{array}{lll} \dfrac{1}{2}&\quad \cdots&\quad \dfrac{1}{2} \end{array}\right] . \end{aligned}$$

Proposition 11.1

If for $u$, $v\in {\mathbb C}^3$, and $\ell$ ($=2\omega '\ell '+2\omega ''\ell ''$) $\in \Lambda$, we define

$$\begin{aligned} L(u,v)&:=2{}^t{u}(\eta 'v'+\eta ''v''),\nonumber \\ \chi (\ell )&:=\exp [\pi \sqrt{-1}\big (2({}^t {\ell '}\delta ''-{}^t {\ell ''}\delta ') +{}^t {\ell '}\ell ''\big )] \ (\in \{1,\,-1\}), \end{aligned}$$

the following holds

$$\begin{aligned} \sigma (u + \ell ) = \sigma (u) \exp (L(u + \frac{1}{2}\ell , \ell )) \chi (\ell ). \end{aligned}$$

Definition 11.2

• (1) We define the double coverings of ${\mathcal J}_g$ by

  $$\begin{aligned} {\mathcal J}_g^{(a)} = {\mathbb C}^g / \Lambda ^{(a)}, \end{aligned}$$

  where $\Lambda ^{(0)} := \bigcap _{a=1}^{2g} \Lambda ^{(a)}$,

  $$\begin{aligned} \Lambda ^{(a)} :=&2 {\mathbb Z}\omega '_a + 4 {\mathbb Z}\omega ''_a + \sum _{b=1,\ne a} (2 {\mathbb Z}\omega '_b + 2 {\mathbb Z}\omega ''_b) \quad \hbox {for } a = 1, 3, \ldots , 2g-1,\\ \Lambda ^{(a)} :=&4 {\mathbb Z}\omega '_a +4 {\mathbb Z}\omega ''_a) + \sum _{b=1,\ne a} (2 {\mathbb Z}\omega '_b + 2 {\mathbb Z}\omega ''_b) \quad \hbox {for } a = 2, 4, \ldots , 2g.\\ \end{aligned}$$

• (2) For a point $\Gamma _{P_c,\infty }\in \Gamma _\infty C_g$,

  $$\begin{aligned} \varepsilon _r^{(c)}: \Gamma _{\infty } C_g \rightarrow {\mathbb Z}_2 \end{aligned}$$

  be defined by $\varepsilon _r^{(c)}:=w_r-w_\infty$ for the winding number $w_r$ around $B_a$ in $\kappa _\infty \Gamma _\infty C_g$ and the winding number $w_\infty$ around $\infty$ in $\kappa _\infty \Gamma _\infty C_g$. For a point $(\Gamma _{P_1,\infty }, \Gamma _{P_2,\infty },\ldots ,\Gamma _{P_g,\infty })$ in $S^g \Gamma _{\infty } C_g$, let

  $$\begin{aligned} \varepsilon _r: S^g \Gamma _{\infty } C_g \rightarrow {\mathbb Z}_2, \quad (\varepsilon _r:= \varepsilon _r^{(1)} + \varepsilon _r^{(2)} + \cdots +\varepsilon _r^{(g)}). \end{aligned}$$

• (3) For a point $(\Gamma _{P_1,\infty }, \Gamma _{P_2,\infty },\ldots ,\Gamma _{P_g,\infty })$ in $S^g \Gamma _{\infty } C_g$, let $u=w (\Gamma _{P_1,\infty }, \Gamma _{P_2,\infty },\ldots ,\Gamma _{P_g,\infty })$. The hyperelliptic $\mathrm {al}$ function over ${\mathcal J}^{(r)}$ and $w^{-1}{\mathcal J}^{(r)}$ as a subset of a quotient space of in $S^g \Gamma _{\infty } C_g$, is formally defined by [4, p.340], [31],

  $$\begin{aligned} \mathrm {al}_r(u): = (-1)^{\varepsilon _r( \Gamma _{\infty , P_1}, \Gamma _{\infty , P_2},\ldots ,\Gamma _{\infty , P_g})} \sqrt{F(b_r)} , \end{aligned}$$

  (11.3)

  where

  $$\begin{aligned} F(x):= (x-x_1) \cdots (x-x_g), \end{aligned}$$

  (11.4)

  for a preimage $(\Gamma _{(x_i, y_i),\infty })_{i=1, \ldots , g} \in S^g\Gamma _{\infty }C_g$ of $w((\Gamma _{(x_i, y_i),\infty })_{i=1, \ldots , g}) = u \in {\mathcal J}^{(r)}$ under the Abel map.

Remark 11.3

The definition (11.3) is historically

$$\begin{aligned} \mathrm {al}_r(u) = \tilde{\gamma }_r\sqrt{F(b_r)} , \end{aligned}$$

(11.5)

where $\tilde{\gamma }_r:=\sqrt{-1/P'(b_r)}$. Thus the preimage of $w$ of ${\mathcal J}^{(r)}$ is a quotient space of $S^g\Gamma _{\infty } C_g$. We comment on the sign $(-1)^{\varepsilon }$ in the right-hand side of (11.3). The hyperelliptic curve $C_g$ admits the hyperelliptic involution $\iota _H : (x, y) \rightarrow (x, -y)$. In a neighborhood of the branch point $B_r=(b_r,0)$, $y$ or $t$ such that $t^2 = (x - b_r)$ are local parameters. Thus for $t_i$ such that $t_i^2 := (x_i - b_r)$ $\iota _H^{(a)} t_i = - t_i$. Similarly, $t_1 t_2 \cdots t_g$ is defined in a neighborhood of $B_r$ and $\iota _H$ can be made to act on the product: a circuit around the point produces the factor $(-1)^{\varepsilon _r( \Gamma _{\infty , P_1}, \Gamma _{\infty , P_2},\ldots ,\Gamma _{\infty , P_g})}$.

Further the inverse $1/t_i$ is a local parameter at $\infty$ and thus there is an action $\iota _H^{(\infty )} (1/t_i) = - (1/t_i)$, and a circuit around $\infty$ generates $(-1)^{\varepsilon _r( \Gamma _{\infty , P_1}, \Gamma _{\infty , P_2},\ldots ,\Gamma _{\infty , P_g})}$.

However we claim that we can make sense of $t_1 t_2 \cdots t_g$ globally and (11.5) holds globally by (11.3). In analogy to Jacobi’s sn, cn, dn functions, we need to extend the domain of the Jacobi inversion from ${\mathcal J}_g$ to ${\mathcal J}_g^{(r)}$ and ${\mathcal J}_g^{(0)}$. We show the extension in Proposition 11.10; here we consider the behavior of the right-hand side of (11.3). Let us regard it as a function of $w(P_1)$ by fixing $P_2$, $\ldots$, $P_g$. Then a circuit around $\alpha _b$ (see Fig. 5a) does not have any effect on the sign factor of $t_1$. On the other hand, when we go around $\beta _a$ in Fig. 5a once, $t_1$ acquires a sign and in order to cancel it, we need to go twice around $\beta _a$. Thus the (homotopy) equivalence relation is the same as that which holds for ${\mathcal J}_g^{(a)}$.

Proposition 11.4

Introducing the half-period $\omega _r := \int ^{b_r}_\infty du^{}$, we have the relation [4, 340],

$$\begin{aligned} \mathrm {al}_r(u) =\gamma _r'' \frac{ \exp (-{}^t u \varphi _r) \sigma ( u + \omega _r)}{\sigma (u)}, \quad r=1,2,\ldots ,2g, \end{aligned}$$

(11.6)

where $\gamma _r''$ is a certain constant.

$$\begin{aligned} \varphi _r = \left\{ \begin{array}{ll} {\eta }'{{\omega }'}^{-1}\omega '_r &{} \quad r = 1, 3, \ldots , 2g - 1, \\ {\eta }''{{\omega }''}^{-1}\omega ''_r +{\eta }'{{\omega }'}^{-1}\omega '_r &{} \quad r = 2, 4, \ldots , 2g . \\ \end{array}\right. \end{aligned}$$

Proof

By comparing zeros and poles of both sides, we have the result. $\square$

Proposition 11.5

For a lattice point $\ell$ in $\Lambda ^{(b)}$

$$\begin{aligned} \mathrm {al}_b(u)=\mathrm {al}_b(u+ \ell ). \end{aligned}$$

Proof

We know:

$$\begin{aligned} \frac{\sigma (u + \omega _b+ \ell )}{\sigma (u+\ell )} = \frac{\sigma (u + \omega _b)}{\sigma (u)} \exp (L(\omega _b , \ell )) . \end{aligned}$$

For the $b = 2a -1$ case,

$$\begin{aligned} L( \omega _b' , \ell )&= 2^t\omega _b' (\eta '\ell ' +\eta ''\ell '')\\&=2^t\omega _b' \eta '\ell ' +2^t\omega _b''\eta '\ell ''-\pi {\sqrt{-1}}\ell ''_b,\nonumber \end{aligned}$$

(11.7)

whereas

$$\begin{aligned} 2^t(\omega '\ell ' +\omega ''\ell '')\eta ' \omega ^{\prime -1}\omega _b' = 2^t\ell '{}^t\eta '\omega _b'+2{}^t\ell ''\omega _b''\eta '. \end{aligned}$$

(11.8)

Hence we have the equality. $\square$

As a generalization of the relation $\mathrm {sn}^2 u + \mathrm {cn}^2 u = 1$, we have the following relation.

Proposition 11.6

Let $A_a(x) = P(x) (x-b_a)$ and $a \in \{2, 4, \ldots , 2g\}$.

$$\begin{aligned} \sum _{r = 1, 3, \ldots , 2g-1, a} \frac{\mathrm {al}_r(x)^2}{A'_a(b_r)} =1. \end{aligned}$$

Proof

See [31, p. 292] and also [22, Proposition 3.4]. $\square$

Remark 11.7

The relation implies the $g$ homogeneous identities,

$$\begin{aligned} \sum _{r = 1, 3, \ldots , 2g-1, a} \frac{(\gamma _r'')^2 \mathrm e^{-2{}^t u \varphi _r} }{A'_a(b_r)} \sigma ( u + \omega _r)^2 \equiv \sigma (u)^2, \quad a = 2, 4, \ldots , 2g, \end{aligned}$$

among $2g+1$ homogeneous coordinates, namely, $\sigma ( u + \omega _r)$ ($r=1,2,\ldots ,2g$) and $\sigma (u)$. Noting that the square of each $\mathrm {al}_r$ is a function over the hyperelliptic Jacobi variety ${\mathcal J}_g$, these quadrics cut out the image of the Jacobian, which is a $g$-dimensional variety embedded in $\mathbb {P}^{2g}$.

Remark 11.8

For the genus-one case, the Weierstrass $\wp$ function corresponds to a curve $y^2 = (x - e_1) (x - e_2) (x - e_3)$, whereas the Jacobi $\mathrm {sn}$ functions is defined on:

$$\begin{aligned} w^2 = (z^2 - 1) (z^2 - k^2), \end{aligned}$$

(11.9)

where $w = y / z\sqrt{(e_2-e_1)^3}$, $z = \sqrt{(x - e_1)/(e_2-e_1)}$ and

$$\begin{aligned} \frac{dx}{2 y} =2 \sqrt{e_2-e_1} \frac{ dz }{2 w}. \end{aligned}$$

We have employed a curve (11.1) with $f(x)$ of odd degree (thus a branchpoint at $\infty$), and the associated $\wp _{ij}$ function.

Note that when $g=1$, (11.10) is essentially reduced to (11.9).

Given that the $\mathrm {al}_r$ function is a generalization of the $\mathrm {sn}$-function, we considered a genus $2g-1$ curve $\hat{C}_{2g-1}$ whose affine part is given by

$$\begin{aligned} w^2 = \prod _{i=1, \ne r}^{2g+1} ( z^2 - a_i), \end{aligned}$$

(11.10)

where $a_i = b_i - b_r$, $z = \sqrt{x - b_r}$,    and $w = y/z$.

Let $(b_i, 0)=B_i \in C_g$ be the branch points on the affine plane and $(x, y) \in C_g$ be a general point $P$. For $c_i^2 := a_i$, let $(\pm c_i, 0) \in \hat{C}_{2g-1}$ be $\hat{B}_i^\pm$ as a finite branch point and $( z, w) \in C_g$ be a general point $\hat{P}^\pm$.

There is an involution $\iota _A: (z, w) \mapsto (- z, w)$ as well as the hyperelliptic involution $\hat{\iota }_H: (z, w) \mapsto (z, - w)$ and $\iota _H: (x, y) \mapsto (x, -y)$.

At the point $\infty$ of $\hat{C}_{2g+1}$, acting by $\iota _H$ and $\iota _A$, we identify the actions $\hat{\iota }_H$ and $\iota _A$, i.e.,

$$\begin{aligned} \hat{\iota }_H : \pm \infty \mapsto \mp \infty , \quad \iota _A : \pm \infty \mapsto \mp \infty . \end{aligned}$$

On the other hand $(0, 0) \in \hat{C}_{2g+1}$, which corresponds to $B_r \in C_{g}$ is the fixed point of $\hat{\iota }_H$ and $\iota _A$.

Let us consider the $r=1$ case. Then there is a double covering:

$$\begin{aligned} \varpi _{g}: \hat{C}_{2g-1} \rightarrow C_g, \quad (\hat{P} =(z, w) \mapsto P = (z^2 + b_r, wz)), \end{aligned}$$

and

$$\begin{aligned} (0,0) \mapsto B_1, \quad \hat{B}_i^\pm \rightarrow B_i, \quad (i = 2, 3, \cdots , 2g, 2g+1). \end{aligned}$$

We illustrate this in Fig. 6, which is essentially the same as the picture in [1, p.296].

Fig. 6

$\varpi : \hat{C}_{2g-1} \rightarrow C_g$

The (unnormalized) basis of holomorphic one-forms over $\hat{C}_{2g-1}$ is denoted by

$$\begin{aligned} \hat{{\nu ^{I}}} := \begin{pmatrix} \hat{{\nu ^{I}}}_1 \\ \hat{{\nu ^{I}}}_2 \\ \vdots \\ \hat{{\nu ^{I}}}_{2g-1} \\ \end{pmatrix}, \quad \hat{\nu }^I_j = \frac{z^{j-1} dz}{ w}, \quad (j = 1, 2, \ldots , 2g - 1). \end{aligned}$$

Here we have removed the factor $1/2$ for later convenience. Let us consider the Abel map

$$\begin{aligned} \hat{w} :{S}^k \Gamma _{-\infty } \hat{C}_{2g-1} \longrightarrow {\mathbb C}^{2g-1}, \quad \left( \hat{w}(\Gamma _{(x_1,y_1),-\infty },\ldots ,\Gamma _{(x_k,y_k),-\infty }):= \sum _{i=1}^k \int _{\Gamma _{(x_i,y_i),-\infty }} \hat{{\nu ^{I}}} \right) . \end{aligned}$$

As the contours in Fig. 5b illustrate, the associated periodic matrices are given as,

$$\begin{aligned} (\hat{\omega '}, \hat{\omega ''}) := \frac{1}{2}\left( \left( \int _{\hat{\alpha }_1} \hat{{\nu ^{I}}}, \left( \int _{\hat{\alpha }_i^+} \hat{{\nu ^{I}}}, \int _{\hat{\alpha }_i^-} \hat{{\nu ^{I}}}\right) _{i=2, \ldots , g}\right) , \left( \int _{\hat{\beta }_1} \hat{{\nu ^{I}}}, \left( \int _{\hat{\beta }_i^+} \hat{{\nu ^{I}}}, \int _{\hat{\beta }_i^-} \hat{{\nu ^{I}}}\right) _{i=2, \ldots , g}\right) \right) . \end{aligned}$$

The lattice associated with the curve $\hat{C}_{2g-1}$ is denoted by $\hat{\Lambda }$ and its Jacobian by $\hat{{\mathcal J}}_{2g-1} = {\mathbb C}^{2g-1}/\hat{\Lambda }$.

Direct computations show the following facts:

Proposition 11.9

• (1)
  $$\begin{aligned} \frac{z^{2i-2} dz}{w} = \frac{x^{i-1} dx}{2 y}, \quad (i=1, \ldots , g), \quad \varpi ^*{\nu ^{I}}= \begin{pmatrix} \hat{{\nu ^{I}}}_1\\ \hat{{\nu ^{I}}}_3\\ \vdots \\ \hat{{\nu ^{I}}}_{2g-1}\\ \end{pmatrix}. \end{aligned}$$
• (2)
  $$\begin{aligned} \hat{\iota }_H \varpi ^* {\nu ^{I}}= \varpi ^* \iota _H {\nu ^{I}}= \hat{\iota }_A \varpi ^* {\nu ^{I}}. \end{aligned}$$

• (3) By defining

  $$\begin{aligned} \left( \sum _{i}^g\int _{\Gamma _{(x_i,y_i),-\infty }} \varpi ^*{\nu ^{I}}\right) , \end{aligned}$$

  $\hat{w}_{\varpi ^* {\nu ^{I}}} : S^g \Gamma _{-\infty }\hat{C}_{2g-1} \rightarrow {\mathbb C}^g$ is a surjection.

Figure 5 shows that as half of $\beta _1$ consists of the path from $\infty$ to $B_1$, the path from $\pm \infty$ to $(0,0)$ in $\hat{C}_{2g-1}$ corresponds to a quarter of $\hat{\beta }_1$. Each $\hat{\beta }_a^\pm$ $(a =2, \ldots , g)$ consists of a contour from $\pm \infty$ to $\hat{B}_{2a -1}^\pm$. Similarly we have $\hat{\alpha }_a^\pm$ $(a =1, \ldots , g)$.

Noting that $({B_1} \rightarrow B_2)$ lifts to $(0,0) \rightarrow \hat{B}_2^{(\pm )})$, we find that

$$\begin{aligned} \int ^{B_2}_{B_1} \nu ^I = \frac{1}{2} \int ^{\hat{B}_2}_{(0,0)} \varpi ^*\nu ^I \end{aligned}$$

is a half-period in $\hat{C}_{2g-1}$. The $(2(2g-1) \times g)$ matrix $(\hat{\omega '}, \hat{\omega ''})|_{\varpi {\nu ^{I}}}$ is given by

$$\begin{aligned} (\omega '_1, \omega _2', \omega _2', \ldots , \omega _g', \omega _g', 2\omega ''_1, \omega _2'', \omega _2'', \ldots , \omega _g'', \omega _g''). \end{aligned}$$

The corresponding lattice is denoted by $\hat{\Lambda }$ and the Jacobian by $\hat{{\mathcal J}} = {\mathbb C}^{2g-1}/\hat{\Lambda }$.

Proposition 11.10

Let

$$\begin{aligned} {\hat{{\mathcal J}}}^{\mathrm {al},(1)}_g := \frac{\hat{w}_{\varpi ^* {\nu ^{I}}}(S^g\Gamma _{-\infty } \hat{C}_{2g-1}) }{\hat{\Lambda } \cap \hat{w}_{\varpi ^* {\nu ^{I}}}(S^g \Gamma _{-\infty }\hat{C}_{2g-1}) }. \end{aligned}$$

Then the following function is defined on ${\hat{{\mathcal J}}}^{\mathrm {al},(1)}_g$,

$$\begin{aligned} (z_1 z_2 \cdots z_g)(u), \end{aligned}$$

where $(z_1, z_2, \ldots , z_g)$ in $S^g\Gamma _{-\infty } \hat{C}_{2g-1}$ is any preimage of $u$ under the extended Abel map.

By identifying $w(S^g\Gamma _{\infty } C_g)={\mathbb C}^g$ and $\hat{w}_{\varpi ^* {\nu ^{I}}}(S^g\Gamma _{-\infty }\hat{C}_{2g-1})={\mathbb C}^g$, ${\hat{{\mathcal J}}}^{\mathrm {al},(1)}_g$ and ${{\mathcal J}}^{(1)}_g$ agree, and their $\mathrm {al}_1$ function is expressed by

$$\begin{aligned} \mathrm {al}_1(u) = (z_1 z_2 \cdots z_g)(u). \end{aligned}$$