e_j,\; \lambda_i
\in \mathbb{C}
\label{hyperelliptic}\end{aligned}$$ Let $({\mathfrak{a}};{\mathfrak{b}})
=( \mathfrak{a}_1,\ldots,\mathfrak{a}_g;  \mathfrak{b}_1,\ldots,\mathfrak{b}_g )  $ be canonic homology basis. Introduce co-homology basis ([*Baker co-homology basis*]{}) $$\begin{aligned}
 \begin{split} \mathrm{d}{u} (x,y)=(  \mathrm{d}u_1 (x,y), \ldots,
 \mathrm{d}u_g (x,y))^T,   \mathrm{d}{r} (x,y)=(  \mathrm{d}r_1 (x,y),    \ldots,
    \mathrm{d}r_g (x,y))^T\\
\mathrm{d}u_i(x,y)=\frac{x^{i-1}}{y}\mathrm{d}x,\quad \mathrm{d}r_j(x,y)=\sum_{k=j}^{2g+1-j}(k+1-j)
\frac{x^k}{4y}\mathrm{d}x,\quad i,j=1,\ldots,g\end{split}\label{bakerbasis}\end{aligned}$$ satisfying to the generalized Legendre relation, $$\begin{aligned}
\mathfrak{M}^TJ\mathfrak{M}=-\frac{\imath \pi}{2} J,\qquad \mathfrak{M}=\left(\begin{array}{cc} \omega&\omega'\\
                        \eta& \eta' \end{array}\right), \quad J = \left( \begin{array}{cc}  0_g& 1_g \\ -1_g&0_g
\end{array}  \right)\end{aligned}$$ where $g\times g$ period matrices $\omega,\omega', \eta, \eta'$ are defined as $$\begin{aligned}
2\omega=\left( \oint_{\mathfrak{a}_j}\mathrm{d}u_i \right),\quad
2\omega'=\left( \oint_{\mathfrak{b}_j}\mathrm{d}u_i \right), \quad
2\eta=-\left( \oint_{\mathfrak{a}_j}\mathrm{d}r_i \right),\quad
2\eta'=-\left( \oint_{\mathfrak{b}_j}\mathrm{d}r_i \right)\end{aligned}$$ We also denote $\mathrm{d}v=(\mathrm{d}v_1,\ldots,\mathrm{d}v_g )^T =  (2\omega)^{-1} \mathrm{d}u$ vector of normalized holomorphic differentials.

Define Riemann matrix $\tau= \omega^{-1}\omega'$ belonging to Siegel half-space $\mathcal{S}_g =\{ \tau^T=\tau,\;  \mathrm{Im} \tau>0 \}  $. Define Jacobi variety of the curve $\mathrm{Jac}(\mathcal{C})=\mathbb{C}^g/ 1_g\oplus \tau  $. Canonic Riemann $\theta$-function is defined on $\mathrm{Jac}(\mathcal{C})\times \mathcal{S}_g$ by Fourier series $$\theta({z};\tau) = \sum_{\mathbb{n} \in \mathbb{Z}^n} \mathrm{e}^{ \imath \pi {n}^T\tau {n}
+ 2\imath \pi {z}^T {n}  }\label{thetacan}$$ We will also use $\theta$-functions with half-integer characteristics $[\varepsilon] =
\left[\begin{array}{c} {\varepsilon'}^T\\ {\varepsilon''} \end{array}  \right]$, $\varepsilon_i', \varepsilon_j'' = 0$ or $1$ defined as $$\theta[\varepsilon]({z};\tau) = \sum_{\mathbb{n} \in \mathbb{Z}^n} \mathrm{e}^{ \imath \pi
( {n+\varepsilon'}/2)^T
\tau( {n+\varepsilon'}/2)
+ 2\imath \pi ({z+\varepsilon''}/2)^T( n+\varepsilon'/2)  }\label{thetachar}$$ 