Characteristic is even or odd whenever $ {\varepsilon'}^T\varepsilon'' = 0 $ (mod 2) or 1 (mod 2) and $\theta[\varepsilon](z;\tau)$ as function of $z$ inherits parity of the characteristic.

Derivatives of $\theta$-functions by arguments $z_i$ will be denoted as $$\theta_i[\varepsilon](z;\tau) = \frac{\partial}{\partial z_i}  \theta[\varepsilon](z;\tau),
\quad \theta_{i,j}[\varepsilon](z;\tau) = \frac{\partial^2}{\partial z_i\partial z_j}  \theta[\varepsilon](z;\tau) , \quad\text{etc.}$$

Fundamental bi-differential $\Omega(P,Q)$ is uniquely definite on the product $(P,Q)\in \mathcal{C}\times \mathcal{C}$ by following conditions:

[**i**]{} $\Omega$ is symmetric, $\Omega(P,Q)=\Omega(Q,P)$

[**ii**]{} $\Omega$ is normalized by the condition $$\begin{aligned}
\oint_{\mathfrak{a}_i} \Omega(P,Q)=0, \quad i=1,\ldots,g\end{aligned}$$

[**iii**]{} Let $P=(x,y)$ and $Q=(z,w)$ have local coordinates $\xi_1=\xi(P)$, $\xi_2=\xi(Q)$ in the vicinity of point $R$, $\xi(R)=0$, then $\Omega(P,Q)$ expands to power series as $$\Omega(P,Q)= \frac{\mathrm{d}\xi_1 \mathrm{d}\xi_2}{( \xi_1-\xi_2)^2 } + \; \text{homorphic 2-form}$$

Fundamental bi-differential can be expressed in terms of $\theta$-function [@fay973] $$\Omega(P,Q)= \mathrm{d}_x\mathrm{d}_z\theta\left( \int_{Q}^P\mathrm{d}{v} + {e}\right), \quad P=(x,y), Q=(z,w)$$ where $\mathrm{d}v$ is normalized holomorphic differential and ${e}$ any non-singular point of the $\theta$-divisor $(\theta)$, i.e. $\theta({e})=0$, but not all $\theta$-derivatives, $ \partial_{z_i}\theta({z})\vert_{{z}={e}}  $, $i=1,\ldots,g$ vanish.

In the case of hyperelliptic curve $\Omega(P,Q)$ can be alternatively constructed as $$\Omega(P,Q) =\frac12 \frac{\partial}{\partial z} \frac{ y+w }{ y(x-z)} \mathrm{d}x\mathrm{d}z
+ \mathrm{d}{r}(P)^T \mathrm{d}{u} (Q) + 2\mathrm{d}{u}^T (P)\varkappa
\mathrm{d}{u} (Q)\label{omega1}$$ where first two terms are given as rational functions 