$\sigma$-function, $$\sigma({u}) = C \theta[{K}_{\infty}]( (2\omega)^{-1}{u})
\mathrm{e}^{ {u}^T\varkappa{u}   },$$ where $[{K}_{\infty}]$ is characteristic of the vector of Riemann constants, ${u}= \int_{\infty}^{P_1} \mathrm{d}{u} + \ldots +  \int_{\infty}^{P_g} \mathrm{d}{u}   $ with non-special divisor $P_1+\ldots+P_g$. The constant $C$ is chosen so that expansion $\sigma(\boldsymbol{u})$ near ${u}\sim 0$ starts with a Schur-Weierstrass polynomial [@bel999]. The whole expression is proved to be invariant under the action of symplectic group $\mathrm{Sp}(2g,\mathbb{Z})$. Klein-Weierstrass multi-variable $\wp$-functions are introduced as logarithmic derivatives, $$\begin{aligned}
\wp_{i,j}({u})=-\frac{\partial^2}{\partial u_i\partial u_j}, \quad \wp_{i,j,k}
({u})=-\frac{\partial^3}{\partial u_i\partial u_j\partial u_k},\quad \text{etc.} \quad
i,j,k = 1,\ldots g\end{aligned}$$

For $r\neq s \in \{1,\ldots, g\}$ the following formula is valid $$\sum_{i,j=1}^g\wp_{i,j} \left( \sum_{k=1}^g \int_{\infty}^{(x_k,y_k)} \mathrm{d}{u}   \right)x_s^{i-1} x_r^{j-1}
= \frac{F(x_s,x_r)-2y_sy_r}{ 4(x_s-x_r)^2}$$

Jacobi problem of inversion of the Abel map $\mathcal{D} \rightarrow \mathcal{A}(  \mathcal{D})$ with $\mathcal{D}= (x_1,y_1)+ \ldots+ (x_g,y_g)$ is resolved as $$\begin{aligned}
\begin{split}
&x^g-\wp_{g,g}({u})x^{g-1} - \ldots -\wp_{g,1}({u})=0\\
&y_k= \wp_{g,g,g}({u})x_k^{g-1}+ \ldots +  \wp_{g,g,1}({u})
, \quad k=1,\ldots,g\end{split}\end{aligned}$$

In this section we present generalization of Weierstrass formulae $$\wp(\omega)=e_1,\; \wp(\omega+\omega')=e_2,\; \wp(\omega')=e_3$$ to the case of genus $g$ hyperelliptic curve (\[hyperelliptic\]). To do that introduce partitions $$\begin{aligned}
\begin{split} \{1,\ldots, 2g+1\} = \mathcal{I}_0\cup \mathcal{J}_0, \quad   \mathcal{I}_0\cap \mathcal{J}_0=\emptyset\\
\mathcal{I}_0=\{ i_1,\ldots, i_g \}, \quad \mathcal{J}_0=\{ j_1,\ldots, j_{g+1} \} \end{split}
 \end{aligned}$$ Then any non-singular even half-period $\Omega_{\mathcal{I}}$ is given as 