modular form representation

$$\begin{aligned}
\varkappa=\frac{1}{8 N_g}\Lambda_g-\frac{1}{2N_g} {(2\omega)^{-1}}^T . \left[
\sum_{N_g\;\text{even}\;[\varepsilon]}   \frac{1}{\theta[\varepsilon]}
\left(\begin{array}{ccc}
\theta_{1,1}[\varepsilon]&\cdots&\theta_{1,g}[\varepsilon]\\
\vdots&\cdots&\vdots\\
\theta_{1,g}[\varepsilon]&\cdots&\theta_{g,g}[\varepsilon]\end{array}\right)\right].(2\omega)^{-1}
\label{formula2}\end{aligned}$$

[*where $2\omega$ is matrix of $\mathfrak{a}$-periods of holomorhphic differentials and $ \theta_{i,j}[\varepsilon]= \partial^2_{z_i,z_j}\theta[\varepsilon]({z})_{{z}=0} $.*]{}

Note that modular form representation of period matrices $\eta, \eta'$ follows from the above formula,

$$\eta=2\varkappa\omega, \qquad \eta'=2\varkappa \omega'-\imath\pi{(2\omega)^T}^{-1}\label{etamodular}$$

At $g=2$ for the curve $y^2=4x^5+\lambda_4x^4+\ldots+\lambda_0$ the following representation of $\varkappa$-matrix is valid $$\begin{aligned}
\varkappa= \frac{1}{80}\left( \begin{array}{cc} 4\lambda_2&\lambda_3\\
\lambda_3&4\lambda_4 \end{array} \right)-\frac{1}{20}\sum_{10\;\text{even}\;[\varepsilon]}\frac{1}{\theta[\varepsilon]}
\left(\begin{array}{cc} \partial^2_{{U}_1^2}\theta[\varepsilon]&   \partial^2_{{U}_1,{U}_2}
\theta[\varepsilon]\\
 \partial^2_{{U}_1,{U}_2}
\theta[\varepsilon]& \partial^2_{{U}_2^2}\theta[\varepsilon]\end{array}\right)\end{aligned}$$ with $\varkappa=\eta(2\omega)^{-1}$, $\mathcal{A}^{-1}=(2\omega)^{-1}=({U}_1,{U}_2 ) $ and directional derivatives $\partial_{{U}_i}$, $i=1,2$.

Representation of $\varkappa$ matrix of genus 2 and 3 hyperelliptic curves in terms of directional derivatives of non-singular odd constant was found in [@eee13]

Co-homologies of Baker and Klein
--------------------------------

Calculations of $\varkappa$-matrix for the hyperelliptic curve (\[hyperelliptic\]) were done in co-homology basis introduced by H.Baker (\[bakerbasis\]). When holomorphic differentials, $ \mathrm{d}{u} (x,y) $ are chosen meromorphic differentials, $ \mathrm{d}{r} (x,y)$ can be find from the symmetry condition $\mathbf{I}$. One can check that symmetry condition also fulfilled if meromorphic differentials will be changed as $$\mathrm{d}{r}(x,y) \rightarrow \mathrm{d}{r}(x,y)+M\mathrm{d}{u}(x,y),$$ where $M$ is arbitrary constant symmetric matrix $M^T=M$. One can then choose $$M=-\frac{1}{8 N_g}\Lambda_g$$ Then $\varkappa$ will change to $$\begin{aligned}
\varkappa=
-\frac1{2}\frac1{N_g}\sum_{N_g\;\text{even}\;\left[\varepsilon_{\mathcal{I}_0}\right]} \frac{1}{\theta[\varepsilon_{\mathcal{I}_0}]({0})} \left(\partial_{{U}_i}\partial_{{U_j}} \theta[\varepsilon_{\mathcal{I}_0}]({0})\right)_{i,j=1,\ldots,g}  .
