On $\mathrm{R}\mathrm{i}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{n}^{)}\mathrm{s} $ Period matrix of $Y^{2}=x^{2n+}1-1$

We are deeply interested in the theory of Abelian integrals which have an ample data concerning the moduli space of the Riemann surfaces. The theta-function is a clue to connect the defining equation to the moduli space and the properties of the theta-function are well-known, but the concrete examples of such functions are not known so much. On the Riemann’s period matrix of the Riemann surface $y^{2}=x^{2n+1}-1 $ , the determinant of the above matrix of the $\circ\sigma \mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{l} $ type $n $ (see [7]) and the above matrix of the type $n=.3 $ (see [8]) are done. In this paper, we have the Riemann’s period matrix of the Riemann surface $y^{2}=$ $x^{9_{\mathcal{R}+1}}\sim-1 $.