The geometry of generalized Lam\'{e} equation, I

In this paper, we prove that the spectral curve ${\mathrm{\Gamma }}_{\mathbf{n}}$ of the generalized Lam'{e} equation with the Treibich-Verdier potential \begin{equation*} y^{\prime \prime }(z)=\bigg[ \sum_{k=0}^{3}n_{k}(n_{k}+1)\wp(z+\tfrac{% \omega_{k}}{2}|\tau)+B\bigg] y(z),$$n_{k}\in $\mathbb{Z}${\geq0} \end{equation*} can be embedded into the symmetric space Sym${}^N{E}_{\tau }$ of the $N$-th copy of the torus $E_{\tau }$, where $N=\sum n_k$. This embedding induces an addition map ${\sigma }_{\mathbf{n}}(\cdot \mathrm{\mid }\tau )$ from ${\mathrm{\Gamma }}_{\mathbf{n}}$ onto $E_{\tau }$. The main result is to prove that the degree of $\sigma {% $\mathbf{n}$}(\cdot|\tau)$ is equal to% \begin{equation*} \sum{k=0}^{3}n{k}(n_{k}+1)/2. \end{equation*} This is the first step toward constructing the premodular form associated with this generalized Lam'{e} equation.