The geometry of generalized Lamé equation, I

Abstract

In this paper, we prove that the spectral curve ${\Gamma }_{\mathbf{n}}$ of the generalized Lamé 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{% ω_{k}}{2}|τ)+B\bigg] y(z),\text{ \ }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 ${\sigma }_{\mathbf{n}}(\cdot |\tau )$ from ${\Gamma }_{\mathbf{n}}$ onto $E_{\tau }$. The main result is to prove that the degree of $σ_{% \mathbf{n}}(\cdot|τ)$ 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 Lamé equation.

References in corpus (1)

• Unitary monodromy implies the smoothness along the real axis for some Painlevé VI equation, I