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

TL;DR

This paper studies the spectral curve of a generalized Lamé equation with Treibich-Verdier potential, embedding it into a symmetric space of a torus and analyzing an addition map to understand its structure, aiding in constructing related premodular forms.

Contribution

It proves the embedding of the spectral curve into a symmetric space and determines the degree of the associated addition map, advancing the understanding of the spectral theory of generalized Lamé equations.

Findings

Spectral curve can be embedded into Sym^N E_τ.

The degree of the addition map σ_n is sum of n_k(n_k+1)/2.

Lays groundwork for constructing premodular forms.

Abstract

In this paper, we prove that the spectral curve $\Gamma_{\mathbf{n}}$ 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),\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 $\sigma _{% \mathbf{n}}(\cdot|\tau)$ 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.