### Abstract

In this paper, we prove that the spectral curve Γ_{n} of the generalized Lamé equation with the Treibich–Verdier potential [Fourmula presented] can be embedded into the symmetric space Sym^{N}E_{τ} of the N-th copy of the torus E_{τ}, where N=∑n_{k}. This embedding induces an addition map σ_{n}(⋅|τ) from Γ_{n} onto E_{τ}. The main result is to prove that the degree of σ_{n}(⋅|τ) is equal to ∑_{k=0} ^{3}n_{k}(n_{k}+1)/2. This is the first step toward constructing the pre-modular form associated with this generalized Lamé equation.

Original language | English |
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Pages (from-to) | 89-120 |

Number of pages | 32 |

Journal | Journal des Mathematiques Pures et Appliquees |

Volume | 127 |

DOIs | |

Publication status | Published - 2019 Jul |

### Keywords

- Degree of the addition map
- Generalized Lamé equation
- Spectral curve

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

Chen, Z., Kuo, T. J., & Lin, C. S. (2019). The geometry of generalized Lamé equation, I.

*Journal des Mathematiques Pures et Appliquees*,*127*, 89-120. https://doi.org/10.1016/j.matpur.2018.08.004