### Abstract

In this paper, the second in a series, we continue to study the generalized Lamé equation with the Treibich-Verdier potential y^{″}(z)=[∑k=03n_{k}(n_{k}+1)℘(z+ [Formula presented] |τ)+B]y(z),n_{k}∈Z_{≥0} from the monodromy aspect. We prove the existence of a pre-modular form Z_{r,s} ^{n}(τ) of weight [Formula presented] ∑n_{k}(n_{k}+1) such that the monodromy data (r,s) is characterized by Z_{r,s} ^{n}(τ)=0. This generalizes the result in [17], where the Lamé case (i.e. n_{1}=n_{2}=n_{3}=0) was studied by Wang and the third author. As applications, we prove among other things that the following two mean field equations Δu+e^{u}=16πδ_{0}andΔu+e^{u}=8π∑k=13δ_{ [Formula presented] } on a flat torus has the same number of even solutions. This result is quite surprising from the PDE point of view.

Original language | English |
---|---|

Pages (from-to) | 251-272 |

Number of pages | 22 |

Journal | Journal des Mathematiques Pures et Appliquees |

Volume | 132 |

DOIs | |

Publication status | Published - 2019 Dec |

### Fingerprint

### Keywords

- Generalized Lamé equation
- Mean field equation
- Pre-modular form

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Journal des Mathematiques Pures et Appliquees*,

*132*, 251-272. https://doi.org/10.1016/j.matpur.2019.05.004