The geometry of generalized Lamé equation, I

Zhijie Chen; Ting Jung Kuo; Chang Shou Lin

doi:10.1016/j.matpur.2018.08.004

The geometry of generalized Lamé equation, I

Zhijie Chen
, Ting Jung Kuo
, Chang Shou Lin^*

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

11 Citations (Scopus)

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^NE_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 ³n_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
Pages (from-to)	89-120
Number of pages	32
Journal	Journal des Mathematiques Pures et Appliquees
Volume	127
DOIs	https://doi.org/10.1016/j.matpur.2018.08.004
Publication status	Published - 2019 Jul

Keywords

Degree of the addition map
Generalized Lamé equation
Spectral curve

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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10.1016/j.matpur.2018.08.004

Cite this

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

abstract = "In this paper, we prove that the spectral curve Γn of the generalized Lam{\'e} equation with the Treibich–Verdier potential [Fourmula presented] can be embedded into the symmetric space SymNEτ of the N-th copy of the torus Eτ, where N=∑nk. This embedding induces an addition map σn(⋅|τ) from Γn onto Eτ. The main result is to prove that the degree of σn(⋅|τ) is equal to ∑k=0 3nk(nk+1)/2. This is the first step toward constructing the pre-modular form associated with this generalized Lam{\'e} equation.",

keywords = "Degree of the addition map, Generalized Lam{\'e} equation, Spectral curve",

author = "Zhijie Chen and Kuo, \{Ting Jung\} and Lin, \{Chang Shou\}",

note = "Publisher Copyright: {\textcopyright} 2018 Elsevier Masson SAS",

year = "2019",

month = jul,

doi = "10.1016/j.matpur.2018.08.004",

language = "English",

volume = "127",

pages = "89--120",

journal = "Journal des Mathematiques Pures et Appliquees",

issn = "0021-7824",

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TY - JOUR

T1 - The geometry of generalized Lamé equation, I

AU - Chen, Zhijie

AU - Kuo, Ting Jung

AU - Lin, Chang Shou

PY - 2019/7

Y1 - 2019/7

N2 - In this paper, we prove that the spectral curve Γn of the generalized Lamé equation with the Treibich–Verdier potential [Fourmula presented] can be embedded into the symmetric space SymNEτ of the N-th copy of the torus Eτ, where N=∑nk. This embedding induces an addition map σn(⋅|τ) from Γn onto Eτ. The main result is to prove that the degree of σn(⋅|τ) is equal to ∑k=0 3nk(nk+1)/2. This is the first step toward constructing the pre-modular form associated with this generalized Lamé equation.

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

KW - Degree of the addition map

KW - Generalized Lamé equation

KW - Spectral curve

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DO - 10.1016/j.matpur.2018.08.004

M3 - Article

AN - SCOPUS:85051112868

SN - 0021-7824

VL - 127

SP - 89

EP - 120

JO - Journal des Mathematiques Pures et Appliquees

JF - Journal des Mathematiques Pures et Appliquees

ER -

The geometry of generalized Lamé equation, I

Abstract

Keywords

ASJC Scopus subject areas

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