Componentwise Geometry and Monodromy of Generalized Lamé Equations
Abstract
We develop a componentwise geometric and monodromy theory for the one-support generalized Lamé equation on an elliptic curve, with singularities at 0 and ± p. Its log-free curve decomposes canonically into irreducible even and non-even components, the former being governed by elliptic Painlevé~VI. For the non-even component, we construct the hyperelliptic spectral curve, Baker--Akhiezer functions, and addition map, and prove that °σn,p(1)=n(n+1). After quotienting by the involution T↦ −T, we identify the non-even spectral curve with the classical Lamé spectral curve of weight n, compatibly with the addition map and the rational function κ. This identification is realized by B̃=T2−n(n+1)℘(p), and associates every non-even generalized Lamé equation with a unique classical Lamé equation on the same elliptic curve having equivalent period monodromy. Fixing B̃ yields an isomonodromic deformation with τ fixed. Together with the Painlevé-VI deformation on the even component, it gives a componentwise interpretation of the collision p→0, and yields a finite descent on the admissible completely reducible locus. The classical spectral, finite-gap, finite-monodromy, and curvature theories consequently transfer to the non-even component. Finally, within the symmetric family (n0,n1,n2,n3,1/2,1/2), the one-support case forms an affine genus-zero hierarchy, whereas for (1,1,0,0,1/2,1/2), the non-even normalization is generically elliptic and becomes rational on the discriminant locus, while the full compactified log-free curve retains arithmetic genus two. This first genus jump marks the boundary of the affine theory and motivates a genus-dependent componentwise geometry.
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BibTeX
@misc{kuo2026componentwise,
title = {{Componentwise Geometry and Monodromy of Generalized Lamé Equations}},
author = {Ting-Jung Kuo and Xuanpu Liang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.22243},
eprint = {2608.22243},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.22243},
}
RIS
TY - UNPB
TI - Componentwise Geometry and Monodromy of Generalized Lamé Equations
AU - Ting-Jung Kuo
AU - Xuanpu Liang
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2608.22243
UR - https://arxiv.org/abs/2608.22243
AB - We develop a componentwise geometric and monodromy theory for the one-support generalized Lamé equation on an elliptic curve, with singularities at 0 and ± p. Its log-free curve decomposes canonically into irreducible even and non-even components, the former being governed by elliptic Painlevé~VI. For the non-even component, we construct the hyperelliptic spectral curve, Baker--Akhiezer functions, and addition map, and prove that °σ_n,p^(1)=n(n+1). After quotienting by the involution T↦ −T, we identify the non-even spectral curve with the classical Lamé spectral curve of weight n, compatibly with the addition map and the rational function κ. This identification is realized by B̃=T^2−n(n+1)℘(p), and associates every non-even generalized Lamé equation with a unique classical Lamé equation on the same elliptic curve having equivalent period monodromy. Fixing B̃ yields an isomonodromic deformation with τ fixed. Together with the Painlevé-VI deformation on the even component, it gives a componentwise interpretation of the collision p→0, and yields a finite descent on the admissible completely reducible locus. The classical spectral, finite-gap, finite-monodromy, and curvature theories consequently transfer to the non-even component. Finally, within the symmetric family (n_0,n_1,n_2,n_3,1/2,1/2), the one-support case forms an affine genus-zero hierarchy, whereas for (1,1,0,0,1/2,1/2), the non-even normalization is generically elliptic and becomes rational on the discriminant locus, while the full compactified log-free curve retains arithmetic genus two. This first genus jump marks the boundary of the affine theory and motivates a genus-dependent componentwise geometry.
ER -