A simple proof of Dahmen's conjectures

TL;DR

This paper provides a straightforward proof of Dahmen's conjectures concerning Lame equations with finite monodromy, utilizing recent geometric insights into spherical tori with conical singularities.

Contribution

It offers an alternative proof of Dahmen's conjectures by applying recent geometric results on spherical tori, connecting differential equations and geometric structures.

Findings

Confirmed Dahmen's conjectures for Lame equations with finite monodromy.

Linked monodromy properties of Lame equations to geometric structures of spherical tori.

Provided a new proof approach using geometric triangulation techniques.

Abstract

The number of Lame equations with finite (ordinary or projective) monodromy has been conjectured by S. R. Dahmen, and a few proofs have been proposed. It is known that Lame equations with unitary monodromy are corresponding to spherical tori with one conical singularity, and the geometry of such surfaces had been studied with triangulation recently. In this paper, we will apply the results on spherical tori to give an alternative proof of Dahmen's conjectures.