A simple proof of Dahmen’s conjectures

Po-Sheng Wu

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.

1. Introduction

Given a lattice $\Lambda=\mathbb{Z}\omega_{1}+\mathbb{Z}\omega_{2}$ on $\mathbb{C}$ with $(\omega_{1},\omega_{2})=(1,\tau),\textnormal{Im}(\tau)>0$ , the Lamé equation on the elliptic curve $E=\mathbb{C}/\Lambda$ is a second order ordinary differential equation:

(1.1)

\frac{\partial^{2}w}{\partial z^{2}}-\Big{(}n(n+1)\wp(z)+B\Big{)}w=0,

where $B\in\mathbb{C}$ and $\wp$ is the Weierstrass elliptic function. We consider the case $n\in\mathbb{Z}_{>0}$ and study number of Lamé equations with given finite monodromy groups.

It is known that all the finite monodromy groups of Lamé equations are cyclic. The following two conjectures are proposed by S.R. Dahmen, and later he proved the first conjecture using dessin d’enfant [S1][S2].