Halphen's transform and middle convolution

TL;DR

This paper demonstrates that the Halphen transform of a Lamé equation can be expressed through symmetric square and Euler transforms, enabling the classification of certain Lamé equations with arithmetic Fuchsian monodromy groups.

Contribution

It establishes a novel connection between the Halphen transform and symmetric square plus Euler transforms, and classifies Lamé equations with specific arithmetic monodromy groups.

Findings

Identifies all Lamé equations with quaternion algebra A over Q as their monodromy group.

Provides a classification of geometric braid group orbits in SL_2(Z)^4.

Includes all cases where the associated quaternion algebra is over Q.

Abstract

We show that the Halphen transform of a Lam\'e equation can be written as the symmetric square of the Lam\'e equation followed by an Euler transform. We use this to compute a list of Lam\'e equations with arithmetic Fuchsian monodromy group. It contains all those Lam\'e equations where the quaternion algebra $A$ over $k$ associated to the arithmetic Fuchsian group is a quaternion algebra $A$ over $\mathbb{Q}$. Further we classify all geometric braid group orbits in $\rm{SL}_2(\ZZ)^4$ with the possible exception of three orbits.