Halphen's transform and middle convolution

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.