Lame operators with projective octahedral and icosahedral monodromies

Abstract

We show that there exists a Lame operator $L_n$ with projective octahedral monodromy for each $n\in 1/2(\mathbf{N}+1/2)\cup 1/3(\mathbf{N}+1/2)$, and with projective icosahedral monodromy for each $n\in 1/3(\mathbf{N}+1/2)\cup 1/5(\mathbf{N}+1/2)$. To this end, we construct Grothendieck's dessin d'enfants corresponding to the Belyi morphisms which pull-back hypergeometric operators into Lame operators $L_n$ with the desired monodromies.

23 pages, 34 figures; some spell-mistakes are fixed, and detailed explanations on the tables are added