Algebraic Heun Operators with Tetrahedral Monodromy

Abstract

Our work adds to the picture of second order differential operators with a full set of algebraic solutions, which we will call algebraic. We see algebraic Heun operators as pull-backs of algebraic hypergeometric operators via Belyi functions. We focus on the case when the hypergeometric one has a tetrahedral monodromy group. We find arithmetic conditions for the pull-back functions to exist. For each distribution of the singular points in the ramified fibers, we identify the minimal values of the exponent differences and we explicitly construct the dessin d’enfant corresponding to the pull-back function in the minimal cases. Then by allowing some parameters to vary, we find infinite families of such graphs, hence of Heun operators with tetrahedral monodromy.

Keywords

• algebraic solutions of differential equations
• belyi functions
• dessins d’enfants
• planar graphs
• primary 11g32, 34m15
• secondary 34m35, 05c10, 05c65, 33c99