<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>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.</p>
</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.

Algebraic Heun Operators with Tetrahedral Monodromy

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"Algebraic Heun Operators with Tetrahedral Monodromy"}

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...

Algebraic Heun Operators with Tetrahedral Monodromy

Published Online: Jun 02, 2022

Page range: 209 - 230

Received: Oct 18, 2021

Accepted: Dec 21, 2021

DOI: https://doi.org/10.2478/auom-2022-0027

Keywords
Algebraic solutions of differential equations, Belyi Functions, Dessins d’enfants, Planar graphs

© 2022 Iulia-Cătălina Pleşca, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Algebraic Heun Operators with Tetrahedral Monodromy

Published Online: Jun 02, 2022

Page range: 209 - 230

Received: Oct 18, 2021

Accepted: Dec 21, 2021

DOI: https://doi.org/10.2478/auom-2022-0027

KeywordsAlgebraic solutions of differential equations, Belyi Functions, Dessins d’enfants, Planar graphs

© 2022 Iulia-Cătălina Pleşca, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
Algebraic solutions of differential equations, Belyi Functions, Dessins d’enfants, Planar graphs