Resolving Thinness of Degree-Six Symplectic Hypergeometric Monodromy Groups

Abstract

We prove arithmeticity for two degree-six symplectic hypergeometric monodromy groups and thinness for a third case. These are the three open cases called C-32, C-47, and C-55 in the paper [BDN25] by Bajpai--Dona--Nitsche. This hence settles thinness of primitive integral symplectic hypergeometric monodromy groups of degree-six, according to the tabulation by Bajpai--Dona--Singh--Singh [BDSS21]. The arithmeticity certificates were found with AlphaEvolve and the thinness certificate is a ping pong table found by a harness that was configured to run ChatGPT 5.5. These certificates are then independently verified with exact matrix arithmetic over Q using a computer. We also include illustrations of limit sets of several degree-six symplectic hypergeometric monodromy groups.

arXiv comment13 pages, 4 figures