Certifying Arithmeticity for Degree-Six Symplectic Hypergeometric Monodromy Groups

It is easy to write down a subgroup of a matrix group: pick a few matrices and take everything they generate. It is hard to say anything about the result. If the generators have integer entries the group is discrete, and the real question is whether it has finite index in the integer points of its Zariski closure, making it an arithmetic lattice, or infinite index, making it what Sarnak named a thin group. Thin groups came into focus through the affine sieve of Bourgain, Gamburd and Sarnak, which extended sieve methods from arithmetic groups to the orbits of thin ones. In his 2014 notes on thin groups, Sarnak called monodromy “the oldest and perhaps most natural source of finitely generated linear groups,” and for the classical hypergeometric equation posed the question directly: is each monodromy group of finite or infinite index?

The hypergeometric differential equation has three singular points, and carrying its solutions around them produces a group generated by two explicit integer matrices. Beukers and Heckman showed these groups are Zariski dense in a symplectic or orthogonal group, so each is either arithmetic or thin. In degree six there are 458 symplectic cases. By 2025 all but three had been settled, and the three left over are labelled C-32, C-47 and C-55.

The proof uses a criterion of Bajpai, Dona and Nitsche: a Zariski-dense subgroup of $\mathrm{Sp}_\Omega(\mathbb{Z})$ is a lattice exactly when it contains two commuting transvections with independent, $\Omega$-orthogonal directions. The group always contains one transvection, so what is needed is a word in the generators that conjugates it to a second. Those words, of length 93 and 49, were found with the assistance of AlphaEvolve, Google DeepMind’s evolutionary search system, which uses a language model to propose candidate words and keeps the ones that score best against a reward measuring the two conditions. Once a word is in hand the proof is a finite computation in exact rational arithmetic; one of my contributions was writing independent verifiers that confirm each certificate separately from the code that found it.

For C-32 we suspected the opposite answer, and the suspicion came from a picture. The limit set of a thin group is a fractal proper subset of projective space, while an arithmetic group’s fills it. Drawn with a renderer I wrote for exploring these groups, the limit set of C-32 looked like the thin examples. A ping-pong argument proving it thin, a cone with 33 facets carried into itself by the group, was found with the help of OpenAI’s models after AlphaEvolve’s searches stalled, and then verified exactly.

With these three cases, the thinness question for primitive integral symplectic hypergeometric groups of degree six is closed. We see this as a test case for AI-assisted mathematics at scale: the machines propose certificates, and the mathematics is the deterministic check that a certificate works, so every result is as auditable as any other computer-assisted proof. Much of my work on the project has been exactly that checking, going through the arguments the machines produced with Max Riestenberg and repairing or improving them. We are now working to resolve arithmeticity or thinness for all hypergeometric monodromy groups in degree six and below.

Status. The arithmeticity of C-47 and C-55 is on arXiv. The expanded paper, Resolving thinness of degree-six symplectic hypergeometric monodromy groups, adds C-32 and is in preparation. The program covering all of degree six and below is underway.