Arithmeticity of degree-six symplectic hypergeometric monodromy groups

Are the degree-six symplectic hypergeometric monodromy groups C-47 and C-55 arithmetic, meaning finite index in their ambient integral symplectic groups?

Why it matters and what counts as progress

Why it matters: These are specific remaining classification cases where explicit group words can turn a computational search into an arithmeticity certificate.

What counts as progress: Explicit group words certifying arithmeticity in unresolved cases, or a proof that a candidate group has infinite index.

How the problem developed

2021 · Classification [1]
Bajpai, Dona, Singh and Singh Enumerated 40 degree-six symplectic hypergeometric groups with maximally unipotent monodromy and studied their arithmetic-or-thin dichotomy.
2025 · Partial result [2]
Bajpai, Dona and Nitsche Classified the remaining maximally-unipotent cases as arithmetic or thin; other degree-six cases were outside this restricted family.
25 May 2026 · Reported claim [3]
Riestenberg, Taha, Trettel and Wienhard, with AlphaEvolve Reported exact certificates proving arithmeticity of the degree-six symplectic groups C-47 and C-55, two cases left open by the cited classification.

AI contribution

Resolving Thinness of Degree-Six Symplectic Hypergeometric Monodromy Groups

J. Maxwell Riestenberg, Diaaeldin Taha, Steve Trettel, Anna Wienhard

AlphaEvolve found explicit witness words that Riestenberg, Taha, Trettel and Wienhard use to prove that the degree-six symplectic hypergeometric monodromy groups C-47 and C-55 have finite index in their ambient integral symplectic groups and are arithmetic.

AI involvementAI-centered

ImportanceNotable

Technical verificationFormal (exact rational matrix arithmetic + human)

Human understandingFull