A Hypergeometric RM Surface: Integral Monodromy and Level-11 Covers

Abstract

We study a genus-two hypergeometric family whose Jacobians have real multiplication by $\Q(\sqrt5)$. We identify the relevant primitive factor and integral rank-two monodromy; at one real embedding the projective image is $Δ(2,5,\infty)$, while the conjugate image is nondiscrete. At the primes above $11$, the degree-$12$ Borel covers are quadratic pullbacks of genus-one triangular modular curves. The same rank-two systems have finite-field hypergeometric Frobenius traces, yielding the local Euler factors and a quadratic transformation twisted by the character of $2$.