Conjecture on canonical divisors of nonrational Hilbert modular surfaces

Let $K$ be the real quadratic field in the paper, let $\rho$ range over the resolution curves of the cusps of the Hilbert modular surface $Y_{cGamma({\mathscr{O}}_K,\id{a})}$, and let $F_N$ denote the Hirzebruch–Zagier divisors defined from primitive integral skew-hermitian matrices of determinant $N/A$. If $Y_{cGamma({\mathscr{O}}_K,\id{a})}$ is not rational, then the canonical-divisor conjecture. the canonical divisor can be written as a rational positive linear combination of the resolution curves and the divisors $F_N$. This conjecture gives a positive-divisor description of the canonical class that is intended to support the study of the geometry and modular-form bounds of Hilbert modular surfaces; the supplied text does not indicate whether it has been resolved.