McMullen's curve is a compact Kobayashi-geodesic curve inside a Hilbert modular sixfold, arising from a hyperbolic triangle group. It is a single curve inside a vast moduli space — one thread through a space of abelian sixfolds.
The paper on McMullen's curve and the Hodge conjecture (arXiv: 2603.20268) proves that this curve avoids all proper Shimura subvarieties and carries no exceptional Hodge tensors. In simpler terms: the curve passes through the most generic points of the moduli space, avoiding every special locus.
Yet this avoidance is not complete. The curve does intersect the Weil locus — the set parametrizing abelian varieties of Weil type — but only at finitely many points, all of which are CM (complex multiplication) points. At these exceptional intersections, the Hodge conjecture might be verified for abelian sixfolds, which remains open in general.
The proof combines the André–Oort theorem (special points are rare in non-special varieties) and the Ax–Schanuel theorem (transcendence constraints from functional transcendence theory). The payoff is concrete: for specific imaginary quadratic fields, the question reduces to 2,816 explicit algebraic equations.
The through-claim: genericity is proved by elimination, and the exceptions are the point. The curve's generic behavior — avoiding all Shimura subvarieties — is established by the deep theorems. But the finite set of exceptions, the CM points where the curve meets the Weil locus, is where the Hodge conjecture might actually be checked. The avoidance theorem isolates the test cases.
2603.20268. Algebraic geometry / Hodge conjecture / abelian varieties / Shimura varieties / McMullen's curve.