McMullen's Curve, the Weil Locus, and the Hodge Conjecture for Abelian Sixfolds

Abstract

McMullen's compact Kobayashi-geodesic curve $V \subset X_L$, arising from the hyperbolic triangle group $\Delta(14,21,42)$ via a modular embedding into the Hilbert modular sixfold $X_L = \mathbb{H}^6/\mathrm{SL}_2(\mathcal{O}_L)$ attached to the totally real cyclic field $L = \mathbb{Q}(\cos\tfrac{\pi}{21})$, is not contained in any proper Shimura subvariety of $X_L$, and the generic fiber $A_v$ satisfies $\mathrm{MT}(A_v) = \mathrm{Res}_{L/\mathbb{Q}}\,\mathrm{SL}_2$, hence carries no exceptional Hodge tensors. The Weil locus $\mathcal{W}_K \subset X_L$ parametrizing abelian sixfolds of Weil type for $K = \mathbb{Q}(\sqrt{-d})$ has codimension $3$ and $20$ irreducible components; the expected dimension $1 + 3 - 6 = -2$ makes any non-empty $V \cap \mathcal{W}_K$ super-atypical in the sense of Zilber-Pink. We prove that $V \cap \mathcal{W}_K$ is finite, possibly empty: every intersection point is a CM point with $\mathrm{End}^0(A_v) = M = KL$, a degree-$12$ CM field with $\mathrm{Gal}(M/\mathbb{Q}) \cong \mathbb{Z}/2 \times \mathbb{Z}/2 \times \mathbb{Z}/3$, established by two independent methods: the Andr\'e-Oort theorem for $\mathcal{A}_6$ and the Ax-Schanuel theorem for period maps. The Hodge-Weil classes in $H^{3,3}$ at intersection points are absolute Hodge yet inaccessible to all existing algebraicity theorems, due to three independent obstructions: CM isolation, absence of a $K$-secant structure, and uncontrolled discriminant. For $d \in \{3,7\}$, so that $M = \mathbb{Q}(\zeta_{42})$, we reduce non-emptiness of $V \cap \mathcal{W}_K$ to $44 \times 64 = 2816$ explicit algebraic equations for the prime $\ell = 43$ via Hecke correspondences on $X_L$, and isolate the remaining open steps toward a new case of the Hodge conjecture for abelian sixfolds.