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

TL;DR

This paper investigates the intersection of a special modular curve with the Weil locus in a Hilbert modular sixfold, proving finiteness of intersection points and exploring implications for the Hodge conjecture in abelian sixfolds.

Contribution

It establishes the finiteness of intersections between McMullen's curve and Weil locus, and reduces the non-emptiness problem to explicit algebraic equations, advancing understanding of the Hodge conjecture.

Findings

Intersection points are CM points with specific endomorphism fields.

The intersection is finite, possibly empty, and contains only CM points.

Explicit algebraic equations are derived for certain cases to test non-emptiness.

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.