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

Amir Mostaed

Abstract

McMullen's compact Kobayashi-geodesic curve $V \subset X_L$, arising from the hyperbolic triangle group $Δ(14,21,42)$ 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 $L = \mathbb{Q}(\cos\tfracπ{21})$, 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 André-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 $M = \mathbb{Q}(ζ_{42})$, 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.

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

Abstract

McMullen's compact Kobayashi-geodesic curve , arising from the hyperbolic triangle group via a modular embedding into the Hilbert modular sixfold attached to the totally real cyclic field , is not contained in any proper Shimura subvariety of , and the generic fiber satisfies , hence carries no exceptional Hodge tensors. The Weil locus parametrizing abelian sixfolds of Weil type for has codimension and irreducible components; the expected dimension makes any non-empty super-atypical in the sense of Zilber-Pink. We prove that is finite, possibly empty: every intersection point is a CM point with , a degree- CM field with , established by two independent methods: the André-Oort theorem for and the Ax-Schanuel theorem for period maps. The Hodge-Weil classes in at intersection points are absolute Hodge yet inaccessible to all existing algebraicity theorems, due to three independent obstructions: CM isolation, absence of a -secant structure, and uncontrolled discriminant. For , so that , we reduce non-emptiness of to explicit algebraic equations for the prime via Hecke correspondences on , and isolate the remaining open steps toward a new case of the Hodge conjecture for abelian sixfolds.
Paper Structure (30 sections, 29 theorems, 63 equations)

This paper contains 30 sections, 29 theorems, 63 equations.

Table of Contents

  1. McMullen's curve
  2. Triangle groups and matrix models
  3. The field $L$ and the Hilbert modular variety
  4. The quadrilateral subgroup $\Delta_0$
  5. The modular embedding and McMullen's curve
  6. Zariski density and the Mumford--Tate group
  7. The Weil locus
  8. Abelian sixfolds of Weil type
  9. The Weil locus and its codimension
  10. Dimensional analysis
  11. Hodge classes on fibers over $V\cap\mathcal{W}_K$
  12. Examples
  13. Finiteness
  14. Every point of $V\cap\mathcal{W}_K$ is a CM point
  15. Finiteness via André--Oort
  16. ...and 15 more sections

Key Result

Theorem 1

Every finite-index subgroup of $\Delta(14,21,42)$ admits a matrix model over the ring of integers of its trace field. In particular, $\Delta_0:=\Delta_0(14,21,42)$ embeds in $\mathop{\mathrm{SL}}\nolimits_2(\mathcal{O}_L)$ as a cocompact subgroup of infinite index (being a finite-index subgroup of $

Theorems & Definitions (68)

  • Theorem 1: McMullen McM23
  • Theorem 2: McMullen McM23
  • Theorem 3: Generic Mumford--Tate group
  • Proposition 4: Structure of $\mathcal{W}_K$
  • Theorem 5: Finiteness of intersection
  • Corollary 6: Isogeny classes and height bound
  • Theorem 7: Inaccessibility of Weil classes
  • Proposition 8: Hecke criterion
  • Proposition 9: Constant-sign obstruction
  • Theorem 10: Reduction to finite computation
  • ...and 58 more