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 $Δ(14,21,42)$ via a modular embedding into the Hilbert modular sixfold $$X_L = \mathbb{H}<sup>6/\mathrm{SL}_2(\mathcal{O}_L)$$ 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 $$\mathrm{MT}(A_v) = \mathrm{Res}<em>{L/\mathbb{Q}}\,\mathrm{SL}_2$$, 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 $H<sup>{3,3}$ 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 $M = \mathbb{Q}(ζ</em>{42})$, 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.

• The paper proves that the intersection of McMullen’s non-Shimura curve with the Weil locus is finite and that every intersection point has complex multiplication by the degree-12 field KL.
• It establishes that the associated Hodge–Weil classes are absolute Hodge classes outside known algebraicity methods because the relevant CM points are isolated and lack required secant or discriminant structures.
• It reduces non-emptiness to at most 2,816 explicit fixed-point equations for the prime ℓ=43, while leaving verification of the Weil signature and algebraicity of the classes open.

Overview

This paper by Amir Mostaed studies the intersection of two rigid subvarieties of the Hilbert modular sixfold $X_L = \mathbb{H}^6/\mathrm{SL}_2(O_L)$, where $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$ is the totally real sextic field with cyclic Galois group $\mathbb{Z}/6\mathbb{Z}$ and class number one. The first object is McMullen's compact Kobayashi-geodesic curve $V \subset X_L$, arising from the non-arithmetic triangle group $\Delta(14,21,42)$ via a modular embedding. The second is the Weil locus $W_K$ of abelian sixfolds carrying an imaginary quadratic field $K=\mathbb{Q}(\sqrt{-d})$ in their endomorphism algebra with Weil signature $(3,3)$. The expected dimension of $V \cap W_K$ is $1+3-6=-2$, so any point of intersection is super-atypical in the Zilber–Pink sense. The paper proves that this intersection is finite, that every point carries CM by the degree-12 compositum $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$0, that the resulting Hodge–Weil classes are absolute Hodge but inaccessible to all known algebraicity methods, and that non-emptiness reduces to 2816 explicit algebraic equations for the prime $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$1.

McMullen's curve and its Mumford–Tate group

The geometric input comes from McMullen's "Hilbert series" of eleven cocompact triangle groups whose invariant quaternion algebra $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$2 splits at every infinite place of the invariant trace field — behavior opposite to Takeuchi's 76 arithmetic triangle groups, where $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$3 splits at exactly one real place. For $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$4, the invariant trace field is $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$5, and every finite-index subgroup admits a matrix model over the ring of integers of its trace field. This places the index-two subgroup $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$6 inside $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$7 as a cocompact subgroup of infinite index — a split structure normally characteristic of arithmetic groups but achieved here by a non-arithmetic group.

Via the Cohen–Wolfart construction, this embedding yields a modular embedding $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$8, equivariant for $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$9 on $\mathbb{Z}/6\mathbb{Z}$0 and the six Galois-twisted actions on $\mathbb{Z}/6\mathbb{Z}$1. After passing to a finite-index subgroup $\mathbb{Z}/6\mathbb{Z}$2, this descends to a holomorphic immersion $\mathbb{Z}/6\mathbb{Z}$3. The paper records two structural facts from McMullen: $\mathbb{Z}/6\mathbb{Z}$4 is compact and Kobayashi-geodesic, and it is not contained in any proper Shimura subvariety of $\mathbb{Z}/6\mathbb{Z}$5. Notably, in dimension two every non-Shimura geodesic curve on a Hilbert modular surface must have a cusp, so compactness here is exceptional to dimension six.

The generic Mumford–Tate group is computed as $\mathbb{Z}/6\mathbb{Z}$6 for Zariski-generic $\mathbb{Z}/6\mathbb{Z}$7, using McMullen's Zariski-density criterion (invariant trace field equal to $\mathbb{Z}/6\mathbb{Z}$8 plus non-virtual-solvability). Consequently the Hodge ring of a generic fiber is generated by divisor classes and the polarization alone. Points of $\mathbb{Z}/6\mathbb{Z}$9 carry a strictly smaller Mumford–Tate group — a torus — and hence a richer Hodge ring containing exceptional Weil classes; this transition from generic to special is the arithmetic core of the paper.

Each component map $V \subset X_L$0 satisfies the same hypergeometric equation $V \subset X_L$1, differing only in monodromy representation. This common ODE is not merely analytic: it explains why primes $V \subset X_L$2 are the natural candidates for the Hecke search, since such primes make all 42nd roots of unity available over $V \subset X_L$3 (for $V \subset X_L$4, $V \subset X_L$5), permitting pre-screening of fixed-point equations before any high-precision computation.

Structure of the Weil locus

An abelian sixfold $V \subset X_L$6 with $V \subset X_L$7-multiplication is of Weil type for $V \subset X_L$8 if $V \subset X_L$9 commutes with the $\Delta(14,21,42)$0-action, is anti-invariant under Rosati, and acts on $\Delta(14,21,42)$1 with eigenvalues $\Delta(14,21,42)$2 each of multiplicity 3. By Cattani–Deligne–Kaplan, the Weil locus $\Delta(14,21,42)$3 is a countable union of closed algebraic subvarieties. The paper establishes:

• Each irreducible component is smooth of codimension 3 at every non-CM point, proved by an explicit transversality computation showing the three defining equations have differentials supported on distinct summands of the tangent space.
• Components are indexed by sign-assignments $\Delta(14,21,42)$4 with $\Delta(14,21,42)$5: exactly $\Delta(14,21,42)$6 components forming four $\Delta(14,21,42)$7-orbits of sizes $\Delta(14,21,42)$8. The orbit of size 2 consists of the alternating assignments $\Delta(14,21,42)$9, stabilized as a set by the index-2 subgroup.
• At every non-CM point of every component, the Hodge–Weil space $W_K$0 is a 2-dimensional $W_K$1-subspace of $W_K$2 contained in $W_K$3, consisting entirely of exceptional Hodge classes in the sense of Moonen–Zarhin: they lie outside the algebra generated by divisor classes.

Since $W_K$4, any non-empty intersection exceeds the expected dimension by at least 2 and cannot be produced by general-position arguments. The paper notes that the Baldi–Klingler–Ullmo framework describes these points as super-atypical intersections of zero period dimension, though their main algebraicity theorem does not apply because the variation of Hodge structures has level 1 rather than level at least 3.

Finiteness and the CM structure of the intersection

The central result is that every point of $W_K$5 is a CM point with $W_K$6, a CM field of degree 12. The proof is short and decisive: membership in $W_K$7 gives $W_K$8, membership in $W_K$9 gives $K=\mathbb{Q}(\sqrt{-d})$0, and since $K=\mathbb{Q}(\sqrt{-d})$1, the containment $K=\mathbb{Q}(\sqrt{-d})$2 forces equality, making the Mumford–Tate group a torus.

Finiteness follows by contradiction through Tsimerman's André–Oort theorem for $K=\mathbb{Q}(\sqrt{-d})$3: infinitely many CM points on $K=\mathbb{Q}(\sqrt{-d})$4 would be Zariski-dense in the curve $K=\mathbb{Q}(\sqrt{-d})$5, forcing $K=\mathbb{Q}(\sqrt{-d})$6 special; since the forgetful morphism $K=\mathbb{Q}(\sqrt{-d})$7 is finite onto its image, $K=\mathbb{Q}(\sqrt{-d})$8 would then be a Shimura subvariety of $K=\mathbb{Q}(\sqrt{-d})$9, contradicting McMullen's theorem. An independent proof uses Wolfart's theorem (algebraic values of the uniformizing function of a non-arithmetic Fuchsian group occur only at CM points) together with the Ax–Schanuel theorem of Blázquez-Sanz–Casale–Freitag–Nagloo. The two routes are logically independent and illuminate different aspects of the rigidity.

Two caveats are stated plainly. First, the finiteness theorem is purely qualitative and gives no bound on $(3,3)$0; an effective version would require an effective André–Oort theorem for $(3,3)$1, which is not currently available. Second, the height bound derived from von Känel–Kret,

$(3,3)$2

(for $(3,3)$3, where $(3,3)$4 and only $(3,3)$5 ramify), is astronomically large but effective. Combined with Faltings' theorem, it implies the fibers lie in finitely many explicitly bounded isogeny classes over $(3,3)$6, defined over number fields of degree at most 12. This converts the existence question into a very large finite search.

For $(3,3)$7 and $(3,3)$8 the compositum coincides: both $(3,3)$9 embed in $V \cap W_K$0, so $V \cap W_K$1 in both cases, while the Weil loci remain distinct subvarieties of $V \cap W_K$2 because the eigenspace condition depends on $V \cap W_K$3. Whether a single fiber can satisfy the Weil condition for both fields simultaneously is left open.

Inaccessibility by known methods

The Hodge–Weil classes at points of $V \cap W_K$4 are absolute Hodge by Deligne's theorem, yet the paper argues they resist every existing algebraicity method, for three structurally independent reasons:

1. CM isolation: $V \cap W_K$5 is isolated in every positive-dimensional deformation space, so Markman's semiregularity argument — which deforms a secant sheaf over the 9-dimensional moduli space of Weil-type sixfolds of discriminant $V \cap W_K$6 — collapses to a point.
2. Absent $V \cap W_K$7-secant geometry: Markman's construction is intrinsic to triples $V \cap W_K$8 arising from a $V \cap W_K$9-secant line on an abelian threefold; the sixfolds $1+3-6=-2$0 do not arise this way.
3. Uncontrolled discriminant: the discriminant of the $1+3-6=-2$1-Hermitian form on $1+3-6=-2$2 is not prescribed by the intersection conditions and is not generically $1+3-6=-2$3, so no discriminant-specific theorem (Schoen, Markman) applies.

The Hazama–Murty route also fails maximally: it requires the Hodge group to equal the full centralizer $1+3-6=-2$4, whereas for a CM variety the Hodge group is a torus. Any proof of algebraicity would therefore need either a method producing cycles at isolated CM points without deformation, or a new criterion independent of discriminant — a genuinely new case of the Hodge conjecture.

On the arithmetic side, of the $1+3-6=-2$5 CM types of $1+3-6=-2$6, all are compatible with the $1+3-6=-2$7-real-multiplication structure, but exactly $1+3-6=-2$8 satisfy the Weil signature condition, forming 10 conjugate pairs; these are in natural bijection with the 20 components of $1+3-6=-2$9. Which of these types is actually realized at points of $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$00 is not determined by the construction.

The Hecke program and reduction to a finite computation

Non-emptiness of $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$01 remains open. The paper formulates a decidable strategy via Hecke correspondences. A key proposition shows that any fixed point of $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$02 on $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$03 (for $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$04 of prime norm $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$05) yields a CM abelian sixfold with $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$06: the Hecke endomorphism $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$07 satisfies $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$08, cannot lie in $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$09 (else $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$10), and generates a CM field of degree 12 over $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$11.

However, the paper is explicit that this does not establish membership in $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$12. Two further conditions are required and neither follows from the Hecke construction alone: (a) the quadratic extension generated by $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$13 must be specifically $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$14, encoded as the trace constraint $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$15; and (b) the sign vector $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$16 for $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$17 must have exactly three positive entries. Two worked examples show rational solutions always fail: for $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$18, $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$19 solves $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$20 but gives sign pattern $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$21; likewise $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$22 for $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$23. Conversely, an element like $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$24 achieves a Weil-compatible sign pattern $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$25 but fails the norm equation. The two constraints are jointly non-trivial and demand genuine computation in $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$26.

For $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$27 — the smallest prime $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$28, splitting completely in $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$29 — the search reduces to at most $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$30 algebraic equations in $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$31, each of degree at most 2 in the values $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$32. By Wolfart's theorem each equation has either no solution or finitely many CM solutions, so the computation terminates regardless of outcome. An explicit generator $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$33 of norm 43 for the relevant prime ideal is provided, exploiting $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$34.

Limitations and open problems

The paper concedes several gaps candidly. The monodromy-to-embedding assignment for the six component maps requires numerical verification against McMullen's matrix generators that has not been carried out. The reflex-field computations determining which Weil-compatible CM types have trivial stabilizer are deferred. Most importantly, three open problems structure the remainder of the program:

• (O1) Execute the $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$35 computation: enumerate solutions to the 2816 fixed-point systems with $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$36 and Weil-compatible sign vector. This is a finite computation requiring no new theory, and has not been performed.
• (O2) Verify the Weil signature: compute the CM type of $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$37 directly from the period point. The sign vector is necessary but not sufficient; this step is not accessible from the Hecke construction alone.
• (O3) Prove algebraicity of the Hodge–Weil classes, overcoming the three obstructions above. This requires genuinely new Hodge-theoretic methods.

The logical structure is serial: O1 is computational, O2 is classical CM theory, and O3 is the substantive open question. Even a complete resolution of O1 and O2 would establish only existence of points in $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$38, not the Hodge conjecture for them.

Conclusion

The paper identifies a geometrically rigid family of abelian sixfolds of Weil type — the fibers over the super-atypical intersection of McMullen's non-Shimura geodesic curve with the Weil locus in the Hilbert modular sixfold for $$L = \mathbb{Q}(\cos\pi/21) = \mathbb{Q}(\zeta_{42})^+$$39 — and establishes three results about them: the intersection is finite with every point CM by a degree-12 field (proved twice, independently); the associated Hodge–Weil classes are absolute Hodge yet obstructed from all known algebraicity criteria by three independent mechanisms; and the existence question is equivalent to a finite, explicit computation of 2816 algebraic systems for the prime 43. What the work provides is a precise, computable location for a potential new case of the Hodge conjecture for abelian sixfolds, together with a clear account of why current methods do not reach it.