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

This paper establishes that the intersection between McMullen's curve and the Weil locus in a Hilbert modular sixfold consists solely of CM points, thereby isolating specific algebraic conditions and explicit equations that represent a critical step toward proving a new case of the Hodge conjecture for abelian sixfolds.

Original authors: Amir Mostaed

Published 2026-03-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Amir Mostaed

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an explorer trying to find a hidden treasure on a vast, complex map. This paper is a detailed expedition report about a specific, incredibly difficult search for a mathematical "treasure" called the Hodge Conjecture.

Here is the story of the search, broken down into simple concepts and analogies.

1. The Map and the Treasure

The Map (The Hilbert Modular Sixfold): Imagine a giant, 6-dimensional landscape. In this landscape, every point represents a complex geometric shape called an "Abelian sixfold" (think of it as a highly twisted, multi-dimensional donut).
The Treasure (The Hodge Conjecture): Mathematicians believe that certain patterns found inside these shapes (called "Hodge classes") must be made of actual, physical building blocks (algebraic cycles). Proving this for all shapes is the "Hodge Conjecture." It's like proving that every shadow cast by a complex object is actually a solid object you can touch.
The Problem: We know this is true for simple shapes, but for these complex 6-dimensional donuts, it's a mystery. We have a hunch that some specific "shadows" exist, but we can't prove they are solid objects.

2. The Two Paths (The Curve and the Locus)

The author, Amir Mostaed, sets up a search by defining two specific paths on this map where the treasure might be hiding:

Path A: McMullen's Curve (The Rigid Road): This is a very specific, narrow, winding road discovered by mathematician Curtis McMullen. It's special because it's "rigid"—it doesn't wiggle or bend easily. It's like a train track laid down on a mountain; it goes exactly where it goes, and nothing else can be on it.
Path B: The Weil Locus (The Special Zone): This is a large, 3-dimensional region on the map where shapes have a special property called "Weil type." Think of this as a "Gold Zone" where the shapes have extra symmetry.

3. The Impossible Intersection

Here is the crazy part:

Path A is 1-dimensional (a line).
Path B is 3-dimensional (a volume).
The whole map is 6-dimensional.

If you drop a 1D line and a 3D volume into a 6D room, they should never touch. Mathematically, the "expected" place for them to meet is empty space. It's like trying to find a single grain of sand that is simultaneously on a specific thread of a spiderweb and inside a specific bubble floating in a different room.

The Discovery: The paper argues that if they do touch, it's a "super-atypical" event. It's a miracle of arithmetic. If they touch, it's not by accident; it's because the universe forced them to meet.

4. The Search Strategy: The "Hecke" Net

Since we can't just look for the intersection with our eyes, the author builds a "net" to catch it.

The Net (Hecke Correspondences): Imagine throwing a giant net over the map. The net is made of mathematical rules (equations) that only catch points with very specific properties.
The Target (CM Points): The net is designed to only catch "CM points." These are points where the shape has a very special, rigid internal structure (like a crystal).
The Calculation: The author calculates that if the treasure exists, it must be one of a finite number of specific points. He writes down 2,816 specific equations.
- Analogy: It's like saying, "If the treasure exists, it must be in one of these 2,816 locked boxes."
- The paper provides the keys (the equations) to open them.

5. Why This is a Big Deal (The Obstacles)

The paper explains why previous attempts to find this treasure failed. It's like trying to open a safe with three different locks, and we only have keys for two of them:

Isolation: The treasure is hidden in a "dead end." You can't wiggle your way to it by moving slightly; it's completely isolated. Old methods required you to wiggle around to prove the treasure was real.
Wrong Geometry: The treasure isn't built using the standard "bricks" (geometric constructions) that mathematicians usually use to prove things.
The "Discriminant" Lock: The shape of the treasure is slightly different (mathematically speaking) from all the other treasures we've proven before. The old keys don't fit this specific lock.

Because of these three locks, no one currently knows how to prove the treasure is real.

6. The Conclusion: A Finite Quest

So, what does the paper actually achieve?

It doesn't solve the Hodge Conjecture yet. Instead, it turns a vague, impossible dream into a finite, checkable to-do list.

The Good News: We know the treasure is either in one of those 2,816 boxes, or it doesn't exist at all. We don't have to search the whole universe anymore.
The Challenge: We need a computer (or a very brave human) to run the equations for the number 43 (the smallest number that fits the rules) and see if any of the 2,816 boxes open.
The Final Hurdle: Even if we find the box and open it, we still need a new kind of key (a new mathematical theory) to prove that what's inside is actually the "treasure" (algebraic) and not just a ghost (transcendental).

Summary

This paper is a roadmap for a future discovery.

It identifies a specific, unlikely place where a mathematical miracle might happen.
It proves that if the miracle happens, it's a finite, countable event.
It writes down the exact equations needed to check for the miracle.
It admits that we don't yet have the final tool to confirm the miracle, but it has cleared the path so that someone else can finish the job.

It's the difference between saying, "Maybe there's a dragon in the cave," and saying, "There are exactly 2,816 caves to check. Here are the maps. If you check them all and find nothing, there is no dragon. If you find one, here is the specific problem you need to solve to prove it's a dragon."

1. Problem Statement

The paper addresses a specific, high-profile open case of the Hodge Conjecture: the algebraicity of exceptional Hodge classes on abelian sixfolds of Weil type.

Context: The Hodge Conjecture posits that rational cohomology classes of type $(p,p)$ on smooth complex projective varieties are algebraic (generated by algebraic cycles). For abelian varieties, Deligne proved these classes are absolute Hodge, but their algebraicity remains open in many cases.
The Specific Case: The focus is on abelian sixfolds ( $g=3$ ) of Weil type over an imaginary quadratic field $K = \mathbb{Q}(\sqrt{-d})$ . These varieties possess a distinguished 2-dimensional subspace of Hodge classes in $H^{3,3}$ (the "Weil classes") that are exceptional (not generated by divisor classes).
The Gap: While algebraicity has been proven for fourfolds ( $g=2$ ) and specific cases of sixfolds (e.g., trivial discriminant), the general case for sixfolds remains completely open. Existing methods (deformation theory, secant geometry, discriminant-specific criteria) fail for the specific geometric family studied here.

2. Methodology

The author constructs a rigid arithmetic intersection between two geometric objects within a Hilbert modular sixfold $X_L$ (where $L = \mathbb{Q}(\cos \frac{\pi}{21})$ is a totally real field of degree 6).

A. The Geometric Setup

McMullen's Curve ( $V$ ):
- Derived from the hyperbolic triangle group $\Delta(14, 21, 42)$ .
- This group is non-arithmetic but belongs to McMullen's "Hilbert series," meaning its invariant quaternion algebra splits at all infinite places of its trace field $L$ .
- This splitting allows an embedding of the group into $SL_2(\mathcal{O}_L)$ , yielding a compact Kobayashi-geodesic curve $V \subset X_L$ .
- Crucially, $V$ is not contained in any proper Shimura subvariety of $X_L$ . Its generic fiber has the maximal Mumford-Tate group allowed by the real multiplication ( $Res_{L/\mathbb{Q}} SL_2$ ).
The Weil Locus ( $W_K$ ):
- The locus of points in $X_L$ corresponding to abelian sixfolds of Weil type for a fixed imaginary quadratic field $K$ .
- $W_K$ is a countable union of closed algebraic subvarieties.
- It has codimension 3 in $X_L$ (dimension 3) and consists of 20 irreducible components indexed by sign assignments of the 6 real embeddings of $L$ .

B. The Intersection Problem

The paper investigates the intersection $V \cap W_K$ .

Dimensional Analysis: $\dim(V) = 1$ , $\dim(W_K) = 3$ , $\dim(X_L) = 6$ .
Expected Dimension: $1 + 3 - 6 = -2$ .
Significance: An intersection with expected dimension $-2$ is super-atypical. In the sense of the Zilber-Pink conjecture, such an intersection should be empty or extremely sparse (finite). The paper proves it is indeed finite and characterizes its points.

3. Key Contributions and Results

A. Finiteness and CM Structure (Theorems E, 3.4)

The paper proves that the intersection $V \cap W_K$ is finite (possibly empty).

Mechanism: Every point $v_0 \in V \cap W_K$ forces the endomorphism algebra of the fiber $A_{v_0}$ to contain the compositum $M = KL$. Since $[M:\mathbb{Q}] = 12 = 2 \dim(A_{v_0})$ , the fiber must be a CM abelian variety (Complex Multiplication).
Proof Strategy: Two independent proofs are provided:
1. André-Oort Theorem: If the intersection were infinite, $V$ would contain a Zariski-dense set of CM points, implying $V$ is a Shimura subvariety. This contradicts Theorem B (McMullen's result that $V$ is not Shimura).
2. Transcendence Theory: Using Wolfart's theorem and the Ax-Schanuel theorem, showing that non-CM points on $V$ cannot satisfy the algebraic conditions of the Weil locus.
Result: The intersection consists of finitely many CM points, each with CM field $M = L(\sqrt{-d})$ (degree 12).

B. Inaccessibility by Known Methods (Theorem G)

The paper demonstrates that the Hodge classes at these points are absolute Hodge but resist all known algebraicity proofs:

CM Isolation: The points are isolated in the moduli space; they do not belong to a positive-dimensional family, rendering deformation arguments (like those of Buchweitz-Flenner) inapplicable.
Absence of Secant Geometry: The varieties do not arise from the specific $X \times \hat{X}$ constructions (K-secant lines) used by Markman to prove algebraicity for other cases.
Uncontrolled Discriminant: The discriminant of the Hermitian form associated with the Weil structure is not fixed to $\pm 1$ or trivial, preventing the application of discriminant-specific theorems (Schoen, Markman).

C. Reduction to Finite Computation (Theorems J, 5.3)

Despite the theoretical obstacles, the non-emptiness of the intersection is decidable via a finite algorithm.

Hecke Program: The author reduces the search for points in $V \cap W_K$ to solving fixed-point equations for Hecke correspondences $T_\ell$ on $X_L$ .
Specific Candidate: The prime $\ell = 43$ (the smallest prime $\equiv 1 \pmod{42}$ ) is identified as the optimal candidate.
The System: The problem reduces to solving 2,816 explicit algebraic equations in the upper half-plane $\mathbb{H}$ $H$ .
- $44$ coset representatives ( $\ell+1$ ).
- $64$ sign choices for the roots of the quadratic equations.
Pre-screening: The hypergeometric structure of the modular embedding allows these equations to be pre-screened modulo 43, drastically reducing the computational load before high-precision numerical work.

D. Explicit Bounds (Corollary F, 3.6)

The paper provides an explicit Faltings height bound for any potential solution:
$h_F(A_{v_0}) \leq 18900 \cdot 42^{30}$
This confirms that the search space is finite and effectively bounded, transforming the geometric existence question into a massive but finite computational task.

4. Significance

New Frontier for the Hodge Conjecture: This work isolates a specific, explicitly defined family of abelian sixfolds where the Hodge conjecture is currently out of reach for all known methods. Proving the algebraicity of the Weil classes here would constitute a genuinely new case of the conjecture, requiring new Hodge-theoretic machinery.
Bridging Geometry and Arithmetic: It connects the theory of non-arithmetic triangle groups (McMullen's work) with the arithmetic of Shimura varieties and the André-Oort conjecture, showing how "super-atypical" intersections can be rigorously analyzed.
Decidability: By reducing the existence question to a finite system of 2,816 equations, the paper moves the problem from the realm of pure existence to computational verification. It provides a clear roadmap (Open Problems O1–O3) for future researchers:
- O1: Execute the Hecke search for $\ell=43$ .
- O2: Verify the Weil signature (CM type) of any found solution.
- O3: Prove the algebraicity of the resulting classes (the ultimate Hodge conjecture challenge).

In summary, the paper does not solve the Hodge conjecture but precisely locates a potential counter-example or a new proof case, proving that the search space is finite and providing the explicit equations necessary to find the solutions. It establishes that if the Hodge conjecture fails or requires a new method for abelian sixfolds, it will likely happen at these specific points.