CM Theory for Cubic Fourfolds

• The paper develops CM theory for cubic fourfolds by constructing moduli stacks with level structures and algebraic period maps to orthogonal Shimura varieties.
• It demonstrates that CM cubic fourfolds, akin to singular K3 surfaces, are defined over abelian extensions of their reflex fields, showcasing deep arithmetic properties.
• Rank-21 cubic fourfolds yield two-dimensional Galois representations and CM newforms, linking Hodge theory to modularity through explicit Shimura–CM methods.

Complex multiplication (CM) theory for cubic fourfolds extends the arithmetic and Hodge-theoretic paper of these four-dimensional smooth cubic hypersurfaces in $\mathbb{P}^5$, drawing a deep analogy with the theory for K3 surfaces. Key ingredients include moduli stacks of cubic fourfolds with level structure, the construction of period maps to orthogonal Shimura varieties, and the demonstration that CM cubic fourfolds are defined over abelian extensions of their reflex fields. Special attention is given to the arithmetic and modular properties of “rank-21” cubic fourfolds, whose transcendental cohomology corresponds to that of singular K3 surfaces.

1. Moduli of Cubic Fourfolds with Level Structure

The foundational moduli problem considers the stack $\mathcal{C}$ over $\operatorname{Spec}\mathbb{Z}$ whose $S$-points classify families $X\to S$ of smooth cubic hypersurfaces in $\mathbb{P}^5$; this stack is smooth, separated, Deligne–Mumford, and of finite type over $\mathbb{Z}$. Via Benoist’s results on Hilbert schemes, one has

$\mathcal{C} \simeq [\mathrm{PGL}_6 \setminus H],$

where $H$ is the open subset of the Hilbert scheme parameterizing smooth cubics. The cohomology lattice $H^4(X,\mathbb{Z}(2))$ has intersection form of signature $(21,2)$ and is isometric to $$L_0 := \langle +1\rangle^2 \oplus \langle -1\rangle^{21}$$. Fixing a primitive vector $v$ with $(v,v)=-3$ and setting $L := v^\perp \subset L_0$ yields a lattice of rank $23$, signature $(2,20)$.

For an integer $N\geq 1$, a level-$N$ structure on a family $(X\to S,\lambda)$ over $S\in \mathrm{Sch}/\mathbb{Z}[1/N]$ is an isometry

$$\alpha : R^4_{et}\pi_*\mathbb{Z}/N(2) \to (L_0\otimes \mathbb{Z}/N)_S,$$

with $\alpha(h^2)=v$. One obtains the Deligne–Mumford stack $\mathcal{C}^{[N]}$ parameterizing cubic fourfolds with level-$N$ structure and its open substack $\widetilde{\mathcal{C}^{[N]}}$ for those structures compatible with primitive cohomology:

$$K_N := \{g\in\mathrm{SO}(L)(\widehat{\mathbb{Z}})\mid g\equiv 1 \bmod N,\, g(v)=v\},$$

an open-compact subgroup fixing $v$.

Theorem (Ito): For each $N\geq 1$, $\widetilde{\mathcal{C}^{[N]}}$ is a Deligne–Mumford stack, locally of finite type over $\mathbb{Z}[1/N]$. If $N \geq 3$ is coprime to $2310=2 \cdot 3 \cdot 5 \cdot 7 \cdot 11$, then $\widetilde{\mathcal{C}^{[N]}}$ is a smooth affine scheme over $\mathbb{Z}[1/N]$ (Ito, 12 Dec 2025).

2. The Arithmetic Period Map and Shimura Variety Structure

Given $V = L\otimes \mathbb{Q}$ and $G = \mathrm{SO}(V)$, define the period domain

$$X_V = \{ h: \mathbb{S} \to G_{\mathbb{R}}\mid h \text{ defines a } \mathbb{Q}\text{-Hodge structure of K3 type} \}.$$

This can be realized as

$$X_V \simeq \{ [\omega]\in \mathbb{P}(V_\mathbb{C})\mid (\omega,\omega)=0,\, (\omega, \bar{\omega})>0 \}$$

with two components $X_V^\pm$. For any open compact $K\subset G(\mathbb{A}_f)$, the corresponding Shimura variety is

$$\mathrm{Sh}_K(L)(\mathbb{C}) = G(\mathbb{Q}) \backslash ( X_V \times G(\mathbb{A}_f) / K ).$$

For $N \geq 3$ coprime to $2310$, the period map on $\mathbb{C}$-points of $\widetilde{\mathcal{C}^{[N]}}$,

$$j_{N, \mathbb{C}}: \widetilde{\mathcal{C}^{[N]}}_\mathbb{C} \to \mathrm{Sh}_{K_N}(L)_\mathbb{C},$$

is defined using markings $a: H^4(X,\mathbb{Z}(2))\cong L_0$, $a(h^2)=v$ and compatible level structures. The image is independent of these choices up to $G(\mathbb{Q})$-action.

Theorem (Ito): $j_{N,\mathbb{C}}$ is algebraic and an open immersion. If $N\geq 3$ is coprime to $2310$, $j_{N,\mathbb{C}}$ descends to a morphism $$j_{N,\mathbb{Q}}: \widetilde{\mathcal{C}^{[N]}}_\mathbb{Q} \to \mathrm{Sh}_{K_N}(L)_\mathbb{Q}$$ (Ito, 12 Dec 2025).

3. Complex Multiplication for Cubic Fourfolds

A $\mathbb{Q}$-Hodge structure $h:\mathbb{S}\to \mathrm{GL}(V_\mathbb{R})$ of type $\{(-1,1),(0,0),(1,-1)\}$ is of “K3 type” if $\dim V^{1,-1}=1$. For irreducible, polarized K3-type Hodge structures, $\mathrm{End}_{HS}(V)\simeq K$, a totally real or CM field. In the CM case, the Mumford–Tate group is a one-dimensional torus.

Definition: A cubic fourfold $X/\mathbb{C}$ is of CM type if the primitive Hodge structure on $PH^4(X,\mathbb{Q})(2)$ has commutative Mumford–Tate group, equivalently, $\mathrm{End}_{HS}(PH^4(X))$ is a CM field $K$ of degree $2$ over $\mathbb{Q}$.

For CM cubic fourfolds, $(\mathrm{MT}(h_X), \{h_X\})$ forms a Shimura datum. The associated reflex field $E=E(X)$, defined via the cocharacter $\mu_h$, is the subfield over which the action on the unique $(1,-1)$ line is defined, $E = \epsilon(K) \subset \mathbb{C}$.

Corollary (Ito): If $X/\mathbb{C}$ is a CM-type cubic fourfold with reflex field $E$, then $X$ admits a model over an abelian extension of $E$. This is realized via the image of $X$ as a special point of $\mathrm{Sh}_{K_N}(L)$, whose canonical model is defined over $E^{ab}$ (Ito, 12 Dec 2025).

4. Rank-21 Cubic Fourfolds, Singularity, and Density

Given $A(X)$, the sublattice of $H^4(X,\mathbb{Z}(2))$ generated by algebraic classes, $1 \leq \operatorname{rk}A(X)\leq 21$. If $\operatorname{rk}A(X) = 21$, $X$ is called rank-21 or “singular.” In this case, $PH^4(X,\mathbb{Z})(2)$ is rank 2, and the transcendental lattice $T(X)\simeq T(S)$ for a singular (Picard rank 20) K3 surface $S$. Such $X$ are automatically of CM type, with reflex field an imaginary quadratic extension $K$.

Theorem (Ito): For each imaginary quadratic $K \subset \mathbb{C}$, the set of rank-21 cubic fourfolds with reflex field $K$ is Zariski-dense in $\mathcal{C}_\mathbb{C}$. This is shown by constructing corresponding CM points in the Shimura variety and using strong approximation to establish density (Ito, 12 Dec 2025).

5. Modularity of Rank-21 Cubic Fourfolds

Let $X/\mathbb{C}$ be rank-21 and defined over $\mathbb{Q}$. The Galois action on the $\ell$-adic transcendental lattice gives a two-dimensional representation:

$$\rho_\ell: \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \mathrm{GL}_2(\mathbb{Z}_\ell)$$

which, restricted to $\mathrm{Gal}(\overline{\mathbb{Q}}/K)$, factors through the torus $$\mathrm{MT}(T(X))(\mathbb{A}_f)\simeq \{z\in K^\times \mid z\bar z=1\}$$ via class field theory. The representation is thus induced by a Hecke character $\psi$ of $K$ of infinity type $(2,0)$. The corresponding $q$-expansion

$$g(q)=\sum_{\mathfrak{a}\subset\mathcal{O}_K,\, (\mathfrak{a},\mathfrak{f})=1} \psi(\mathfrak{a}) q^{N(\mathfrak{a})}$$

defines a weight-3 cusp form, and there exists a unique newform $f$ giving the same traces as $\rho_\ell$, so that

$L(\rho_\ell,s) = L(f,s-1).$

Theorem (Ito): Any rank-21 cubic fourfold $X/\mathbb{Q}$ has

$L(H^4(X)_{tr},s) = L(f,s-1),$

with $f$ a CM newform of weight 3 with rational integer Fourier coefficients, matching Livné’s modularity by purely Shimura–CM methods (Ito, 12 Dec 2025).

6. Significance and Future Directions

The construction of arithmetic period maps, explicit moduli with level structures, and precise statements of CM and modularity phenomena for cubic fourfolds parallels the well-developed theory for K3 surfaces, but is new in higher dimension. The demonstration that every CM-type cubic fourfold arises over abelian extensions of its reflex field and the modularity of rank-21 cases provide alternative, Hodge-theoretic arguments that do not rely on $p$-adic Hodge theory. This suggests further lines of inquiry regarding the arithmetic of irrationality loci, the construction of motives with prescribed Galois representations, and possible generalizations to other classes of higher-dimensional varieties with rich Hodge-theoretic and automorphic properties.