Bhargava Cubes: Theory, Invariants, and Composition

• Bhargava cubes are 2×2×2 arrays of integers that connect classical Gauss composition with modern arithmetic invariant theory.
• Their relative invariants, including the Cayley hyperdeterminant, classify orbits and encapsulate key arithmetic data like discriminants.
• The group actions on these cubes illuminate the structure of Shintani zeta functions and Weyl group multiple Dirichlet series.

A Bhargava cube is a central object in the arithmetic invariant theory of prehomogeneous vector spaces, specifically the space of $2 \times 2 \times 2$ cubes of integers or elements over a field. Bhargava’s construction connects classical Gauss composition of binary quadratic forms, the structure of orbits under natural group actions, the theory of Shintani zeta functions, and the arithmetic of Weyl group multiple Dirichlet series, notably type $A_3$. The cube formalism extends and unifies earlier arithmetic constructions, providing new moduli interpretations and illuminating the geometry and representation theory underpinning higher composition laws (Wen, 2013, Devalapurkar, 15 Apr 2024, Gan et al., 2013).

1. The Space of Bhargava Cubes and Group Actions

Let $$V = \mathbb{Q}^2 \otimes \mathbb{Q}^2 \otimes \mathbb{Q}^2$$ be the 8-dimensional space of $2 \times 2 \times 2$ "cubes" with rational entries. An element $A \in V$ has coordinates corresponding to the vertices of a physical $2 \times 2 \times 2$ cube. This space receives a natural group action: $$G(\mathbb{Q}) = B_2(\mathbb{Q}) \times B_2(\mathbb{Q}) \times \mathrm{GL}_2(\mathbb{Q})$$, where $B_2(\mathbb{Q})$ is the lower-triangular Borel and $\mathrm{GL}_2(\mathbb{Q})$ acts on the different "faces" of the cube (Wen, 2013).

On the integral lattice $V(\mathbb{Z})$, the corresponding group is $$G(\mathbb{Z}) = B_2(\mathbb{Z}) \times B_2(\mathbb{Z}) \times \mathrm{SL}_2(\mathbb{Z})$$. The action on cubes can also be realized as an action on three pairs of $2 \times 2$ matrices extracted by slicing the cube in each dimension.

In the representation-theoretic context, Bhargava cubes arise as the quotient $N/[N,N]$ for the maximal parabolic $P = MN$ in the split, simply connected Chevalley group of type $D_4$, so $V \simeq V_2 \otimes V_2 \otimes V_2$ (with $V_2$ the standard two-dimensional representation of $\mathrm{SL}_2$) (Gan et al., 2013, Devalapurkar, 15 Apr 2024).

2. Relative Invariants, Slices, and the Cayley Hyperdeterminant

Bhargava identified three fundamental relative invariants for the group action on cubes:

• $m(A) = -\det M^1(A)$,
• $n(A) = -\det M^2(A)$,
• $\Delta(A) = \operatorname{disc}(A)$, a degree–2 polynomial in the cube entries given by $(bg+cf-de)^2 - 4(ad-bc)(eh-fg)$ for an appropriately labeled cube (Wen, 2013).

Each $M^i, N^i$ is a $2 \times 2$ matrix determined by slicing the cube in a coordinate direction. These invariants are algebraically independent and generate the ring of $G$-relative invariants.

The discriminant $\Delta(A)$ coincides (up to scaling) with the Cayley hyperdeterminant of the cube. The Cayley hyperdeterminant is a quartic form in the eight entries and serves as the shared discriminant of the three quadratic forms obtained from the cube slices:

$q_i(x, y) = -\det(M_i x + N_i y), \quad i=1,2,3,$

All $q_i$ have discriminant equal to $\Delta(A)$ (or equivalently Det$(A)$ in the invariant-theoretic language) (Devalapurkar, 15 Apr 2024).

3. Classification of Orbits and Arithmetic Correspondence

Two integral cubes $A, A' \in V(\mathbb{Z})$ are in the same $G(\mathbb{Z})$-orbit if and only if they share the same values of $(m, n, \Delta)$, up to certain congruence conditions (e.g., $0 \leq$ "middle coefficients" $< 2|m|$ or $< 2|n|$) (Wen, 2013).

In the setting of orbits with nonvanishing invariants (the "semi-stable" locus $V^{ss}(\mathbb{Z})$), there is a bijection:

$$G(\mathbb{Z}) \backslash V^{ss}(\mathbb{Z}) \longleftrightarrow \{\text{triples } (m, n, D): D \neq 0,\, D \equiv 0,1 \pmod{4},\, \text{congruence data}\}$$

Thus, the orbits of integer cubes can be classified purely in terms of the values of the three invariants.

A deep arithmetic feature is the correspondence between semi-stable orbits and pairs of oriented ideals in a quadratic ring $R = \mathbb{Z}[\tau]$ of discriminant $\Delta(A)$ (Wen, 2013). Explicitly, each cube $A$ determines a pair $(R; I_1, I_2)$ where $I_i$ are ideals associated to slices of the cube, and the map is finite and surjective onto the moduli space of such pairs.

4. Shintani Zeta Functions and the $A_3$ Weyl Group Multiple Dirichlet Series

The Shintani zeta function attached to the prehomogeneous space $(G, V)$ is defined:

$$Z(s_1, s_2, s_3) = \sum_{A \in G(\mathbb{Z}) \backslash V^{ss}(\mathbb{Z})} |m(A)|^{-s_1} |n(A)|^{-s_2} |\Delta(A)|^{-s_3}$$

By summing over the invariants $(m,n,D)$, and interpreting the counting function $B(D;m,n)$ (number of cubes with given invariants), one arrives at a functional equation and structure closely matching the quadratic $A_3$ multiple Dirichlet series (Wen, 2013).

For odd discriminants, a precise comparison yields:

$$Z^{\mathrm{odd}}(s_1, s_2, s_3) = 4 \frac{\zeta(s_1)}{\zeta(2s_1)} \frac{\zeta(s_2)}{\zeta(2s_2)} Z_{A_3}(s_1, s_2, s_3)$$

where $Z_{A_3}$ is the quadratic $A_3$-multiple Dirichlet series. The local factors of these series are realized as rational functions invariant under the Weyl group $W(A_3)$, aligning with the Chinta–Gunnells construction.

This identification interprets the Shintani zeta function as the quadratic metaplectic Whittaker series of type $A_3$ and demonstrates that the combinatorics of integer cubes encode deep properties of automorphic $L$-functions.

5. Geometric Realization: Weyl Group, Derived Satake, and Hyperdeterminant

Bhargava cubes are tightly connected to advanced geometric representation theory. The vector space $V = (\mathbb{A}^2)^{\otimes 3}$ equipped with its $\mathrm{SL}_2^3$-action forms the setting for the moment map:

$$\mu: V \to \mathfrak{sl}_2^* \times \mathfrak{sl}_2^* \times \mathfrak{sl}_2^*,$$

where the moment map outputs the three binary quadratic forms attached to the slices of the cube (Devalapurkar, 15 Apr 2024). Bhargava’s composition law emerges as the categorical pullback along $\mu$ in the derived geometric Satake correspondence for the quotient $PGL_2^3/PGL_2^{\text{diag}}$.

The Cayley hyperdeterminant ($\mathrm{Det}(C)$) serves as the common discriminant of the three forms, and the invariant theory of $V$ under $\mathrm{SL}_2^3$ is generated by this quartic polynomial. Orbits are classified by the value of the discriminant, with a unique open orbit ($\mathrm{Det} \neq 0$).

Explicitly, Bhargava’s bijection shows that the map

$$\mu: V \to \mathfrak{sl}_2^* \times_{\mathfrak{sl}_2^*/SL_2} \mathfrak{sl}_2^*$$

is bijective at the level of orbits, and the third quadratic form produced by slicing is (up to sign) the Gauss-composed form of the first two.

6. Twisted Bhargava Cubes and Generalizations

The "twisted Bhargava cube" construction arises when considering forms over general fields and as orbits under quasi-split forms of $D_4$. For a field $F$ of $\mathrm{char} \neq 2,3$ and an étale cubic $F$-algebra $E$, the space $V_E(F) \cong E \oplus E \oplus E \oplus F$ carries an action of $M_E$, with three "legs" permuted via $S_3$ triality (Gan et al., 2013).

The quartic invariant generalizes to a form $A_E(v)$, quasi-invariant under $M_E$. For generic orbits ($A_E(v) \neq 0$), there is a natural bijection with

• isomorphism classes of $E$-twisted composition algebras of $E$-dimension $2$,
• isomorphism classes of pairs $(J, i)$ with $J$ a 9-dimensional Freudenthal–Jordan $F$-algebra and $i: E \to J$ an $F$-algebra embedding.

The classification of orbits reflects new phenomena not present in the split case, such as the appearance of twisted composition laws and associated moduli spaces involving cubic and quadratic structures.

7. Moduli Interpretations and Arithmetic Applications

Every semi-stable orbit of an integral cube $A$ canonically determines a pair $(R; I_1, I_2)$, where $R$ is an oriented quadratic ring of discriminant $\Delta(A)$, and $I_i$ are oriented ideals corresponding to two of the cube’s faces. For three-way slices (in the integer lattice $V=\mathbb{Z}^8$), the correspondence is with triples of oriented ideals $(I_1, I_2, I_3)$ in an order of discriminant $D$ such that $I_1 I_2 I_3 = \mathcal{O}$ (Gan et al., 2013). The assignment is $G(\mathbb{Z})$-invariant and surjective, leading to a finite-to-one map onto the appropriate moduli space.

In this way, the Bhargava cube framework unifies the structure of orbits, Shintani zeta functions, multiple Dirichlet series, invariant theory, and moduli spaces of ideals in orders, structuring the arithmetic of composition laws and providing a bridge between classical and higher composition phenomena (Wen, 2013, Gan et al., 2013, Devalapurkar, 15 Apr 2024).

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References:

• (Wen, 2013) Bhargava Integer Cubes and Weyl Group Multiple Dirichlet Series
• (Gan et al., 2013) Twisted Bhargava Cubes
• (Devalapurkar, 15 Apr 2024) Derived geometric Satake for $$\mathrm{PGL}_2^{\times 3}/\mathrm{PGL}_2^\mathrm{diag}$$