Bhargava cube

An expression of the form ${\displaystyle Q(x,y)=ax^{2}+bxy+cy^{2}}$, where a, b and c are fixed integers and x and y are variable integers, is called an integer binary quadratic form. The discriminant of the form is defined as

${\displaystyle D=b^{2}-4ac.}$

The form is said to be primitive if the coefficients a, b, c are relatively prime. Two forms

${\displaystyle Q(x,y)=ax^{2}+bxy+cy^{2},\quad Q^{\prime }(x,y)=a^{\prime }x^{2}+b^{\prime }xy+c^{\prime }y^{2}}$

are said to be equivalent if there exists a transformation

${\displaystyle x\mapsto \alpha x+\beta y,\quad y\mapsto \gamma x+\delta y}$

with integer coefficients satisfying ${\displaystyle \alpha \delta -\beta \gamma =1}$ which transforms ${\displaystyle Q(x,y)}$ to ${\displaystyle Q^{\prime }(x,y)}$. This relation is indeed an equivalence relation in the set of integer binary quadratic forms and it preserves discriminants and primitivity.

Let ${\displaystyle Q(x,y)}$ and ${\displaystyle Q^{\prime }(x,y)}$ be two primitive binary quadratic forms having the same discriminant and let the corresponding equivalence classes of forms be ${\displaystyle [Q(x,y)]}$ and ${\displaystyle [Q^{\prime }(x,y)]}$. One can find integers ${\displaystyle p,q,r,s,p^{\prime },q^{\prime },r^{\prime },s^{\prime },a^{\prime \prime },b^{\prime \prime },c^{\prime \prime }}$ such that

${\displaystyle X=px_{1}x_{2}+qx_{1}y_{2}+ry_{1}x_{2}+sy_{1}y_{2}}$

${\displaystyle Y=p^{\prime }x_{1}x_{2}+q^{\prime }x_{1}y_{2}+r^{\prime }y_{1}x_{2}+s^{\prime }y_{1}y_{2}}$

${\displaystyle Q^{\prime \prime }(x,y)=a^{\prime \prime }x^{2}+b^{\prime \prime }xy+c^{\prime \prime }y^{2}}$

${\displaystyle Q^{\prime \prime }(X,Y)=Q(x_{1},y_{1})Q^{\prime }(x_{2},y_{2})}$

The class ${\displaystyle [Q^{\prime \prime }(x,y)]}$ is uniquely determined by the classes [Q(x, y)] and [Q^′(x, y)] and is called the composite of the classes ${\displaystyle [Q(x,y)]}$ and ${\displaystyle [Q^{\prime }(x,y)]}$.[3] This is indicated by writing

${\displaystyle [Q^{\prime \prime }(x,y)]=[Q(x,y)]\ast [Q^{\prime }(x,y)]}$

The set of equivalence classes of primitive binary quadratic forms having a given discriminant D is a group under the composition law described above. The identity element of the group is the class determined by the following form:

${\displaystyle Q_{Id}^{(D)}(x,y)={\begin{cases}x^{2}-{\frac {D}{4}}y^{2}&D\equiv 0{\pmod {4}}\\x^{2}+xy+{\frac {1-D}{4}}y^{2}&D\equiv 1{\pmod {4}}\end{cases}}}$

The inverse of the class ${\displaystyle [ax^{2}+hxy+by^{2}]}$ is the class ${\displaystyle [ax^{2}-hxy+by^{2}]}$.

Let (M, N) be the pair of 2 × 2 matrices associated with a pair of opposite sides of a Bhargava cube; the matrices are formed in such a way that their rows and columns correspond to the edges of the corresponding faces. The integer binary quadratic form associated with this pair of faces is defined as

${\displaystyle Q=-\det(Mx+Ny)}$

The quadratic form is also defined as

${\displaystyle Q=-\det(Mx-Ny)}$

However, the former definition will be assumed in the sequel.

The three forms

Let the cube be formed by the integers a, b, c, d, e, f, g, h. The pairs of matrices associated with opposite edges are denoted by (M₁, N₁), (M₂, N₂), and (M₃, N₃). The first rows of M₁, M₂ and M₃ are respectively [a b], [a c] and [a e]. The opposite edges in the same face are the second rows. The corresponding edges in the opposite faces form the rows of the matrices N₁, N₂, N₃ (see figure).

Bhargava cube showing the pair of opposite faces M1 and N1.

Bhargava cube showing the pair of opposite faces M2 and N2.

Bhargava cube showing the pair of opposite faces M3 and N3.

The quadratic form associated with the faces defined by the matrices ${\displaystyle M_{1}={\begin{bmatrix}a&b\\c&d\end{bmatrix}},N_{1}={\begin{bmatrix}e&f\\g&h\end{bmatrix}}}$ (see figure) is

${\displaystyle Q_{1}=-\det(M_{1}x+N_{1}y)=-(\det(M_{1})x^{2}+(ah+ed-bg-fc)xy+\det(N_{1})y^{2})}$

The discriminant of a quadratic form Q₁ is

${\displaystyle D_{1}=(ah+ed-bg-fc)^{2}-4\det(M_{1})\det(N_{1}).}$

The quadratic form associated with the faces defined by the matrices ${\displaystyle M_{2}={\begin{bmatrix}a&c\\e&g\end{bmatrix}},N_{2}={\begin{bmatrix}b&d\\f&h\end{bmatrix}}}$ (see figure) is

${\displaystyle Q_{2}=-\det(M_{2}x+N_{2}y)=-(\det(M_{2})x^{2}+(ah+bg-fc-ed)xy+\det(N_{2})y^{2})}$

The discriminant of a quadratic form Q₂ is

${\displaystyle D_{2}=(ah+bg-fc-ed)^{2}-4\det(M_{2})\det(N_{2}).}$

The quadratic form associated with the faces defined by the matrices ${\displaystyle M_{3}={\begin{bmatrix}a&e\\b&f\end{bmatrix}},N_{3}={\begin{bmatrix}c&g\\d&h\end{bmatrix}}}$ (see figure) is

${\displaystyle Q_{3}=-\det(M_{3}x+N_{3}y)=-(\det(M_{3})x^{2}+(ah+fc-ed-bg)xy+\det(N_{3})y^{2})}$

The discriminant of a quadratic form Q₃ is

${\displaystyle D_{3}=(ah+fc-ed-bg)^{2}-4\det(M_{3})\det(N_{3}).}$

Manjul Bhargava's surprising discovery can be summarised thus:[2]

If a cube A gives rise to three primitive binary quadratic forms Q₁, Q₂, Q₃, then Q₁, Q₂, Q₃ have the same discriminant, and the product of these three forms is the identity in the group defined by Gauss composition. Conversely, if Q₁, Q₂, Q₃ are any three primitive binary quadratic forms of the same discriminant whose product is the identity under Gauss composition, then there exists a cube A yielding Q₁, Q₂, Q₃.

An example of Bhargava cube

The three quadratic forms associated with the numerical Bhargava cube shown in the figure are computed as follows.

${\displaystyle {\begin{aligned}Q_{1}(x,y)=-\det(M_{1}x+N_{1}y)&=-\det \left({\begin{bmatrix}1&0\\0&-2\end{bmatrix}}x+{\begin{bmatrix}0&3\\4&5\end{bmatrix}}y\right)\\&=-{\begin{vmatrix}x&3y\\4y&-2x+5y\end{vmatrix}}=2x^{2}-5xy+12y^{2}\\\\Q_{2}(x,y)=-\det(M_{2}x+N_{2}y)&=-\det \left({\begin{bmatrix}1&0\\0&4\end{bmatrix}}x+{\begin{bmatrix}0&3\\-2&5\end{bmatrix}}y\right)\\&=-{\begin{vmatrix}x&3y\\-2y&4x+5y\end{vmatrix}}=-4x^{2}-5xy-6y^{2}\\\\Q_{3}(x,y)=-\det(M_{3}x+N_{3}y)&=-\det \left({\begin{bmatrix}1&0\\0&3\end{bmatrix}}x+{\begin{bmatrix}0&4\\-2&5\end{bmatrix}}y\right)\\&=-{\begin{vmatrix}x&4y\\-2y&3x+5y\end{vmatrix}}=-3x^{2}-5xy-8y^{2}\\{}\end{aligned}}}$

The composition ${\displaystyle [Q_{1}(x,y)]\ast [Q_{2}(x,y)]}$ is the form ${\displaystyle [Q(x,y)]}$ where ${\displaystyle Q(x,y)=-3x^{2}+5xy-8y^{2}}$ because of the following:

${\displaystyle X=-2x_{1}x_{2}+4y_{1}x_{2}+y_{1}y_{2}}$

${\displaystyle Y=x_{1}x_{2}+3y_{1}y_{2}}$

${\displaystyle Q(X,Y)=Q_{1}(x_{1},y_{1})Q_{2}(x_{2},y_{2})}$

Also ${\displaystyle [Q_{3}(x,y)]^{-1}=Q(x,y)}$. Thus ${\displaystyle [Q_{1}(x,y)]\ast [Q_{2}(x,y)]\ast [Q_{3}(x,y)]}$ is the identity element in the group defined by the Gauss composition.