Equivalence

Definition 2.1   The modular group $\SL_2(\mathbb{Z})$ is the group of all $2\times 2$ integer matrices with determinant $+1$.

If $g=\left( \begin{smallmatrix}p&q\\ r&s\end{smallmatrix}\right)\in\SL_2(\mathbb{Z})$ and $f(x,y) = ax^2 + bxy + cy^2$ is a quadratic form, let

$$f\vert _g (x,y)= f(px+qy, rx+sy) = f\left(\left( \begin{matrix}p&q\\ r&s \end{matrix}\right)\vtwo{x}{y}\right),$$

where for simplicity we will sometimes write $f\left(\vtwo{x}{y}\right)$ for $f(x,y)$.

Proposition 2.2   The above formula defines a right action of the group $\SL_2(\mathbb{Z})$ on the set of binary quadratic forms, in the sense that

$$f\vert _{gh} = (f\vert _g)\vert _h.$$

Proof.

$$f\vert _{gh}(x,y) = f\left(gh\vtwo{x}{y}\right) = f\vert _g\left(h\left(\vtwo{x}{y}\right)\right) = (f\vert _g)\vert _h(x,y).$$

$\qedsymbol$

Proposition 2.3   Let $g\in\SL_2(\mathbb{Z})$ and let $f(x,y)$ be a binary quadratic form. The set of integers represented by $f(x,y)$ is exactly the same as the set of integers represented by $f\vert _g(x,y)$.

Proof. If $f(x_0, y_0)=n$ then since $g^{-1}\in\SL_2(\mathbb{Z})$, we have $g^{-1}\vtwo{x_0}{y_0}\in\mathbb{Z}^2$, so

$$f\vert _g\left(g^{-1}\vtwo{x_0}{y_0}\right) = f (x_0, y_0) = n.$$

Thus every integer represented by $f$ is also represented by $f\vert _g$. Conversely, if $f\vert _g(x_0,y_0)=n$, then $f\left(g\vtwo{x_0}{y_0}\right) = n$, so $f$ represents $n$. $\qedsymbol$

Define an equivalence relation $\sim$ on the set of all binary quadratic forms by declaring that $f$ is equivalent to $f'$ if there exists $g\in\SL_2(\mathbb{Z})$ such that $f\vert _g = f'$.

For simplicity, we will sometimes denote the quadratic form $ax^2 + bxy+cy^2$ by $(a,b,c)$. Then, for example, since $g=\left( \begin{smallmatrix}0&-1\\ 1&\hfill 0\end{smallmatrix}\right)\in\SL_2(\mathbb{Z})$, we see that $(a,b,c)\sim (c,-b,a)$, since if $f(x,y) = ax^2 + bxy + cy^2$, then $f(-y,x) = ay^2 - bxy + cx^2$.

Example 2.4   Consider the binary quadratic form

$$f(x,y) = 458x^2 + 214xy + 25y^2.$$

Solving the representation problem for $f$ might, at first glance, look hopeless. We find $f(x,y)$ for a few values of $x$ and $y$:

$$f(-1,-1) =17\cdot 41$$

$$f(-1,0) =2\cdot 229$$

$$f(0,-1) =5^2$$

$$f(1,1) =269$$

$$f(-1,2) =2\cdot 5\cdot 13$$

$$f(-1,3) =41$$

Each number is a sum of two squares! Letting $g=\left( \begin{smallmatrix}\hfill4&-3\\ -17&13\end{smallmatrix}\right)$, we have

$$f\vert _g = 458(4x-3y)^2 + 214(4x-3y)(-17x+13y) + 25(-17x+13y)^2 = \cdots = x^2 + y^2!!$$

By Proposition 2.3, $f$ represents an integer $n$ if and only if $n$ is a sum of two squares.