# On binary forms

After months of silence, I am finally able to share the research I’ve been doing in the last few months. I’ve dropped hints before in this post and this other one. These are all part of a big project I’ve been working on jointly with my advisor on the representation of integers by binary forms. More recently, I have been working on a project with Cindy Tsang on counting binary quartic forms with small Galois groups. These are all connected by an insight into binary forms essentially due to Hooley.

Let $F$ be a binary form of degree $d$, integer coefficients, and non-zero discriminant $\Delta(F)$. Put $R_F(Z)$ for the number of integers $n$ in the interval $[-Z,Z]$ for which the equation $F(x,y) = n$ has a solution in integers $x,y$. Put $N_F(Z) = \# \{(x,y) \in \mathbb{Z}^2 \text { s.t. } |F(x,y)| \leq Z\}$. When $d = 2$ and $F$ is positive definite, Gauss proved that $N_F(Z) \sim A_1 Z$. He conjectured, and then Landau proved, that $R_F(Z) \sim A_2 Z (\log Z)^{-1}$ in this case. Thus most integers cannot be represented by $F$, and for each integer that can be represented, there are many representations on average.

However, this is very atypical behaviour. Indeed, quadratic forms are complete norm forms of degree 2. For incomplete norm forms, i.e., binary forms of degree at least 3, one should expect a totally different behaviour in that $N_F(Z)$ and $R_F(Z)$ are not that different. This was confirmed by Hooley in a significant paper in 1967 [Hoo1]. Indeed, he showed that when $F$ is an irreducible binary cubic form such that $\Delta(F)$ is not a perfect integer square, then $N_F(Z) \sim R_F(Z)$. We shall refer to this as the easy cubic case’. In 1986 [Hoo2], he went on to obtain the asymptotic formula for $R_F(Z)$ for binary quartic forms of the shape $F(x,y) = Ax^4 + Bx^2 y^2 + Cy^4$. In this case, one has

$R_F(Z) \sim \frac{1}{4} N_F(Z)$

when $A/C$ is not a perfect 4-th power of a rational number, and

$\displaystyle R_F(Z) \sim \frac{1}{4} \left(1 - \frac{1}{2|AC|}\right) N_F(Z)$

otherwise. We refer to this as the easy quartic case’. Finally, he went on to deal with the case when $F$ is a binary cubic form such that $\Delta(F)$ is a square in $\mathbb{Z}$. In this case, he showed that there is a positive integer $m$ which can be determined explicitly in terms of the coefficients of $F$ such that

$\displaystyle R_F(Z) \sim \left(1 - \frac{2}{3m} \right) N_F(Z).$

We will refer to this as the `hard cubic case’.

There is a general theory which applies to all binary forms (of any degree at least three) which allows one recover both the $|AC|$ term in the easy quartic case and the $m$ in the hard cubic case. This will be fully explained in an joint paper by my advisor Professor Cameron Stewart and I. However, in the cubic and quartic cases specifically, one can resolve the problem more finely, in that not only can one describe the relationship between $R_F(Z)$ and $N_F(Z)$, but show that there is a positive rational number $W_F$ which can be given explicitly in terms of the coefficients of $F$ when $F$ is a binary cubic or quartic form. My contribution to the cubic case is that we can find this rational constant even when $F$ is not assumed to be irreducible. Indeed, we find that the most interesting behaviour actually occurs when $F$ is completely reducible (but still with non-zero discriminant)! The reducibility of $F$ seems to have no effect in the case of quartic forms.

The key observation is that the behaviour of the constant $W_F$ is determined completely by the automorphism group of the binary form $F$ in $\text{GL}_2(\mathbb{Q})$. This appears to be an extraordinary insight made by Hooley in his investigation of the hard cubic case. It appears that almost all authors either explicitly or implicitly assumed that it suffices to look only at the smaller group $\text{GL}_2(\mathbb{Z})$.

More details will be posted later.