2.1 An elementary height machine on stacky curves

We define our algebraic stacks in terms of a base variety along with some extra data which are enough to uniquely construct an algebraic stack. A stacky curve defined over a number field K is determined by the following data: A smooth variety X defined over K, and a finite number of stacky points $P_1,\dots ,P_r$ along with integer multiplicities $m_{P_i}=m_i>1$ attached to each point $P_i$ . We use the notation

$$\begin{align*}\mathfrak{X}=(X:(P_1,m_1), \ldots, (P_r,m_r))\end{align*}$$

to denote the stacky curve with multiplicities $m_{P_i}=m_i$ at the points $P_i$ . We will write $\mathfrak {X}(X:({\mathbf {a}}, {\mathbf {m}}))$ as an abbreviation. We identify the rational points of the stack $\mathfrak {X}$ with those of the coarse space, which is just the variety X. That is, we require

(2.1) $$ \begin{align} \mathfrak{X}(K)=X(K).\end{align} $$

In [Reference Ellenberg, Satriano and Zureick-Brown6], some care is taken to work with the locus of stacky points on an algebraic stack. In particular, one must contend with the accumulation of infinitely new stacky points. We ignore such difficulties since they are not important in our context.

To obtain an E-S-ZB height on a stacky curve $\mathfrak {X}(X:({\mathbf {a}},{\mathbf {m}}))$ , we must choose a vector bundle on $\mathfrak {X}(X:({\mathbf {a}},{\mathbf {m}}))$ . We will be primarily interested in the stacky curves $\mathfrak {X}({\mathbb {P}}^1_{\mathbb {Q}}: (a_1,m_1),\dots ,(a_r,m_r))$ and the vector bundle being a line bundle. Unless otherwise mentioned, stacky curve $\mathfrak {X}$ will now be of this form. Associated with each stacky point, $a_i$ is a line bundle ${\mathcal {L}}_{a_i}$ and it suffices to consider line bundles of the form

$$\begin{align*}\mathcal{L}=L\otimes \prod_{i=1}^s{\mathcal{L}}_{a_i}^{\otimes c_i},\end{align*}$$

where L is a divisor on ${\mathbb {P}}^1$ and $0\leq c_i\leq m_i-1$ . To associate a height on such a divisor, we associate a height to each ${\mathcal {L}}_{a_i}^{\otimes c_i}$ and extend linearly. Motivated by [Reference Ellenberg, Satriano and Zureick-Brown6], we develop the following construction. For each stacky point $a_i = [\alpha _i : \beta _i]$ with $\alpha _i,\beta _i$ coprime integers, we associate to it the linear form $\ell _i(x,y) = \alpha _i y- \beta _i x$ . For each $m_i$ , define $\phi _{m_i}(n)$ is defined to be the smallest positive integer such that $n \phi _{m_i}(n)$ is a perfect $m_i$ -th power. The height function associated with ${\mathcal {L}}_{a_i}^{\otimes c_i}$ is then

$$\begin{align*}H_{{\mathcal{L}}_{a_i}^{\otimes c_i}}(x,y)=\phi_{m_i}(\ell_i(x,y)^{c_i})^{\frac{1}{m_i}}.\end{align*}$$

The linear form $\ell _i$ takes into account the point $a_i$ and $\phi _{m_i}$ accounts for the multiplicity of $a_i$ , while the power $c_i$ accounts for the multiple of ${\mathcal {L}}_{a_i}$ . The introduction of the functions $\phi _{m_i}$ is due to [Reference Ellenberg, Satriano and Zureick-Brown6] and working with these functions is a key feature of stacky curves with coarse space ${\mathbb {P}}^1$ . We define a height function for any divisor $\mathcal {L}=L\otimes \prod _{i=1}^s{\mathcal {L}}_{a_i}^{\otimes c_i}$ on $\mathfrak {X}$ as

(2.2) $$ \begin{align} H_{\mathcal{L}}(x,y)=\max\{\vert x\vert,\vert y\vert\}^{\deg {\mathcal{L}}}\cdot \prod_{i=1}^r\phi_{m_i}(\ell(x,y)^{c_i})^{\frac{1}{m_i}} \end{align} $$

whenever $x,y$ are coprime integers. The Euler characteristic of the stacky curve is defined to be the degree of the anti-canonical divisor. If we wish to emphasize in our situation that $\chi (\mathfrak {X})$ only depends on the vector of multiplicities ${\mathbf {m}}$ and that the anti-canonical height only depends on $({\mathbf {a}},{\mathbf {m}})$ , we write

(2.3) $$ \begin{align} 2 - \sum_{i=1}^n \left(1 - \frac{1}{m_i}\right)=\chi(\mathfrak{X}({\mathbb{P}}^1_{\mathbb{Q}},(a_1,m_1),\ldots,(a_r,m_r)))=\delta({\mathbf{m}})\end{align} $$

and

(2.4) $$ \begin{align} \max\{\vert x\vert,\vert y\vert \}^{\delta({\mathbf{m}})}\prod_{i=1}^r\phi_{m_i}(\ell_i(x,y))^{\frac{1}{m_i}}=H_{-K_{\mathfrak{X}}}([x:y])=\mathcal{H}_{({\mathbf{a}}, {\mathbf{m}})}(x,y). \end{align} $$

2.2 Properties of the anti-canonical E-S-ZB height $H_{-K_{\mathfrak {X}}}$

The Northcott property of the naive height on ${\mathbb {P}}^1$ implies that the ESZ-B anti-canonical height $H_{-K_{\mathfrak {X}}}$ has the Northcott property whenever $\chi (\mathfrak {X})=\delta ({\mathbf {m}})>0$ . On the other hand, if $\chi (\mathfrak {X})\leq 0$ , then it is not at all obvious whether $H_{-K_{\mathfrak {X}}}$ should have the Northcott property. The following question is fundamental: Let ${\mathcal {L}}$ be a line bundle on $\mathfrak {X}(X: ({\mathbf {a}},{\mathbf {m}}))$ and let $H_{{\mathcal {L}}}$ be the associated ESZ-B height. When does $H_{\mathcal {L}}$ have the Northcott property? We will tackle this question when ${\mathcal {L}}=-K_{\mathfrak {X}}$ and $X={\mathbb {P}}^1$ leaving the general case for future study.

Theorem 2.1 Let $\{a_1, \ldots , a_n\} \subset {\mathbb {P}}_{\mathbb {Q}}^1$ and ${\mathbf {m}} = (m_1, \ldots , m_n)$ be a vector of multiplicities. Then whenever

$$\begin{align*}\chi(\mathfrak{X}({\mathbb{P}}^1 : ({\mathbf{a}}, {\mathbf{m}}))\leq 0,\end{align*}$$

the anti-canonical height $H_{-K_{\mathfrak {X}}}$ given by (2.2) on the curve $\mathfrak {X}({\mathbb {P}}^1 : (a_1, m_1), \ldots , (a_n, m_n))$ fails to have the Northcott property.

If we assume that the ESZ-B theory should behave roughly like its classical counterpart, we can argue the converse: When $\chi (\mathfrak {X})\leq 0$ , one should have that $H_{-K_{\mathfrak {X}}}$ should fail to have the Northcott property. In particular, Theorem 2.1 shows that our arithmetic and geometric intuition prove to be correct when $\mathfrak {X}$ has coarse space ${\mathbb {P}}^1$ . This answers a question posed by Ellenberg.

The proof of Theorem 2.1 uses the following theorem about elliptic curves.

Theorem 2.2 Let $F \in {\mathbb {Z}}[x,y]$ be a non-singular binary quartic form. Then there exists square-free $d \in {\mathbb {Z}}$ such that the curve

$$\begin{align*}dz^2 = F(u,v)\end{align*}$$

has a rational point and such that its Jacobian has positive rank as an elliptic curve defined over ${\mathbb {Q}}$ .

The proof of Theorem 2.2 is provided to us by Shnidman in [Reference Shnidman15], and we graciously acknowledge his assistance.

Combining these results gives the following uniform statement.

In cases where we can prove that the Northcott property fails, we expect, according to [Reference Ellenberg, Satriano and Zureick-Brown6], that there should be a stacky Vojta conjecture. In particular, we would like to know whether the anti-canonical height $\mathcal {H}_{({\mathbf {a}}, {\mathbf {m}})}$ can be modified to recover the Northcott property. Difficulties arise because the ESZ-B height machine is not functorial in the usual sense; in the setting of algebraic varieties, one can work with a linear spaces of divisors and then apply the height machine which by functoriality will respect the linear structure. Such methods are not immediately available to us. Instead, we will apply the height machine, and then apply linear operations. We ask that for $\mathfrak {X}=\mathfrak {X}({\mathbb {P}}^1:({\mathbf {a}},{\mathbf {m}}))$ with $\chi (\mathfrak {X})\leq 0$ , what can be said about the quantity

$$\begin{align*}\inf\{t\in {\mathbb{R}}_{\geq 0}\colon H_{{\mathbb{P}}^1}^tH_{-K_{\mathfrak{X}}}\ \mathrm{has\ the\ Northcott\ property}\}.\end{align*}$$

Clearly, if we change the exponent in the classical part of the height so that it is positive, then we will recover the Northcott property. In fact, we expect that something far less drastic suffices.

For a real number $\delta $ and the curve $\mathfrak {X}({\mathbb {P}}^1 : ({\mathbf {a}}, {\mathbf {m}}))$ , define the height

(2.5) $$ \begin{align} \mathcal{H}_{({\mathbf{a}}, {\mathbf{m}})}^\delta(x,y) = \prod_{i=1}^n \phi_{m_i}(\ell_i(x,y))^{1/m_i} \max\{|x|, |y|\}^{\delta}. \end{align} $$

We then see that $\mathcal {H}_{({\mathbf {a}}, {\mathbf {m}})} = \mathcal {H}_{({\mathbf {a}}, {\mathbf {m}})}^{\chi (\mathfrak {X})}$ . Next, put

(2.6) $$ \begin{align} \gamma(\mathfrak{X}) = \inf \{\delta \in {\mathbb{R}}: \mathcal{H}_{({\mathbf{a}}, {\mathbf{m}})}^\delta \text{ has the Northcott property on } \mathfrak{X} \}. \end{align} $$

In fact, $\gamma (\mathfrak {X})$ depends only on ${\mathbf {m}}$ , so we may also write it as $\gamma ({\mathbf {m}})$ . We make the following conjecture.

Conjecture 2.4 is in fact a version of Vojta’s conjecture for stacky curves, and agrees with a conjecture of Ellenberg, Satriano, and Zureick-Brown in [Reference Ellenberg, Satriano and Zureick-Brown6]. Toward this conjecture, we have the following.

Combined with Theorem 2.1, the conjecture predicts that the set of $\delta \in {\mathbb {R}}$ such that $\mathcal {H}^\delta _{{\mathbf {a}},{\mathbf {m}}}$ has the Northcott property is an interval of the form $(\chi (\mathfrak {X}),\infty )$ when $\chi (\mathfrak {X})<0$ and $(0,\infty )$ when $\chi (\mathfrak {X})\geq 0$ . Therefore, while Theorem 2.1 tells us we cannot count points with $\mathcal {H}_{{\mathbf {a}},{\mathbf {m}}}$ , Conjecture 2.4 predicts that we can count points using $\mathcal {H}^{\chi (\mathfrak {X})+\varepsilon }_{{\mathbf {a}},{\mathbf {m}}}$ for any $\varepsilon>0$ .

Next, we prove that Conjecture 2.4 is a consequence of the $abc$ -conjecture. However, it seems that we are very far from being able to prove such a result as strong as Conjecture 2.4 unconditionally.

In fact, Conjecture 2.4 is equivalent to the $abc$ -conjecture (see Theorem 1.3). The proof of the converse is quite different and so we give it in a separate subsection.

2.3 Quantitative arithmetic on stacky curves

In the positive Euler characteristic case, we consider a particular family of stacky curves, which includes an important example suggested by J. EllenbergFootnote ¹ and show that our theory of heights matches [Reference Ellenberg, Satriano and Zureick-Brown6] in this instance. Finally, we verify a specific instance of the main conjecture in [Reference Ellenberg, Satriano and Zureick-Brown6] given by Ellenberg, Satriano, and Zureick-BrownFootnote ¹ using analytical methods. We remark that P. Le Boudec had obtained the same result as us in independent work (private communication).

We study the expression (2.3) a bit more carefully. It is easy to deduce that $\delta ({\mathbf {m}}) \geq 0$ if and only if $n \leq 4$ , and $\delta ({\mathbf {m}})> 0$ only if $n \leq 3$ . We will not consider the case $n \leq 2$ in this paper.

If we assume $m_1 \leq m_2 \leq m_3$ , then the only cases when we have positive Euler characteristic are when $m_1 = m_2 = 2$ , $m_1 = 2, m_2 = m_3 = 3$ or $m_1 = 2, m_2 = 3, m_3 = 4$ . In each of these three cases, the Northcott property for $\mathcal {H}_{({\mathbf {a}}, {\mathbf {m}})}$ holds trivially.

We now focus on the simplest cases, where $m_1 = m_2 = 2$ and $m_3 = m, m \geq 2$ . Using that $\operatorname {PGL}_2$ acts 3-transitively on ${\mathbb {P}}^1$ , we reduce to the case $\{a_1, a_2, a_3\} = \{0,-1,\infty \}$ . For $[x,y] \in {\mathbb {P}}^1$ , we may then set

$$\begin{align*}x = x_1 x_2^2, y = y_1 y_2^2\end{align*}$$

with $x_1, y_1$ square-free, and

$$\begin{align*}x + y = z_1 z_2^2 \cdots z_{m-1}^{m-1} z_m^m,\end{align*}$$

with $z_1, \ldots , z_{m-1}$ square-free. In this notation, the E-S-ZB height is given by

(2.7) $$ \begin{align} \mathcal{H}_{{\mathbf{m}}}(x,y) = |x_1|^{1/2} |y_1|^{1/2} |z_1^{m-1} \cdots z_{m-1}|^{1/m} \max\{|x_1 x_2^2|,|y_1 y_2^2|\}^{1/m}. \end{align} $$

We normalize the height so that the exponent of the “classical part” is equal to one, to obtain the normalized height

(2.8) $$ \begin{align} H_m(x,y) = |x_1|^{m/2} |y_1|^{m/2} |z_1^{m-1} \cdots z_{m-1}| \max\{|x_1 x_2^2|, |y_1 y_2^2|\}. \end{align} $$

We now put

(2.9) $$ \begin{align} N_m(T) &= \# \{(x_1, x_2), (y_1, y_2), (z_1, \ldots, z_m) : \gcd(x_1 y_1, x_2 y_2)\nonumber\\& \qquad = 1, x_1, y_1, z_1, \ldots, z_{m-1} \text{ square-free and pairwise co-prime}, \end{align} $$

$$\begin{align*}x_1 x_2^2 + y_1 y_2^2 = z_1 z_2^2 \cdots z_{m-1}^{m-1} z_m^m, |x_1|^{m/2} |y_1|^{m/2} |z_1^{m-1} \cdots z_{m-1}| \max\{|x_1 x_2^2|, |y_1 y_2^2|\} \leq T\}.\end{align*}$$

We prove the following theorem, which gives a crude upper bound for $N_m(T)$ .

Theorem 2.7 Let $\mathfrak {X} = \mathfrak {X}({\mathbb {P}}^1 : (0,2), (\infty , 2), (-1, m))$ , and let $H_m$ be the height function on $\mathfrak {X}$ defined by (2.8). Then, for any $\varepsilon> 0$ , we have

$$\begin{align*}T^{1/m} \ll N_m(T) \ll_{\varepsilon} T^{2/m + \varepsilon}.\end{align*}$$

When $m = 2$ , the upper bound of Theorem 2.7 is essentially the trivial bound, but it is nontrivial as soon as $m> 2$ . In general, we expect the exponent in Theorem 2.7 to be equal to the lower bound. Indeed, this can be verified when $m = 2$ . Even more, we can give an exact order of magnitude for $N_2(T)$

Theorem 2.8 There exist positive numbers $c_1, c_2, c_3$ such that

$$\begin{align*}c_1 T^{1/2} (\log T)^3 < N_2(T) < c_2 T^{1/2} (\log T)^3 \end{align*}$$

for all $T> c_3$ .

In particular, we confirm the stacky Batyrev–Manin conjecture [Reference Ellenberg, Satriano and Zureick-Brown6, Main Conjecture] for $\mathfrak {X}({\mathbb {P}}^1_{\mathbb {Q}},(a,2),(b,2),(c,2))$ . For this stacky curve, [Reference Ellenberg, Satriano and Zureick-Brown6, Main Conjecture] predicts that $N_2(T) = O_{\varepsilon } \left (T^{1/2 + \varepsilon }\right )$ .Footnote ¹ Our theorem gives an exact order of magnitude for $N_2(T)$ . We remark, once again, that P. Le Boudec had obtained the same result. Further, our counting arguments are similar to those obtained by Le Boudec in [Reference le Boudec13] which studies the equation (7.4).

The other cases with positive Euler characteristic do not yield to the simple analytic counting arguments used to prove Theorem 2.8, though in principle counting rational points by height is a well-posed problem. We plan on returning to this issue in the future.

We illustrate how the stacky curve height machine (equation (2.2)) allows one to detect integral points on stacky curves. In this case, the standard height is given by $H_s(a,b) = \max \{|a|,|b|\}$ and the stacky height is given by (2.7). They are equal precisely when

$$\begin{align*}|\operatorname{sqf}(a) \operatorname{sqf}(b) \operatorname{sqf}(a+b)| = 1, \end{align*}$$

or in the notation of (7.4), that $|x_1| = |x_2| = |x_3| = 1$ . (7.4) then turns into

$$\begin{align*}\pm y_1^2 \pm y_2^2 = \pm y_3^2, \end{align*}$$

and up to rearranging we are essentially counting points on the conic

(2.10) $$ \begin{align} y_1^2 + y_2^2 = y_3^2.\end{align} $$

Therefore, if we denote by ${\mathcal {N}}(T)$ the number of integral points (in the sense of Definition 3.19) on ${\mathbb {P}}_{2,2,2}^1$ , then:

Corollary 2.9 There exist positive numbers $c_1, c_2, c_3$ such that for all $T> c_3$ we have

$$\begin{align*}c_1 T^{1/2} < {\mathcal{N}}(T) <c_2 T^{1/2}.\end{align*}$$

The proof is elementary, since the curve can be explicitly parametrized by

$$\begin{align*}y_1 = u^2 - v^2, y_2 = 2uv, y_3 = u^2 + v^2.\end{align*}$$

The condition $\max \{|y_1|, |y_2\} \leq T^{1/2}$ is subsumed by $u^2 + v^2 \leq 4T^{1/2}$ say, so the number of possible $u,v$ ’s is $\asymp T^{1/2}$ as desired.

Theorem 2.8 and Corollary 2.9 imply that asymptotically $0\%$ of the rational points on $\mathfrak {X}({\mathbb {P}}^1 :(0,2),,(-1,2),(\infty ,2))({\mathbb {Q}})$ are integral, in the sense of Darmon (Definition 3.19).

To close off this subsection, we note that in [Reference Bhargava and Poonen2], Bhargava and Poonen study situations where the rational and integral points of a stacky curve satisfy the Hasse Principle. Motivated by this work, we prove that the integral points on $\mathfrak {X}({\mathbb {P}}^1 : (a_1, 2), \ldots , (a_n, 2))$ satisfy Hasse’s principle.

Theorem 2.10 Let

$$\begin{align*}\mathfrak{X} = \mathfrak{X}({\mathbb{P}}_{\mathbb{Q}}^1 : ([a_1 : -b_1], 2), ([a_2 : -b_2], 2), ([a_3, -b_3] : 2) ).\end{align*}$$

Then $\mathfrak {X}$ has integral points if and only if the ternary quadratic form

$$\begin{align*}Q_{\mathbf{a}}(u,v,w) = \det \begin{pmatrix} u^2 & v^2 & w^2 \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{pmatrix} \end{align*}$$

defines a conic with a rational point.

Notation

We denote by $d_k(n)$ for the number of ways of writing n as a product of k (not necessarily distinct) positive integers, and write $d(n) = d_2(n)$ for the usual divisor function. We will also use the big-O notation as well as Landau’s notation. In particular, we will denote in the subscripts any dependencies; if there are no subscripts, then the implied constants are absolute.

3.1 Construction of heights

We will give an alternative construction of heights on a stacky curve associated with line bundles. Our construction only depends on the coarse space, and the multiplicities of points. The ideas exposited in this section can be viewed as a “bottom up” construction, similar to the work of Geraschenko and Satriano in [Reference Geraschenko and Satriano10]. We then show that our height construction corresponds to the heights associated with line bundles in [Reference Ellenberg, Satriano and Zureick-Brown6] when the coarse space is ${\mathbb {P}}^1_{\mathbb {Q}}$ . As in the classical setting, we will work with height functions up to some bounded function. Line bundles on a stacky curve can be described as follows.

Lemma 3.3 ([Reference Fantechi, Mann and Nironi7, Section 1.3])

Let $\mathfrak {X}=\mathfrak {X}(X:(P_1,\cdots ,P_r),(m_1,\cdots ,m_r))$ . Let ${\mathcal {O}}_{X}(P)$ be the line bundle associated with the divisor P on X. Then there are line bundles ${\mathcal {L}}_{P_i}$ on $\mathfrak {X}$ such that

(3.1) $$ \begin{align} {\mathcal{L}}_{P_i}^{\otimes m_i}\cong \pi_{\mathfrak{X}}^*{\mathcal{O}}_X(P), \end{align} $$

where $\pi _{\mathfrak {X}}\colon \mathfrak {X}\rightarrow X$ is the coarse space map. Moreover, we have that any line bundle ${\mathcal {L}}$ on $\mathfrak {X}$ can be uniquely written as

(3.2) $$ \begin{align} {\mathcal{L}}\cong \pi_{\mathfrak{X}}^*M\otimes \prod_{i=1}^r{\mathcal{L}}_{P_i}^{\otimes d_i}, \end{align} $$

where $0\leq d_i<m_i$ and M is a line bundle on X.

We will use the definition of the degree of a line bundle on a stacky curve.

Definition 3.4 Let $\mathfrak {X}=\mathfrak {X}(X:(P_1,\ldots ,P_r),(m_1, \ldots , m_r)))$ and $\mathcal {L}=\pi _{\mathfrak {X}}^* D\otimes \prod _{i=1}^r{\mathcal {L}}_{P_i}^{\otimes d_i}$ . Then we define

$$\begin{align*}\deg_{\mathfrak{X}} \mathcal{{\mathcal{L}}}=\deg_X D+\sum_{i=1}^r\dfrac{d_i}{m_i}.\end{align*}$$

In [Reference Ellenberg, Satriano and Zureick-Brown6], the height is broken down into two parts: a so-called stable part and a local part. We now define the stable part in our setting.

Definition 3.5 Let $\mathfrak {X}=\mathfrak {X}(X:(P_1,\ldots ,P_r),(m_1, \ldots , m_r)))$ , and let ${\mathcal {L}}$ be a line bundle on $\mathfrak {X}$ with

$$\begin{align*}{\mathcal{L}}\cong \pi_{\mathfrak{X}}^*M\otimes \prod_{i=1}^r{\mathcal{L}}_{P_i}^{\otimes d_i},\end{align*}$$

where $0\leq d_i<m_i$ and M is a line bundle on X with $\pi _{\mathfrak {X}}$ being the coarse space map. We define the stable height associated with ${\mathcal {L}}$ as

$$\begin{align*}H_{{\mathcal{L}}}^{\mathrm{st}}=H_M\cdot\prod_{i=1}^rH_{P_i}^{\frac{d_i}{m_i}}.\end{align*}$$

Later in Proposition 4.3, we show that this definition matches the one given in [Reference Ellenberg, Satriano and Zureick-Brown6]. The stable height should be thought of as the part of the height consisting of classical height functions.

We will use the notions introduced in Section 3 to define our heights. We further choose a finite set of primes S of ${\mathcal {O}}_K$ containing all the primes of bad reduction and all infinite places of K. We further choose a smooth and proper model $\underline {X}$ of X over ${\mathcal {O}}_{K,S}$ . Let $P,Q$ be distinct points in $X(K)$ and place $\nu $ a place in K with $\nu \notin S$ . Take $\mathfrak {p}_\nu \subset {\mathcal {O}}_K$ to be the prime ideal associated with $\nu $ .

Definition 3.6 (Darmon [Reference Darmon4])

We define the intersection multiplicity of P and Q at $\nu $ as follows:

$$\begin{align*}(P\cdot Q)_\nu:=\max\{m:\ \mathrm{the\ images\ of}\ P,Q\ \mathrm{in}\ \underline{X}({\mathcal{O}}_{K,S}/\mathfrak{p}_\nu^m)\ \mathrm{are\ equal.}\},\end{align*}$$

where the maximum over the empty set is defined to be 0.

We now package all the intersection multiplicities together while taking into account the arithmetic of the field extension $K\mid {\mathbb {Q}}$ . We shall use the following notation for the remainder of this section.

The integer $\lambda (P,t)$ is an exponential version of the familiar looking intersection product

$$ \begin{align*} \sum_{\nu\notin S}(t\cdot P)_\nu \log(p)=\sum_{p}\left(\sum_{\nu\notin S,\nu \mid p}\mathfrak{f}_\nu\cdot (t\cdot P)_\nu\log(p)\right)=\sum_p(t\cdot P)_p\log (p). \end{align*} $$

We will also require the following basic functions.

Definition 3.8 For each integer $m\geq 1$ , we let $[0], \ldots ,[m-1]$ be a set of representatives of the equivalence classes of ${\mathbb {Z}}/m{\mathbb {Z}}$ . Define

(3.3) $$ \begin{align} N_{m,\mathrm{can}}([r])=r\end{align} $$

for $0\leq r<m$ and

(3.4) $$ \begin{align} N_{m,d}([r])=N_{m,\mathrm{can}}([-rd]) \end{align} $$

for any $d\in {\mathbb {Z}}$ . With this notation, $N_{m,-}=N_{m,1}$ .

These functions are used to make the following definition of the height function associated with ${\mathcal {L}}_{P_i}^{d_i}$ .

Definition 3.9 The stacky height function associated with ${\mathcal {L}}_{P_i}^{d_i}$ is a function

$$\begin{align*}H_{{\mathcal{L}}_{P_i}^{d_i}}\colon X(K)\setminus{P_i}\rightarrow {\mathbb{Z}}_{\geq 0}\end{align*}$$

defined by

(3.5) $$ \begin{align} H_{{\mathcal{L}}_{P_i}^{d_i}}(t)=\left(\prod_p p^{N_{m_i,d_i}((t\cdot P_i)_p)}\right)^{\frac{1}{m_i}},\end{align} $$

where $N_{m_i,d_i}\colon {\mathbb {Z}}_{\geq 0}\rightarrow {\mathbb {Z}}_{\geq 0}$ is the function defined by equation (3.4).

Putting this all together, we obtain the following.

Definition 3.10 (Definition of heights)

Let ${\mathcal {L}}$ be the line bundle ${\mathcal {L}}\cong \pi _{\mathfrak {X}}^*M\otimes \prod _{i=1}^r{\mathcal {L}}_{P_i}^{\otimes d_i}$ on $\mathfrak {X}$ . The stacky height associated with ${\mathcal {L}}$ is defined to be

$$\begin{align*}H_{\mathcal{L}}(t)=H_{{\mathcal{L}}}^{\mathrm{st}}\cdot \prod_{i=1}^rH_{\mathfrak{X},{\mathcal{L}}_{P_i}^{\otimes d_i}}.\end{align*}$$

Unwinding the definitions, we obtain

(3.6) $$ \begin{align} H_{\mathcal{L}}(t)&=H_{{\mathcal{L}}}^{\mathrm{st}}\cdot \prod_{i=1}^rH_{\mathfrak{X},{\mathcal{L}}_{P_i}^{\otimes d_i}} \end{align} $$

(3.7) $$ \begin{align} & =H_M\cdot \prod_{i=1}^r\left(H_{P_i}^{d_i}\cdot \prod_{p}p^{N_{m_i,d_i}((t\cdot P_i)_p)}\right)^{\frac{1}{m_i}} \end{align} $$

(3.8) $$ \begin{align} & = H_M\cdot \prod_{i=1}^r\left(H_{P_i}^{d_i}\cdot \prod_{p}p^{-d_i(t\cdot P_i)_p \quad\mod m_i}\right)^{\frac{1}{m_i}}. \end{align} $$

This decomposition allows us to define what we call the classical and stacky part of a height function.

Definition 3.11 We call

$$\begin{align*}H_{{\mathcal{L}}}^{\mathrm{st}}=H_M\cdot \prod_{i=1}^rH_{P_i}^{\frac{d_i}{m_i}}\end{align*}$$

the classical part of the height $H_{\mathcal {L}}$ and

$$\begin{align*}H_{{\mathcal{L}},\mathrm{stacky}}=\prod_{i=1}^rH_{\mathfrak{X},{\mathcal{L}}_{P_i}^{\otimes d_i}}\end{align*}$$

the stacky part of the height $H_{\mathcal {L}}$ .

We will primarily work explicitly with stacky heights on ${\mathbb {P}}^1$ , the formulas in that case are as follows.

Corollary 3.12 Use the notation of Notation 3.7. The canonical height function may be computed as

$$\begin{align*}H_{K_{\mathfrak{X}}}=H_{{\mathbb{P}}^1}^{\deg K_{\mathfrak{X}}}\prod_{i=1}^r\left(\prod_pp^{(t\cdot P_i)_p \quad\mod m_i}\right)^{\frac{1}{m_i}}.\end{align*}$$

The anti-canonical height function may be computed as

$$\begin{align*}H_{-K_{\mathfrak{X}}}=H_{{\mathbb{P}}^1}^{-\deg K_{\mathfrak{X}}}\prod_{i=1}^r\left(\prod_pp^{-(t\cdot P_i)_p \quad\mod m_i}\right)^{\frac{1}{m_i}}.\end{align*}$$

Proof This follows directly from the definition, Corollary 4.4, and the fact that $K_{\mathfrak {X}}$ corresponds to the line bundle

$$\begin{align*}\pi^*_{\mathfrak{X}} {\mathcal{O}}_{{\mathbb{P}}^1}(K_{{\mathbb{P}}^1})\otimes \prod_{i=1}^r{\mathcal{L}}_{P_i}^{\otimes m_i-1}.\\[-39pt] \end{align*}$$

We now introduce two multiplicative functions $\phi _m$ and $r_m$ that depend on an integer $m\geq 1$ . The functions $\phi _m$ and $r_m$ are dual to one another in a certain sense. This duality is key to understanding the nonlinear aspects of heights on stacky curves.

Let x be a positive integer with prime factorization $x=\prod _{p}p^{\operatorname {ord}_p(x)}$ . We will work with the following functions.

1. (1) Using the division algorithm, we define integers $q_{p,m}(x),r_{p,m}(x)$ by the equation $\operatorname {ord}_p(x)=q_{p,m}(x)m+r_{p,m}(x)$ where $0\leq r_{p,m}(x)<m$ .

2. (2) $q_m(x)=\prod _{p}p^{q_{p,m}(x)}$ and $r_m(x)=\prod _{p}p^{r_{m,p}(x)}$ .

3. (3) Set $\phi _m(x)$ to be the least positive integer such that $x\phi _m(x)$ is an mth power.

4. (4) We define the m-radical of x to be the product of all prime divisors of x whose order is not a factor of m. In other words,

   $$\begin{align*}\operatorname{rad}_m(x)=\prod_{p\ \mathrm{s.t.}\ \operatorname{ord}_p(x)\neq 0 \quad\mod m}p.\end{align*}$$

The $r_m$ is related to $N_{m,\mathrm {can}}$ and $\phi _m$ is related to $N_{m,-}$ because of the following.

Proposition 3.13 Let $x\in {\mathbb {Z}}_{\geq 0}$ . Then we have

$$ \begin{align*} r_m(x)&=\prod_{p}p^{r_{p,m}(x)}=\prod_{p}p^{N_{m,\mathrm{can}}(\operatorname{ord}_p(x))},\\ \phi_m(x)&=\prod_{p\ \mathrm{s.t.}\ \operatorname{ord}_p(x)\neq 0 \quad\mod m}p^{m-r_{p,m}(x)}=\prod_{p\mid \operatorname{rad}_m(x)}p^{m-r_{p,m}(x)}=\prod_pp^{N_{-,m}(\operatorname{ord}_p(x))}. \end{align*} $$

In particular, we have

$$ \begin{align*} H_{\mathfrak{X},{\mathcal{L}}_{P_i}^{-1}}(t)&=r_{m_i}(\lambda(P_i,t))^{\frac{1}{m_i}},\\ H_{\mathfrak{X},P,{\mathcal{L}}_{P_i}^{\otimes d_i}}(t)&=\phi_m(\lambda(P_i,t)^{d_i})^{\frac{1}{m_i}}. \end{align*} $$

From the above formulas, we obtain the following.

With these definitions in hand, we relate our height functions on the stacky curve $\mathfrak {X}$ to a function from the functions $\phi _{m_P}(\lambda (P,t))^{\frac {1}{m_P}},$ and the classical Weil heights on the coarse space X.

Corollary 3.15 Use the notation of Notation 3.7. Consider the line bundle

$$ \begin{align*} {\mathcal{L}}\cong \pi_{\mathfrak{X}}^*M\otimes \prod_{i=1}^r{\mathcal{L}}_{P_i}^{\otimes d_i} \end{align*} $$

with $0\leq d_i\leq m_i-1$ . Then

(3.9) $$ \begin{align} H_{\mathcal{L}}=H_M\cdot \prod_{i=1}^rH_{P_i}^{\frac{d_i}{m_i}}\cdot \prod_{i=1}^r \phi_{m_i}(\lambda(P_i,t)^{d_i})^{\frac{1}{m_i}}. \end{align} $$

In particular, when $X={\mathbb {P}}^1$ , we have

(3.10) $$ \begin{align} H_{K_{\mathfrak{X}}}(t)&=H_{{\mathbb{P}}^1}^{-\chi(\mathfrak{X})}(t)\cdot\prod_{i=1}^rr_{m_i}(\lambda(P_i,t))^{\frac{1}{m_i}}, \end{align} $$

(3.11) $$ \begin{align} H_{-K_{\mathfrak{X}}}(t)&=H_{{\mathbb{P}}^1}^{\chi(\mathfrak{X})}(t)\cdot\prod_{i=1}^r\phi_{m_i}(\lambda(P_i,t))^{\frac{1}{m_i}}, \end{align} $$

where $\chi (\mathfrak {X})=-\deg K_{\mathfrak {X}} $ is the Euler characteristic of $\mathfrak {X}$ .

One interesting feature of the heights given by (3.10) is that they differentiate between rational and integral points on stacky curves. The connection to [Reference Ellenberg, Satriano and Zureick-Brown6] and our heights is the following, which is proved in 4.1.

Theorem 3.16 Fix a stacky curve $\mathfrak {X}=({\mathbb {P}}^1_{\mathbb {Q}}:(P_1,m_1), \ldots , (P_r,m_r))$ . Choose S to be the set of all finite primes of ${\mathbb {Z}}$ , and let ${\mathbb {P}}^1_{\mathbb {Z}}$ be the canonical model of ${\mathbb {P}}^1_{\mathbb {Q}}$ over ${\mathbb {Z}}$ . Let ${\mathcal {L}}$ be a line bundle on $\mathfrak {X}$ . Let $H^{\mathrm{ESZ}\text{-}\mathrm{B}}_{\mathcal {L}}$ be the height constructed in [Reference Ellenberg, Satriano and Zureick-Brown6] associated with ${\mathcal {L}}$ . Then there is some constant $C>0$ with

(3.12) $$ \begin{align} C^{-1}\cdot H^{\mathrm{ESZ}\text{-}\mathrm{B}}_{\mathcal{L}}\leq H_{\mathcal{L}}\leq C\cdot H^{\mathrm{ESZ}\text{-}\mathrm{B}}_{\mathcal{L}}. \end{align} $$

That is to say, up to a constant, the stacky heights from Definition 3.10 agree with the ESZ-B heights in [Reference Ellenberg, Satriano and Zureick-Brown6] when the coarse space is ${\mathbb {P}}^1_{\mathbb {Q}}$ .

We now explain how the functions $\phi _m$ and $r_m$ can be used to understand the difference between $H_{{\mathcal {L}}}$ and $H_{{\mathcal {L}}^{\otimes n}}$ .

Proposition 3.17 Let $m\in {\mathbb {Z}}_{\geq 1}$ and choose an integer $d\geq 0$ .

$$\begin{align*}\phi_m(x^{-d \quad\mod m})=r_m(x^{d \quad\mod m}) .\end{align*}$$

The theory of heights is different from the classical theory of heights, as $H_{{\mathcal {L}}^{-1}}\neq \frac {1}{H_{\mathcal {L}}}+O(1)$ and $H_{{\mathcal {L}}^{\otimes n}}\neq (H_{\mathcal {L}})^n+O(1)$ . The functions $\phi _m$ and $r_m$ can be used to compute these quantities.

Theorem 3.18 (Duality theorem)

Let $\mathfrak {X}=\mathfrak {X}(X:(P_1,m_1), \ldots ,(P_r,m_r))$ be a stacky curve, and let

$$\begin{align*}{\mathcal{L}}\cong \pi_{\mathfrak{X}}^*M\otimes \prod_{i=1}^r{\mathcal{L}}_{P_i}^{\otimes d_i}\end{align*}$$

with $0\leq d_i\leq m_i-1$ . Fix an integer $n\neq 0$ and write $n_i=nd_i\ \ \mod m_i$ .

1. (1) Then we always have

   $$ \begin{align*} H_{{\mathcal{L}}^{\otimes n}}&=(H_{{\mathcal{L}}}^{\mathrm{st}})^n\cdot \prod_{i=1}^r\phi_{m_i}(\lambda(P_i,t)^{n_i})^{\frac{1}{m_i}}\\ &=(H_{{\mathcal{L}}}^{\mathrm{st}})^n\cdot \prod_{i=1}^r\phi_{m_i}(\lambda(P_i,t)^{nd_i\quad \mod m_i})^{\frac{1}{m_i}}. \end{align*} $$

2. (2) If $n>0$ , then

   $$ \begin{align*} H_{{\mathcal{L}}^{\otimes -n}}=(H_{{\mathcal{L}}}^{\mathrm{st}})^{-n}\cdot \prod_{i=1}^rr_{m_i}(\lambda(P_i,t)^{nd_i \quad\mod m_i})^{\frac{1}{m_i}}. \end{align*} $$
   In particular,
   $$\begin{align*}H_{{\mathcal{L}}^{-1}}=(H_{{\mathcal{L}}}^{\mathrm{st}})^{-1}\cdot \prod_{i=1}^rr_{m_i}(\lambda(P_i,t)^{d_i})^{\frac{1}{m_i}}.\end{align*}$$

Proof Write $nd_i=m_iq_i+r_i$ with $0\leq r_i<m_i$ . Note that ${\mathcal {L}}_{P_i}^{\otimes m_i}=\pi ^*_{\mathfrak {X}} {\mathcal {O}}_X(P_i)$ . So we have that

(3.13) $$ \begin{align} {\mathcal{L}}^{\otimes n}=\pi^*_{\mathfrak{X}}(M^{\otimes n})\cdot\prod_{i=1}^r\pi_{\mathfrak{X}}^*({\mathcal{O}}_X(q_iP_i))\cdot \prod_{i=1}^r{\mathcal{L}}_{P_i}^{\otimes r_i}. \end{align} $$

Therefore, we have

$$ \begin{align*} H_{{\mathcal{L}}^{\otimes n}}^{\mathrm{st}}&=H_m^n\cdot\prod_{i=1}^rH_{P_i}^{q_i}\cdot \prod H_{P_i}^{\frac{r_i}{m_i}}=H_m^n\cdot\prod_{i=1}^rH_{P_i}^{\frac{m_iq_i+r_i}{m_i}}\\ &=H_m^n\cdot\prod_{i=1}^rH_{P_i}^{\frac{nd_i}{m_i}}=\left(H_M\cdot \prod_{i=1}^rH_{P_i}^{\frac{d_i}{m_i}}\right)^n=(H_{{\mathcal{L}}}^{\mathrm{st}})^n. \end{align*} $$

Now we have that

$$ \begin{align*} H_{{\mathcal{L}}^{\otimes n}}&=H_{{\mathcal{L}}^{\otimes n}}^{\mathrm{st}}\prod_{i=1}^r\phi_{m_i}(\lambda(P_i,t)^{r_i})^{\frac{1}{m_i}}\\ &=H_{{\mathcal{L}}^{\otimes n}}^{\mathrm{st}}\prod_{i=1}^r\phi_{m_i}(\lambda(P_i,t)^{nd_i \quad\mod m_i})^{\frac{1}{m_i}} \end{align*} $$

by the definition of the $r_i$ . Now let $n>0$ . By Proposition 3.17, we have $\phi _m(\lambda (P_i,t)^{-dn \ \ \mod m})=r_m(\lambda (P_i,t)^{nd_i \ \ \mod m_i})$ giving the desired conclusion of (2).

3.2 Integral points on stacky curves

Here, we show that the height functions defined in Definition 3.10) can be used to obtain information about integral points on $\mathfrak {X}$ . Fix a stacky curve $\mathfrak {X}=\mathfrak {X}(X:({\mathbf {p}},{\mathbf {m}}))$ .

Let $H_{\mathfrak {X},P_i}$ be the height function associated with the line bundle ${\mathcal {L}}_{P_i}$ and put ${\mathcal {D}}_{\mathfrak {X}}=\prod _{i=1}^r{\mathcal {L}}_{P_i}.$ Recall that we have $H_{\mathfrak {X},P_i}(t)=\phi _{m_i}(\lambda (P_i,t))^{1/m_i}$ . We will work with the height

$$\begin{align*}H_{{\mathcal{D}}_{\mathfrak{X}}}(t)=\prod_{i=1}^r\phi_{m_i}(\lambda(P_i,t))^{1/m_i}.\end{align*}$$

This is the stacky part of the anti-canonical height. We will prove that the set of integral points is contained in the set of points where $H_{{\mathcal {D}}_{\mathfrak {X}}}(t)=1$ . When we take $K={\mathbb {Q}}$ , we see that this condition is sufficient. In other words, the S-integral points are those where the stacky part of the height is trivial. Following Darmon [Reference Darmon4], we have the following notion of integral points on an stacky curve.

Definition 3.19 (Darmon)

Let $\mathfrak {X}=(X:(P_1,m_1), \ldots , (P_r,m_r))$ be a stacky curve over a number field K, S a finite set of places of K containing all primes of bad reduction. Let $\underline {X}$ be a smooth proper model for X over ${\mathcal {O}}_{K,S}$ . The $(\underline {X},S)$ -integral points of $\mathfrak {X}$ (usually abbreviated to S-integral points of $\mathfrak {X}$ ) are the points $t\in X(K)$ such that

(3.14) $$ \begin{align} (t\cdot P)_\nu\equiv 0\quad\mod m_P \end{align} $$

for all $P\in X(K)$ and $\nu \notin S$ .

We shall prove the following theorem.

Fix a prime $\nu \notin S$ and write $(t\cdot P)_\nu =m_{P}^{e_{\nu ,P}(t)}\cdot q_{\nu ,P}(t)$ where $e_{\nu ,P}(t)\geq 0$ and $q_{\nu ,P}(t)\geq 0$ is not divisible by $m_P$ . In other words, $q_{\nu ,P}(t)$ is the $m_P$ -free part of $(t\cdot P)_\nu $ . Set $\mathrm {N}(\mathfrak {p}_\nu )=p_\nu ^{f(\nu )}$ . Then

(3.15) $$ \begin{align} \lambda_\nu(P,t)=p_\nu^{f(\nu)(t\cdot P)_\nu}=p_\nu^{m_{P}^{e_{\nu,P}(t)}\cdot q_{\nu,P}(t)\cdot f(\nu)} \end{align} $$

and

(3.16) $$ \begin{align} \lambda(P_i,t)=\prod_{\nu\notin S}p_\nu^{m_{P_i}^{e_{\nu,P_i}(t)}\cdot q_{\nu,P_i}(t)\cdot f(\nu)}. \end{align} $$

Using the functions $\lambda (P,t)$ , we can find subsets of the rational points that contain all integral points.

Proposition 3.21 Suppose that $m_P>1$ . Define $\mathfrak {X}(P;K)=\{t\in X(K)\colon H_{\mathfrak {X},P}(t)=1\}$ . Then

$$\begin{align*}\mathfrak{X}({\mathcal{O}}_{K,S,\underline{X}})\subseteq \mathfrak{X}(P;K).\end{align*}$$

Proof Note that $H_{\mathfrak {X},P}(t)=1$ is equivalent to $\phi _{m_P}(\lambda (P,t))=1$ , which is then equivalent to $r_{m_P}(\lambda (P,t))=1$ . Suppose now that t is an S-integral point. Then $(t\cdot P)_\nu \equiv 0\ \ \mod m_{P}\Rightarrow e_{\nu ,P}(t)>0$ for all $\nu \notin S$ . Thus,

$$\begin{align*}\lambda(P,t)=\prod_{\nu\notin S}p_\nu^{m_{P_i}^{e_{\nu,P_i}(t)}\cdot q_{\nu,P_i}(t)\cdot f(\nu)}=\left(\prod_{v\notin S}p_\nu^{m_{P_i}^{e_{\nu,P_i}(t)-1}\cdot q_{\nu,P_i}(t)\cdot f(\nu)}\right)^{m_P},\end{align*}$$

whence $H_{\mathfrak {X},P}(t)=1$ as $\lambda (P,t)$ is an $m_P$ -power.

We see that each point P with multiplicity $m_P>1$ imposes a height dropping condition on the set of integral points. Thus, to study integral points, it suffices to study

$$\begin{align*}\mathfrak{X}({\mathcal{O}}_{K,S,\underline{X}})\subseteq \bigcap_{m_P>1}\mathfrak{X}(P;K).\end{align*}$$

We obtain the following, that the stacky part of the anti-canonical height cuts out the integral points.

Corollary 3.22 Let $\mathfrak {X}=\mathfrak {X}(X:({\mathbf {p}},{\mathbf {m}}))$ be a stacky curve and ${\mathcal {D}}_{\mathfrak {X}}=\prod _{i=1}^r{\mathcal {L}}_{P_i}$ . Then

$$\begin{align*}\mathfrak{X}({\mathcal{O}}_{K,S})\subseteq \{t\in X(K)\colon H_{{\mathcal{D}}_{\mathfrak{X}}}(t)=1\}.\end{align*}$$

In other words, the integral points are precisely those points where the stacky part of the anti-canonical height vanishes.

Proof of Theorem 3.20

We have already shown part $(1)$ of Theorem 3.20 in (3.21). We turn to part $(2)$ and assume that $K={\mathbb {Q}}$ . We know that $\bigcap _{m_P>1}\mathfrak {X}(P;{\mathbb {Q}})\subseteq \mathfrak {X}({\mathcal {O}}_{{\mathbb {Q}},S,\underline {X}})$ by (3.21). We now show the reverse inclusion. Let $t\in X({\mathbb {Q}})$ with $H_{\mathfrak {X},P}(t)=1$ for all P with $m_P>1$ . Since $K={\mathbb {Q}}$ , we have that $\mathrm {N}(\mathfrak {p}_\nu )=p_\nu $ and $f(\nu )=1$ for all finite places $\nu $ . Fix P with $m_P>1$ . Toward a contradiction suppose that $(t\cdot P)_{\nu _0}\neq 0 \ \ \mod m_P$ for some $\nu _0\notin S$ . Then $e_{\nu _0,P}(t)=0$ . Notice that $H_{\mathfrak {X},P}(t)=1$ means that $\lambda (P,t)$ is an $m_P$ -power. Since if $\nu \neq \nu ^\prime $ we have that $p_\nu \neq p_{\nu ^{\prime }}$ , we have by unique factorization of integers that

$$\begin{align*}\lambda(P,t)=\prod_{\nu\notin S}p_\nu^{m_{P}^{e_{\nu,P}(t)}\cdot q_{\nu,P}(t)}=(\prod_{\nu\notin S}p_\nu^{z_\nu(t)})^{m_P}\end{align*}$$

for some integers $z_\nu (t)$ . In particular, for $\nu _0$ , we have

$$\begin{align*}p_{\nu_0}^{m_{P}^{e_{\nu_0,P}(t)}\cdot q_{\nu_0,P}(t)}=p_{\nu_0}^{{q_{\nu_0,P}(t)}}=p_{\nu_0}^{z_{\nu_0}(t)m_P}.\end{align*}$$

Thus, $z_{\nu _0}(t)m_P=q_{\nu _0,P}(t)$ , which contradicts $q_{\nu _0,P}(t)$ being indivisible by $m_P$ . Thus, for all $m_P>1$ and $\nu \notin S$ , we have $(t\cdot P)_\nu \equiv 0 \ \ \mod m_P$ and t is an S-integral point of $\mathfrak {X}$ by definition.

4.1 Proof of Theorem 3.16

We follow the argument given in [Reference Ellenberg, Satriano and Zureick-Brown6, p. 45]. Write ${\mathcal {L}}\cong \pi _{\mathfrak {X}}^*M\otimes \prod _{i=1}^r{\mathcal {L}}_{P_i}^{\otimes d_i}$ . From [Reference Ellenberg, Satriano and Zureick-Brown6, Section 2.3], we have a decomposition

(4.2) $$ \begin{align} \frac{H_{{\mathcal{L}}}^{\mathrm{ESZ}\text{-}\mathrm{B}}}{H_{{\mathcal{L}}}^{\mathrm{st},\mathrm{ESZ}\text{-}\mathrm{B}}} =\prod_p\delta_{{\mathcal{L}},p}, \end{align} $$

where the $\delta _{{\mathcal {L}},p}$ ’s are the local discrepancies associated with $H_{{\mathcal {L}}}^{\mathrm{ESZ}\text{-}\mathrm{B}}$ . On the other hand, we have that

(4.3) $$ \begin{align} \frac{H_{\mathcal{L}}(P)}{H^{\mathrm{st}}_{\mathcal{L}}(P)} =\prod_{i=1}^r\phi_{m_i}(\ell_i(P)^{d_i})^{\frac{1}{m_i}}. \end{align} $$

By Proposition 4.3, we have that

(4.4) $$ \begin{align} H_{{\mathcal{L}}}^{\mathrm{st},\mathrm{ESZ}\text{-}\mathrm{B}}=H_{{\mathcal{L}}}^{\mathrm{st}} \end{align} $$

up to some positive constant. Therefore, it suffices to show that

$$\begin{align*}\prod_p\delta_{{\mathcal{L}},p}(P)=\prod_{i=1}^r \phi_{m_i}(\ell_i(P)^{d_i})^{\frac{1}{m_i}}.\end{align*}$$

Let $x\colon \mathrm {spec}\ {\mathbb {Q}}\rightarrow \mathfrak {X}$ be a rational point whose image is not any of the stacky points $P_i$ . Then, in [Reference Ellenberg, Satriano and Zureick-Brown6], there is a one-dimensional stack ${\mathcal {C}}$ called the tuning stack and a diagram:

Moreover, the local discrepancies can be computed at a prime p by

$$\begin{align*}p^{-\deg \pi_*\bar{x}^* {\mathcal{L}}^{-1}-(-\deg \bar{x}^*{\mathcal{L}}^{-1})}. \end{align*}$$

These degrees can be computed locally on $\mathfrak {X}$ in terms of the stacky points $P_i$ . In other words,

$$ \begin{align*} \deg \bar{x}^* {\mathcal{L}}^{-1}&=\sum_{i=1}^r\deg_{P_i}\bar{x}^*{\mathcal{L}}^{-1},\\ \deg\pi_*\bar{x}^*{\mathcal{L}}^{-1}&=\sum_{i=1}^r\deg_{P_i}\pi_*\bar{x}^*{\mathcal{L}}^{-1}. \end{align*} $$

The local degree of ${\mathcal {L}}$ at $P_i$ at ${\mathbb {Q}}/{\mathbb {Z}}$ is $\frac {d_i}{m_i}$ . Following [Reference Ellenberg, Satriano and Zureick-Brown6], we have the local degrees

$$ \begin{align*} \deg_{P_i} \bar{x}^*{\mathcal{L}}^{-1}&=-\frac{d_i\operatorname{ord}_p(\ell_i(x))}{m_i},\\ \deg_{P_i} \pi_*\bar{x}^*{\mathcal{L}}^{-1}&=\left\lfloor-\frac{d_i\operatorname{ord}_p(\ell_i(x))}{m_i}\right\rfloor. \end{align*} $$

We obtain that the contribution at $P_i$ to the local discrepancy at p can be written as

$$\begin{align*}p^{\left\lceil \frac{d_i\ell_i(x)}{m_i}\right\rceil-\frac{d_i\ell_i(x)}{m_i}}.\end{align*}$$

Now write $d_i\ell _i(x)=q_im_i+r_i$ with $0\leq r_i\leq m_i-1$ . First, suppose that $r_i=0$ . Then

$$\begin{align*}\left\lceil \frac{d_i\ell_i(x)}{m_i}\right\rceil-\frac{d_i\ell_i(x)}{m_i}=\left\lceil\frac{q_im_i}{m_i} \right\rceil-\frac{q_im_i}{m_i}=q_i-q_i=0.\end{align*}$$

Now suppose that $r_i\neq 0$ . Then we have that

$$ \begin{align*} \left\lceil \frac{d_i\ell_i(x)}{m_i}\right\rceil-\frac{d_i\ell_i(x)}{m_i}&=\left\lceil \frac{q_im_i+r_i}{m_i}\right\rceil-\frac{q_im_i+r_i}{m_i}\\ &=q_i+\left\lceil\frac{r_i}{m_i}\right\rceil-\frac{r_i}{m_i}-q_i\\ &=1-\frac{r_i}{m_i}=\frac{m_i-r_i}{m_i}. \end{align*} $$

In other words, we have shown that

$$ \begin{align*} \left\lceil \frac{d_i\ell_i(x)}{m_i}\right\rceil-\frac{d_i\ell_i(x)}{m_i}&=\begin{cases} 0, & d_i\ell_i(x)=0\quad\mod m_i, \\ -d_i\ell_i(x) \quad\mod m_i, & d_i\ell_i(x)\neq 0 \quad\mod m_i, \end{cases}\\ &= \frac{-d_i\ell_i(x) \quad\mod m_i}{m_i}=\frac{N_{m_i,d_i}(\operatorname{ord}_p(\ell_i(x)))}{m_i}. \end{align*} $$

Thus, the local discrepancies are given by

$$ \begin{align*} \prod_{p}\delta_{{\mathcal{L}},p}(x)&=\prod_{i=1}^r\cdot \left(\prod_p p^{\frac{N_{m_i,d_i}(\operatorname{ord}_p(\ell_i(x)))}{m_i}}\right)\\ &=\prod_{i=1}^r\phi_{m_i}(\ell_i(x)^{d_i})^{\frac{1}{m_i}}. \end{align*} $$

Combining equations 4.2–4.4 gives the desired conclusion.

Question 4.6 Consider a stacky curve $\mathfrak {X}=(X:(P_1,m_1), \ldots ,(P_r,m_r))$ . A line bundle M on X with a chosen height function $H_M$ and integers $0\leq d_i<m_i$ . Then does

$$\begin{align*}H_{\mathcal{L}}(t)=H_M(t)\cdot \prod_{i=1}^rH_{P_i}^{\frac{d_i}{m_i}}(t)\cdot \prod_{i=1}^r \phi_{m_i}(\lambda(P_i,t)^{d_i})^{\frac{1}{m_i}}\end{align*}$$

agree with the height constructed in [Reference Ellenberg, Satriano and Zureick-Brown6] up to a bounded constant when $X\neq {\mathbb {P}}^1_{\mathbb {Q}}$ . In other words, does our stacky height machine recover the heights in [Reference Ellenberg, Satriano and Zureick-Brown6] for all stacky curves over all number fields? If not, can one define different size functions $\tilde {N}_{m_i,d_i}\colon {\mathbb {Z}}_{\geq 0}\rightarrow {\mathbb {Z}}_{\geq 0}$ so that the ESZ-B height associated with ${\mathcal {L}}$ is of the form

$$\begin{align*}H_{\mathcal{L}}(t)= \left(H_M(t)\cdot\prod_{i=1}^rH_{P_i}^{\frac{d_i}{m_i}}(t)\right)\cdot\left(\prod_{i=1}^r\left(\prod_p p^{\tilde{N}_{m_i,d_i}(t\cdot P_i)_p}\right)^{\frac{1}{m_i}}\right).\end{align*}$$

This result will follow provided one can show that the local degree of $\bar {x}^*{\mathcal {L}}$ with respect to $P_i$ over a prime p is

$$\begin{align*}\frac{d_i\operatorname{ord}_p(\lambda(P_i,t))}{m_i}=\frac{d_i(t\cdot P_i)_p}{m_i}.\end{align*}$$

In this case, the argument given in Theorem 3.16 would give the desired result. Further, one might ask if these methods could be extended to compute the height functions of line bundles certain higher dimensional analogues of stacky curves.

4.2 Morphisms of stacky curves

We will require some results on morphisms between stacky curves.

Definition 4.7 (Darmon [Reference Darmon4])

Let $\mathfrak {X}_1=(X_1,(P,m_P)),\mathfrak {X}_2=(X_2,(Q,m_Q))$ be M curves defined over a number field K. A morphism of stacky curves over K is a morphism of algebraic curves $\pi \colon X_1\rightarrow X_2$ defined over K such that for all $P\in X_1(K)$ with $\pi (P)=Q$ , we have that

$$\begin{align*}m_Q\mid e_\pi(P)m_P,\end{align*}$$

where $e_\pi (P)$ is the ramification index of $\pi $ at P. We also define $e_{\underline {\pi }}(P)=\frac {e_{\pi }(P)m_P}{m_{Q}}$ the ramification index of $\underline {\pi }$ at P.

Now let $\mathfrak {X}=(X:{\mathbb {Q}};(P_1,m_1), \ldots ,(P_r,m_r))$ be an M curve. For any $s<r$ , choose positive divisors $d_i$ of $m_i$ for $i=1,...,s$ . Then there is a multiplicity lowering morphism

$$\begin{align*}\underline{\pi}(d_1,\ldots,d_s)\colon \mathfrak{X}(X:(P_1,m_1), \ldots ,(P_r,m_r))\rightarrow \mathfrak{X}(X:(P_1,d_1), \ldots ,(P_s,d_s))\end{align*}$$

defined by the identity morphism on X. The usefulness of this notion can be seen by the following.

Proposition 4.8 (Darmon [Reference Darmon4])

Let $\underline {\pi }\colon \mathfrak {X}_1\rightarrow \mathfrak {X}_2$ be a morphism of stacky curves defined over K. Then

$$\begin{align*}\underline{\pi}(\mathfrak{X}_1({\mathcal{O}}_{K,S}))\subseteq \mathfrak{X}_2({\mathcal{O}}_{K,S}).\end{align*}$$

In other words, a morphism of stacky curves preserves the notion of S-integral points.

Lemma 4.9 Let K be a field and

$$\begin{align*}U=\begin{bmatrix} 0 & 1\\ -1 &0 \end{bmatrix}.\end{align*}$$

Define a bilinear form $L({\mathbf {v}},{\mathbf {x}})={\mathbf {v}}^TU^T{\mathbf {x}}$ . Let T be a non-singular matrix with entries in K. Then

$$\begin{align*}L(T{\mathbf{v}},T{\mathbf{x}})=(\det T)L({\mathbf{v}},{\mathbf{x}}).\end{align*}$$

Proof We have that

$$\begin{align*}L(T{\mathbf{v}},T{\mathbf{x}})=(T{\mathbf{v}})^TU^TT{\mathbf{x}}={\mathbf{v}} T^TU^TT{\mathbf{x}}={\mathbf{v}}(UT)^TT{\mathbf{x}}.\end{align*}$$

Direct computation shows that $(UT)^TT=(\det T)U^T$ . We then have

$$\begin{align*}L(T{\mathbf{v}},T{\mathbf{x}})={\mathbf{v}}(UT)^TT{\mathbf{x}}={\mathbf{v}}^T(\det T)U^T{\mathbf{x}}=(\det T)L({\mathbf{v}},{\mathbf{x}}).\\[-35pt] \end{align*}$$

Note that if $P=[a:b]$ and $t=[x,y]$ where $a,b$ and $x,y$ are coprime integers, then $\lambda (P,t)=ay-bx$ by (4.1). In other words, $\lambda (P,t)=L(P,t)$ as defined in (4.9).

Lemma 4.10 Let $P=[a:b],t=[x,y]\in {\mathbb {P}}^1_{\mathbb {Q}}$ with $a,b$ and $x,y$ coprime integers. Fix an integer $m>1$ . Let $\alpha \colon {\mathbb {P}}^1_{\mathbb {Q}}\rightarrow {\mathbb {P}}^1_{\mathbb {Q}}$ be an automorphism. Let $\det \alpha $ be the smallest possible nonnegative determinant of an integral representation of $\alpha $ and similarly for $\det \alpha ^{-1}$ . Then we have

(4.5) $$ \begin{align} (\operatorname{rad}_m(\det \alpha^{-1}))^{-(m-1)}\phi_m(\lambda(P,t))\leq \phi_m(\lambda(\alpha(P),\alpha(t))\leq\operatorname{rad}_m(\det\alpha)^{m-1}\phi_m(\lambda(P,t)). \end{align} $$

Proof Let L be as (4.1). Then we have that

(4.6) $$ \begin{align} \lambda(P,t)=L\left(\dfrac{\alpha(P)}{d_1},\dfrac{\alpha(t)}{d_2}\right)=\dfrac{(\det \alpha)}{d_1d_2}L(P,t)=\dfrac{(\det \alpha)}{d_1d_2}\lambda(P,t) \end{align} $$

for some integers $d_1,d_2$ which account for common factors of $\alpha (P)$ and $\alpha (t)$ . Let $n_1=q_1m+r_1,n_2=q_2m+r_2$ be integers with $0\leq r_i<m$ .

(4.7) $$ \begin{align} \phi_m(p^{n_1+n_2})= \begin{cases} p^{m-r_1-r_2}=\dfrac{\phi_m(p^{n_1})\phi_m(p^{n_2})}{p^m}, & \mathrm{if}\ r_1+r_2<m,\\ p^{m-(r_1+r_2-m)}=p^{2m-r_1-r_2}=\phi_m(p^{n_1})\phi_m(p^{n_2}), &\mathrm{if}\ r_1+r_2\geq m. \end{cases} \end{align} $$

Since $\phi _m$ is multiplicative, we have that $\phi _m(zw)\leq \phi _m(z)\phi _m(w)\leq \operatorname {rad}_m(z)^{m-1}\phi _m(w)$ . Therefore, using (4.6),

$$\begin{align*}\phi_m(\lambda(\alpha(P),\alpha(t)))\leq \operatorname{rad}_m(\det \alpha)^{m-1}\phi_m(\lambda(P,t)).\end{align*}$$

Applying the same reasoning using $\alpha ^{-1}$ , we have that

$$\begin{align*}\phi_m(\lambda(P,t))\leq \operatorname{rad}_m(\det \alpha^{-1})^{m-1}\lambda(\phi(P),\phi(t)). \end{align*}$$

Therefore, we have

(4.8) $$ \begin{align} (\operatorname{rad}_m(\det \alpha^{-1}))^{-(m-1)}\phi_m(\lambda(P,t))\leq \phi_m(\lambda(\alpha(P),\alpha(t))\leq\operatorname{rad}_m(\det\alpha)^{m-1}\phi_m(\lambda(P,t)), \end{align} $$

as required.

Corollary 4.11 Let $\mathfrak {X}=\mathfrak {X}({\mathbb {P}}^1;(P_1,m_1), \ldots ,(P_r,m_r))$ . Let $\alpha \colon {\mathbb {P}}^1_{\mathbb {Q}}\rightarrow {\mathbb {P}}^1_{\mathbb {Q}}$ be an automorphism of ${\mathbb {P}}^1_{\mathbb {Q}}$ . Let $Q_i=\alpha (P_i)$ and $\mathfrak {Y}=\mathfrak {Y}({\mathbb {P}}^1;(Q_1,m_1), \ldots ,(Q_r,m_r))$ . Then $\alpha $ induces an isomorphism $\underline {\alpha }\colon \mathfrak {X}\rightarrow \mathfrak {Y}$ . Let $\det \alpha $ and $\det \alpha ^{-1}$ be as in (4.10). Let $D(\alpha ,\mathfrak {X})=\prod _{i=1}^r\operatorname {rad}_{m_i}(\det \alpha )^{1-1/m_i}$ and similarly define $D(\alpha ^{-1},\mathfrak {X})$ . Suppose that $C_\phi $ is a constant (such a constant always exists) such that

(4.9) $$ \begin{align} C_\alpha^{-1}H_{{\mathbb{P}}^1}(t)^{\chi(\mathfrak{X})}\leq H_{{\mathbb{P}}^1}(\phi(t))^{\chi(\mathfrak{X})}\leq C_\alpha H_{{\mathbb{P}}^1}(t)^{\chi(\mathfrak{X})}. \end{align} $$

Then

(4.10) $$ \begin{align} D(\alpha^{-1},\mathfrak{X})^{-1}C_\alpha^{-1} H_{-K_{\mathfrak{X}}}(t)\leq H_{{\mathbb{T}}_{\mathfrak{Y}}}(\phi(t)) \leq D(\alpha,\mathfrak{X})C_\alpha H_{-K_{\mathfrak{X}}}(t). \end{align} $$

Proof By assumption, for each $i=1, \ldots ,r$ , we have from (4.10) and our assumption that

$$ \begin{align*} H_{{\mathcal{T}}_{\mathfrak{Y}}}(\phi(t))&=H_{{\mathbb{P}}^1}(\phi(t))^{\chi( \mathfrak{Y})}\prod_{i=1}^r\phi_{m_i}(\lambda(Q_i,\phi(t)))^{\frac{1}{m_i}}\\ &\leq C_\alpha H_{{\mathbb{P}}^1}(t)^{\chi(\mathfrak{X})}\prod_{i=1}^r\operatorname{rad}_{m_i}(\det(\alpha))^{1-1/m_i}\phi_{m_i}(\lambda(P_i,t))^{\frac{1}{m_i}}\\&=C_\alpha D(\alpha,\mathfrak{X})H_{-K_{\mathfrak{X}}}(t). \end{align*} $$

Similarly, we have

$$ \begin{align*} H_{{\mathcal{T}}_{\mathfrak{Y}}}(\phi(t))&=H_{{\mathbb{P}}^1}(\phi(t))^{\chi( \mathfrak{Y})}\prod_{i=1}^r\phi_{m_i}(\lambda(Q_i,\phi(t)))^{\frac{1}{m_i}}\\&\geq C_\alpha^{-1} H_{{\mathbb{P}}^1}(t)^{\chi(\mathfrak{X})}\prod_{i=1}^r\operatorname{rad}_{m_i}(\det(\alpha^{-1}))^{-(1-1/m_i)}\phi_{m_i}(\lambda(P_i,t))^{\frac{1}{m_i}}\\&=C_\alpha^{-1} D(\alpha^{-1},\mathfrak{X})H_{-K_{\mathfrak{X}}}(t).\\[-37pt] \end{align*} $$

We can get slightly worse, but more understandable bounds as follows. We always have $\operatorname {rad}_m(x)\leq \operatorname {rad}(x)$ . Note that we have that $\sum _{i=1}^r(1-1/m_i)=2-\chi (\mathfrak {X})=2g(\mathfrak {X})$ . Thus, we in fact have

(4.11) $$ \begin{align} C_\alpha^{-1}\operatorname{rad}(\det\alpha^{-1})^{-2g(\mathfrak{X})}H_{-K_{\mathfrak{X}}}(t)\leq H_{{\mathcal{T}}_{\mathfrak{Y}}}(\phi(t))\leq C_\alpha\operatorname{rad}(\det\alpha)^{2g(\mathfrak{X})}H_{-K_{\mathfrak{X}}}(t). \end{align} $$

Of particular note is that we see that when studying the Northcott property, we may change the height by an automorphism. Thus, the Northcott property is stable under isomorphism as expected.

4.3 Northcott property of the canonical height on stacky curves

We now investigate the properties of the canonical height on stacky curves, given by

(4.12) $$ \begin{align} H_{({\mathbf{a}}, {\mathbf{m}})}(x,y) = \prod_{i=1}^n r_{m_i} (\ell_i(x,y))^{\frac{1}{m_i}} \cdot \max\{|x|, |y|\}^{-\delta({\mathbf{m}})}. \end{align} $$

When $\delta ({\mathbf {m}}) = 0$ , we see that the canonical height exhibits a clear duality with the anti-canonical height, and so the same argument shows that Northcott’s property will fail. When $\delta ({\mathbf {m}}) < 0$ , we then see at once that $H_{({\mathbf {a}}, {\mathbf {m}})}$ will have Northcott’s property as a consequence that the Weil height having Northcott’s property. It remains to consider Northcott’s property when $\delta ({\mathbf {m}})> 0$ . In this case, we have ${\mathbf {m}} = (2, m_2, m_3)$ with

$$\begin{align*}\frac{1}{m_2} + \frac{1}{m_3}> \frac{1}{2}.\end{align*}$$

It suffices to show that for any such pair $(m_2,m_3)$ , there exist integers $a,b,c$ such that the curve

$$\begin{align*}ax^2 + by^{m_2} +c z^{m_3} = 0, \frac{1}{2} + \frac{1}{m_2} + \frac{1}{m_3}> 1\end{align*}$$

has infinitely many primitive integral solutions. This is the content of Beukers’ paper [Reference Beukers1], and we are done.

5.1 Minimally nonnegative tuples with $n = 4$

We begin with the case ${\mathbf {m}} = (2,2,2,2)$ , and we will need Theorem 2.2. By $3$ -transivity of the action of $\operatorname {PGL}_2$ on ${\mathbb {P}}^1$ and Lemma 4.10, we may assume that three of the points are $0, 1, \infty $ , corresponding to the linear forms $x,y, x+y$ in the variables $x,y$ . We then write $\ell (x,y) = ax + by$ for the linear form representing the fourth half-point.

We prove the following as a warm-up.

Proof In this case, the height is given by

$$\begin{align*}H(x,y) = \operatorname{sqf}(x) \operatorname{sqf}(y) \operatorname{sqf}(x+y) \operatorname{sqf}(ax + by),\end{align*}$$

so this is equal to one if and only if each of $x, y, x+y, ax+by$ is a square. To wit, we set

$$\begin{align*}x = x_1^2, y= x_2^2, x+y = x_3^2.\end{align*}$$

This induces the equation

$$\begin{align*}x_1^2 + x_2^2 = x_3^2,\end{align*}$$

which is solvable and whose (primitive) integral solutions are parametrized by

$$\begin{align*}x_1 = 2uv, x_2 = u^2 - v^2, x_3 = u^2 + v^2.\end{align*}$$

Inserting this into $ax + by$ gives

$$\begin{align*}a(2uv)^2 + b(u^2 - v^2)^2 = F_{a,b}(u,v).\end{align*}$$

We then fix $u = u_0, v = v_0$ so that $2u_0 v_0$ and $u_0^2 - v_0^2$ are co-prime, then solve the linear diophantine equation

$$\begin{align*}a (2u_0 v_0)^2 + b (u_0^2 - v_0^2) = 1.\end{align*}$$

Given a solution $(a,b)$ to this diophantine equation, one obtains a genus curve defined by

$$\begin{align*}w^2 = F_{a,b}(u,v)\end{align*}$$

which is isomorphic to an elliptic curve, since it has a rational point given by $w_0^2 = F_{a,b}(u_0, v_0)$ . In particular, it must be isomorphic to its Jacobian. A simple calculation shows that the Jacobian of this genus one curve is given by the equation

(5.1) $$ \begin{align} E_{a,b} : y^2 &= x^3 - \frac{16(a^2 - ab + b^2)}{3} - \frac{64(a+b)(2a-b)(a-2b}{27}\\&= \left( x - \frac{4b - 8a}{3}\right)\left( x - \frac{4a - 8b}{3}\right)\left( x - \frac{4b + 4a}{3}\right), \nonumber \end{align} $$

so it suffices to find $a,b$ such that $E_{a,b}$ has positive rank. We find that setting $u_0 = 1, v_0 = 5$ and $a = 17, b = -118$ that the curve $E_{a,b}$ has positive rank, and therefore $w^2 = F_{a,b}(u,v)$ will have infinitely many integral solutions $(u,v,w)$ . This gives infinitely many pairs $u,v$ such that $F_{17,-118}(u,v)$ is a square. This implies our result, since

$$\begin{align*}&\operatorname{sqf}(x) \operatorname{sqf}(y) \operatorname{sqf}(x+y) \operatorname{sqf}(17x-118y)\\& \quad = \operatorname{sqf}((2uv)^2) \operatorname{sqf}((u^2 - v^2)^2) \operatorname{sqf}((u^2 + v^2)^2) \operatorname{sqf}(F_{17,-118}(u,v)) = 1.\\[-34pt] \end{align*}$$

The general case will follow by applying the same ideas in tandem with Theorem 2.2. Indeed, Theorem 2.2 gives that for any $a,b \in {\mathbb {Z}}$ such that $F_{a,b}(u,v) = a(u^2 - v^2)^2 + 4bu^2 v^2$ is non-singular that there exists $d \in {\mathbb {Z}}$ such that $C_F({\mathbb {Q}}) \ne \emptyset $ and $E_F$ has positive rank. Fixing such a d, we see that there are infinitely many co-prime integers $u,v,z$ such that

$$\begin{align*}dz^2 = F(u,v).\end{align*}$$

Recall that in this setup we have

$$\begin{align*}x = (u^2 - v^2)^2, y = 4u^2 v^2, x + y = (u^2 + v^2)^2,\end{align*}$$

whence

$$\begin{align*}H(x,y) = 1 \cdot 1 \cdot 1 \cdot \operatorname{sqf}(F(u,v)) \leq |d|.\end{align*}$$

This concludes the proof for the ${\mathbf {m}} = (2,2,2,2)$ case.

Note that if ${\mathbf {m}} = (m_1, m_2, m_3, m_4)$ is minimally nonnegative, then $m_1 = m_2 = 2$ , since $\delta ((3,3,3)) = 0$ . Thus, we may write

$$\begin{align*}x = x_1 x_2^2, y = y_1 y_2^2\end{align*}$$

with $x_1, y_1$ square-free. We then write

$$\begin{align*}\ell_3(x,y) = x + y = z_1 z_2^2 \cdots z_{m_3}^{m_3}, \ell_4(x,y) = w_1 w_2^2 \cdots w_{m_4}^{m_4}.\end{align*}$$

Again, we have $z_i, w_j$ are square-free for $1 \leq i \leq m_3 - 1$ and $1 \leq j \leq m_4 - 1$ .

We now specialize to the points where $z_i = w_j = 1$ except for $i = 2$ and $j = 1,2$ , as well as $x_1 = y_1 = 1$ . Applying Theorem 2.2 and using the same argument as in the $(2,2,2,2)$ case, we see that there is a choice of $w_1 = d$ such that there are infinitely many choices of $x_2, y_2, w_2, z_2$ satisfying

$$\begin{align*}x = x_2^2, y = y_2^2, x + y = z_2^2, \ell_4(x,y) = dw_2^2.\end{align*}$$

The height of such a point is given by

$$\begin{align*}\phi_2(x_2^2) \phi_2(y_2^2) \phi_{m_3} (z_2^2) \phi_{m_4} (d w_2^2) \max\{x_2^2, y_2^2\}^{1/m_3 + 1/m_4 - 1} \end{align*}$$

$$\begin{align*}\ll |z_2|^{1 - 2/m_3} |w_2|^{1 - 2/m_4} \max\{|z_2|, |w_2|\}^{(2/m_3 - 1) + (2/m_4 - 1} \ll 1. \end{align*}$$

It follows that there are infinitely many points of bounded height, and so Northcott’s property fails.

5.2 Minimally nonnegative tuples with $n = 3$

To complete the proof of Theorem 2.1, it remains to handle the cases when $n = 3$ and $\chi (\mathfrak {X}) \leq 0$ . We shall assume that $m_1 \leq m_2 \leq m_3$ . We then note that $\delta ({\mathbf {m}}) \leq 0$ if and only if one of the following conditions is satisfied:

1. (1) $m_1 \geq 3$ .

2. (2) $m_1 = 2, m_2 = 3, m_3 \geq 6$ .

3. (3) $m_1 = 2, m_2 \geq 4$ .

We deal with the first case. We then write

(5.2) $$ \begin{align} x = x_{1,1} x_{1,2}^2 \cdots x_{1,m_1}^{m_1}, y = x_{2,1} x_{2,2}^2 \cdots x_{2,m_2}^{m_2}, \end{align} $$

$$\begin{align*}x + y = x_{3,1} x_{3,2}^2 \cdots x_{3,m_3}^{m_3}.\end{align*}$$

Now set

$$\begin{align*}x_{i,j} = 1 \text{ for } (i,j) \not\in \{(1,3), (2,3), (3,3), (3,1)\} \end{align*}$$

and

$$\begin{align*}x_{1,3} = z_1, x_{2,3} = z_2, x_{3,3} = z_3, x_{3,1} = d.\end{align*}$$

Then the value of the height $H_{({\mathbf {a}}, {\mathbf {m}})}(x,y)$ in this case is given by

$$ \begin{align*} H(z_1^3, z_2^3) & = \phi_{m_1}(z_1^3)^{1/m_1} \phi_{m_2}(z_2^3)^{1/m_2} \phi_{m_3}(d z_3^3)^{1/m_3} \max\{|z_1|^3, |z_2|^3\}^{1/m_1 + 1/m_2 + 1/m_3 - 1} \\ & \leq d^{m_3 - 1} |z_1|^{\frac{m_1 - 3}{m_1}} |z_2|^{\frac{m_2 - 3}{m_2}} |z_3|^{\frac{m_3 - 3}{m_3}} \max\{|z_1, |z_2|\}^{\left(\frac{3}{m_1} - 1 \right) + \left(\frac{3}{m_2} - 1 \right) + \left(\frac{3}{m_3} - 1 \right)}. \end{align*} $$

Observe that

$$\begin{align*}|z_i|^{\frac{m_i - 3}{3}} \max\{|z_1|, |z_2|\}^{-1 + \frac{3}{m_i}} \ll 1 \end{align*}$$

for $i = 1,2,3$ .

It remains to choose d so that the plane cubic curve

$$\begin{align*}z_1^3 + z_2^3 = dz_3^3\end{align*}$$

has infinitely many rational points. This is an easy consequence of the seminal work of Stewart and Top [Reference Stewart and Top18, Theorem 7], which in turn depends on the important work of Stewart in [Reference Stewart17]. In particular, they showed that the number of cube-free integers d with $|d| \leq X$ such that the equation

$$\begin{align*}x^3 + y^3 = d\end{align*}$$

defines an elliptic curve with rank at least $2$ is asymptotically greater than $X^{1/3}$ . We, of course, do not need such a strong statement; indeed, we only need one such d. This completes the proof for the case $m_1 \geq 3$ .

We proceed to handle the case $m_1 = 2, m_2 \geq 4$ . Using the same notation as in (5.2), we then set

$$\begin{align*}x_{i,j} = 1 \text{ for all } (i,j) \not\in \{(1,1), (1,2), (2,4), (3,4)\},\end{align*}$$

and set

$$\begin{align*}x_{1,2} = z_1, x_{2,4} = z_2, x_{3,4} = z_3, x_{1,1} = d.\end{align*}$$

This gives a curve

$$\begin{align*}dz_1^2 = z_3^4 - z_2^4.\end{align*}$$

We need to choose square-free d so that this curve has infinitely many integral solutions, and such a d exists by Theorem 2.2. The height $H_{({\mathbf {a}}, Bm)}(x,y)$ is given by

$$ \begin{align*} H_{({\mathbf{a}}, {\mathbf{m}})}(x,y) & = \phi_2(dz_1^2)^{1/2} \phi_{m_2}(z_2^3)^{1/m_i} \phi_{m_3}(d z_3^3)^{1/m_3} \max\{d|z_1|^2, |z_2|^4\}^{1/m_2 + 1/m_3 - 1/2} \\ & \leq d |z_2|^{1 - 4/m_2} |z_3|^{1 - 4/m_3} \max\{d|z_1|^2, |z_2|^4\}^{1/m_2 + 1/m_3 - 1/2}. \end{align*} $$

Note that

$$\begin{align*}|z_3|^4 \asymp \max\{dz_1^2, z_2^4\},\end{align*}$$

so we obtain the upper bound

$$\begin{align*}\left(\frac{|z_2|}{\max\{|z_2|, |z_3|\}}\right)^{1 - \frac{4}{m_2}} \cdot \left(\frac{|z_3|}{\max\{|z_2|, |z_3|\}} \right)^{1 - \frac{4}{m_3}} \ll 1. \end{align*}$$

It follows that there are infinitely many integers $x,y$ such that $H_{({\mathbf {a}}, {\mathbf {m}})}(x,y)$ remains bounded.

Finally, we resolve the case $m_1 = 2, m_2 = 3, m_3 \geq 6$ . In this case, we use the fact that there exist integers $a,b,c$ with a square-free, b cube-free, and c sixth power-free such that the equation

(5.3) $$ \begin{align} ax^2 + by^3 + cz^6 = 0\end{align} $$

has infinitely many primitive solutions (see [Reference Darmon and Granville5, Section 6.3]). Thus, by fixing such a triple $(a,b,c)$ and setting

$$\begin{align*}x_{1,1} = a, x_{2,1} x_{2,2}^2 = b, x_{3,1} \cdots x_{3,5} = c,\end{align*}$$

$$\begin{align*}x_{1,2} = u_1, x_{2,3} = u_2, x_{3,6} = u_3\end{align*}$$

and

$$\begin{align*}x_{5,j} = 1 \text{ for all } j \geq 7,\end{align*}$$

we specialized a point on $\mathfrak {X}({\mathbb {P}}^1 : (0, 2), (\infty , 3), (-1, 6))$ to the curve given by (5.3). The height of such a point is then bounded in terms of $a,b,c$ only, and is thus absolutely bounded. This shows that the Northcott property fails in this case as well.

This concludes the proof of Theorem 2.1.

5.3 Proof of Theorems 2.5

We proceed to prove Theorem 2.5. The claim when $\delta ({\mathbf {m}}) = 0$ is covered in Theorem 2.1, so we will not discuss it again. When $\delta ({\mathbf {m}})> 0$ , we note that $n = 3$ , and that in each such case, there exist integers $a_{\mathbf {m}}, b_{\mathbf {m}}, c_{\mathbf {m}}$ such that the equation

$$\begin{align*}a_{\mathbf{m}} x^{m_1} + b_{{\mathbf{m}}} y^{m_2} + c_{{\mathbf{m}}} z^{m_3} = 0 \end{align*}$$

has infinitely many primitive integral solutions (see, for example, [Reference Beukers1]). This shows that the Northcott property fails for $H^0$ .

We may now work with the case when $\delta ({\mathbf {m}}) < 0$ . We see that the height $H^0$ is bounded below by

(5.4) $$ \begin{align} \phi_{m_1}(x)^{1/m_1} \phi_{m_2}(y)^{1/m_2} \phi_{m_3}(x+y)^{1/m_3}, \end{align} $$

so it suffices to show that this quantity necessarily goes to infinity. We then use the notation from (5.2), to obtain the equation

(5.5) $$ \begin{align} x_{1,m_1}^{m_1} \prod_{j=1}^{m_1 - 1} x_{1,j}^j + x_{2,m_2}^{m_2} \prod_{j=1}^{m_2 - 1} x_{2,j}^{j} = x_{3,m_3}^{m_3} \prod_{j=1}^{m_3 - 1} x_{3,j}^j. \end{align} $$

By convention, we have that $x_{i,j}$ is square-free for $1 \leq i \leq 3$ and $1 \leq j \leq m_i - 1$ . Thus, (5.4) is equal to

$$\begin{align*}\prod_{j=1}^{m_1 - 1} x_{1,j}^{\frac{m_1 - j}{m_1}} \prod_{j=1}^{m_2 - 1} x_{2,j}^{\frac{m_2 - j}{m_2}} \prod_{j=1}^{m_3 - 1} x_{3,j}^{\frac{m_3 - j}{m_3}}.\end{align*}$$

Viewing these products as coefficients in (5.5), we see that if $H^0$ is to be bounded, these coefficients must be bounded. Therefore, it suffices to check that for a fixed triple of integers $a,b,c$ , the equation

$$\begin{align*}ax^{m_1} + b y^{m_2} + c z^{m_3} = 0\end{align*}$$

has finitely many primitive integer solutions when $1/m_1 + 1/m_2 + 1/m_3 < 1$ . But this is exactly the content of Darmon and Granville’s paper [Reference Darmon and Granville5], so we are done.

6.1 A linear program bound

Observe that for each $1 \leq i \leq n$ ,

$$\begin{align*}\prod_{j=1}^{m_i} z_{i, j} \geq \text{rad}\left( \prod_{j=1}^{m_i} z_{i,j} \right).\end{align*}$$

Applying Proposition 6.1 to the binary form

$$\begin{align*}Q_{\mathbf{a}}(x,y) = \prod_{i=1}^n \ell_i(x,y) \end{align*}$$

in conjunction with the above observation, we obtain

(6.4) $$ \begin{align} \prod_{i=1}^n \prod_{j=1}^{m_i} z_{i,j} \geq \operatorname{rad} \left(\prod_{i=1}^n \ell_i(x,y) \right) \gg_{\varepsilon} \max\{|x|, |y|\}^{n - 2 - \varepsilon}. \end{align} $$

Similarly, for each i, we have the bound

(6.5) $$ \begin{align} |\ell_i(x,y)| \ll \max\{|x|, |y|\}. \end{align} $$

Taking logarithms and writing $y_{i,j} = \log |z_{i,j}|$ , we then have an optimization problem:

(6.6) $$ \begin{align} \min \sum_{i=1}^n \frac{1}{m_i} \sum_{j=1}^{m_i - 1} (m_i -j) y_{i,j} \end{align} $$

subject to

(6.7) $$ \begin{align} \sum_{i=1}^n \sum_{j=1}^{m_i} y_{i,j} \geq (n-2 -\varepsilon) \log B \end{align} $$

and

(6.8) $$ \begin{align} \sum_{j=1}^{m_i} j y_{i,j} \ll \log B, \end{align} $$

where $B = \max \{|x|, |y|\}$ . Further, we have $y_{i,j} \geq 0$ for all $i,j$ .

We emphasize that, at this point, integrality no longer plays a role, and neither does the syzygies relating the $z_{i,j}$ ’s. Indeed, we only need to solve the above linear program allowing arbitrary real inputs.

Now put

$$\begin{align*}c_{ij}=\dfrac{m_i-j}{m_i}\end{align*}$$

for $1\leq i\leq n$ and $1\leq j\leq m_i-1$ . Write $c_{i,m_i}=0$ and let ${\mathbf {c}}=(c_{i,j})$ to be the column vector with

$$\begin{align*}{\mathbf{c}}^{T}=[c_{1,1},c_{1,2},\dots,c_{1,m_1-1},0,c_{2,1},c_{2,2},\dots,c_{n,1},\dots c_{n,m_n-1},0].\end{align*}$$

We have that ${\mathbf {c}}\in {\mathbb {R}}^{N}$ where $N=\sum _{i=1}^n n m_i$ .

Let A be the matrix with rows representing the constraints,

(6.9) $$ \begin{align} C_0&:\sum_{i=1}^n \sum_{j=1}^{m_i} y_{i,j} \geq (n-2 -\varepsilon) \log B \end{align} $$

(6.10) $$ \begin{align} C_i&:-\sum_{j=1}^{m_i} j y_{i,j} \gg -\log B.\qquad\quad \end{align} $$

If we have taken ${\mathbf {e}}_{i,j}$ to be a basis of ${\mathbb {R}}^N$ , then we have that the rows of A are given by

(6.11) $$ \begin{align} {\mathbf{a}}_0&=\sum_{i=1}^n\sum_{j=1}^{m_i}{\mathbf{e}}_{i,j}, \end{align} $$

(6.12) $$ \begin{align} {\mathbf{a}}_i&=\sum_{j=1}^{m_i}-j{\mathbf{e}}_{i,j}. \end{align} $$

Finally, let ${\mathbf {b}}$ be the column vector with $n+1$ entries representing the constraints given by (6.7) and (6.8). In other words, we have

(6.13) $$ \begin{align} b_0&=(n-2-\epsilon)\log B,\end{align} $$

(6.14) $$ \begin{align} b_i&=-\log B. \end{align} $$

Our linear programming problem is then the following: Let ${\mathbf {y}}=(y_{ij})$ ordered as above.

(6.15) $$ \begin{align} &\mathrm{Minimize}{:}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\mathbf{c}}^{T}{\mathbf{y}},\\ &\mathrm{subject\ to}{:}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ A{\mathbf{y}}\geq {\mathbf{b}}\ \mathrm{and}\ {\mathbf{y}}\geq 0. \notag \end{align} $$

The dual linear program is

(6.16) $$ \begin{align} &\mathrm{Maximize}{:}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\mathbf{b}}^{T}{\mathbf{x}},\\ &\mathrm{subject\ to}{:}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ A^T{\mathbf{x}}\leq {\mathbf{c}}\ \mathrm{and}\ {\mathbf{x}}\geq 0, \notag \end{align} $$

where ${\mathbf {x}}=[x_0,x_1,...,x_n]$ . We call a vector ${\mathbf {x}}$ dual feasible if $A^T{\mathbf {x}}\leq {\mathbf {c}}$ and vector ${\mathbf {y}}$ primal feasible if $A{\mathbf {y}}\geq {\mathbf {b}}$ . We have the following well-known weak duality statement.

Lemma 6.2 (Weak duality)

Let A be an $m\times n$ matrix with real entries and ${\mathbf {c}}$ an $n\times 1$ real vector and ${\mathbf {b}}$ an $m\times 1$ real vector. Consider the primal linear program

$$ \begin{align*} &\mathrm{Minimize}{:}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\mathbf{c}}^{T},\\ &\mathrm{subject\ to}{:}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ A{\mathbf{y}}\geq {\mathbf{b}}\ \mathrm{and}\ {\mathbf{y}}\geq 0 \end{align*} $$

and the dual linear program

$$ \begin{align*} &\mathrm{Maximize}{:}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\mathbf{b}}^{T}{\mathbf{x}},\\ &\mathrm{subject\ to}{:}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ A^T{\mathbf{x}}\leq {\mathbf{c}}\ \mathrm{and}\ {\mathbf{x}}\geq 0. \end{align*} $$

Let ${\mathbf {y}}$ be any primal feasible vector and ${\mathbf {x}}$ a dual feasible vector. Then

$$\begin{align*}{\mathbf{c}}^{T}{\mathbf{y}}\geq {\mathbf{b}}^{T}{\mathbf{x}}. \end{align*}$$

Proof Let $A=(a_{i,j})$ . Because ${\mathbf {y}}$ is primal feasible, we have $A{\mathbf {y}}\geq {\mathbf {b}}$ . Therefore, for all $1\leq i\leq m$ , we have

$$\begin{align*}\sum_{j=1}^na_{i,j}y_j\geq b_i. \end{align*}$$

Multiplying by $x_i$ and summing over all i, we have

(6.17) $$ \begin{align} \sum_{i=1}^m\sum_{j=1}^na_{i,j}y_jx_i\geq \sum_{i=1}^n b_ix_i={\mathbf{b}}^{T}{\mathbf{x}}. \end{align} $$

On the other hand, because ${\mathbf {x}}$ is dual feasible, we have that $A^T{\mathbf {x}}\leq {\mathbf {c}}$ , so for each $1\leq j\leq n$ , we have

$$\begin{align*}\sum_{i=1}^ma_{j,i}x_i\leq c_j.\end{align*}$$

Multiplying by $y_j$ and summing over all j gives

(6.18) $$ \begin{align} \sum_{j=1}^n\sum_{i=1}^ma_{i,j}x_iy_j\leq \sum_{j=1}^ny_jc_j={\mathbf{c}}^{T}{\mathbf{y}}. \end{align} $$

Combining inequality (6.17) and inequality (6.18) gives

$$\begin{align*}{\mathbf{c}}^{\ T}{\mathbf{y}}\geq \sum_{j=1}^n\sum_{i=1}^ma_{i,j}x_iy_j\geq {\mathbf{b}}^{T}{\mathbf{x}}.\\[-41pt] \end{align*}$$

Returning to our problem, the weak duality theorem tells us that it suffices to find a dual feasible solution ${\mathbf {x}}=[x_0,\dots x_n]$ such that ${\mathbf {b}}^{T}{\mathbf {x}}\geq -\chi (\mathfrak {X})+\epsilon $ . In other words, we seek ${\mathbf {x}}=[x_0,\dots , x_n]$ with

$$ \begin{align*} {\mathbf{b}}^{T}{\mathbf{x}}&=\log B\left((n-2-\epsilon)x_1-\sum_{j=1}^nx_i\right)\geq -\chi(\mathfrak{X})+\epsilon, \\ A^T{\mathbf{x}}&\leq {\mathbf{c}},\\ {\mathbf{x}}&\geq 0. \end{align*} $$

Take ${\mathbf {x}}=[1,\frac {1}{m_1},\frac {1}{m_2},\dots ,\frac {1}{m_n}]$ . We first show that ${\mathbf {x}}$ is dual feasible. In this case, A is an $(n+1)\times \sum _{i=1}^n m_i$ matrix. So a row of $A^T$ is indexed by a pair $(i,j)$ with $1\leq i\leq n$ and $1\leq j\leq m_i$ . We have that the $(i,j)$ entry of $A^T{\mathbf {x}}={\mathbf {x}}^{T}A$ can be computed as

$$\begin{align*}x_0-jx_i.\end{align*}$$

Therefore, to show that ${\mathbf {x}}$ is dual feasible for an arbitrary ${\mathbb {X}}$ , we need that

$$\begin{align*}x_0-jx_i\leq c_{i,j}=\dfrac{m_i-j}{m_i}=1-\dfrac{j}{m_i}.\end{align*}$$

In our case, the $(i,j)$ entry of $A^T{\mathbf {x}}$ is given by

$$\begin{align*}1-\dfrac{j}{m_i}=c_{i,j},\end{align*}$$

so ${\mathbf {x}}$ is dual feasible. We then compute

$$ \begin{align*} {\mathbf{b}}^{T}{\mathbf{x}}&=\log B(n-2-\varepsilon)x_0-\sum_{i=1}^nx_i\log B\\ &=\log B\left(n-2-\varepsilon-\sum_{i=1}^n\dfrac{1}{m_i}\right)\\ &=\log B\left( -(2-\sum_{i=1}^n(1-\dfrac{1}{m_i})-\epsilon )\right)\\ &=\log B(-\chi(\mathfrak{X})-\epsilon). \end{align*} $$

Therefore, ${\mathbf {x}}=\left (1,\dfrac {1}{m_1},\dots ,\dfrac {1}{m_n} \right )$ is a dual feasible solution and

$$\begin{align*}{\mathbf{b}}^{T}{\mathbf{x}}=\log B(-\chi(\mathfrak{X})-\epsilon). \end{align*}$$

By the weak duality theorem, we have that

$$\begin{align*}\sum_{i=1}^n \frac{1}{m_i} \sum_{j=1}^{m_i - 1} (m_i -j) y_{i,j}\geq \log B(-\chi(\mathfrak{X})-\epsilon). \end{align*}$$

Exponentiating gives

$$\begin{align*}\prod_{i=1}^n\prod_{j=1}^{m_i-1}z_{i,j}^{\frac{m_i-j}{m_i}}\geq B^{(-\chi(\mathfrak{X})-\epsilon)}. \end{align*}$$

As $B=\max \{\vert x\vert ,\vert y\vert \}$ , we have verified inequality (6.1) and consequently we have that conditional on the $abc$ -conjecture that

$$\begin{align*}\gamma(\mathfrak{X})=\chi(\mathfrak{X})\end{align*}$$

when $\chi (\mathfrak {X})\leq 0$ .

6.2 Proof of Theorem 1.3

One direction of the theorem is provided by Theorem 2.6, which we proved in the previous subsection. It suffices to prove the converse.

Actually, for the converse, we only need the assertion that for any $\kappa> 0$ and $m \geq 4$ , the function $H_{-K_{\mathfrak {X}_m}}({\mathbf {x}}) \cdot H({\mathbf {x}})^\kappa $ has Northcott’s property, where $\mathfrak {X}_m = \mathfrak {X}({\mathbb {P}}^1 : ((0, 1, \infty ), (m, m, m))$ . To see this, let us fix $\varepsilon> 0$ . Choose $0 < \kappa < \varepsilon /3$ and choose $m \in {\mathbb {N}}$ sufficiently large so that

$$\begin{align*}\frac{m-3}{m-1} + \frac{\kappa m}{2(m-1)}> \frac{1}{1 + \varepsilon} .\end{align*}$$

By hypothesis, we have

(6.19) $$ \begin{align} H_{-K_{\mathfrak{X}_m}}({\mathbf{x}}) = \phi_m(x)^{1/m} \phi_m(y)^{1/m} \phi_m(x+y)^{1/m}\max\{\vert x\vert,\vert y\vert\}^{\frac{3}{m}-1} \gg_\kappa \max\{|x|, |y|\}^{1 - \frac{3}{m} - \frac{\kappa}{2} }. \end{align} $$

Trivially, we see that

$$\begin{align*}\phi_m(u) \leq \operatorname{rad}(u)^{m-1}\end{align*}$$

for all $u \in {\mathbb {Z}}$ . Hence, (6.19) implies

(6.20) $$ \begin{align} \operatorname{rad}(x)^{\frac{m-1}{m}} \operatorname{rad}(y)^{\frac{m-1}{m}} \operatorname{rad}(x+y)^{\frac{m-1}{m}} \gg_\kappa \max\{|x|, |y|\}^{1 - \frac{3}{m} + \frac{\kappa}{2}}. \end{align} $$

Since $x,y,x+y$ are pairwise co-prime, we have $\operatorname {rad}(x)\operatorname {rad}(y)\operatorname {rad}(x+y) = \operatorname {rad}(xy(x+y))$ ; hence,

$$\begin{align*}\operatorname{rad}(xy(x+y))^{\frac{m-1}{m}} \gg_\kappa \max\{|x|,|y|\}^{1 - \frac{3}{m} + \frac{\kappa}{2}}.\end{align*}$$

Raising both sides to the $m/(m-1)$ power, we have

$$\begin{align*}\operatorname{rad}(xy(x+y)) \gg_\kappa \max\{|x|,|y|\}^{\frac{m-3}{m-1} + \frac{\kappa m}{2(m-1)}} \gg_\kappa \max\{|x|, |y|\}^{\frac{1}{1+\varepsilon}},\end{align*}$$

by our hypotheses on $m, \kappa $ . It follows that

$$\begin{align*}\operatorname{rad}(xy(x+y))^{1 + \varepsilon} \gg_{\varepsilon} \max\{|x|,|y|\},\end{align*}$$

which is plainly equivalent to the $abc$ -conjecture, provided we adjust the implied constant.

7.1 Crude bound for $N_{\mathbf {m}}(T)$ , with ${\mathbf {m}} = (2,2,m)$

Here, we deal with the case ${\mathbf {m}} = (2,2,m)$ . The Euler characteristic is equal to

$$\begin{align*}\delta({\mathbf{m}}) = 2 - \frac{1}{2} - \frac{1}{2} - 1 + \frac{1}{m} = \frac{1}{m}.\end{align*}$$

The height $H(x,y)$ is given by

$$\begin{align*}H(x,y) = |x_1|^{1/2} |y_1|^{1/2} |z_1^{m-1} \cdots z_{m-1}|^{1/m} \max\{|x_1 x_2^2|, |y_1 y_2^2|\}^{1/m}, \end{align*}$$

where $x = x_1 x_2^2, y = y_1 y_2^2$ and

(7.1) $$ \begin{align} x_1 x_2^2 + y_1 y_2^2 = z_1 z_2^2 \cdots z_{m-1}^{m-1} z_m^m,\end{align} $$

with $x_1, y_1, z_1, \ldots , z_{m-1}$ square-free. We normalize the height by raising it to the mth power, obtaining the bound

(7.2) $$ \begin{align} |x_1 y_1|^{m/2} |z_1^{m-1} \cdots z_{m-1}| \max\{|x_1 x_2^2|, |y_1 y_2^2|\} \leq T. \end{align} $$

From here, we see that

$$\begin{align*}|z_1 z_2^2 \cdots z_m^m| \ll \max\{|x_1 x_2^2|, |y_1 y_2^2|\} \ll \frac{T}{|x_1 y_1|^{m/2} |z_1^{m-1} \cdots z_{m-1}|}, \end{align*}$$

whence we conclude that

$$\begin{align*}|z_m| \ll \frac{T^{1/m}}{|x_1 y_1|^{1/2} |z_1 \cdots z_{m-1}|}.\end{align*}$$

This bound and $|z_m| \geq 1$ implies that

$$\begin{align*}|x_1 y_1|^{1/2} |z_1 \cdots z_{m-1}| \ll T^{1/m}.\end{align*}$$

From here, we obtain a crude upper bound for $N_{\mathbf {m}}(T)$ , which proves Theorem 2.7. Indeed, having chosen $x_1, y_1, z_1, \ldots , z_{m-1}$ , there are then $O(T^{1/m}/(|x_1 y_1|^{1/2} |z_1 \cdots z_{m-1}|)$ possibilities for $z_m$ . Having chosen $z_m$ as well, there are then $O_{\varepsilon } (T^\varepsilon )$ possibilities for $x_2, y_2$ , since $x_2, y_2$ are polynomially bounded, so they will be determined by the norm-equation (7.1) up to a log factor. Thus, there are

(7.3) $$ \begin{align} \sum_{|x_1 y_1|^{1/2} |z_1 \cdots z_{m-1}| \leq T^{1/m}} O_{\varepsilon} \left(\frac{T^{1/m + \varepsilon}}{|x_1 y_1|^{1/2} |z_1 \cdots z_{m-1}|} \right) \end{align} $$

possible solutions to (7.1) satisfying the height bound (7.2). We evaluate this as

$$ \begin{align*} & \sum_{|x_1 y_1|^{1/2} |z_1 \cdots z_{m-1}| \leq T^{1/m}} O_{\varepsilon} \left(\frac{T^{1/m + \varepsilon}}{|x_1 y_1|^{1/2} |z_1 \cdots z_{m-1}|} \right) \\ & \quad = \sum_{|x_1 y_1| \leq T^{2/m}} \frac{1}{|x_1 y_1|^{1/2}} \sum_{|z_1 \cdots z_{m-1}| \leq T^{1/m}/|x_1 y_1|^{1/2}} O_{\varepsilon} \left(\frac{T^{1/m + \varepsilon}}{|z_1 \cdots z_{m-1}|} \right) \\ & \quad \ll_{\varepsilon} \sum_{|x_1 y_1| \leq T^{2/m}} \frac{T^{1/m + \varepsilon}}{|x_1 y_1|^{1/2}} \ll_{\varepsilon} T^{2/m + \varepsilon}. \end{align*} $$

To give a lower bound, we choose square-free integers $a,b,c$ so that the curve

$$\begin{align*}ax^2 + by^2 = cz^m\end{align*}$$

has a primitive integral solution. Such a triple is guaranteed to exist (see [Reference Beukers1]). Then we can parametrize (some) of the solutions by a triple of integral binary forms $(F,G, h)$ where $\deg F = \deg G = m$ and $\deg h = 2$ . By

$$\begin{align*}x = F(u,v), y = G(u,v), z = h(u,v).\end{align*}$$

The height is

$$\begin{align*}|a|^{m/2} |b|^{m/2} |c|^{m-1} \max\{|ax^2|, |by^2|\},\end{align*}$$

so if we treat $a,b,c$ as constants, then

$$\begin{align*}\max\{|x|, |y|\} \ll_{a,b,c} T^{1/2}.\end{align*}$$

Therefore, we are looking for solutions to the Thue inequality

$$\begin{align*}\max\{|F(u,v)|, |G(u,v)|\} \ll_{a,b,c} T^{1/2}.\end{align*}$$

If we restrict $u,v$ so that

$$\begin{align*}\max\{|u|, |v|\} \ll_{a,b,c} T^{1/(2m)},\end{align*}$$

then we see that the above height bound is satisfied. Thus, $N_m(T) \gg T^{1/m}$ .

7.2 Proof of Theorem 2.8

In this section, we prove Theorem 2.8. To do so, we will show that $N_2(T) = O \left (T^{1/2} (\log T)^3 \right )$ and give a separate argument to show that $N_2(T) \gg T^{1/2} (\log T)^3$ . The incompatibility of these two arguments represents the main obstacle as to why an asymptotic formula for $N_2(T)$ remains elusive.

We count rational points of bounded height on the curve $\mathfrak {X}({\mathbb {P}}^1_{\mathbb {Q}};(0,2), (-1,2),(\infty ,2))$ with the height on ${\mathbb {P}}^1$ given by (2.7). On writing

$$\begin{align*}a = x_1 y_1^2, b = x_2 y_2^2, x_1, x_2 \text{ square-free}\end{align*}$$

(note that this differs from the notation used elsewhere in the paper), we then have

$$\begin{align*}H(a,b) = |x_1 x_2| \operatorname{sqf}(x_1 y_1^2 + x_2 y_2^2) \max\{|x_1 y_1^2|, |x_2 y_2^2|\},\end{align*}$$

and the max on the right-hand side is dependent only on the relative size of $|a|,|b|$ . If we write

(7.4) $$ \begin{align} x_1 y_1^2 + x_2 y_2^2 = x_3 y_3^2,\end{align} $$

then we further obtain the expression

$$\begin{align*}H(a,b) = \max\{|x_2 x_3 (x_1 y_1)^2|, |x_1 x_3 (x_2 y_2)^2|\}.\end{align*}$$

We may assume without loss of generality that $|x_1 y_1^2| \geq |x_2 y_2|^2$ and $x_1> 0$ , so that

$$\begin{align*}H(a,b) = |x_2 x_3 (x_1 y_1)^2|.\end{align*}$$

We consider the problem of counting integral points on the variety defined by (7.4), subject to the constraint

(7.5) $$ \begin{align} 0 < |x_2 x_3 (x_1 y_1)^2| \leq T, |x_1 y_1^2| \geq |x_2 y_2^2|. \end{align} $$

To obtain the upper bound, we must dissect (7.5) into suitable ranges. When $|x_1 x_2 x_3| \leq T^{1/2}$ , we fix $x_1, x_2, x_3$ and treat (7.4) as a diagonal ternary quadratic form, say $Q_{\mathbf {x}}$ . It is then the case that

(7.6) $$ \begin{align} |y_i| \leq \frac{T}{|x_1 x_2 x_3| \cdot |x_i|}\end{align} $$

for $i = 1,2,3$ , and by Corollary 2 of [Reference Browning and Heath-Brown3], we then have the estimate

$$\begin{align*}O \left(d(x_1 x_2 x_3) \left(\frac{T^{1/2}}{|x_1 x_2 x_3|} + O(1) \right) \right) \end{align*}$$

for the number of ${\mathbf {y}} \in {\mathbb {Z}}_{\ne 0}^3$ satisfying (7.5) and (7.4) provided that the quadratic form $Q_{\mathbf {x}}$ has a rational zero. Otherwise, it is clear that there will be no contribution. Thus, we must estimate

$$\begin{align*}\sum_{\substack{1 \leq |x_1 x_2 x_3| \leq T^{1/2} \\ Q_{\mathbf{x}} \text{ has a rational zero}}} d(x_1 x_2 x_3). \end{align*}$$

This is similar to the work of Guo in [Reference Guo12], except he counted with respect to the height $\lVert {\mathbf {x}} \rVert _\infty $ . Nevertheless, the techniques are similar, and again this may be of independent interest.

Next, we must deal with the case when $|x_1 x_2 x_3| \geq T^{1/2}$ . For this, it suffices to observe from (7.6) that $|x_1 x_2 x_3| \geq T^{1/2}$ implies

$$\begin{align*}|y_1 y_2 y_3| \leq \frac{T^{3/2}}{(x_1 x_2 x_3)^2} \leq T^{1/2}.\end{align*}$$

We then treat (7.4) as a linear form $L_{\mathbf {y}}$ in ${\mathbf {x}}$ . We use this to show that the contribution for each ${\mathbf {y}}$ is $O\left (T^{1/2} |y_1 y_2 y_3|^{-1} + 1 \right )$ , which gives an acceptable contribution upon summing over ${\mathbf {y}}$ .

For the lower bound, we first restrict $y_1, y_2, y_3 \in {\mathbb {Z}}_{\ne 0}$ satisfying

$$\begin{align*}|y_1 y_2 y_3| \leq T^\delta \end{align*}$$

for some explicit $\delta> 0$ to be specified later. We note that to obtain the correct order of magnitude, it is permissible to choose any $\delta> 0$ .

Having fixed ${\mathbf {y}} = (y_1, y_2, y_3)$ , we consider the simultaneous conditions (7.4) and (7.5). This gives rise to a binary form inequality of the shape

(7.7) $$ \begin{align} |x_1^2 x_2 (y_1^2 x_1 + y_2^2 x_2)| \leq T y_3^2 y_1^{-2}. \end{align} $$

Because $|y_1 y_2 y_3|$ is small, we can count the number of solutions ${\mathbf {x}}$ to this inequality with reasonable precision. However, even with $|y_1 y_2 y_3|$ counting the number of solutions ${\mathbf {x}}$ with enough uniformity appears to still be a challenging task, because the binary form in (7.7) is singular. This difficulty is exacerbated by the fact that we will need to apply a square-free sieve eventually to produce triples ${\mathbf {x}}$ with each coordinate square-free.

To get around this issue, we simply count solutions to (7.7) with $x_1, x_2$ satisfying the inequalities

$$\begin{align*}|x_i y_i^2| \leq c_i T^{1/4} |y_1 y_2 y_3|^{1/2}, i = 1,2\end{align*}$$

for some positive numbers $c_1, c_2$ . This has the effect that the long cusps inherent in (7.7) are removed, and reduce the problem to a more straightforward geometry of numbers question.

7.2.1 Upper bounds

To obtain upper bounds, it is crucial to view (7.4) as a plane in $x_1, x_2, x_3$ when $|y_1 y_2 y_3| \leq T^{1/2}$ and viewing (7.4) as a conic in $y_1, y_2, y_3)$ when $|x_1 x_2 x_3| \leq T^{1/2}$ . We call the former the linear case and the latter the quadratic case. We proceed to deal with the linear case below.

We shall first suppose that $|y_1 y_2 y_3| \leq T^{1/2}$ is fixed, and count the triples $(x_1, x_2, x_3)$ and $(y_1, y_2, y_3)$ for which (7.4) holds.

The key is the following lemma on counting points in sublattices of ${\mathbb {Z}}^2$ .

Lemma 7.1 Let $\Lambda \subset {\mathbb {Z}}^2$ be a lattice. Then, for all positive real numbers $R_1, R_2$ , the number of primitive integral points ${\mathbf {x}} \in \Lambda $ satisfying $|x_i| \leq R_i, i = 1,2$ is at most $O \left (R_1 R_2/\det (\Lambda ) + 1 \right )$ .

Proof If the rectangle $[-R_1, R_2] \times [-R_2, R_2]$ contains at least two primitive vectors in $\Lambda $ , say ${\mathbf {x}}_1, {\mathbf {x}}_2$ , then since this rectangle is convex it contains the parallelogram with end points $\pm {\mathbf {x}}_1, \pm {\mathbf {x}}_2$ . The area of this parallelogram is at least as large as $\det \Lambda $ , since the lattice spanned by ${\mathbf {x}}_1, {\mathbf {x}}_2$ is a sublattice of $\Lambda $ . It thus follows that

$$\begin{align*}R_1 R_2 \gg \det \Lambda.\end{align*}$$

Otherwise, the rectangle $[-R_1, R_1] \times [-R_2, R_2]$ contains at most one primitive vector in $\Lambda $ . This completes the proof.

The strength of this lemma is that it gives a strong upper bound even in lopsided boxes.

Given (7.4), it follows that there is at least one $i \in \{2,3\}$ such that

$$\begin{align*}|x_i y_i^2|/2 \leq x_1 y_1^2 \leq 2|x_i y_i^2|, \end{align*}$$

whence

$$\begin{align*}\frac{x_1 y_1^2}{2 y_i^{-2}} \leq |x_i| \leq \frac{2 x_1 y_1^2}{ y_i^{2}}.\end{align*}$$

Without loss of generality, we assume that this holds for $i = 2$ . Suppose that $M_1 \leq x_1 < 2M_1$ . By (7.5), we have

$$\begin{align*}|x_3| \leq \frac{T}{|x_2 x_1^2 y_1^2|},\end{align*}$$

whence

$$ \begin{align*} |x_3| & \leq T \cdot \frac{2 y_2^2}{(x_1 y_1^2)(x_1^2 y_1^2)} \\ & \leq \frac{2y_2^2 T}{M_1^3 y_1^4}. \end{align*} $$

Applying Lemma 7.1 to the lattice defined by the congruence $y_1^2 x_1 - y_3^2 x_3 \equiv 0 \pmod {y_2^2}$ which has determinant equal to $y_2^2$ , there are

$$\begin{align*}O \left(M_1 \cdot \frac{T y_2^2}{M_1^3 y_1^4} \cdot \frac{1}{y_2^2} + 1 \right) = O \left(\frac{T}{M_1^2 y_1^4} + 1 \right)\end{align*}$$

possibilities for $x_1, x_3$ , which then determines $x_2 = (y_1^2 x_1 - y_3^2 x_3)/y_2^2$ . Similarly, applying Lemma 7.1 to the lattice defined by $y_1^2 x_1 + y_2^2 x_2 \equiv 0 \pmod {y_3^2}$ , with determinant equal to $y_3^2$ , gives the estimate

$$\begin{align*}O \left(M_1 \cdot \frac{y_1^2 M_1}{y_2^2} \frac{1}{y_3^2} + 1\right) = O \left(\frac{M_1^2 y_1^2}{y_2^2 y_3^2} + 1 \right) \end{align*}$$

for the number of $x_1, x_2$ , which then also determine $x_3$ . The two bounds coincide when

$$\begin{align*}M_1 = \frac{T^{1/4} |y_2 y_3|^{1/2}}{|y_1|^{3/2}}, \end{align*}$$

and we get the bound

$$\begin{align*}O \left(\frac{T^{1/2} |y_2 y_3| y_1^2}{y_2^2 y_3^2 |y_1|^3} + 1 \right) = O \left(\frac{T^{1/2}}{|y_1 y_2 y_3|} + 1\right) \end{align*}$$

for the number of $x_1, x_2, x_3$ given $y_1, y_2, y_3$ . Thus, we obtain an acceptable estimate whenever $|y_1 y_2 y_3| \ll T^{1/2}$ , since

$$ \begin{align*} \sum_{1 \leq |y_1 y_2 y_3| \leq T^{1/2}} \frac{T^{1/2}}{|y_1 y_2 y_3|} + 1 & \ll T^{1/2} \sum_{n \leq T^{1/2}} \frac{d_3(n)}{n} + \sum_{n \leq T^{1/2}} d_3(n) \end{align*} $$

It is well known that

$$\begin{align*}\sum_{n \leq Z} d_3(n) = Z (\log Z)^2 + O(Z \log Z). \end{align*}$$

By partial summation, we have

$$ \begin{align*} \sum_{n \leq Z} \frac{d_3(n)}{n} & = Z^{-1} \sum_{n \leq Z} d_3(n) + \int_1^Z \left(\sum_{n \leq t} d_3(n) \right) \frac{dt}{t^2} \\ & \ll (\log Z)^2 + \int_1^Z \frac{(\log t)^2 dt}{t} \\ & \ll (\log Z)^3. \end{align*} $$

It follows that

$$\begin{align*}T^{1/2} \sum_{n \leq T^{1/2}} \frac{d_3(n)}{n} + \sum_{n \leq T^{1/2}} d_3(n) \ll T^{1/2} (\log T)^3.\end{align*}$$

It remains to deal with the case when $|y_1 y_2 y_3| \gg T^{1/2}$ , where we instead fiber over ${\mathbf {x}}$ and consider zeros of the corresponding diagonal quadratic forms $Q_{\mathbf {x}}$ . Since

$$\begin{align*}|x_i y_i^2| \ll x_1 y_1^2 \end{align*}$$

for $i = 1,2$ by assumption, it follows that

$$\begin{align*}|x_1 x_2 x_3 y_1^2 y_2^2 y_3^2| \leq x_1^3 y_1^6;\end{align*}$$

hence,

$$\begin{align*}|y_1^2 y_2^2 y_3^2| \ll \frac{x_1^3 y_1^6}{x_1 |x_2 x_3|}.\end{align*}$$

If $|x_1 x_2 x_3| \gg T^{1/2}$ , then

$$\begin{align*}x_1^3 y_1^6 \gg T^{3/2} \Leftrightarrow x_1 y_1^2 \gg T^{1/2}.\end{align*}$$

This implies that

$$\begin{align*}|x_1 x_2 x_3| \cdot x_1 y_1^2 \gg T,\end{align*}$$

which violates (7.5) if the implied constants are sufficiently large. It thus follows that we must have $|x_1 x_2 x_3| \ll T^{1/2}$ in this case.

We now fix $x_1, x_2, x_3$ and consider (7.4) as a ternary quadratic form in $y_1, y_2, y_3$ . We shall require the following version of Corollary 2 in [Reference Browning and Heath-Brown3], which is an analogue of Lemma 7.1.

Lemma 7.2 Let $x_1, x_2, x_3$ be pairwise co-prime square-free integers. Let $R_1, R_2, R_3$ be positive real numbers. Then the number of primitive solutions $y_1, y_2, y_3$ to the equation

$$\begin{align*}x_1 y_1^2 + x_2 y_2^2 = x_3 y_3^2\end{align*}$$

with $|y_i| \leq R_i$ is bounded by

$$\begin{align*}O \left( d(x_1 x_2 x_3) \left( \left(\frac{R_1 R_2 R_3}{|x_1 x_2 x_3|} \right)^{1/3} + 1 \right) \right).\end{align*}$$

Since $|x_i y_i^2| \ll x_1 y_1^2$ for $i = 1,2$ , it follows that

$$\begin{align*}|x_1 x_2 x_3 (x_i y_i^2)| \ll |x_1 x_2 x_3 (x_1 y_1^2)| \leq T\end{align*}$$

for $i = 1,2$ , whence

$$\begin{align*}|(x_1 y_1)^2 x_2 x_3|, |(x_2 y_2)^2 x_1 x_3|, |(x_3 y_3)^2 x_1 x_2| \ll T.\end{align*}$$

This implies that

$$\begin{align*}(y_1 y_2 y_3)^2 (x_1 x_2 x_3)^4 \ll T^3;\end{align*}$$

hence

$$\begin{align*}|y_1 y_2 y_3| \ll \frac{T^{3/2}}{(x_1 x_2 x_3)^2}.\end{align*}$$

Lemma 7.2 then implies that for fixed $x_1, x_2, x_3$ , the number of primitive ${\mathbf {y}} = (y_1, y_2, y_3)$ satisfying (7.4) is

$$\begin{align*}O \left(d(x_1 x_2 x_3) \left(\frac{T^{1/2}}{|x_1 x_2 x_3|} + 1 \right) \right). \end{align*}$$

We now sum over primitive ${\mathbf {x}} \in {\mathbb {Z}}^3$ satisfying $|x_1 x_2 x_3| \ll T^{1/2}$ , with the property that the quadratic form $Q_{\mathbf {x}}$ given by (7.4) has a rational zero. By the Hasse–Minkowski theorem, this is tantamount to the form $Q_{\mathbf {x}}({\mathbf {y}}) = x_1 y_1^2 + x_2 y_2^2 - x_3 y_3^2$ being everywhere locally soluble. The estimation of this is interesting on its own right and will be handled in a separate subsection.

7.2.2 Counting soluble ternary quadratic forms

In this section, we consider the set

$$\begin{align*}{\mathcal{S}} = &\{(x_1, x_2, x_3) \in {\mathbb{Z}}^3 : x_1, x_2, x_3> 0, \gcd(x_1, x_2) = \gcd(x_1, x_3) = \gcd(x_2, x_3) = 1,\\& \quad x_i \text{ square-free for } i = 1,2,3, x_1 y_1^2 + x_2 y_2^2 - x_3 y_3^2 \text{ is everywhere locally soluble}\}. \end{align*}$$

By a well-known theorem of Legendre (see [Reference Guo12]), the indicator function for ${\mathcal {S}}$ is given by

(7.8) $$ \begin{align} f_{\mathcal{S}}(x_1, x_2, x_3) = \left(2^{-\omega(x_1)} \sum_{a_1 | x_1} \left(\frac{x_2 x_3}{a_1} \right) \right) \left(2^{-\omega(x_2)} \sum_{a_2 | x_2} \left(\frac{x_1 x_3}{a_2} \right) \right) \left(2^{-\omega(x_3)} \sum_{a_3 | x_3} \left(\frac{-x_1 x_2}{a_3} \right) \right). \end{align} $$

We will now combine the ideas given in [Reference Guo12] and those in [Reference Fouvry and Kluners8].

Put

$$ \begin{align*} {\mathcal{S}}(X) & = \sum_{1 \leq x_1 x_2 x_3 \leq X} \sum_{(x_1, x_2, x_3) \in {\mathcal{S}}} \frac{d(x_1 x_2 x_3)}{x_1 x_2 x_3} \\ & = \sum_{1 \leq |x_1 x_2 x_3| \leq X} \frac{d(x_1 x_2 x_3)}{x_1 x_2 x_3} f_{\mathcal{S}}(x_1, x_2, x_3). \end{align*} $$

Since $x_1, x_2, x_3$ are pairwise coprime and square-free, it follows that

$$\begin{align*}d(x_1 x_2 x_3) = 2^{\omega(x_1 x_2 x_3)} = 2^{\omega(x_1)} \cdot 2^{\omega(x_2)} \cdot 2^{\omega(x_3)}, \end{align*}$$

where $\omega (n)$ is the number of distinct prime factors of n. It follows that

(7.9) $$ \begin{align} {\mathcal{S}}(X) &= \sum_{1 \leq x_1 x_2 x_3 \leq X} \frac{2^{\omega(x_1 x_2 x_3)}}{x_1 x_2 x_3} f_{\mathcal{S}}(x_1, x_2, x_3)\\&= \sum_{1 \leq x_1 x_2 x_3 \leq X} \frac{1}{x_1 x_2 x_3} \left(1 + \left(\frac{x_2 x_3}{x_1} \right) \left(\frac{x_1 x_3}{x_2} \right) \left(\frac{-x_1 x_2}{x_3} \right) + \sum_g g(x_1, x_2, x_3)\right),\nonumber \end{align} $$

where g expresses a product of Jacobi symbols. The sum

(7.10) $$ \begin{align} {\mathcal{S}}_1(X) = \sum_{1 \leq |x_1 x_2 x_3| \leq X} \frac{1}{x_1 x_2 x_3} \left(1 + \left(\frac{x_2 x_3}{x_1} \right) \left(\frac{x_1 x_3}{x_2} \right) \left(\frac{-x_1 x_2}{x_3} \right) \right) \end{align} $$

is expected to contribute the main term, while the sum

(7.11) $$ \begin{align} {\mathcal{S}}_2(x) = \sum_{1 \leq x_1 x_2 x_3 \leq X} \frac{1}{x_1 x_2 x_3} \sum_g g(x_1, x_2, x_3) \end{align} $$

is expected to be negligible, due to the cancellation of characters.

By partial summation, we obtain

(7.12) $$ \begin{align} {\mathcal{S}}_i(X) = \frac{1}{X} \Sigma_i(X) + \int_1^X \Sigma_i(t) \frac{t}{t^2}, \end{align} $$

where

$$\begin{align*}\Sigma_1(X) = \sum_{\substack{1 \leq |x_1 x_2 x_3| \leq X \\ x_1 x_2 x_3 \text{ square-free} \\ Q_{(x_1, x_2, x_3)} \text{ is soluble} }} \left(1 + \left(\frac{x_2 x_3}{x_1} \right) \left(\frac{x_1 x_3}{x_2} \right) \left(\frac{x_1 x_2}{x_3} \right) \right) \end{align*}$$

and

$$\begin{align*}\Sigma_2(X) = \sum_{1 \leq x_1 x_2 x_3 \leq X} \sum_g g(x_1, x_2, x_3). \end{align*}$$

Our situation differs from that of Guo in [Reference Guo12] since we are counting over triples with $|x_1 x_2 x_3| \leq X$ rather than $\max \{|x_1|, |x_2|, |x_3|\} \leq X$ , which introduces some difficulties. However, this is exactly analogous to the situation encountered by Fouvry and Kluners in [Reference Fouvry and Kluners8].

Our key proposition will be the following.

Proposition 7.3 We have the asymptotic upper bound

$$\begin{align*}{\mathcal{S}}(X) = O\left((\log X)^3 \right).\end{align*}$$

In fact, we can refine Proposition 7.3 to give an asymptotic formula, but this is unnecessary for our purposes.

We proceed to prove Proposition 7.3 in the remainder of the section. We begin by showing that triples $(x_1, x_2, x_3)$ with $\mu ^2(x_1 x_2 x_3) = 1$ and $\omega (x_1 x_2 x_3)$ large contribute negligibly. To wit, put

$$\begin{align*}{\mathcal{S}}_2^{(r)}(X) = \sum_{\substack{1 \leq x_1 x_2 x_3 \leq X \\ \omega(x_1 x_2 x_3) = r}} \frac{1}{x_1 x_2 x_3} \sum_g g(x_1, x_2, x_3).\end{align*}$$

By the triangle inequality, it is clear that

$$\begin{align*}\left \lvert {\mathcal{S}}_2^{(r)}(X) \right \rvert \ll \sum_{\substack{n \leq X \\ \mu^2(n) = 1, \omega(n) = r}} \frac{d_3(n)}{n}. \end{align*}$$

By partial summation, we have

$$\begin{align*}\sum_{\substack{n \leq X \\ \mu^2(n) = 1, \omega(n) = r}} \frac{d_3(n)}{n} = X^{-1} \sum_{\substack{n \leq X \\ \mu^2(n) = 1, \omega(n) = r}} d_3(n) + \int_1^X \left(\sum_{\substack{n \leq t \\ \mu^2(n) = 1, \omega(n) = r}} d_3(n) \right)\frac{dt}{t^2}. \end{align*}$$

To estimate the latter sum, we will need the following result, which is Lemma 11 in [Reference Fouvry and Kluners8].

Lemma 7.4 There exists an absolute constant $B_0 \geq 1$ such that for every $r \geq 0$ , we have

$$\begin{align*}|\{n \leq X : \omega(n) = r, \mu^2(n) = 1\}| \leq B_0 \cdot \frac{X}{\log X} \cdot \frac{(\log \log X + B_0)^r}{r!}. \end{align*}$$

Applying the lemma, we have for $\Omega = 30 (\log \log X + B_0)$

$$ \begin{align*} \sum_{\substack{n \leq X \\ \mu^2(n) = 1, \omega(n) \geq \Omega}} d_3(n) & \ll \frac{X}{\log X} \sum_{r \geq \Omega} 3^r \cdot \frac{(\log \log X + B_0)^r}{r!} \\ & \ll \frac{X}{\log X} \sum_{r \geq \Omega} \left(\frac{3e (\log \log X + B_0)}{r} \right)^r \\ & \ll \frac{X}{\log X} \sum_{r \geq \Omega} \left(\frac{3e}{10} \right)^r, \end{align*} $$

the final sum a convergent geometric series. Hence,

$$\begin{align*}\sum_{r \geq \Omega} \left(\frac{3e}{10} \right)^r \ll \left(\frac{3e}{10} \right)^{\Omega} \ll \frac{1}{\log X}. \end{align*}$$

We thus conclude that

(7.13) $$ \begin{align} \sum_{r \geq \Omega} \left \lvert S_2^{(r)}(X) \right \rvert & \ll 1 + (\log X)^{-2} + \int_1^X \frac{dt}{t (\log t)^2} \\ & = O(1) \notag \end{align} $$

and is thus negligible.

Note that $x_1, x_2, -x_3$ cannot all be the same sign; otherwise, (7.4) will only have a trivial real solution. Hence, the signs of $(x_1, x_2, x_3)$ must be $(+, +, +)$ , or $(+, -, +)$ , since we assumed $x_1> 0$ and $x_1 y_1^2 \geq |x_2 y_2^2|$ . By rearranging, we must thus assume $x_1, x_2, x_3> 0$ .

We then expand (7.11) by writing $x_i = x_{i1}x_{i2}$ for $i = 1,2,3$ , and

$$\begin{align*}&\sum_{\substack{1 \leq x_1 x_2 x_3 \leq X \\ \mu^2(x_1 x_2 x_3) = 1}} \sum_g g(x_1, x_2, x_3)\\& \quad = \sum_{\substack{(x_{11}x_{12})(x_{21}x_{22})(x_{31}x_{32})\leq X \\ 1 < x_{i1} < x_i \text{ for } 1 \leq i \leq 3}} \left(\frac{x_{21}x_{22} x_{31}x_{32} }{x_{11}} \right) \left( \frac{x_{11} x_{12} x_{31} x_{32}}{x_{21}} \right) \left(\frac{x_{11} x_{12} x_{21} x_{22}}{x_{31}} \right). \end{align*}$$

We now follow the strategy outlined in [Reference Fouvry and Kluners8] and break up the set

$$\begin{align*}\{(x_{11}, x_{12}, x_{21}, x_{22}, x_{31}, x_{32}) \in {\mathbb{N}}^6 : x_{11} x_{12} x_{21} x_{22} x_{31} x_{32} \leq X \} \end{align*}$$

by restricting the $x_{ij}$ ’s to intervals of the form

$$\begin{align*}[A_{ij}, \Delta A_{ij}),\end{align*}$$

where

$$\begin{align*}\Delta = 1 + (\log X)^{-3}.\end{align*}$$

For a given $\mathbf {A} = (A_{11}, A_{12}, A_{21}, A_{22}, A_{31}, A_{32})$ , put

$$\begin{align*}{\mathcal{S}}_2(X; \mathbf{A}) = \sum_{\substack{x_{ij} \in [A_{ij}, \Delta A_{ij}) \\ \mu^2(x_{11} x_{12} x_{21} x_{22} x_{31} x_{32}) = 1 \\ \prod_{i,j} x_{ij} \leq X }} \left(\frac{x_{21}x_{22} x_{31}x_{32} }{x_{11}} \right) \left( \frac{x_{11} x_{12} x_{31} x_{32}}{x_{21}} \right) \left(\frac{x_{11} x_{12} x_{21} x_{22}}{x_{31}} \right). \end{align*}$$

We then have the following lemma.

Lemma 7.5 We have the bound

$$\begin{align*}\sum_{\prod A_{ij} \geq \Delta^{-6} X} \left \lvert {\mathcal{S}}_2(X; \mathbf{A}) \right \rvert = O \left(X (\log X)^{-1} \right). \end{align*}$$

Proof We have

$$ \begin{align*} \sum_{\prod A_{ij} \geq \Delta^{-6} X} \left \lvert {\mathcal{S}}_2(X; \mathbf{A}) \right \rvert & \leq \sum_{\substack{\Delta^{-6} X \leq n \leq X \\ \mu^2(n) = 1}} d_3(n) \\ & \ll \sum_{\Delta^{-6} X \leq n \leq X} 3^{\omega(n)} \\ & \ll (1 - \Delta^{-6} ) X (\log X)^2. \end{align*} $$

By Taylor’s theorem, we have

$$\begin{align*}\Delta^{-6} = (1 + (\log X)^{-3})^{-6} = 1 - 6 (\log X)^{-3} + O \left((\log X)^{-6} \right).\end{align*}$$

The proof then follows.

To proceed, we shall require the following well-known lemma regarding character sums.

Lemma 7.6 (Double Oscillation Lemma)

Let $\{\alpha _n\}, \{\beta _m\}$ be two sequences of complex numbers with each term having absolute value bounded by $1$ . Let $M,N$ be positive real numbers. Then we have

$$\begin{align*}&\sum_{m \leq M} \sum_{n \leq N} \alpha_m \beta_n \mu^2(2m) \mu^2(2n) \left(\frac{m}{n} \right)\\& \quad \ll \min \left\{ \left(M^{-1/2} + (N/M)^{-1/2} \right), \left(N^{-1/2} + (M/N)^{-1/2} \right) \right\} \end{align*}$$

and for every $\varepsilon> 0$ ,

$$\begin{align*}\sum_{m \leq M} \sum_{n \leq N} \alpha_m \beta_n \mu^2(2m) \mu^2(2n) \ll_{\varepsilon} MN \left(M^{-1/2} + N^{-1/2} \right) (MN)^\varepsilon. \end{align*}$$

We will also need the following variant of the Siegel–Walfisz theorem.

Lemma 7.7 Let $\chi _q$ be a primitive character modulo $q \geq 2$ . Then, for every $A> 1$ , we have

$$\begin{align*}\sum_{Y \leq p \leq X} \chi_q(p) = O_A \left(\sqrt{q} \cdot X (\log X)^{-A} \right) \end{align*}$$

uniformly for $X \geq Y \geq 2$ .

We now consider, as in [Reference Fouvry and Kluners8], the quantities

(7.14) $$ \begin{align} X^\dagger = (\log X)^9, X^\ddagger = \exp\left( (\log X)^{1/8} \right). \end{align} $$

We now consider those $\mathbf {A}$ with the property that at most two entries larger than $X^\ddagger $ . We dissect the sum according to the number $r \leq 2$ of terms $A_{ij}$ greater than $X^\ddagger $ . Let n be the product of those $x_{ij}$ which are larger than $X^\ddagger $ , and m the product of the remaining ones. We sum over $\mathbf {A}$ with such properties to obtain

$$ \begin{align*} \sideset{}{^{(2)}} \sum_{\mathbf{A}} |{\mathcal{S}}_2(X; \mathbf{A})| & \leq \sum_{r \leq 2} \sum_{m \leq (X^\ddagger)^{6-r}} \mu^2(m) d_{6-r}(m) \sum_{n \leq X/m} \mu^2(n) d_r(n) \\ & \ll \sum_{r \leq 2} \sum_{m \leq (X^\ddagger)^{6-r}} \mu^2(m) d_{6-r}(m) \left(\frac{X}{m} \right) (\log X)^{r-1} \\ & \ll X \left(\sum_{r \leq 2} (\log X)^{r-1} \right) \left(\sum_{m \leq (X^\ddagger)^{6}} \frac{d_{6}(m)}{m} \right) \\ & \ll X (\log X)\left(\log \exp \left((\log X)^{1/8} \right) \right)^7 \\ & \ll X (\log X)^{15/8}. \end{align*} $$

This is sufficiently small for our purposes.

We may now assume that $A_{ij} \geq X^\ddagger $ for at least three pairs $i,j$ with $1 \leq i \leq 3, 1 \leq j \leq 2$ . We now suppose that there exist $a \ne b$ such that

$$\begin{align*}A_{a,2}, A_{b,1} \geq X^\dagger. \end{align*}$$

The sum over $\mathbf {A}$ satisfying these properties can be bounded by

$$ \begin{align*} \sum_{\mathbf{A}} \left \lvert {\mathcal{S}}_2(X; \mathbf{A}) \right \rvert & \leq \sum_{x_{ij}, (i,j) \ne (a,2), (b,1)} \prod_{(i,j) \ne (a,2), (b,1)} \left \lvert \sum_{x_{a,2}} \sum_{x_{b,1}} \alpha_{(a,2)} \beta_{(b,1)} \left(\frac{x_{a,2}}{x_{b,1}} \right) \right \rvert, \end{align*} $$

where $\alpha , \beta $ have modulus at most one. Lemma 7.6 then applies, and since our variables $x_{a,2}, x_{b,1}$ range over intervals exceeding $X^\dagger $ in length, it follows that

$$\begin{align*}|{\mathcal{S}}_2(X;\mathbf{A})| \ll \left(\prod_{(i,j) \ne (a,2), (b,1)} A_{ij} \left(A_{a,2} A_{b,1} \left(A_{a,2}^{-1/3} + A_{b,1}^{-1/3} \right) \right) \right) \ll X (X^\dagger)^{-1/3} = O \left(X (\log X)^{-3} \right), \end{align*}$$

which is again enough.

Next, consider the family where the two previous conditions do not hold, and in addition there exist $a \ne b$ such that $2 \leq A_{b,1} \leq X^\dagger $ and $A_{a,2}> X^\ddagger $ . Under these conditions, we see that

$$\begin{align*}|{\mathcal{S}}_2(X; \mathbf{A})| \ll \sum_{x_{ij}, (i,j) \ne (a,2), (b,1)} \sum_{x_{a,2}} \left \lvert \sum_{x_{b,1}} \mu^2 \left(\prod_{(i,j) \ne (a,2), (b,1)} x_{ij} \right) \left(\frac{x_{a,2}}{x_{b,1}} \right) \right \rvert, \end{align*}$$

where $A_{ij} \leq x_{ij} \leq \Delta A_{ij}$ and $\omega (x_{ij}) \leq \Omega $ for $1 \leq i \leq 3, 1 \leq j \leq 2$ . Now put $\ell = \omega (x_{a,2})$ , writing

$$\begin{align*}x_{a,2} = p_1 \cdots p_\ell \end{align*}$$

with $p_1 < p_2 < \cdots < p_\ell $ , we obtain

$$\begin{align*}|{\mathcal{S}}_2(X; \mathbf{A})| \ll \sum_{\substack{x_{ij} \\ (i,j) \ne (a,2), (b,1)}} \sum_{x_{b,1}} \sum_{0 \leq \ell \leq \Omega} \left \lvert \sum_{\omega(x_{a,2}) = \ell} \mu^2\left(\prod_{i,j} x_{ij} \right) \left(\frac{x_{a,2}}{x_{b,1}} \right) \right \rvert,\end{align*}$$

the inner sum being bounded by

$$\begin{align*}\sum_{p_1 \cdots p_{\ell-1}} \left \lvert \sum_{p_\ell} \left(\frac{p_\ell}{x_{b,1}} \right) \right \rvert, \end{align*}$$

and $p_1, \ldots , p_\ell $ satisfy $A_{a,2} \leq p_1 \cdots p_{\ell } \leq \Delta A_{a,2}$ . Note that

$$\begin{align*}p_\ell \geq A_{a,2}^{1/\ell} \geq \exp \left((\log X)^{1/9} \right). \end{align*}$$

We may now apply Lemma 7.7 to obtain the bound

$$\begin{align*}\left \lvert \sum_{p_\ell} \left(\frac{p_\ell}{x_{b,1}} \right) \right \rvert \ll_A A_{b,1}^{1/2} \frac{A_{a,2}}{p_1 \cdots p_{\ell-1}} (\log X)^{-A/9} + \Omega, \end{align*}$$

with A arbitrarily large. Note that $p_1 \cdots p_{\ell -1} \leq X$ , and hence

$$\begin{align*}\sum_{p_1 \cdots p_{\ell-1} \leq X} (p_1 \cdots p_{\ell-1})^{-1} \ll \sum_{n \leq X} \frac{1}{n} \ll \log X. \end{align*}$$

Hence,

$$\begin{align*}\sideset{}{^{(3)}} \sum_{\mathbf{A}} |{\mathcal{S}}_2(X; \mathbf{A})| \ll A_{b,1}^{1/2}\prod_{i,j} A_{ij} (\log X)^{-A/9 + 1} \ll X (\log X)^{-A/9 + 11/2}. \end{align*}$$

Choosing A large shows that this contribution is negligible.

The remaining case can be summarized by the following properties:

1. (1) $\prod _{i,j} A_{ij} \leq \Delta ^{-6} X$ .

2. (2) $A_{ij} \geq X^\ddagger $ for at least three pairs of indices $(i,j)$ .

3. (3) If $A_{ij}, A_{k\ell } \geq X^\dagger $ , then $j = \ell $ .

4. (4) If $A_{ij} \leq A_{k \ell }$ with $j \ne \ell $ , then either $A_{ij} = 1$ or $2 \leq A_{ij} \leq X^\dagger $ and $A_{k \ell } < X^\ddagger $ .

We now show that the second option in (4) cannot happen. This will imply that we have accounted for all possibilities for (7.11), and hence reduced our problem to estimating ${\mathcal {S}}_1(X)$ .

Suppose, without loss of generality, that $2 \leq A_{11} \leq X^\dagger $ and $A_{22} < X^\ddagger $ . Since $A_{ij} \geq X^\ddagger $ for at least three pairs of indices $(i,j)$ , one of $A_{12}$ or $A_{32}$ must exceed $X^\ddagger $ . We then have $A_{11} \leq X^\dagger $ and $A_{32}$ , say, exceeds $X^\ddagger $ , which means that our earlier estimation covers this case.

The upshot now is that

(7.15) $$ \begin{align} \Sigma_2(X) \ll_A X (\log X)^{15/8} \end{align} $$

for some $\kappa (A)> 0$ . It follows from (7.12) that

$$ \begin{align*} {\mathcal{S}}_2(X) & = X^{-1} \Sigma_2(X) + \int_1^X \Sigma_2(t) \frac{dt}{t^2} \\ & \ll (\log X)^{15/8} + \int_1^X \frac{(\log t)^{15/8} dt}{t} \\ & = (\log X)^{23/8}, \end{align*} $$

which is sufficiently small for our purposes.

Finally, we may evaluate the main term, which is given by (7.10). By the triangle inequality, we have

$$\begin{align*}{\mathcal{S}}_1(X) \ll \sum_{x_1 x_2 x_3 \leq X} \frac{1}{x_1 x_2 x_3} = \sum_{n \leq X} \frac{d_3(n)}{n},\end{align*}$$

which is $O((\log X)^3)$ . This completes the proof of the proposition.