2.1 (Spherical) triangle groups

We follow [Reference Clark and VoightCV19, Section 2]. For more on this topic, see [Reference MagnusMag74, Chapter II].

Let $a,b,c> 1$ be positive integers. We say that the triple $(a,b,c)$ is ${\color {linkcolor}\textsf {spherical}}$ , ${\color {linkcolor}\textsf {Euclidean}}$ , or ${\color {linkcolor}\textsf {hyperbolic}}$ according as the quantity

$$ \begin{align*} \chi(a,b,c) := \tfrac1a + \tfrac1b + \tfrac1c - 1 \end{align*} $$

is positive, zero, or negative.

Definition 2.1 Given integers $a,b,c> 1$ , the ${\color {linkcolor}\textsf {extended triangle group}}$ $\triangle (a,b,c)$ is defined as the group generated by elements $\delta _0, \delta _1, \delta _\infty , -1$ , with $-1$ central in $\triangle (a,b,c)$ , subject to the relations $(-1)^2 = 1$ and

(10) $$ \begin{align} \delta_0^a = \delta_1^b = \delta_\infty^c = \delta_0\delta_1\delta_\infty = -1. \end{align} $$

Define the ${\color {linkcolor}\textsf {triangle group}} \bar \triangle (a,b,c)$ as the quotient of $\triangle (a,b,c)$ by $\left \{ \pm 1 \right \}$ .

The spherical triangle groups are all finite groups. Moreover, they are all finite subgroups of ${\mathrm {PGL}}_2({\bar {\mathbb {Q}}})$ . These were classified by Klein more than a century ago. By [Reference Clark and VoightCV19, Remark 2.2], reordering the signature $(a,b,c)$ to be nondecreasing $a \leqslant b \leqslant c$ does not affect the isomorphism class of $\bar \triangle (a,b,c)$ .

• • For the ${\color {linkcolor}\textsf {dihedral signatures}} (a,b,c) = (2,2,c)$ with $c\geqslant 2$ , the triangle groups $\bar \triangle (2,2,c)$ are isomorphic to the dihedral group $D_{c}$ with $2c$ elements. In particular, $\bar \triangle (2,2,3)$ is isomorphic to the symmetric group in three letters $S_3$ . The group $\bar \triangle (2,2,2)$ is isomorphic to the Klein four group $C_2 \times C_2$ .

• • For the ${\color {linkcolor}\textsf {tetrahedral signature}} (a,b,c) = (2,3,3)$ , the triangle group $\bar \triangle (2,3,3)$ is isomorphic to $A_4$ ; the group of rotational symmetries of the tetrahedron.

• • For the ${\color {linkcolor}\textsf {octahedral signature}} (a,b,c) = (2,3,4)$ , the triangle group $\bar \triangle (2,3,4)$ is isomorphic to $S_4$ ; the group of rotational symmetries of the octahedron.

• • For the ${\color {linkcolor}\textsf {icosahedral signature}} (a,b,c) = (2,3,5)$ , the triangle group $\bar \triangle (2,3,5)$ is isomorphic to $A_5$ ; the group of rotational symmetries of the icosahedron.

2.2 (Spherical) Belyi maps

By ${\color {linkcolor}\textsf {curve,}}$ we mean a separated scheme of finite type over a field of dimension one. We say that a curve is ${\color {linkcolor}\textsf {nice}}$ if it is smooth, projective, and geometrically irreducible.

Definition 2.6 The ${\color {linkcolor}\textsf {signature}}$ of a geometrically Galois k-Belyi map $\phi \colon Z_k \to \mathbb {P}^1_k$ is the triple $(e_0, e_1, e_\infty ),$ where $e_P$ is the ramification index $e_\phi (z)$ of any critical point $z \in Z_k$ with critical value $P \in \left \{ 0,1,\infty \right \}$ . The ${\color {linkcolor}\textsf {Euler characteristic}}$ of $\phi $ is the quantity

(11) $$ \begin{align} \chi(\phi) := \tfrac{1}{e_0} + \tfrac{1}{e_1} + \tfrac{1}{e_\infty} - 1. \end{align} $$

As a consequence of the Riemann Existence Theorem, there exist Galois Belyi maps of any spherical signature. See [Reference Darmon and GranvilleDG95, Proposition 3.1] and [Reference PoonenPoo05, Lemma 2.5] for a proof of the following proposition.

A crucial fact that we will need later is that for every spherical signature $(a,b,c)$ , there exists a geometrically Galois Belyi map defined over $\mathbb {Q}$ with signature $(a,b,c)$ . The reader may find several examples in the Belyi maps LMFDB beta database [LMF25]. The maps presented in Table 2 are adapted from the parameterizations found in [Reference CohenCoh07, Chapter 14]. The original sources are [Reference BeukersBeu98, Reference EdwardsEdw04].

Table 2

Spherical triangle groups.

Table 2 Long description

The table consists of three columns and five rows including the header.

Column 1 header is open parenthesis a comma b comma c close parenthesis.

Column 2 header is triangle bar open parenthesis a comma b comma c close parenthesis.

Column 3 header is chi open parenthesis a comma b comma c close parenthesis.

Row 1: Column 1 is open parenthesis 2 comma 2 comma c close parenthesis. Column 2 is D sub c. Column 3 is 1 all over c.

Row 2: Column 1 is open parenthesis 2 comma 3 comma 3 close parenthesis. Column 2 is A sub 4. Column 3 is 1 all over 6.

Row 3: Column 1 is open parenthesis 2 comma 3 comma 4 close parenthesis. Column 2 is S sub 4. Column 3 is 1 all over 12.

Row 4: Column 1 is open parenthesis 2 comma 3 comma 5 close parenthesis. Column 2 is A sub 5. Column 3 is 1 all over 30.

3.1 The primitivity defect set

Given $(s,t) \in \mathcal {V}(\mathbb {Z})$ , it does not follow that $\tilde \phi (s,t) = (\phi _0(s,t), \phi _\infty (s,t))\in \mathcal {V}(\mathbb {Z})$ . For example, consider the map

$$\begin{align*}\tilde\phi(s,t) = ((s^2-t^2)^2, (s^2+t^2)^2)\end{align*}$$

arising in the parameterization of Pythagorean triples. When s and t have the same parity, $\gcd \tilde \phi (s,t) = 4$ . In general, $\tilde \phi \colon \mathcal {V}(\mathbb {Z}) \to \mathbb {Z}^2$ and we have the following commutative diagram of sets:

Define the ${\color {linkcolor}\textsf {primitivity\ defect\ set\ of}\ \phi }$ by

(12) $$ \begin{align} \mathcal{D}(\phi) := \left\{ \gcd \tilde\phi(s,t) : (s,t) \in \mathcal{V}(\mathbb{Z}) \right\}. \end{align} $$

The set $\mathcal {D}(\phi )$ is finite. Indeed, let $R(\phi ) \in \mathbb {Z}$ denote the resultant of the homogeneous polynomials $\phi _0$ and $\phi _\infty $ . Then, every primitivity defect divides $R(\phi )$ .

Proof Let $e \in \mathcal {D}(\phi )$ . By definition, there exists $(s,t)\in \mathcal {V}(\mathbb {Z})$ such that $\gcd \tilde \phi (s,t) = e$ . By standard properties of the resultant, we can find polynomials $g_0,g_\infty \in \mathbb {Z}[\textsf {s},\textsf {t}]$ such that

$$\begin{align*}R(\phi) = g_0(\textsf{s},\textsf{t})\cdot \phi_0(\textsf{s},\textsf{t}) + g_\infty(\textsf{s},\textsf{t})\cdot\phi_\infty(\textsf{s},\textsf{t}).\end{align*}$$

By evaluating the expression above at $(\textsf {s},\textsf {t}) = (s,t)$ , we see that $R(\phi )$ is a multiple of e.

For each $e \in \mathcal {D}(\phi )$ , consider the set

$$ \begin{align*} \mathcal{V}(\mathbb{Z})_e := \left\{ (s,t) \in \mathcal{V}(\mathbb{Z}) : \gcd\tilde\phi(s,t) = e \right\}. \end{align*} $$

We have a partition

(13) $$ \begin{align} \mathcal{V}(\mathbb{Z}) = \bigsqcup_{e\in \mathcal{D}(\phi)} \mathcal{V}(\mathbb{Z})_e. \end{align} $$

For each $e\in \mathcal {D}(\phi )$ , consider the subsets

$$ \begin{align*} \mathbb{Z}\cdot \mathcal{V}(\mathbb{Z})_e := \left\{ (ns,nt) : n\in \mathbb{Z}, (s,t)\in \mathcal{V}(\mathbb{Z})_e \right\} \subset \mathbb{Z}^2. \end{align*} $$

From the partition Figure 1 of primitive points, we obtain the partition

(14) $$ \begin{align} \mathbb{Z}^2 = \bigsqcup_{e\in \mathcal{D}(\phi)}\mathbb{Z}\cdot \mathcal{V}(\mathbb{Z})_e. \end{align} $$

Figure 1

Partition $\mathcal {V}(\mathbb {Z}) = \mathcal {V}(\mathbb {Z})_1 \sqcup \mathcal {V}(\mathbb {Z})_4$ with respect to the Galois map $\phi (s:t) = ((s^2-t^2)^2:(s^2+t^2)^2)$ , with primitivity defect set $\mathcal {D}(\phi ) = \left \{ 1,4 \right \}$ .

Figure 1 Long description

The visualization consists of two square panels side-by-side, each containing a grid of light blue and orange-red dots.

* The left panel shows a sparse grid where the majority of the dots are light blue. Orange-red dots are distributed in a symmetric, repeating pattern that forms a larger square lattice within the grid. The density of dots is lower compared to the right panel, with significant white space between individual points.

* The right panel shows a much denser grid of dots. The pattern is more complex, with orange-red and light blue dots alternating to create a textured, woven appearance. A distinct X-shaped cross of light blue dots intersects at the center of the panel, extending toward the four corners. The orange-red dots are concentrated in the quadrants created by this central cross, forming a highly regular and dense mathematical field.

3.2 Proof of Proposition 3.2 and Corollary 3.3

We start with the proof of the asymptotic bounds. We will abbreviate

$$\begin{align*}\max \tilde\phi(s,t) := \max\left\{ |\phi_0(s,t)|, |\phi_\infty(s,t)| \right\}.\end{align*}$$

Proof of Proposition 3.2

We may apply the principle of Lipschitz [Reference DavenportDav51] to obtain

(15) $$ \begin{align} \notag\widetilde M(h) := \#\left\{ (s,t) \in \mathbb{Z}^2 : \max\tilde\phi(s,t) \leqslant h \right\} \\ = {\mathrm{vol}}(\mathcal{R}_1)\cdot h^{2/d} + O\left(h^{1/d}\right), \end{align} $$

where ${\mathrm {vol}}\left ( \mathcal {R}_1 \right )$ is the Lebesgue measure of the compact region $\mathcal {R}_1$ in $\mathbb {R}^2$ given by $\max \left \{ |\phi _0(s,t)|,|\phi _\infty (s,t)| \right \} \leqslant 1$ .

In light of the partition Equation (14), we see that, for each $e\in \mathcal {D}(\phi ),$ the set $\mathbb {Z}\cdot \mathcal {V}(\mathbb {Z})_e$ has a density $\delta _e\in [0,1]$ , and $\sum _{e\in \mathcal {D}(\phi )}\delta _e = 1$ . Moreover, if we define

$$ \begin{align*} \widetilde M_e(h) := \#\left\{ (s,t) \in \mathbb{Z}\cdot \mathcal{V}(\mathbb{Z})_e : \max\tilde\phi(s,t) \leqslant h \right\}, \end{align*} $$

then $\widetilde M_e(h) = \delta _e\cdot \widetilde M(h) + O(1)$ .

We apply a standard Möbius sieve to Equation (15) to obtain, for every $\varepsilon> 0$ , the asymptotic

(16) $$ \begin{align} \notag \widetilde N(h) := \#\left\{ (s,t) \in \mathcal{V}(\mathbb{Z}) : \max\tilde\phi(s,t) \leqslant h \right\} \\ = \frac{6}{\pi^2}\cdot{\mathrm{vol}}(\mathcal{R}_1)\cdot h^{2/d} + O_{e,\varepsilon}\left(h^{1/d+\varepsilon}\right). \end{align} $$

Moreover, if we define

$$ \begin{align*} \widetilde N_e(h) := \#\left\{ (s,t) \in \mathcal{V}(\mathbb{Z})_e : \max\tilde\phi(s,t) \leqslant h \right\}, \end{align*} $$

then $\widetilde N_e(h) = \delta _e\cdot \widetilde N(h) + O(1)$ . Consider the counting function

$$ \begin{align*} N(h) := \#\left\{ (s:t) \in \mathbb{P}^1(\mathbb{Q}) : {\mathrm{Ht}}(\phi(s:t)) \leqslant h \right\}, \end{align*} $$

which counts all $\mathbb {Q}$ -rational points on $\mathbb {P}^1$ with respect to the height ${\mathrm {Ht}}$ pulled back by $\phi $ . In general, we have the inequalities

(17) $$ \begin{align} \tfrac1d \cdot N(h) \leqslant N(\Omega(\phi);h) \leqslant N(h), \end{align} $$

which arise from the fact that a point $Q = \phi (P) \in \Omega (\phi )$ has at least one rational point in the fiber $\phi ^{-1}(Q)$ , and at most $d = \deg \phi $ .

To conclude, we relate $N(h)$ to the counting functions $\widetilde N_e(h)$ . By the definition of ${\mathrm {Ht}}$ , we see that

(18) $$ \begin{align} N(h) &= \frac12 \sum_{e\in\mathcal{D(\phi)}}\widetilde N_e(eh) \notag\\ &= \frac12 \sum_{e \in \mathcal{D}(\phi)}\left( \frac{6}{\pi^2}\cdot{\mathrm{vol}}(\mathcal{R}_1)\cdot\delta_e\cdot (eh)^{2/d} + O\left((eh)^{1/d+\varepsilon}\right) \right) \notag \\ &= \frac{3}{\pi^2}{\mathrm{vol}}(\mathcal{R}_1)\left( \sum_{e\in\mathcal{D}(\phi)} \delta_e\cdot e^{2/d} \right) \cdot h^{2/d} + O\left(h^{1/d+\varepsilon}\right). \end{align} $$

In particular, the leading constant is

(19) $$ \begin{align} \delta(\phi) = \frac{3}{\pi^2}{\mathrm{vol}}(\mathcal{R}_1)\left( \sum_{e\in\mathcal{D}(\phi)} \delta_e\cdot e^{2/d} \right).\\[-34pt]\nonumber \end{align} $$

We will use Proposition 3.2 in the special case of a geometrically Galois $\mathbb {Q}$ -Belyi map $\phi $ .

Proof of Corollary 3.3

Suppose that $\phi $ is geometrically Galois, with Galois group ${\mathrm {Gal}}(\phi ) = {\mathrm {Aut}}(\phi _{\bar {\mathbb {Q}}})$ . Then, ${\mathrm {Gal}}(\phi )$ acts transitively and without stabilizers on the fibers of unramified points $Q\in \mathbb {P}^1(\mathbb {Q})$ . Since there are finitely many points that ramify, they do not influence the asymptotic count, so we ignore them. We claim that for every $Q \in \phi (\mathbb {P}^1(\mathbb {Q})) = \Omega (\phi )$ , we have that

$$ \begin{align*} \# \phi^{-1}(Q)(\mathbb{Q}) = \#{\mathrm{Aut}}(\phi). \end{align*} $$

Indeed ${\mathrm {Aut}}(\phi ) = {\mathrm {Aut}}(\phi _{{\bar {\mathbb {Q}}}})^{{\mathrm {Gal}}_{\bar {\mathbb {Q}}}}$ , and for every $P \in \phi ^{-1}(Q)(\mathbb {Q})$ and $\gamma \in {\mathrm {Aut}}(\phi )$ , we have that $\gamma (P) \in \phi ^{-1}(Q)(\mathbb {Q})$ as well. On the other hand, given $P,P' \in \phi ^{-1}(Q)(\mathbb {Q})$ , there exists $\gamma \in {\mathrm {Aut}}(\phi _{\bar {\mathbb {Q}}})$ such that $\gamma (P') = P$ . For any $\sigma \in {\mathrm {Gal}}_{\mathbb {Q}}$ , we see that $\gamma ^\sigma (P') = \gamma (\sigma ^{-1}P') = \gamma (P')$ . Therefore, $\gamma ^{-1}\gamma ^\sigma $ stabilizes $P'$ , which implies that $\gamma ^{-1}\gamma ^\sigma =1$ , and therefore $\gamma \in {\mathrm {Aut}}(\phi )$ . It follows that $N_\phi (h) = \#{\mathrm {Aut}}(\phi )\cdot N(\Omega (\phi );h)$ , and the proof is complete. In particular, the leading constant is

(20) $$ \begin{align} \kappa(\phi) = \frac{3}{\pi^2}\dfrac{{\mathrm{vol}}(\mathcal{R}_1)}{\#{\mathrm{Aut}}(\phi)}\left( \sum_{e\in\mathcal{D}(\phi)} \delta_e\cdot e^{2/d} \right).\\[-34pt]\nonumber \end{align} $$

Example 3.5 (Pythagorean constant)

In Section 1.4, we concluded that for $F\colon \textsf {x}^2 + \textsf {y}^2 - \textsf {z}^2 = 0$ , we have the identity $\Omega (F) = \Omega (\phi )$ , where $\phi \colon Z := {\mathrm {Proj}} \mathbb {Q}[\textsf {x},\textsf {y},\textsf {z}]/(\textsf {x}^2 + \textsf {y}^2 - \textsf {z}^2) \to \mathbb {P}^1_{\mathbb {Q}}$ is the Galois Belyi map $(x:y:z) \mapsto (x^2:z^2)$ . Take the isomorphism $\mathbb {P}^1 \cong Z$ given by $(s:t) \mapsto (s^2-t^2:2st:s^2+t^2)$ , and rename $\phi $ to be the composition $\mathbb {P}^1 \cong Z \to \mathbb {P}^1$ , $(s:t) \mapsto ((s^2-t^2)^2: (s^2+t^2)^2)$ .

• • Since $\max \left \{ |s^2-t^2|^2, |s^2+t^2|^2 \right \} = (s^2+t^2)^2$ , the region $\mathcal {R}_1$ is the unit disc, and ${\mathrm {vol}}(\mathcal {R}_1) = \pi $ .

• • The primitivity defect set $\mathcal {D}(\phi ) = \left \{ 1,4 \right \}$ . The densities are $\delta _1 = 2/3$ and $\delta _4 = 1/3$ .

Putting this data into Equation (19), we see that

$$ \begin{align*} \delta(\phi) = \frac{3}{\pi^2}\cdot \pi\left( \dfrac{2}{3} + \dfrac{4^{2/4}}{3} \right) = \dfrac{4}{\pi}. \end{align*} $$

Finally, since ${\mathrm {Aut}}(\phi ) \cong G \cong C_2\times C_2$ , we obtain $\kappa (\phi ) = \delta (\phi )/4 = \tfrac {1}{\pi }$ .

4.1 Covering

The covering is a geometrically Galois $\mathbb {Q}$ -Belyi map $\phi \colon \mathbb {P}^1_{\mathbb {Q}} \to \mathbb {P}^1_{\mathbb {Q}}$ with signature $(a,b,c)$ . For instance, we can always start with one of the maps described by the rational functions in Table 3 and, since we are not assuming that $a \leqslant b \leqslant c$ , compose with an appropriate permutation $\gamma \in {\mathrm {PGL}}_2(\mathbb {Q})$ of $\left \{ 0,1,\infty \right \}$ .

Table 3

Examples of geometrically Galois $\mathbb {Q}$ -Belyi maps for the spherical signatures.

Table 3 Long description

The table consists of three columns: open parenthesis a comma b comma c close parenthesis, triangle group delta bar open parenthesis a comma b comma c close parenthesis, and Example.

* Row 1: Signature open parenthesis 2 comma 2 comma c close parenthesis. Group D sub c. Example: numerator open parenthesis s to the c power plus t to the c power close parenthesis squared all over denominator 4 open parenthesis s t close parenthesis to the c power.

* Row 2: Signature open parenthesis 2 comma 3 comma 3 close parenthesis. Group A sub 4. Example: numerator open parenthesis s squared minus 2 s t minus 2 t squared close parenthesis squared open parenthesis s to the 4 power plus 2 s cubed t plus 6 s squared t squared minus 4 s t cubed plus 4 t to the 4 power close parenthesis squared all over denominator 2 to the 6 power t cubed open parenthesis s minus t close parenthesis cubed open parenthesis s squared plus s t plus t squared close parenthesis cubed.

* Row 3: Signature open parenthesis 2 comma 3 comma 4 close parenthesis. Group S sub 4. Example: numerator negative open parenthesis 4 s t close parenthesis squared open parenthesis s squared minus 3 t squared close parenthesis squared open parenthesis s to the 4 power plus 6 s squared t squared plus 81 t to the 4 power close parenthesis squared open parenthesis 3 s to the 4 power plus 2 s squared t squared plus 3 t to the 4 power close parenthesis squared all over denominator open parenthesis s squared plus 3 t squared close parenthesis to the 4 power open parenthesis s to the 4 power minus 18 s squared t squared plus 9 t to the 4 power close parenthesis to the 4 power.

* Row 4: Signature open parenthesis 2 comma 3 comma 5 close parenthesis. Group A sub 5. Example: numerator negative open parenthesis 3 to the 4 power s to the 10 power plus 2 to the 8 power t to the 10 power close parenthesis squared open parenthesis 3 to the 8 power s to the 20 power minus 2 to the 7 3 to the 10 power s to the 15 power t to the 5 power minus 2 to the 18 power 3 to the 10 power s to the 10 power t to the 10 power plus 2 to the 12 power 3 to the 10 power s to the 5 power t to the 15 power plus 2 to the 16 power t to the 20 power close parenthesis squared all over denominator open parenthesis 12 s t close parenthesis to the 5 power open parenthesis 81 s to the 10 power minus 1584 s to the 5 power t to the 5 power minus 256 t to the 10 power close parenthesis to the 5 power.

4.2 Twisting

By [Reference Arango-PiñerosAra25, Lemma 3.23], there exists a finite set of primes $\mathcal {S}$ for which the map $\phi $ admits an R-model $\Phi \colon \mathbb {P}^1_R \to \mathbb {P}^1_R$ such that $\mathbb {P}^1(a,b,c)_R \cong [\mathbb {P}^1_R/{\textbf {Aut}}(\Phi )]$ . Descent theory gives the partition

$$ \begin{align*} \mathbb{P}^1(a,b,c)\langle R \rangle &= \bigsqcup_{\tau\in \mathrm{H}^1(R, {\textbf{Aut}}(\Phi))} \Phi_\tau(\mathbb{P}^1_\tau(R)) \\ & = \bigsqcup_{\tau \in \mathrm{H}^1_{\mathcal{S}}(\mathbb{Q},{\mathrm{Gal}}(\phi))} \phi_\tau(\mathbb{P}^1_\tau(\mathbb{Q})). \end{align*} $$

Here, $\mathrm {H}^1(R,{\textbf {Aut}}(\Phi ))$ denotes the fppf Čech cohomology pointed set. It is in bijection with isomorphism classes of fppf ${\textbf {Aut}}(\Phi )$ -torsor schemes $T \to {\mathrm {Spec}}\ R$ . Restriction to the generic fiber induces an isomorphism

$$ \begin{align*} \mathrm{H}^1(R,{\textbf{Aut}}(\Phi)) \cong \mathrm{H}^1_{\mathcal{S}}(\mathbb{Q},{\mathrm{Gal}}(\phi)) \end{align*} $$

of pointed sets. Note that ${\mathrm {Gal}}(\phi ) \cong {\textbf {Aut}}(\phi )({\bar {\mathbb {Q}}}) = {\mathrm {Aut}}(\phi _{\bar {\mathbb {Q}}})$ , so the action of the absolute Galois group ${\mathrm {Gal}}_{\mathbb {Q}}$ is the natural one. In general, $\mathrm {H}^1_{\mathcal {S}}(\mathbb {Q},{\mathrm {Gal}}(\phi ))$ is only a pointed set and not a group, since ${\mathrm {Gal}}(\phi ) \cong \bar \triangle (a,b,c)$ as abstract groups, and the only abelian spherical triangle group is $\bar \triangle (2,2,2) \cong C_2 \times C_2$ . Crucially, the set $\mathrm {H}^1_{\mathcal {S}}(\mathbb {Q}, {\mathrm {Gal}}(\phi ))$ is finite and classifies twists of the Belyi map $\phi $ . It is worth noting that in some cases, the source curve of a twist $\phi _\tau \colon \mathbb {P}^1_\tau \to \mathbb {P}^1$ might be a pointless conic. Nevertheless, since the equations

$$ \begin{align*} \textsf{x}^2 + \textsf{y}^2 - \textsf{z}^c &= 0, \quad (c \geqslant 2) \\ \textsf{x}^2 + \textsf{y}^3 - \textsf{z}^3 &= 0,\\ \textsf{x}^2 + \textsf{y}^3 - \textsf{z}^4 &= 0,\\ \textsf{x}^2 + \textsf{y}^3 - \textsf{z}^5 &= 0, \end{align*} $$

all have primitive integral solutions, we know that $\Omega (a,b,c) \neq \varnothing $ , and there will always be at least one twist for which $\mathbb {P}^1_\tau (\mathbb {Q}) \neq \varnothing $ .

4.3 Sieving

Combining the partition above with Corollary 3.3, we obtain

$$ \begin{align*} N(\Omega_{\mathcal{S}}(a,b,c); h) = \sum_\tau N(\Omega(\phi_\tau);h), \end{align*} $$

where the sum ranges over all the $\tau \in \mathrm {H}^1_{\mathcal {S}}(\mathbb {Q}, {\mathrm {Gal}}(\phi ))$ for which $\mathbb {P}^1_\tau $ is isomorphic to $\mathbb {P}^1_{\mathbb {Q}}$ . To sieve out the excess of elements in $\mathbb {P}^1(a,b,c)\langle R \rangle $ not corresponding to points in $\Omega (a,b,c) = \mathbb {P}^1(a,b,c)\langle \mathbb {Z} \rangle $ , we show that we can restrict to certain subsets $T(F) \subset T(a,b,c) \subset \mathrm {H}^1_{\mathcal {S}}(\mathbb {Q},{\mathrm {Gal}}(\phi ))$ to cover all of $\Omega (F)$ and $\Omega (a,b,c)$ . The proofs of both Theorems 1.2 and 1.3 (in the special case of simplified equations (Definition 1.5)) will follow immediately from the following lemma.

Lemma 4.2 Fix a possibly empty subset $\mathcal {T} \subset \mathcal {S}$ . Take a $\mathcal {T}$ -simplified Fermat equation $F\colon A\textsf {x}^{a} + B\textsf {y}^{b} + C\textsf {z}^{c} = 0$ . Then, there is a finite subset $T(F) \subseteq \mathrm {H}^1_{\mathcal {S}}(\mathbb {Q},{\mathrm {Gal}}(\phi ))$ such that

(21) $$ \begin{align} \Omega(F) = \bigsqcup_{\tau \in T(F)}\phi_\tau(\mathbb{P}^1_\tau(\mathbb{Q})). \end{align} $$

Moreover, defining $T(a,b,c)$ as the disjoint union of the sets $T(F)$ , as F ranges over all $\varnothing $ -simplified Fermat equations of signature $(a,b,c)$ , we have

(22) $$ \begin{align} \Omega(a,b,c) = \bigsqcup_{\tau \in T(a,b,c)}\phi_\tau(\mathbb{P}^1_\tau(\mathbb{Q})). \end{align} $$

Proof Any geometrically Galois $\mathbb {Q}$ -Belyi map $\phi \colon \mathbb {P}^1_{\mathbb {Q}} \to \mathbb {P}^1_{\mathbb {Q}}$ of signature $(a,b,c)$ is given by a rational function

$$ \begin{align*} \dfrac{\phi_0}{\phi_\infty} = 1 + \dfrac{\phi_1}{\phi_\infty} \in {\textbf k}(\mathbb{P}^1_{\mathbb{Q}}), \end{align*} $$

where

1. (i) $\phi _0, \phi _1,\phi _\infty \in \mathbb {Z}[\textsf {s},\textsf {t}]$ are homogeneous of degree $\#\bar \triangle (a,b,c)$ ,

2. (ii) $\gcd (\phi _0,\phi _\infty ) = \gcd (\phi _1, \phi _\infty ) = 1$ , and

3. (iii) we can write

   $$ \begin{align*} \phi_0(\textsf{s},\textsf{t}) &= C_0\cdot X(\textsf{s},\textsf{t})^a, \\ \phi_1(\textsf{s},\textsf{t}) &= C_1\cdot Y(\textsf{s},\textsf{t})^b,\\ \phi_\infty(\textsf{s},\textsf{t}) &= C_\infty\cdot Z(\textsf{s},\textsf{t})^c, \end{align*} $$
   for unique polynomials $X,Y,Z \in \mathbb {Z}[\textsf {s},\textsf {t}]$ , and a unique triple $(C_0,C_1,C_\infty )$ of $\mathcal {S}(\phi )$ -simplified Fermat coefficients, where $\mathcal {S}(\phi )$ is an explicit set of bad primes.

We denote this triple by $\textbf {sfc}(\phi )$ . Observe that for any $Q\in \mathbb {P}^1(\mathbb {Q})$ , we have that $\textbf {sfc}(\phi ) = \textbf {sfc}(\phi (Q))$ .

Returning to the situation of this section, to each cohomology class $\tau ,$ we can associate the $\mathcal {S}$ -simplified Fermat coefficient triple $ \textbf {sfc}(\phi _\tau )$ . If F is $\mathcal {T}$ -simplified, then it is also $\mathcal {S}$ -simplified. Moreover, for every primitive integral solution $(x,y,z)$ to F, the point $j(x,y,z) \in \mathbb {P}^1(\mathbb {Q})$ is in $\Omega _{\mathcal {S}}(a,b,c)$ . Define

$$ \begin{align*} T(F) := \left\{ \tau \in \mathrm{H}^1_{\mathcal{S}}(\mathbb{Q},{\mathrm{Gal}}(\phi)) : \textbf{sfc}(\phi_\tau) = (A,B,C) \right\}.\\[-34pt] \end{align*} $$

To finish the proof of Theorem 1.3, we must consider the case of non-simple equations. To guide our intuition, consider the equation $F'\colon 25\textsf {x}^2 + \textsf {y}^2 = \textsf {z}^2$ . Our strategy is to use the simplification $F \colon \textsf {x}^2 + \textsf {y}^2 = \textsf {z}^2$ to deduce the asymptotic result for $F'$ from that of F. In this case, the $\mathbb {Q}$ -isomorphism of nice curves $C \to C', \quad (x:y:z) \mapsto (x/5:y:z)$ enables this translation. The idea is that the congruence condition $x \equiv 0 \operatorname{mod} 5$ cuts out a positive proportion of the primitive integral solutions to the Pythagorean equation, and only the constant term in the asymptotic will change.

Start with a non-simple equation $F'\colon A'\textsf {x}^a + B'\textsf {y}^b + C'\textsf {z}^c = 0$ . Without loss of generality, we may assume that $\gcd (A',B',C') = 1$ . In this case, we can write

$$ \begin{align*} A' = A\cdot A_0^a, \quad B' = B\cdot B_1^b, \quad C' = C\cdot C_\infty^c, \end{align*} $$

to obtain a $\mathcal {T}$ -simplified coefficient triple $(A,B,C)$ , where $\mathcal {T}$ is the set of primes dividing $A'\cdot B'\cdot C'$ . The Fermat equation $F\colon A\textsf {x}^{a} + B\textsf {y}^{b} + C\textsf {z}^{c} = 0$ is the ${\color {linkcolor}\textsf {simplification}}$ of $F'$ . From Lemma 4.2, we have a partition

(23) $$ \begin{align} \Omega(F) = \bigsqcup_{\tau \in T(F)} \Omega(\phi_\tau), \end{align} $$

where each $\phi _\tau $ is a geometrically Galois $\mathbb {Q}$ -Belyi map $\mathbb {P}^1_{\mathbb {Q}} \to \mathbb {P}^1_{\mathbb {Q}}$ of signature $(a,b,c)$ . Let $\phi $ be one of these maps. We have seen that $\phi $ corresponds to a rational function

$$ \begin{align*} \phi = \dfrac{A\cdot X(\textsf{s},\textsf{t})^a}{C\cdot Z(\textsf{s},\textsf{t})^c} = 1 + \dfrac{B\cdot Y(\textsf{s},\textsf{t})^b}{C\cdot Z(\textsf{s},\textsf{t})^c}. \end{align*} $$

To conclude, we use a clever argument of Beukers [Reference BeukersBeu98, Proof of Theorem 1.5]. Consider the polynomial map

(24) $$ \begin{align} \alpha\colon \mathbb{Q}^2 \to \mathbb{Q}^3, \quad (s,t) \mapsto \left( \dfrac{X(s,t)}{A_0}, \dfrac{Y(s,t)}{B_1}, \dfrac{Z(s,t)}{C_\infty} \right). \end{align} $$

We use $\alpha $ to define a lattice of rank two generated by the points whose image is integral

$$ \begin{align*} \Lambda(\alpha) := \mathrm{Span}_{\mathbb{Z}}\left\{ (s,t)\in \mathbb{Q}^2 : \alpha(s,t)\in \mathbb{Z}^3 \right\}. \end{align*} $$

Choose an integral basis $\left \{ \vec \alpha _1, \vec \alpha _2 \right \}$ for $\Lambda (\alpha )$ , and define

$$ \begin{align*} \phi' = \dfrac{A\cdot X(\textsf{s}\vec\alpha_1 + \textsf{t}\vec\alpha_2)^a}{C\cdot Z(\textsf{s}\vec\alpha_1 + \textsf{t}\vec\alpha_2)^c} = 1 + \dfrac{B\cdot Y(\textsf{s}\vec\alpha_1 + \textsf{t}\vec\alpha_2)^b}{C\cdot Z(\textsf{s}\vec\alpha_1 + \textsf{t}\vec\alpha_2)^c}. \end{align*} $$

Applying this construction to every $\phi _\tau $ appearing in Equation (23), we obtain the partition

$$ \begin{align*} \Omega(F') = \bigsqcup_{\tau \in T(F)} \Omega(\phi^{\prime}_\tau), \end{align*} $$

from which we conclude the proof.