is nonzero modulo $p$. This shows that $D^{*}$ can indeed not have vertical
prime divisors, which completes the proof of the lemma. ∎

Next, we combine the above result with Lemmas 11.8, 11.9 and 11.10 to compute
the divisors $D_{i}$ in terms of the modular curves $gC$ constructed in
$\mathsection$11.3.2.

######  Lemma 11.12.

The following statements hold.

  * (i)

The support of $D_{i}$ is the union of four disjoint $gC$ with
$g\in\textnormal{SL}_{2}(\mathcal{O}/2\mathcal{O})$.

  * (ii)

The intersection $\cap_{i}D_{i}$ is empty.

###### Proof.

To prove (i) we first determine the support of $D_{i}$ over $\mathbb{Q}$. In
Lemma 11.10 we computed the constant term of the Fourier expansion of our
Eisenstein series $E_{i}$ in each cusp $j$ of $M_{2}^{*}(\mathbb{C})$: It is
nonzero if and only if $j=i$. Thus our $E_{i}$ is up to a constant factor the
Eisenstein series $E_{i+1}$ in [vdG88, p.190]. Then [vdG88, p.190] implies

$\textnormal{supp}D_{i,\mathbb{C}}=\cup F^{ij}_{\mathbb{C}}$ (11.18)

since by construction the Eisenstein series $E_{i}$ corresponds to the global
section of $\pi_{*}\omega^{\otimes 2}$ defining the divisor $D_{i}$. Here the
union is taken over the four cusps $j\neq i$ and we recall from (11.12) that
the curve $F_{1}(\mathbb{C})=\cup F^{ij}_{\mathbb{C}}$ has 10 irreducible
components $F^{ij}_{\mathbb{C}}$. Lemma 11.9 gives
$g_{ij}\in\textnormal{SL}_{2}(\mathcal{O}/2\mathcal{O})$ such that
$F^{ij}_{\mathbb{C}}=(g_{ij}C)_{\mathbb{C}}$. Then (11.18) leads to

$\textnormal{supp}D_{i,\mathbb{Q}}=\cup F^{ij}_{\mathbb{Q}},\quad
F^{ij}=g_{ij}C$ (11.19)

with the union taken over the four cusps $j\neq i$. We now compute the closure
inside $M_{2}$. As the moduli stack $\mathcal{E}_{2}$ of elliptic curves with
symplectic level 2-structures is irreducible and $F^{ij}=g_{ij}C$ is the
scheme-theoretic image of $\pi\tau_{g_{ij}}\phi:\mathcal{E}_{2}\to M_{2}$, we
obtain that $F^{ij}=g_{ij}C$ is an integral closed subscheme of $M_{2}$ with
generic fiber $F^{ij}_{\mathbb{Q}}$ nonempty. Hence $F^{ij}$ is the closure of
$F^{ij}_{\mathbb{Q}}$ in $M_{2}$. Further, the closure of
$\textnormal{supp}D_{i,\mathbb{Q}}$ in $M_{2}$ is given by
$\textnormal{supp}D_{i}$ since each prime divisor of $D_{i}$ is the closure of
its generic fiber by Lemma 11.11. Thus (11.19) shows

$\textnormal{supp}D_{i}=\cup F^{ij}$ (11.20)

with the union taken over the four cusps $j\neq i$. Next, we determine the
intersection $F^{ij}\cap F^{kl}$ for arbitrary cusps $i,j,k,l$. We claim that
either $F^{ij}\cap F^{kl}$ is empty or $F^{ij}=F^{kl}$. To prove this claim,
we assume that $F^{ij}\cap F^{kl}$ is nonempty. Then Lemma 11.8 gives that
$g=g_{kl}^{-1}g_{ij}$ lies in the image of
$\textnormal{GL}_{2}(\mathbb{Z}/2\mathbb{Z})$ in
$\textnormal{GL}_{2}(\mathcal{O}/2\mathcal{O})$ via
$\mathbb{Z}/2\mathbb{Z}\otimes\mathcal{O}\cong\mathcal{O}/2\mathcal{O}$. It
follows that $gC=C$, since $C$ is the scheme-theoretic image of the morphism
$\pi\phi:\mathcal{E}_{2}\to M_{2}$ and $\phi$ is induced by tensoring with
$\mathcal{O}$. This implies that $F^{ij}=g_{ij}C=g_{kl}C=F^{kl}$ as claimed.
We now consider the four subschemes $F^{ij}\subset M_{2}$ appearing in
(11.20). They are pairwise distinct since their base changes
$F^{ij}_{\mathbb{C}}$ are pairwise distinct by (11.12), and thus the claim
below (11.20) assures that these four $F^{ij}$ are disjoint. This completes
the proof of (i).

It remains to prove (ii). Suppose that (ii) does not hold, that is $\cap
D_{i}$ is nonempty. Then (11.20) gives that $\cap_{i}\cup F^{ij}$ is nonempty
and hence there are $j_{i}\neq i$ such that $\cap_{i}F^{ij_{i}}$ is nonempty.
But, as the five $F^{ij_{i}}$ can not all be equal by (11.12), the claim below
(11.20) shows that there exist at least two disjoint $F^{ij_{i}}\subset M_{2}$
which implies that $\cap_{i}F^{ij_{i}}$ is empty. This contradiction proves
(ii) and completes the proof of Lemma 11.12. ∎

In the proof of Lemma 11.12 (i) we obtain in addition a description in (11.20)
of those four $g\in\textnormal{SL}_{2}(\mathcal{O}/2\mathcal{O})$ with $gC$ a
prime divisor of $D_{i}$. The description is in terms of the intersection
behaviour of the cusp resolutions in the minimal desingularization of
$M_{2}^{*}(\mathbb{C})$. This is crucial for the proof of (ii) and might be of
independent interest.

####  11.3.5 Proof of Proposition 11.6

We continue our notation. Recall that
$\bar{\pi}:{\overline{\mathcal{M}}_{2}}\to{M_{2}^{*}}$ is the canonical
morphism from (11.5) and that $s_{i}$ is the global section of
$\omega^{\otimes 2}$ corresponding to the Eisenstein series $E_{i}$ defined in
$\mathsection$11.3.3 where $\omega$ denotes the Hodge bundle on
${\overline{\mathcal{M}}_{2}}$. In the following proof of Proposition 11.6, we
combine Lemmas 11.10 and 11.12 with fundamental results for ${M_{2}^{*}}$
obtained by Rapoport [Rap78] and Faltings–Chai [FC90, Cha90].

###### Proof of Proposition 11.6.

We first show that the sections $s_{i}$ have no common zeroes on the boundary
${\overline{\mathcal{M}}_{2}}\setminus\mathcal{M}_{2}$. It follows for example
from [Cha90, Main Thm] that
${\overline{\mathcal{M}}_{2}}\setminus\mathcal{M}_{2}$ has five connected
components which we label such that
$\bar{\pi}:{\overline{\mathcal{M}}_{2}}\to{M_{2}^{*}}$ sends the component $i$
to the cusp $i$ of $M_{2}^{*}$, where the five cusps of ${M_{2}^{*}}$ are
labelled by $0,\dotsc,4$. The section $s_{i}$ corresponds to the Eisenstein
series $E_{i}$, and Lemma 11.10 gives that the constant term of the Fourier
expansion of $E_{i}$ at the cusp $i$ of $M_{2}^{*}$ is invertible in $B$. Then
the $q$-expansion principle implies that $s_{i}$ does not vanish in the
connected component $i$ of
${\overline{\mathcal{M}}_{2}}\setminus\mathcal{M}_{2}$. Thus the sections
$s_{i}$ have no common zeroes on the boundary
${\overline{\mathcal{M}}_{2}}\setminus\mathcal{M}_{2}$ as desired.

Further, it is a rather direct consequence of Lemma 11.12 (ii) that the
sections $s_{i}$ have no common zeroes in $\mathcal{M}_{2}$. The details are
as follows. Let $p:X\to\mathcal{M}_{2}$ be an étale scheme cover of
$\mathcal{M}_{2}$. To show that the sections $s_{i}$ generate $\omega^{\otimes
2}$ over $\mathcal{M}_{2}$, it suffices to prove that the intersection
$\cap_{i}\textnormal{div}(p^{*}s_{i})$ is empty. Corollary 5.6 implies that
the initial morphism $\pi:\mathcal{M}_{2}\to M_{2}$ is étale and hence
$q=p\pi$ is étale. Then we compute

$\textnormal{div}(p^{*}s_{i})=\textnormal{div}(q^{*}t_{i})=q^{*}D_{i}$

where $D_{i}$ is the divisor associated to the global section $t_{i}$ of
$\pi_{*}\omega^{\otimes 2}$ defined by $s_{i}$. Here we identified
$\pi^{*}t_{i}$ with $s_{i}$ via the adjunction isomorphism
$\pi^{*}\pi_{*}\omega^{\otimes 2}\overset{\sim}{\to}\omega^{\otimes 2}$ in
Lemma 11.7. Now, as the intersection $\cap_{i}D_{i}$ is empty by Lemma 11.12
(ii), we deduce that $\cap_{i}\textnormal{div}(p^{*}s_{i})=\cap_{i}q^{*}D_{i}$
is empty as desired. This completes the proof of Proposition 11.6. ∎

###  11.4 Proof of Proposition 11.5

We continue the notation and terminology of $\mathsection$11.2 and
$\mathsection$11.3. Recall that ${M_{2}^{*}}$ denotes the minimal
compactification of the coarse moduli scheme $M_{2}$. In this section we use
the above results to construct the isomorphisms
${M_{2}^{*}}\overset{\sim}{\to}X$ and $M_{2}\overset{\sim}{\to}Y$ in
Proposition 11.5.

###### Proof of Proposition 11.5.

As in the statement of Proposition 11.5, we consider the objects over the base
scheme $B=\textnormal{Spec}(\mathbb{Z}[\tfrac{1}{30}])$. The following proof
consists of four steps.

##### 1.

We first construct a morphism ${M_{2}^{*}}\to\mathbb{P}^{4}_{B}$ using the
Hodge bundle $\omega$ on the toroidal compactification
${\overline{\mathcal{M}}_{2}}$ of $\mathcal{M}_{2}$. Proposition 11.6 gives
that $\omega^{\otimes 2}$ is globally generated by the five global sections
$s_{i}$ corresponding to the five Eisenstein series $E_{i}$ defined in
$\mathsection$11.3.3. Hence these sections $s_{i}$ define a morphism
${\overline{\mathcal{M}}_{2}}\to\mathbb{P}^{4}_{B}$ which is automatically
proper since ${\overline{\mathcal{M}}_{2}}$ is proper over $B$. Then Stein
factorization decomposes ${\overline{\mathcal{M}}_{2}}\to\mathbb{P}^{4}_{B}$
into two morphisms:

${\overline{\mathcal{M}}_{2}}\to^{\bar{\pi}}{M_{2}^{*}}\to^{f}\mathbb{P}^{4}_{B},$

where the canonical morphism
$\mathcal{O}_{{M_{2}^{*}}}\overset{\sim}{\to}\bar{\pi}_{*}\mathcal{O}_{{\overline{\mathcal{M}}_{2}}}$
is an isomorphism and the morphism $f$ is finite. Here we used the
construction of ${M_{2}^{*}}$ which gives moreover that the restriction of
$\bar{\pi}$ to $\mathcal{M}_{2}$ is an initial morphism $\mathcal{M}_{2}\to
M_{2}$, see [Cha90, Main Thm (vi)].

##### 2.

Next, we prove that the scheme theoretic image $X^{\prime}$ of $f$ equals $X$.
The projection formula together with $\omega^{\otimes
2}\cong\bar{\pi}^{*}f^{*}\mathcal{O}(1)$ and
$\bar{\pi}_{*}\mathcal{O}_{{\overline{\mathcal{M}}_{2}}}\cong\mathcal{O}_{{M_{2}^{*}}}$
implies that $\bar{\pi}_{*}\omega^{\otimes 2}\cong f^{*}\mathcal{O}(1)$. In
particular $\bar{\pi}_{*}\omega^{\otimes 2}$ is an invertible sheaf on
${M_{2}^{*}}$ which is globally generated by the five sections defined by the
$s_{i}$. The section $s_{i}$ corresponds to the Eisenstein series $E_{i}$
which coincides up to a constant factor with the Eisenstein series $E_{i+1}$
in van der Geer [vdG88, p.191]. Thus the arguments in [vdG88, p.190-191],
which can be modified so that they work over $\mathbb{Q}$ as explained by
Shepherd-Barron–Taylor [SBT97, p.289], show that

$f_{\mathbb{Q}}:{M^{*}_{2,\mathbb{Q}}}\hookrightarrow\mathbb{P}^{4}_{\mathbb{Q}}$

is a closed immersion with scheme theoretic image
$X_{\mathbb{Q}}\subset\mathbb{P}^{4}_{\mathbb{Q}}$. Hence
$X^{\prime}_{\mathbb{Q}}$ equals $X_{\mathbb{Q}}$ inside
$\mathbb{P}^{4}_{\mathbb{Q}}$. The schemes $X^{\prime}$ and $X$ are both
integral closed subschemes of $\mathbb{P}^{4}_{B}$. Indeed the scheme-
theoretic image $X^{\prime}$ of the finite morphism
$f:{M_{2}^{*}}\to\mathbb{P}^{4}_{B}$ has this property since ${M_{2}^{*}}$ is
integral, while a computation (using Macaulay2) with the defining equations of
$X$ gives that $X$ is integral as well. Furthermore
$X^{\prime}\subset\mathbb{P}^{4}_{B}$ and $X\subset\mathbb{P}^{4}_{B}$ have
the same generic point, since $X^{\prime}_{\mathbb{Q}}=X_{\mathbb{Q}}$ inside
$\mathbb{P}^{4}_{B}$ and since the generic points of $X^{\prime}$ and $X$ are
the generic points of $X^{\prime}_{\mathbb{Q}}$ and $X_{\mathbb{Q}}$
respectively. It follows that $X^{\prime}=X$ as desired.

##### 3.

We now apply Zariski’s main theorem to deduce that
$f:{M_{2}^{*}}\to\mathbb{P}^{4}_{B}$ induces an isomorphism on its image. The
noetherian schemes ${M_{2}^{*}}$ and $X$ are both integral, and a computation
(using Macaulay2) with the defining equations of $X$ shows that $X$ is normal.
As $X$ is the scheme-theoretic image of $f$ by step 2, we obtain that $f$
induces a morphism

$f:{M_{2}^{*}}\to X.$

This is a birational morphism of integral schemes, since it extends
$f_{\mathbb{Q}}:M^{*}_{2,\mathbb{Q}}\to X_{\mathbb{Q}}$ which is an
isomorphism of $\mathbb{Q}$-schemes as explained in step 2. Then, as $X$ is
normal, we can apply Zariski’s main theorem to deduce that
$\mathcal{O}_{X}\overset{\sim}{\to}f_{*}\mathcal{O}_{{M_{2}^{*}}}$. This
proves that the finite morphism $f:{M_{2}^{*}}\to X$ has to be an isomorphism.

##### 4.

It remains to show that $f:{M_{2}^{*}}\overset{\sim}{\to}X$ restricts to an
isomorphism $M_{2}\overset{\sim}{\to}Y$. Write $V$ for the boundary
${M_{2}^{*}}\setminus M_{2}$, and recall that $Z$ is the union of the images
of the five $B$-points of $X$ obtained by permuting the coordinates of
$(1,0,0,0,0)$. We now prove that

$f(V)=Z.$

It follows from [vdG88, p.190-191] and [SBT97, p.289] that $V_{\mathbb{Q}}$ is
given by five $\mathbb{Q}$-points of ${M_{2}^{*}}$ and
$f(V_{\mathbb{Q}})=Z_{\mathbb{Q}}$. Further, $Z$ is the closure of
$Z_{\mathbb{Q}}$ in $X$ and [Cha90, Main Thm (vi)] implies that $V$ is the
closure of $V_{\mathbb{Q}}$ in ${M_{2}^{*}}$. As $f$ is a homeomorphism, we
then deduce that $f(V)=\overline{f(V_{\mathbb{Q}})}=Z$. Thus the isomorphism
$f:{M_{2}^{*}}\overset{\sim}{\to}X$ restricts to an isomorphism
$M_{2}={M_{2}^{*}}\setminus V\overset{\sim}{\to}X\setminus Z=Y$ as desired.
This completes the proof of Proposition 11.5. ∎

Proposition 11.5 implies that $Y$ is a coarse Hilbert moduli scheme over $B$
of the moduli problem $\mathcal{P}$ on $\mathcal{M}_{2}$ defined in (11.6).
For what follows it will be crucial to specify an initial morphism
$\mathcal{M}_{2}\to Y$. In the above proof of Proposition 11.5, we constructed
in step 1 the morphisms
${\overline{\mathcal{M}}_{2}}\to^{\bar{\pi}}{M_{2}^{*}}\to^{f}\mathbb{P}^{4}_{B}$
and we showed in step 4 that the restriction to
$\mathcal{M}_{2}\subset{\overline{\mathcal{M}}_{2}}$ of the composition
$f\bar{\pi}:{\overline{\mathcal{M}}_{2}}\to\mathbb{P}^{4}_{B}$ defines an
initial morphism

$\pi:\mathcal{M}_{2}\to Y.$ (11.21)

Indeed $f$ restricts to an isomorphism $M_{2}\overset{\sim}{\to}Y$ by step 4
and the restriction of $\bar{\pi}$ to
$\mathcal{M}_{2}\subset{\overline{\mathcal{M}}_{2}}$ is an initial morphism
$\mathcal{M}_{2}\to M_{2}$ by [Cha90, Main Thm (vi)]. By construction
$\pi:\mathcal{M}_{2}\to Y$ is given by the five global sections of
$\omega^{\otimes 2}$ on $\mathcal{M}_{2}$ corresponding via the $q$-expansion
principle to the five Eisenstein series $E_{i}$ for $\Gamma(2)$ of weight 2
defined in $\mathsection$11.3.3.

###  11.5 Heights

Let $Y$ be as in (1.2) and define $B=\mathbb{Z}[\tfrac{1}{30}]$. Proposition
11.5 gives that $Y_{B}$ is a coarse moduli scheme over $B$ of the arithmetic
moduli problem $\mathcal{P}$ on $\mathcal{M}_{B}$ of principal level
2-structures with (11.6). The goal of this section is to prove the following
result which explicitly bounds the logarithmic Weil height $h$ on
$Y\subset\mathbb{P}^{4}_{\mathbb{Z}}$ in terms of the height $h_{\phi}$.

######  Proposition 11.13.

Any $P\in Y(\bar{\mathbb{Q}})$ satisfies $h(P)\leq
2h_{\phi}(P)+8^{8}\log(h_{\phi}+8)$.

Here $h_{\phi}$ is the height on $Y(\bar{\mathbb{Q}})\cong
Y_{B}(\bar{\mathbb{Q}})$ defined with respect to the initial morphism
$\pi:\leavevmode\nobreak\ \mathcal{M}_{\mathcal{P}}\to Y_{B}$ in (11.21), and
recall that the logarithmic Weil height $h$ on projective space is normalized
as in [BG06, p.16]. We shall discuss the bound at the end of Section 11.5.

In the remaining of this section we prove Proposition 11.13 as follows. We
first relate the height $h$ to a Theta height $h_{\theta}$ on $Y$ by using the
work of Hirzebruch [Hir76] and computations for Hilbert theta functions due to
Götzky [G2̈8], Gundlach [Gun63] and Lauter–Naehrig–Yang [LNY16]. Then we bound
$h_{\theta}$ in terms of $h_{\phi}$ via the explicit height comparison of
Pazuki [Paz12] which is based on ideas of Bost–David. Throughout this section
we work over $\bar{\mathbb{Q}}$ and we write $Y$ for $Y_{\bar{\mathbb{Q}}}$
and $\mathbb{P}^{n}$ for $\mathbb{P}^{n}_{\bar{\mathbb{Q}}}$ where
$n\in\mathbb{Z}_{\geq 1}$.

##### Theta height on $Y$.

We can define a Theta height on $Y$ by pulling back a Theta height on the
coarse moduli space $A_{2,2}$ over $\bar{\mathbb{Q}}$ of principally polarized
abelian varieties with a symplectic level two structure [FC90, p.19]. More
precisely, we define the Theta height

$h_{\theta}=\theta^{*}h:\ Y(\bar{\mathbb{Q}})\to\mathbb{R}$

where $\theta:Y\to\mathbb{P}^{9}$ is the composition of the following
morphisms: The morphism $Y\to A_{2,2}$ induced on coarse moduli schemes by
forgetting $\iota:\mathcal{O}\to\textup{End}(A)$ and the morphism
$A_{2,2}\to\mathbb{P}^{9}$ defined by the fourth powers of the even Theta
functions $\theta_{m}=\theta_{m}(\tau,0)$ as in [vdG88, p.217]. The heights
$h_{\theta}$ and $h$ on $Y$ are closely related.

######  Lemma 11.14.

Any $P\in Y(\bar{\mathbb{Q}})$ satisfies $h(P)\leq h_{\theta}(P).$

###### Proof.

Let $x_{i}$ be the pullback along $Y\hookrightarrow\mathbb{P}^{4}$ of the
$i$-th coordinate function on $\mathbb{P}^{4}$, and let $z_{j}$ be the
pullback along $\theta:Y\to\mathbb{P}^{9}$ of the $j$-th coordinate function
on $\mathbb{P}^{9}$. Lemma 11.14 is a direct consequence of the following
claim: On $Y(\bar{\mathbb{Q}})$ it holds

$x_{i}^{4}=\pm\prod_{j\in J_{i}}z_{j}$ (11.22)

where $J_{i}\subset\\{0,\dotsc,9\\}$ with $|J_{i}|=4$. To prove this claim, we
need to introduce some notation. Let $\mathbb{H}_{2}$ be the Siegel upper half
space of genus 2 on which the symplectic group
$\textnormal{Sp}_{2}(\mathbb{Z})$ acts as usual and write
$\mathbb{H}=\\{z\in\mathbb{C},\textnormal{im}(z)>0\\}$. We denote by
$\Gamma_{2}$ the kernel of the projection
$\textnormal{Sp}_{2}(\mathbb{Z})\to\textnormal{Sp}_{2}(\mathbb{Z}/2\mathbb{Z})$
and we denote by $\Gamma(2)$ the kernel of the projection
$\textnormal{SL}_{2}(\mathcal{O})\to\textnormal{SL}_{2}(\mathcal{O}/2\mathcal{O})$.
Then we consider the commutative diagram

$\textstyle{Y(\bar{\mathbb{Q}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y(\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{[\mathcal{M}_{2}(\bar{\mathbb{C}})]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\mathbb{H}^{2}/\Gamma(2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{\epsilon}}$$\textstyle{A_{2,2}(\bar{\mathbb{Q}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{2,2}(\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{[\mathcal{A}_{2,2}(\bar{\mathbb{C}})]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\mathbb{H}_{2}/\Gamma_{2}}$
(11.23)

which is obtained as follows: The map $\phi_{\epsilon}$ is the modular
morphism of Lauter–Yang [LNY16, p.153] with $\epsilon=\tfrac{1+\sqrt{5}}{2}$.
The two bijections in the middle come from the coarse moduli spaces
$\pi_{\bar{\mathbb{Q}}}:\mathcal{M}_{2}\to Y$ and $\mathcal{A}_{2,2}\to
A_{2,2}$, where
$\mathcal{M}_{2}=(\mathcal{M}_{\mathcal{P}})_{\bar{\mathbb{Q}}}$ and
$\mathcal{A}_{2,2}$ is the moduli stack over $\bar{\mathbb{Q}}$ of principally
polarized abelian schemes with a symplectic level two structure as in [FC90,
p.19]. The two bijections on the right come from the usual uniformization of
complex abelian varieties (with real multiplications), where
$[\mathcal{M}_{2}(\mathbb{C})]\overset{\sim}{\to}\mathbb{H}^{2}/\Gamma(2)$
factors through the inverse of the map
$\phi_{0}:\mathbb{H}^{2}\to\mathbb{H}^{2}$ defined in [LNY16, p.153].

We now use the diagram to compute $x_{i}$ in terms of the $z_{j}$. The
discussion right after (11.21) provides that
$\pi_{\bar{\mathbb{Q}}}:\mathcal{M}_{2}\to Y$ is given by the five global
sections of the square $\omega^{\otimes 2}$ of the Hodge bundle on
$\mathcal{M}_{2}$ corresponding via the $q$-expansion principle to the five
Eisenstein series $E_{i}$ for $\Gamma(2)$ of weight 2 which are defined in
$\mathsection$11.3.3. Then the inverse
$\mathbb{H}^{2}/\Gamma(2)\overset{\sim}{\to}Y(\mathbb{C})$ of the composition
of the isomorphisms
$Y(\mathbb{C})\overset{\sim}{\to}[\mathcal{M}_{2}(\mathbb{C})]\overset{\sim}{\to}\mathbb{H}^{2}/\Gamma(2)$
in the diagram (11.23) is given by the five Eisenstein series $E_{i}$.
Moreover, it holds

$Y_{\mathbb{C}}\cong\textnormal{Proj}\bigl{(}\mathbb{C}[E_{i}]/(\sigma_{2},\sigma_{4})\bigl{)}\setminus
Z_{\mathbb{C}},\quad{\textnormal{and}}\quad
E_{0}=\theta_{01}\theta_{02}\theta_{03}\theta_{04}$ (11.24)

by Hirzebruch’s computation in [Hir76, p.166]. Here $Z_{\mathbb{C}}$ is given
by the five permutations of $(1,0,\dotsc,0)$ and $\theta_{ij}$ is the theta
function (viewed on $\mathbb{H}^{2}$) of Götzky [G2̈8, (19)] and Gundlach
[Gun63, p.233]. We next express the $\theta_{ij}$, and then all Eisenstein
series $E_{i}$, in terms of the Hilbert theta functions $\theta_{a,b}$ for
some $(a,b)\in(\mathcal{O}/2\mathcal{O})^{2}$, where we write $\theta_{a,b}$
for the function $\theta_{a,b}^{\epsilon}$ of Lauter–Naehrig–Yang [LNY16,
(2.5)] defined with respect to the fundamental unit
$\epsilon=\tfrac{1+\sqrt{5}}{2}$ and where $\mathcal{O}$ denotes the ring of
integers of $\mathbb{Q}(\sqrt{5})$. In the table displayed in [Gun63, p.232],
Gundlach computed all pairs $(\alpha,\beta)$ which are relevant for the
definition of the functions $\theta_{ij}$. For any pair $(\alpha,\beta)$ in
this table, it holds that $\alpha\beta\in\\{0,1,-1\\}$ and thus the field
trace $t=\textnormal{Tr}(\tfrac{\epsilon}{\sqrt{5}}\tfrac{ab}{4})$ is zero for
$(a,b)=(\alpha,\beta/\epsilon)$. Then unwinding the definitions of the
functions $\theta_{ij}$ and $\theta_{a,b}$ on $\mathbb{H}^{2}$ leads to

$\theta_{ij}=e^{-2\pi it}\theta_{a,b}=\theta_{a,b}$ (11.25)

for some $(a,b)\in(\mathcal{O}/2\mathcal{O})^{2}$. The group
$\textnormal{SL}_{2}(\mathcal{O}/2\mathcal{O})$ acts transitively on the set
of cusps of $\mathbb{H}^{2}/\Gamma(2)$ and recall from (11.13) that the
Eisenstein series $E_{i}=E_{0}|_{A_{i}}$ is obtained from $E_{0}$ via the
slash operator for some $A_{i}\in\textnormal{SL}_{2}(\mathcal{O})$. Thus, on
combining (11.24) and (11.25) with the transformation formula for
$\theta_{ij}^{8}$ established by Götzky [G2̈8, Lem 6], we obtain

$E_{i}^{4}=\pm\prod\theta_{a,b}^{4}.$ (11.26)

Here the product is taken over four distinct pairs
$(a,b)\in(\mathcal{O}/2\mathcal{O})^{2}$. These four pairs are all even in the
sense of [LNY16, p.154] since $E_{i}\neq 0$. For any even
$(a,b)\in(\mathcal{O}/2\mathcal{O})^{2}$, Lauter–Naehrig–Yang [LNY16,
$\mathsection$2.5, $\mathsection$3] showed that each Hilbert theta function
$\theta_{a,b}$ is (up to a sign) the pullback along
$\phi_{\epsilon}:\mathbb{H}^{2}/\Gamma(2)\to\mathbb{H}_{2}/\Gamma_{2}$ of an
even Theta function $\theta_{m}$. Thus the commutative diagram (11.23) gives
that $\theta_{a,b}^{4}$ identifies on
$Y(\mathbb{C})\cong\mathbb{H}^{2}/\Gamma(2)$ with the pullback
$z_{j}=\theta^{*}t_{j}$ of a coordinate function $t_{j}$ on $\mathbb{P}^{9}$
along the composition $\theta:Y\to A_{2,2}\to\mathbb{P}^{9}$. Then (11.23) and
(11.26) imply the claim (11.22) since $x_{i}$ identifies on
$Y(\mathbb{C})\cong\mathbb{H}^{2}/\Gamma(2)$ with $E_{i}$ by (11.24). This
completes the proof of the lemma. ∎

One can determine the sign of $E_{i}^{4}=\pm\prod\theta_{a,b}^{4}$ in (11.26),
and thus of $x_{i}^{4}=\pm\prod_{j\in J_{i}}z_{j}$ in (11.22), by going into
the proof of [G2̈8, Lem 6] and computing therein the sign in the
transformation formula for five explicit
$A_{i}\in\textnormal{SL}_{2}(\mathcal{O})$ defining the five
$E_{i}=E_{0}|_{A_{i}}$.

##### Proof of Proposition 11.13.

To prove an explicit height bound, we need to compare our normalizations of
the heights with other normalizations. Recall from Section 3.4 that
$h_{\phi}:Y(\bar{\mathbb{Q}})\to\mathbb{R}$ is the pullback of
$h_{F}:\underline{A}_{2}(\bar{\mathbb{Q}})\to\mathbb{R}$ along the forgetful
map

$\phi_{\bar{\mathbb{Q}}}:Y(\bar{\mathbb{Q}})\overset{\sim}{\to}^{\pi^{-1}}[\mathcal{M}_{\mathcal{P}}(\bar{\mathbb{Q}})]\to\underline{A}_{2}(\bar{\mathbb{Q}})$
(11.27)

of the coarse Hilbert moduli scheme $Y$ defined right after (3.2), where
$h_{F}$ is the stable Faltings height with the metric normalized as in
Faltings [Fal83, p.354]. Further, we recall that the logarithmic Weil height
$h$ on $\mathbb{P}^{n}$ is defined as usual in Diophantine approximation with
respect to the infinity norm at the infinite places ([BG06, p.16]). It holds

$h_{F}=h^{\prime}_{F}-\log\pi,\quad h\leq h^{\prime}\leq
h+\tfrac{1}{2}\log(n+1),$ (11.28)

where $h_{F}^{\prime}$ denotes the stable Faltings height with the metric
normalized as in Deligne [Del85, p.27] and where $h^{\prime}$ denotes the
logarithmic Weil height on $\mathbb{P}^{n}$ defined with respect to the usual
Euclidean or Hermitian $l^{2}$ norm at the infinite places.

We now deduce Proposition 11.13 by combining Lemma 11.14 with the explicit
height comparison of Pazuki [Paz12, Cor 1.3.2] which is based on ideas of
Bost–David.

###### Proof of Proposition 11.13.

To prove the statement, it suffices to bound the Theta height $h_{\theta}$ on
$Y$ since $h_{\theta}\geq h$ by Lemma 11.14. We first relate $h_{\theta}$ to
another Theta height $h^{\prime}_{\theta}$ on $Y$ which is the pullback of the
Theta height defined in Pazuki [Paz12]. More precisely, applying [Paz12, Def
2.6] with $g=2$ and $r=2$ gives a height
$h_{\Theta}:A_{2}(\bar{\mathbb{Q}})\to\mathbb{R}$ on the coarse moduli scheme
$A_{2}$ over $\bar{\mathbb{Q}}$ of principally polarized abelian surfaces.
Then we define

$h^{\prime}_{\theta}=f^{*}h_{\Theta}:Y(\bar{\mathbb{Q}})\to\mathbb{R}$

where $f:Y\to A_{2}$ is the composition of the morphism $Y\to A_{2,2}$ induced
on coarse moduli schemes by forgetting $\iota:\mathcal{O}\to\textup{End}(A)$
and the morphism $A_{2,2}\to A_{2}$ induced on coarse moduli schemes by
forgetting the symplectic level 2-structure. The Theta height $h_{\Theta}$ is
determined by the 10 even Theta functions $\theta_{m}$, since any Theta
function $\theta_{m}$ is nonzero if and only if the Theta characteristic $m$
is even. Thus (the proof of) (11.28) leads to

$h_{\theta}\leq 4h^{\prime}_{\theta}\leq 4h_{\theta}+2\log 10.$ (11.29)

We next compare the heights $h^{\prime}_{\theta}$ and $h_{\phi}$ on $Y$. It
follows from (11.27) that the map
$\phi_{\bar{\mathbb{Q}}}:Y(\bar{\mathbb{Q}})\to\underline{A}_{2}(\bar{\mathbb{Q}})$
factors as $Y(\bar{\mathbb{Q}})\to
A_{2}(\bar{\mathbb{Q}})\to\underline{A}_{2}(\bar{\mathbb{Q}})$ where
$A_{2}(\bar{\mathbb{Q}})\to\underline{A}_{2}(\bar{\mathbb{Q}})$ is induced by
forgetting the polarization. Then, on combining (11.28) and (11.29) with the
bound for $|h_{\Theta}-\tfrac{1}{2}h_{F}^{\prime}|$ in Pazuki [Paz12, Cor
1.3.2], we deduce

$4h^{\prime}_{\theta}\leq 2h_{\phi}+8^{8}\log(h_{\phi}+8).$ (11.30)

To simplify the shape of this estimate, we used here in addition Bost’s
explicit lower bound $h^{\prime}_{F}\geq-\log(2\pi)$ in [Bos96a]. Finally for
any $P\in Y(\bar{\mathbb{Q}})$ the inequalities (11.29) and (11.30) give an
upper bound for $h_{\theta}(P)\geq h(P)$ as claimed in Proposition 11.13. ∎

We now discuss the bounds. The constants $8^{8}$ and $8$ appearing in (11.30)
and Proposition 11.13 crucially depend on the value at $g=2$ and $r=2$ of the
function $C_{2}(g,r)$ in Pazuki’s general result [Paz12, Cor 1.3]. We
simplified both constants $8^{8}$ and $8$ and thus they can be improved up to
a certain extent. However, the factor 2 in front of $h_{\phi}$ is optimal in
(11.30) by the height comparison [Paz12, Cor 1.3] and hence the factor 2 might
also be optimal in Proposition 11.13 in view of the relation
$x_{i}^{4}=\pm\prod z_{j}$ in (11.22).

The bound in Proposition 11.13 is linear in $h_{\phi}$ which is best possible
in the sense that for any exponent $e<1$ the bound $h\ll h^{e}_{\phi}$ can not
hold on $Y(\bar{\mathbb{Q}})$. Indeed on combining [Paz12, Cor 1.3] and
(11.29) with the relation $x_{i}^{4}=\pm\prod z_{j}$ in (11.22), we see that
such a bound already fails on the subset $Y^{0}(\mathbb{Q})\subset
Y(\bar{\mathbb{Q}})$ which is an infinite set by (11.3), where
$Y^{0}=Y\setminus\cup V_{+}(x_{i})$ for $x_{i}$ the $i$-th coordinate function
on $\mathbb{P}^{4}_{\mathbb{Z}}=\textnormal{Proj}(\mathbb{Z}[x_{i}])$.

###  11.6 Proof of Theorem 11.3 and the Corollary

In this section we first prove Theorem 11.3 by combining Propositions 11.5 and
11.13. Then we deduce from Theorems 4.1 and 11.3 the Corollary stated in the
introduction.

We continue the notation of Section 11.1. Let $Y$ be the variety over
$\mathbb{Z}$ defined in (1.2). We denote by $h$ the logarithmic Weil height on
$Y\subset\mathbb{P}^{4}_{\mathbb{Z}}$, we write $\mathcal{M}$ for the Hilbert
moduli stack associated to $L=\mathbb{Q}(\sqrt{5})$ and we define
$B=\mathbb{Z}[\tfrac{1}{30}]$.

###### Proof of Theorem 11.3.

Let $\mathcal{P}$ be the moduli problem on $\mathcal{M}_{B}$ of principal
level 2-structures satisfying the condition (11.6). Proposition 11.5 gives
that $Y_{B}$ is a coarse moduli scheme over $B$ of $\mathcal{P}$. Let
$\pi:(\mathcal{M}_{B})_{\mathcal{P}}\to Y_{B}$ be the initial morphism given
in (11.21) and let $h_{\phi}$ be the height on $Y(\bar{\mathbb{Q}})\cong
Y_{B}(\bar{\mathbb{Q}})$ defined with respect to $\pi$. We claim that

$B_{\mathcal{P}}=\emptyset,\quad|\mathcal{P}|_{\bar{\mathbb{Q}}}\leq 16\
\textnormal{ and }\ h(P)\leq 2h_{\phi}(P)+8^{8}\log(h_{\phi}(P)+8),\,P\in
Y(\bar{\mathbb{Q}}).$

Indeed on using exactly the same arguments as in the proof of Corollary 5.6,
we deduce from Proposition 5.5 that the branch locus $B_{\mathcal{P}}$ is
empty. Further, as in (10.8) we find that the maximal number of level
$\mathcal{P}$-structures over $\bar{\mathbb{Q}}$ is bounded by
$|\mathcal{P}|_{\bar{\mathbb{Q}}}\leq 16$, while the height bound is provided
by Proposition 11.13. Now, the claim implies Theorem 11.3 after extending
$Y_{B}$ to a coarse moduli scheme over $\mathbb{Z}$. The details are as
follows: We extend $Y_{B}$ to a coarse moduli scheme over $\mathbb{Z}$ via the
naive extension $\mathcal{P}^{\prime}$ on $\mathcal{M}$ defined right above
(5.6) and we use the initial morphism
$\pi^{\prime}:\mathcal{M}_{\mathcal{P}^{\prime}}\to Y_{B}$ given by the
composition

$\mathcal{M}_{\mathcal{P}^{\prime}}\overset{\sim}{\to}(\mathcal{M}_{B})_{\mathcal{P}}\to^{\pi}Y_{B}.$
(11.31)

Here
$\mathcal{M}_{\mathcal{P}^{\prime}}\overset{\sim}{\to}(\mathcal{M}_{B})_{\mathcal{P}}$
is induced by the identity. Thus the branch locus $B_{\mathcal{P}^{\prime}}$
equals $B_{\mathcal{P}}$, the height defined with respect to
$\pi^{\prime}:\mathcal{M}_{\mathcal{P}^{\prime}}\to Y_{B}$ coincides on
$Y(\bar{\mathbb{Q}})\cong Y_{B}(\bar{\mathbb{Q}})$ with the height $h_{\phi}$
defined with respect to $\pi:(\mathcal{M}_{B})_{\mathcal{P}}\to Y_{B}$, and
$\mathcal{P}^{\prime}$ is an arithmetic moduli problem on $\mathcal{M}$ since
$\mathcal{P}$ is an arithmetic moduli problem on $\mathcal{M}_{B}$. We also
notice that
$|\mathcal{P}^{\prime}|_{\bar{\mathbb{Q}}}=|\mathcal{P}|_{\bar{\mathbb{Q}}}$.
Hence the claim proves that Theorem 11.3 holds with
$\mathcal{P}=\mathcal{P}^{\prime}$ and $\pi=\pi^{\prime}$. ∎

We now deduce from Theorems 4.1 and 11.3 the Corollary stated in the
introduction.

###### Proof of Corollary.

Let $S$ be a finite set of rational primes. Theorem 11.3 gives that $Y_{B}$ is
a coarse moduli scheme of an arithmetic moduli problem $\mathcal{P}$ on
$\mathcal{M}$ with $|\mathcal{P}|_{\bar{\mathbb{Q}}}\leq 16$ such that the
branch locus $B_{\mathcal{P}}$ is empty and such that each $P\in
Y(\bar{\mathbb{Q}})$ satisfies

$h(P)\leq 2h_{\phi}(P)+8^{8}\log(h_{\phi}(P)+8).$

Here $h_{\phi}$ is the height on $Y(\bar{\mathbb{Q}})\cong
Y_{B}(\bar{\mathbb{Q}})$ defined in (3.2) with respect to the initial morphism
$\pi:\mathcal{M}_{\mathcal{P}}\to Y_{B}$ given in (11.21). Base change
properties of the height $h_{\phi}$ assure that $h_{\phi}$ coincides on
$Y(\mathbb{Z}_{S})\subset Y(\bar{\mathbb{Q}})$ with the height $h_{\phi}$ on
$Y(\mathbb{Z}_{S})$ obtained by applying the definition at the end of Section
3.4 with $\pi:\mathcal{M}_{\mathcal{P}}\to Y_{B}$; notice that
$\textnormal{Spec}(\mathbb{Z})-S=\textnormal{Spec}(\mathbb{Z}_{S})$ is a
nonempty open subscheme of $\textnormal{Spec}(\mathbb{Z})$. Then an
application of Theorem 4.1 with $T=B$, $S=\textnormal{Spec}(\mathbb{Z}_{S})$
and $Z=\emptyset$ gives that any $P\in Y(\mathbb{Z}_{S})$ satisfies

$h_{\phi}(P)\leq
c_{1}N_{U}^{24},\quad\textnormal{and}\quad|Y(\mathbb{Z}_{S})|\leq
c_{2}|\mathcal{P}|_{\bar{\mathbb{Q}}}N_{U}^{e_{2}}\Delta\log(3\Delta)^{3}.$

Here the constants are $c_{1}=7^{7^{14}}$, $c_{2}=9^{9^{18}}$ and
$e_{2}=4^{16}$, while $\Delta=5$ and $N_{U}=\prod_{p\in S^{\prime}}p$ for
$S^{\prime}=S\cup\\{2,3,5\\}$. As $|\mathcal{P}|_{\bar{\mathbb{Q}}}\leq 16$
and $N_{U}\leq 30N_{S}$, the displayed bounds show that the Corollary holds
with $e=10^{12}$ and $c=10^{e}$ as claimed. This completes the proof. ∎

###  11.7 Diophantine equations and the Clebsch–Klein surfaces

In this section we first collect basic results for the Clebsch–Klein surfaces
and the Diophantine equations (1.1) and (1.3). Then we combine these results
with the Corollary stated in the introduction to deduce Corollaries 11.1 and
11.2 giving explicit bounds for the solutions of (1.1) and (1.3). Finally, we
discuss lower bounds for rational points of bounded height
($\mathsection$11.7.2) and the degeneration of $S$-integral points
($\mathsection$11.7.3).

We continue the notation introduced in Section 11.1. Let $Y$ be the variety
over $\mathbb{Z}$ defined in (1.2) and let $U$ be the variety over
$\mathbb{Z}$ defined in (1.6). For any finite set $S$ of rational primes, we
recall that $\mathbb{Z}_{S}=\mathbb{Z}[1/N_{S}]$ where $N_{S}=\prod_{p\in
S}p$. We first state basic lemmas relating the solutions of (1.1) and (1.3)
with the $\mathbb{Z}_{S}$-points of $Y$ and $U$ respectively. Sending $x$ to
$-x$ defines an action of the group $\\{\pm 1\\}$ on the set of solutions of
(1.3).

######  Lemma 11.15.

The set of solutions of (1.3) modulo $\\{\pm 1\\}$ identifies with
$Y(\mathbb{Z}_{S})$.

Here the identification comes from the usual description of
$\mathbb{Z}_{S}$-points on projective space. The identification in the
following lemma also comes from this description.

######  Lemma 11.16.

The set of solutions of (1.1) identifies with $U(\mathbb{Z}_{S})$.

We write $\mathbb{P}^{4}$ for $\mathbb{P}^{4}_{\mathbb{Z}}$. In Section 11.1
we defined the logarithmic Weil height $h$ on the sets of solutions of (1.1)
and (1.3) and also on the subsets $Y(\mathbb{Z}_{S})$ and $U(\mathbb{Z}_{S})$
of $\mathbb{P}^{4}(\bar{\mathbb{Q}})$.

######  Lemma 11.17.

The identifications in Lemmas 11.15 and 11.16 are compatible with $h$.

In the remaining of this section we prove inter alia the above lemmas and we
explicitly construct the open immersion $U\hookrightarrow Y$ over
$\mathbb{Z}[1/3]$ appearing in Lemma 11.4. The proofs use standard
computations. In particular we shall use the usual identification

$\mathbb{P}^{4}(\mathbb{Z}_{S})\overset{\sim}{\to}\Sigma_{\mathbb{P}^{4}}/\mathbb{Z}_{S}^{\times}.$
(11.32)

Here the group $\mathbb{Z}_{S}^{\times}$ acts diagonally on the set
$\Sigma_{\mathbb{P}^{4}}$ of $x\in\mathbb{Z}_{S}^{5}$ with
$\gcd(x_{i})\in\mathbb{Z}_{S}^{\times}$ and the bijection (11.32) comes from
the standard description of $\mathbb{P}^{4}(\mathbb{Z}_{S})$ in terms of
isomorphism classes of surjections $\mathbb{Z}_{S}^{5}\to\mathbb{Z}_{S}$ using
that $\textnormal{Pic}(\mathbb{Z}_{S})=0$. We first prove Lemma 11.15.

###### Proof of Lemma 11.15.

Let $P$ a $\mathbb{Z}_{S}$-point of $\mathbb{P}^{4}$ corresponding via (11.32)
to some $[x]$ in $\Sigma_{\mathbb{P}^{4}}/\mathbb{Z}_{S}^{\times}$, and let
$Q$ be the $\mathbb{Z}_{S}$-point of $\mathbb{P}^{4}$ given by
$e_{1}=(1,0,\dotsc,0)$. For any closed point
$p\in\textnormal{Spec}(\mathbb{Z}_{S})$, the images in $\mathbb{P}^{4}$ of the
sections $P$ and $Q$ have empty intersection in the fiber $\mathbb{P}^{4}_{p}$
if and only if the image of $x$ in $\mathbb{F}_{p}^{5}$ does not lie in the
line $L_{p}=\mathbb{F}_{p}\cdot e_{1}$ where
$\mathbb{F}_{p}=\mathbb{Z}_{S}/p\mathbb{Z}_{S}$. It follows that the
identification (11.32) restricts to a bijection

$Y(\mathbb{Z}_{S})\overset{\sim}{\to}\Sigma_{Y}/\mathbb{Z}_{S}^{\times}$

for $\Sigma_{Y}$ the set of $x\in\mathbb{Z}_{S}^{5}$ with
$\gcd(x_{i})\in\mathbb{Z}_{S}^{\times}$ such that
$\sigma_{2}(x)=0=\sigma_{4}(x)$ and such that for all rational primes $p\notin
S$ the image of $x$ in $\mathbb{F}_{p}^{5}$ does not lie in any of the five
permutations of the line $L_{p}$. Let $\Sigma_{\sigma}$ be the set of
$x\in\Sigma$ with $\sigma_{2}(x)=0=\sigma_{4}(x)$ for $\Sigma$ as in (1.3).
Multiplying with the denominators of the $x_{i}$ and dividing by the $\gcd$
defines a bijection

$\Sigma_{Y}/\mathbb{Z}_{S}^{\times}\overset{\sim}{\to}\Sigma_{\sigma}/\\{\pm
1\\}$ (11.33)

which composed with
$Y(\mathbb{Z}_{S})\overset{\sim}{\to}\Sigma_{Y}/\mathbb{Z}_{S}^{\times}$ gives
an identification $Y(\mathbb{Z}_{S})\overset{\sim}{\to}\Sigma_{\sigma}/\\{\pm
1\\}$ as desired. The remaining of this proof contains the details of the
definition of the bijection (11.33). Suppose that $x=(x_{i})$ lies in
$\Sigma_{Y}$. As the $x_{i}$ are in $\mathbb{Z}_{S}$ with
$\gcd(x_{i})\in\mathbb{Z}_{S}^{\times}$, there exists
$u\in\mathbb{Z}_{S}^{\times}$ which is unique up to $\pm 1$ such that $y=ux$
lies in $\mathbb{Z}^{5}$ with $\gcd(y_{i})=1$. The scalar multiple $y=ux$ of
$x$ satisfies again the homogeneous equations $\sigma_{2}=0=\sigma_{4}$.
Moreover, if $p\notin S$ is a rational prime then $\textnormal{ord}_{p}(u)=0$
and thus our assumption $x\in\Sigma_{Y}$ assures that the image of $y$ in
$(\mathbb{Z}/p\mathbb{Z})^{5}\cong\mathbb{F}_{p}^{5}$ is not a scalar multiple
of a standard basis vector of $(\mathbb{Z}/p\mathbb{Z})^{5}$. Hence $y$ lies
in $\Sigma_{\sigma}$. On using that $u$ is unique up to $\pm 1$, we see that
$x\mapsto y$ defines an injective map
$\Sigma_{Y}/\mathbb{Z}_{S}^{\times}\to\Sigma_{\sigma}/\\{\pm 1\\}$ which is
surjective since $\Sigma_{\sigma}\subset\Sigma_{Y}$. Now, we define (11.33) by
$x\mapsto y$. This completes the proof of Lemma 11.15. ∎

We now deduce Lemma 11.16 from the identification
$\mathbb{P}^{4}(\mathbb{Z}_{S})\overset{\sim}{\to}\Sigma_{\mathbb{P}^{4}}/\mathbb{Z}_{S}^{\times}$
in (11.32).

###### Proof of Lemma 11.16.

Recall that $U=V\setminus\cup V_{+}(z_{i})$ for $V\subset\mathbb{P}^{4}$ as in
(1.6). We first compute the image of
$U(\mathbb{Z}_{S})\subset\mathbb{P}^{4}(\mathbb{Z}_{S})$ under the
identification (11.32). Let $P$ be a $\mathbb{Z}_{S}$-point of
$\mathbb{P}^{4}=\textnormal{Proj}(\mathbb{Z}[z_{i}])$ corresponding via
(11.32) to $[x]$ in $\Sigma_{\mathbb{P}^{4}}/\mathbb{Z}_{S}^{\times}$. Then
$P$ is a $\mathbb{Z}_{S}$-point of $\mathbb{P}^{4}\setminus V_{+}(z_{i})$ if
and only if $x_{i}$ lies in $\mathbb{Z}_{S}^{\times}$. This implies that
(11.32) restricts to a bijection

$U(\mathbb{Z}_{S})\overset{\sim}{\to}\Sigma_{U}/\mathbb{Z}_{S}^{\times}$

where $\mathbb{Z}_{S}^{\times}$ acts diagonally on the set $\Sigma_{U}$ of
$x\in(\mathbb{Z}^{\times}_{S})^{5}$ satisfying $\sum x_{i}=0=\sum x_{i}^{3}$.
Let $\Sigma^{\prime}$ be the set of solutions of (1.1). To identify
$\Sigma_{U}/\mathbb{Z}_{S}^{\times}$ with $\Sigma^{\prime}$, we take
$x\in\Sigma_{U}$ and we define $y_{i}=-x_{i}/x_{4}$ for $i\leq 3$. As $\sum
y_{i}=1=\sum y_{i}^{3}$, we see that $y=(y_{i})$ lies in $\Sigma^{\prime}$. It
follows that $x\mapsto y$ defines a bijection
$\Sigma_{U}/\mathbb{Z}_{S}^{\times}\overset{\sim}{\to}\Sigma^{\prime}$ which
composed with
$U(\mathbb{Z}_{S})\overset{\sim}{\to}\Sigma_{U}/\mathbb{Z}_{S}^{\times}$ gives
an identification $U(\mathbb{Z}_{S})\overset{\sim}{\to}\Sigma^{\prime}$ as
desired. This completes the proof of Lemma 11.16. ∎

Next, we show that the two identifications constructed in the above proofs are
compatible with $h$. For each
$[x]\in\Sigma_{\mathbb{P}^{4}}/\mathbb{Z}_{S}^{\times}$ we define $h([x])$ as
the logarithmic Weil height of $x\in\mathbb{P}^{4}(\bar{\mathbb{Q}})$. Then
the identification
$\mathbb{P}^{4}(\mathbb{Z}_{S})\overset{\sim}{\to}\Sigma_{\mathbb{P}^{4}}/\mathbb{Z}_{S}^{\times}$
in (11.32) is compatible with $h$.

###### Proof of Lemma 11.17.

In the proof of Lemma 11.15 we identified (modulo $\pm 1$) the set of
solutions $\Sigma_{\sigma}$ of (1.3) with $Y(\mathbb{Z}_{S})$ via the inverse
$\Sigma_{\sigma}/\\{\pm 1\\}\overset{\sim}{\to}Y(\mathbb{Z}_{S})$ of the
composition

$Y(\mathbb{Z}_{S})\overset{\sim}{\to}\Sigma_{Y}/\mathbb{Z}_{S}^{\times}\overset{\sim}{\to}\Sigma_{\sigma}/\\{\pm
1\\}.$

Here $Y(\mathbb{Z}_{S})\overset{\sim}{\to}\Sigma_{Y}/\mathbb{Z}_{S}^{\times}$
is compatible with $h$ since it is the restriction of (11.32), while
$\Sigma_{Y}/\mathbb{Z}_{S}^{\times}\overset{\sim}{\to}\Sigma_{\sigma}/\\{\pm
1\\}$ is induced by $x\mapsto y$ which is compatible with $h$ since $x$ and
$y=ux$ are equal in $\mathbb{P}^{4}(\bar{\mathbb{Q}})$ and since
$h(x)=\max_{i}\log|x_{i}|$ for any $x\in\mathbb{P}^{4}(\bar{\mathbb{Q}})$ with
$x_{i}\in\mathbb{Z}$ and $\gcd(x_{i})=1$. Thus the identification
$\Sigma_{\sigma}/\\{\pm 1\\}\overset{\sim}{\to}Y(\mathbb{Z}_{S})$ is
compatible with $h$ as desired.

We now go into the proof of Lemma 11.16. Therein we identified the set
$\Sigma^{\prime}$ of solutions of (1.1) with $U(\mathbb{Z}_{S})$ via the
inverse $\Sigma^{\prime}\overset{\sim}{\to}U(\mathbb{Z}_{S})$ of the
composition

$U(\mathbb{Z}_{S})\overset{\sim}{\to}\Sigma_{U}/\mathbb{Z}_{S}^{\times}\overset{\sim}{\to}\Sigma^{\prime}.$

Here again, the bijection
$U(\mathbb{Z}_{S})\overset{\sim}{\to}\Sigma_{U}/\mathbb{Z}_{S}^{\times}$ is
compatible with $h$ since it is the restriction of (11.32). The bijection
$\Sigma_{U}/\mathbb{Z}_{S}^{\times}\overset{\sim}{\to}\Sigma^{\prime}$ is
induced by $x\mapsto y$ which is compatible with $h$ by the following
computation: As $x$ and $y^{\prime}=(y,-1)$ are equal in
$\mathbb{P}^{4}(\bar{\mathbb{Q}})$, we obtain that
$h(x)=h(y^{\prime})=h((y,1))=h(y)$. Hence the identification
$\Sigma^{\prime}\overset{\sim}{\to}U(\mathbb{Z}_{S})$ is compatible with $h$
as desired. This completes the proof of Lemma 11.17. ∎

Finally we construct the open immersion $\varphi:U\hookrightarrow Y$ over
$\mathbb{Z}[1/3]$ of Lemma 11.4.

###### Proof of Lemma 11.4.

Consider the open subscheme $Y^{0}=Y\setminus\cup V_{+}(x_{i})$ of $Y$. Any
isomorphism $U\overset{\sim}{\to}Y^{0}$ induces an open immersion
$U\hookrightarrow Y$. To construct an isomorphism $U\overset{\sim}{\to}Y^{0}$
over $\mathbb{Z}[1/3]$, we first explicitly describe the schemes
$U=V\setminus\cup V_{+}(z_{i})$ and $Y^{0}$ inside
$\mathbb{A}^{4}_{\mathbb{Z}}$. On using that $U=D_{+}(\prod z_{i})\subset V$,
we see that $t_{i}=z_{i}/z_{0}$ satisfies

$U=\textnormal{Spec}\bigl{(}\mathbb{Z}[t_{i},t_{i}^{-1}]/(u,v)\bigl{)}\subset\mathbb{A}^{4}_{\mathbb{Z}}.$
(11.34)

Here $u=\sigma_{1}(t_{i})=\sum t_{i}$ and $v=\sum t_{i}^{3}$ with both sums
taken over all $i\in\\{0,\dotsc,4\\}$. Analogously, on using that
$Y^{0}=D_{+}(\prod x_{i})\subset Y$, we deduce that $s_{i}=x_{i}/x_{0}$
satisfies

$Y^{0}=\textnormal{Spec}\left(\mathbb{Z}[s_{i},s_{i}^{-1}]/\bigl{(}\sigma_{2}(s_{i}),\sigma_{4}(s_{i})\bigl{)}\right)\subset\mathbb{A}^{4}_{\mathbb{Z}}.$

In the remaining of this proof, we work over $R=\mathbb{Z}[1/3]$ and we use
the descriptions of $U\subset\mathbb{A}^{4}_{\mathbb{Z}}$ and
$Y^{0}\subset\mathbb{A}^{4}_{\mathbb{Z}}$ to show that $s_{i}\mapsto 1/t_{i}$
defines an isomorphism of $R$-schemes:

$\varphi:U_{R}\overset{\sim}{\to}Y^{0}_{R}.$

Let $I$ be the ideal of $A=R[s_{i},s_{i}^{-1}]$ generated by
$\sigma_{2}(s_{i}),\sigma_{4}(s_{i})$ and let $J$ be the ideal of
$B=R[t_{i},t_{i}^{-1}]$ generated by $u,v$. To see that $s_{i}\mapsto 1/t_{i}$
defines an isomorphism $A/I\overset{\sim}{\to}B/J$ of $R$-algebras, we recall
the Newton identity $3\sigma_{3}=\sigma_{1}\sigma_{2}-\sigma_{1}p_{2}+p_{3}$
for $p_{n}=\sum y_{i}^{n}$ and we observe that
$\tau\sigma_{2}(s_{i})=\sigma_{3}(t_{i})$ and $\tau\sigma_{4}(s_{i})=u$ where
$s_{i}=1/t_{i}$ and $\tau=\prod t_{i}$. Thus $s_{i}\mapsto 1/t_{i}$ sends $I$
inside $J$ since $3$ is invertible in $R$, and $t_{i}\mapsto 1/s_{i}$ sends
$J$ inside $I$. It follows that $s_{i}\mapsto 1/t_{i}$ indeed defines an
isomorphism $A/I\overset{\sim}{\to}B/J$ of $R$-algebras, and hence an
isomorphism $\varphi:U_{R}\overset{\sim}{\to}Y^{0}_{R}$ of $R$-schemes, whose
inverse is given by $t_{i}\mapsto 1/s_{i}$.

It remains to prove the height inequalities for any point $P\in
U(\mathbb{Q})$. We view $P$ inside $\mathbb{P}^{4}(\mathbb{Q})$ via
$U\hookrightarrow V\hookrightarrow\mathbb{P}^{4}$. Then the description of $U$
in (11.34) shows that $P\in\mathbb{P}^{4}(\mathbb{Q})$ is defined by some
$x\in\mathbb{Q}^{5}$ with $x_{1}=1$ and $x_{i}\neq 0$. There exists
$a\in\mathbb{Z}^{5}$ with $\gcd(a_{i})=1$ and $a_{i}\neq 0$ such that
$x_{i}=a_{i}/a_{0}$. It holds that $h(P)=\log\max_{i}|a_{i}|$ and then we
compute

$h(\varphi(P))\leq\max_{i}\log|a_{i}^{-1}\prod a_{j}|\leq 4h(P).$

To verify the converse height inequality, we now use the displayed description
of $Y^{0}$. Then an application of the above arguments with
$\varphi^{-1}:Y^{0}_{R}\overset{\sim}{\to}U_{R}$ and $Q=\varphi(P)$ shows that
$h(P)=h(\varphi^{-1}(Q))$ is at most $4h(Q)=4h(\varphi(P))$. Thus the height
inequalities hold for any $P\in U(\mathbb{Q})$ as desired. This completes the
proof of Lemma 11.4. ∎

####  11.7.1 Proof of Corollaries 11.1 and 11.2

We continue the notation of the previous section. For any finite set $S$ of
rational primes, the (proof of the) Corollary stated in the introduction gives
that each $P\in Y(\mathbb{Z}_{S})$ satisfies

$4h(P)\leq c(\tfrac{1}{3}N_{S})^{24}\quad\textnormal{ and
}\quad|Y(\mathbb{Z}_{S})|\leq(\tfrac{c}{3}N_{S})^{e}$ (11.35)

for $e=10^{12}$ and $c=10^{e}$. We now deduce Corollaries 11.1 and 11.2 by
combining the results in (11.35) with the basic lemmas collected in the
previous section.

###### Proof of Corollary 11.1.

In Lemma 11.15 we identified (modulo $\pm 1$) the set of solutions of (1.3)
with the set $Y(\mathbb{Z}_{S})$, and Lemma 11.17 assures that this
identification is compatible with the Weil height $h$. Thus (11.35) proves
Corollary 11.1. ∎

###### Proof of Corollary 11.2.

Recall that $\Sigma^{\prime}$ denotes the set of solutions of (1.1) and define
$T=S\cup\\{3\\}$. Lemma 11.17 gives that the identification
$\Sigma^{\prime}\overset{\sim}{\to}U(\mathbb{Z}_{S})$ constructed in Lemma
11.16 is compatible with the Weil height $h$. Then we deduce

$|\Sigma^{\prime}|=|U(\mathbb{Z}_{S})|\leq|Y(\mathbb{Z}_{T})|\quad\textnormal{and}\quad\sup_{x\in\Sigma^{\prime}}h(x)=\sup_{P\in
U(\mathbb{Z}_{S})}h(P)\leq 4\sup_{Q\in Y(\mathbb{Z}_{T})}h(Q).$

Here we exploited that $U(\mathbb{Z}_{S})\subseteq
U(\mathbb{Z}_{T})=U_{\mathbb{Z}[1/3]}(\mathbb{Z}_{T})$ which follows since $U$
is separated over the terminal object $\mathbb{Z}$ and $3$ is invertible in
the dense open
$\textnormal{Spec}(\mathbb{Z}_{T})\subseteq\textnormal{Spec}(\mathbb{Z}_{S})$,
and we used the open immersion $\varphi:U_{\mathbb{Z}[1/3]}\to
Y_{\mathbb{Z}[1/3]}$ of Lemma 11.4 with $h(P)\leq 4h(\varphi(P))$ for all
$P\in U(\mathbb{Q})$. An application of (11.35) with $S=T$ gives explicit
bounds for $|Y(\mathbb{Z}_{T})|$ and $\sup_{Q\in Y(\mathbb{Z}_{T})}h(Q)$ in
terms of $N_{T}$. As $N_{T}\leq 3N_{S}$, these explicit bounds combined with
the displayed inequalities show that Corollary 11.2 holds with $e=10^{12}$ and
$c=10^{e}$. ∎

####  11.7.2 Lower bounds for the number of rational points of bounded height

On using a secant-tangent process, Slater–Swinnerton-Dyer [SSD98] established
for certain smooth cubic surfaces the lower bound predicted by Manin’s
conjecture ([FMT89]) on rational points of bounded height. In this section we
give the basic computations which show that their result directly implies the
lower bounds in (11.3).

We continue our notation. Let $U\subset V$ be as in (1.6) and let $\rho$ be
the Picard rank of the smooth projective surface $V_{\mathbb{Q}}$. We first
consider the $\mathbb{Q}$-points of $U$ of bounded height.

_Claim: For any $b\in\mathbb{R}$ with $b\to\infty$ it holds_ $|\\{P\in
U(\mathbb{Q}),h(P)\leq\log b\\}|\gg b(\log b)^{\rho-1}.$

Here the implied constant depends only on $V_{\mathbb{Q}}$. The claim has the
following two consequences. Firstly, it proves the lower bounds in (11.3) when
combined with the isomorphism
$U_{\mathbb{Q}}\overset{\sim}{\to}Y^{0}_{\mathbb{Q}}$ constructed in the proof
of Lemma 11.4 for $Y^{0}=Y\setminus\cup_{i}V_{+}(x_{i})$. Secondly, the claim
and (the proofs of) Lemmas 11.16 and 11.17 show for each $b\in\mathbb{R}$ with
$b\to\infty$ that

$|\\{x\in\mathcal{S},h(x)\leq\log b\\}|\gg b(\log b)^{\rho-1},\
\mathcal{S}=\\{x\in(\mathbb{Q}^{\times})^{4},\sum x_{i}^{3}=1=\sum x_{i}\\}.$
(11.36)

Here for each $x\in\mathcal{S}$ we denote by $h(x)$ the logarithmic Weil
height of $x\in\mathbb{A}^{4}_{\mathbb{Z}}(\bar{\mathbb{Q}})$ and the implied
constant depends again only on $V_{\mathbb{Q}}$. The following proof shows
that the lower bounds in the claim and in (11.36) are both optimal if Manin’s
conjecture holds for the $\mathbb{Q}$-points of the smooth cubic surface
defined by $\sum x_{i}^{3}-(\sum x_{i})^{3}$ inside
$\mathbb{P}^{3}_{\mathbb{Q}}$.

###### Proof of the Claim.

To identify $V_{\mathbb{Q}}$ with a smooth cubic surface in
$\mathbb{P}^{3}_{\mathbb{Q}}$, we consider $f=\sum x_{i}^{3}-(\sum x_{i})^{3}$
in $A=\mathbb{Z}[x_{1},\dotsc,x_{4}]$ and $p_{3}=\sum z_{i}^{3}$ in
$B=\mathbb{Z}[z_{0},\dotsc,z_{4}]$. Then $x_{i}\mapsto z_{i}$ defines an
isomorphism of rings $A/(f)\overset{\sim}{\to}B/(\sigma_{1},p_{3})$ with
inverse given by $z_{i}\mapsto x_{i}$ for $i\geq 1$ and $z_{0}\mapsto-\sum
x_{i}$. This induces an isomorphism of varieties over $\mathbb{Z}$:

$\tau:V^{\prime}=\textnormal{Proj}\bigl{(}B/(f)\bigl{)}\overset{\sim}{\to}V,\quad
h(P)\leq h(\tau(P))\leq h(P)+\log 8,\ P\in V^{\prime}(\bar{\mathbb{Q}}),$
(11.37)

where $h$ denotes the logarithmic Weil height on
$V^{\prime}\subset\mathbb{P}^{3}_{\mathbb{Z}}$ and
$V\subset\mathbb{P}^{4}_{\mathbb{Z}}$. We now work over $\mathbb{Q}$. To
simplify notation we write $V$ for $V_{\mathbb{Q}}$ and $V^{\prime}$ for
$V^{\prime}_{\mathbb{Q}}$. As $V^{\prime}\subset\mathbb{P}^{3}_{\mathbb{Q}}$
is a smooth cubic surface with two rational skew lines, it satisfies all
assumptions in [SSD98, Thm 1] of Slater–Swinnerton-Dyer. Thus for each
$b\in\mathbb{R}$ with $b\to\infty$ their result gives

$|\\{P\in U^{\prime}(\mathbb{Q}),h(P)\leq\log b\\}|\gg b(\log b)^{\rho-1},$
(11.38)

where $U^{\prime}\subset V^{\prime}$ is obtained by removing all the lines
from $V^{\prime}$ and where the implied constant depends only on $V\cong
V^{\prime}$. To relate $U^{\prime}$ with $U$, we consider the 15 lines
$\Delta_{n}$ on $V$ given by the permutations of the line $z_{0}+z_{1}=0$,
$z_{2}+z_{3}=0$, $z_{4}=0$ and we consider for
$\epsilon=\tfrac{1+\sqrt{5}}{2}$ the 12 lines $L_{i}$ on $V$ given by the
permutations (fixing $z_{0}$) of the line

$z_{1}+\epsilon z_{2}+z_{3}=0,\ \epsilon z_{1}+z_{2}+z_{4}=0,\
-\epsilon(z_{1}+z_{2})+z_{0}=0.$

Notice that $\tau^{-1}(L_{i})$ and $\tau^{-1}(\Delta_{n})$ are both lines
inside $V^{\prime}$, the lines $L_{i}$ have no $\mathbb{Q}$-points since
$\epsilon$ does not lie in $\mathbb{Q}$, and the variety $V^{\prime}$ contains
exactly 27 lines since it is a smooth cubic surface. Then, on using the Newton
identity for $p_{3}$, we compute

$U=V\setminus(\cup\Delta_{n})\quad\textnormal{and}\quad\tau:U^{\prime}(\mathbb{Q})\overset{\sim}{\to}(U\setminus\cup
L_{i})(\mathbb{Q})=U(\mathbb{Q}).$ (11.39)

Thus (11.37) and (11.38) prove the claim. Furthermore, if Manin’s conjecture
holds for $V^{\prime}(\mathbb{Q})$, then the lower bound (11.38) is optimal
and hence the claim is optimal as well; indeed (11.37) assures that the
bijection $\tau^{-1}:U(\mathbb{Q})=(U\setminus\cup
L_{i})(\mathbb{Q})\overset{\sim}{\to}U^{\prime}(\mathbb{Q})$ injects the set
$\\{P\in U(\mathbb{Q}),h(P)\leq\log b\\}$ into $\\{P\in
U^{\prime}(\mathbb{Q}),h(P)\leq\log b\\}$ for all $b\in\mathbb{R}_{>0}$. ∎

####  11.7.3 Degeneration of $S$-integral points

Corvaja–Zannier [CZ04, CZ06, CZ10] proved in particular the degeneration of
$S$-integral points on smooth cubic surfaces minus two completely reducible
hyperplane sections. In this section we give the basic computations which show
that their result directly implies the degeneration of $S$-integral points on
the affine variety defined in (11.4).

We continue our notation. Denote by $V=V_{\mathbb{Q}}$ the projective variety
over $\mathbb{Q}$ defined by (1.6), and for $i\neq j$ consider the divisor
$D=V_{+}(z_{i})\cup V_{+}(z_{j})$ of $V$. We again identify $V$ with the
smooth cubic surface $V^{\prime}\subset\mathbb{P}^{3}_{\mathbb{Q}}$ defined by
(11.37). After restricting the inverse
$\tau^{-1}:V\overset{\sim}{\to}V^{\prime}$ of the isomorphism in (11.37) to
the open subvariety $V\setminus D$, we obtain

$\tau^{-1}:V\setminus D\overset{\sim}{\to}V^{\prime}\setminus D^{\prime},\quad
D^{\prime}=\tau^{-1}(D).$

To determine the divisor $D^{\prime}$ of $V^{\prime}$, we denote by
$\\{\Delta_{n}\\}$ the 15 lines defined in $\mathsection$11.7.2. On using the
same arguments as in (11.39), we compute that $V_{+}(z_{i})$ is the union of
the three lines in $\\{\Delta_{n}\\}$ with $z_{i}=0$. Then $D=\cup\Delta_{n}$
is the union of six lines $\Delta_{n}$ and we find that

$D^{\prime}=\cup\tau^{-1}(\Delta_{n})$

consists of six lines $\tau^{-1}(\Delta_{n})$ lying in two planes such that no
three of them intersect. Hence an application of Corvaja–Zannier [CZ10, Thm 1]
with $V^{\prime}$ and $D^{\prime}$ gives the following: For any number field
$K$ and each finite set $S$ of places of $K$, the set of $S$-integral points
of $V^{\prime}\setminus D^{\prime}$, and thus of $V\setminus D$, is not
Zariski dense. Moreover [CZ10, Thm 9] of Corvaja–Zannier shows inter alia that
this statement is optimal in the following sense: For any line $\Delta_{n}$
inside $D$ define $D_{0}=D\setminus\Delta_{n}$ and
$D^{\prime}_{0}=D^{\prime}\setminus\tau^{-1}(\Delta_{n})$. Then there exist
$K$ and $S$ such that the set of $S$-integral points of $V^{\prime}\setminus
D^{\prime}_{0}$, and thus of $V\setminus D_{0}$, is in fact Zariski dense.

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Rafael von Känel, IAS Tsinghua University, Beijing

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Arno Kret, University of Amsterdam, Amsterdam

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