The split torsor method for Manin's conjecture

TL;DR

This paper introduces the split torsor method to count rational points on Fano varieties and proves Manin's conjecture for specific nonsplit quartic del Pezzo surfaces using o-minimal structures.

Contribution

The paper develops the split torsor method and applies it to establish Manin's conjecture for certain nonsplit quartic del Pezzo surfaces over any number field.

Findings

Proves Manin's conjecture for all nonsplit quartic del Pezzo surfaces of type A3+A1

Introduces the split torsor method for counting rational points

Utilizes o-minimal structures to solve the counting problem

Abstract

We introduce the split torsor method to count rational points of bounded height on Fano varieties. As an application, we prove Manin's conjecture for all nonsplit quartic del Pezzo surfaces of type $\mathbf A_3+\mathbf A_1$ over arbitrary number fields. The counting problem on the split torsor is solved in the framework of o-minimal structures.

Equations

$$N_{U,H}(B):=|\{\mathbf{x}\in U(K):H(\mathbf{x})\le B\}|=c_{S,H}B(\log B{)}^{\rho (\tilde{S})-1}(1+o(1))$$

$$\hat{T}:={\mathrm{Hom}}_{\mathrm{Gal}(\overline{K}/K)}(T_{\overline{K}},\text{mathds}{G}_{\mathrm{m},\overline{K}})\to \mathrm{Pic}(X_{\overline{K}})$$

$$\pi :\mathcal{Y}({\mathcal{O}}_K)\to \mathcal{X}({\mathcal{O}}_K)\subseteq X(K)$$

$${\mathcal{T}}_a(\text{mathds}Z)=\{(z_1,z_2)\in \text{mathds}{Z}^2:z_1^2-a{z}_2^2=1\}.$$

$${\pi }^{\prime }:{\mathcal{Y}}^{\prime }({\mathcal{O}}_K)\to \mathcal{X}({\mathcal{O}}_K).$$

$$S_a:x_0{x}_4+x_1^2-a{x}_3^2=x_2{x}_3-x_4^2=0,$$

$$\eta (\mathfrak{q};a):=\left|\left\{\rho (\mathrm{mod}\mathfrak{q}{\mathfrak{g}}^{-1}{\mathfrak{g}}^{\prime })|\begin{array}{rl}& \rho {\mathcal{O}}_K+\mathfrak{q}{\mathfrak{g}}^{-1}{\mathfrak{g}}^{\prime }={\mathfrak{g}}^{\prime },{\rho }^2\equiv a(\mathrm{mod}\mathfrak{q}),\\ & \mathfrak{g}=\mathfrak{q}+a{\mathcal{O}}_K,{\mathfrak{g}}^{\prime }=\prod _{\mathfrak{p}}{\mathfrak{p}}^{\lceil v_{\mathfrak{p}}(\mathfrak{g})/2\rceil }\end{array}\right\}\right|.$$

$$\left(\frac{a}{\mathfrak{p}}\right)=\{\begin{array}{ll}1,& \text{a is a quadratic residue mod p},\\ -1,& \text{a is a quadratic nonresidue mod p}.\end{array}$$

$$\left(\frac{a/{\pi }_{\mathfrak{p}}^{v_{\mathfrak{p}}(a)}}{\mathfrak{p}}\right)$$

$$N_{U,H}(B)=c_{S_a,H}B(\log B{)}^4+O_{ϵ,a,K}(B(\log B{)}^{4-\frac{1}{d}+ϵ}).$$

$$c_{S_a,H}:={\alpha }_{S_a}{\left(\frac{2^{r_1}(2\pi {)}^{r_2}{h}_K{R}_K}{|{\mu }_K|\cdot \sqrt{|{\Delta }_K|}}\right)}^5\frac{1}{|{\Delta }_K|}\left(\prod _{v\in {\Omega }_K}{\omega }_v(S_a)\right),$$

$${\alpha }_{S_a}$$

$${\omega }_v(S_a)$$

$$r_a(\mathfrak{p})$$

$$s_a(\mathfrak{p})$$

$$N_v(y_5,y_6,y_7):=\max \{|y_6(a{y}_6^2-y_7^2){|}_v,|y_5{y}_6{y}_7{|}_v,|y_5^3{|}_v,|y_5{y}_6^2{|}_v,|y_5^2{y}_6{|}_v\}.$$

$${\omega }_p(S_a)={\left(1-\frac{1}{p}\right)}^5\cdot \{\begin{array}{ll}1+\frac{5}{p},& p\mid a,\\ 1+\frac{6}{p}+\frac{1}{p^2},& p\ne 2,\left(\frac{a}{p}\right)=1,\\ 1+\frac{4}{p}+\frac{1}{p^2},& p\ne 2,\left(\frac{a}{p}\right)=-1,\\ \frac{17}{4},& p=2,a\equiv 1(\mathrm{mod}8),\\ \frac{15}{4},& p=2,a\equiv 5(\mathrm{mod}8),\\ \frac{7}{2},& p=2,a\equiv 3,7(\mathrm{mod}8).\end{array}$$

$$\text{mathds}{Z}^5\cong \mathrm{Pic}({\tilde{S}}_a)\hookrightarrow \mathrm{Pic}(({\tilde{S}}_a{)}_{\overline{K}})\cong \text{mathds}{Z}^6.$$

$$a_1{a}_8+a_7^2-a{a}_2^4{a}_3^2{a}_4^6{a}_6^2=0,$$

$$N_{U,H}(B)\sim \frac{{\rho }_K}{3|{\Delta }_K|}\cdot \prod _{v\mid \infty }{\omega }_v(S_a)\cdot B\cdot \sum _{{\mathfrak{a}}_1,\dots ,{\mathfrak{a}}_4}\frac{{\vartheta }_1({\mathfrak{a}}_1,\dots ,{\mathfrak{a}}_4)}{\mathfrak{N}({\mathfrak{a}}_1\cdots {\mathfrak{a}}_4)};$$

$$3{\rho }_K^4\cdot {\alpha }_{S_a}\cdot (\log B{)}^4\cdot \prod _{\mathfrak{p}}{\omega }_{\mathfrak{p}}(S_a),$$

$${\rho }_K:=\frac{2^{r_1}(2\pi {)}^{r_2}{R}_K{h}_K}{|{\mu }_K|\cdot |{\Delta }_K{|}^{1/2}},$$

$$H(\mathbf{x}):=\prod _{v\in {\Omega }_K}\max \{|x_0{|}_v,\dots ,|x_n{|}_v\}.$$

$$\overline{R}=\overline{k}[{\eta }_1,\dots ,{\eta }_N]/I$$

$$f_1,\dots ,f_t\in \overline{k}[{\eta }_1,\dots ,{\eta }_N]\setminus \sqrt{I}$$

$$R:=A[{\eta }_1,\dots ,{\eta }_N]/(g_1,\dots ,g_s)$$

$$i,j\in \{1,\dots ,t\}$$

$$R[f_i^{-1}]$$

$$\sqrt{C_{\overline{k}}}\cap A[{\eta }_1,\dots ,{\eta }_N]=\sqrt{C_A},$$

$${\left(\frac{\partial g_i}{\partial {\eta }_j}(\underline{a})\right)}_{\begin{array}{c}1\le i\le s\\ 1\le j\le N\end{array}}$$

$$\eta ({\mathfrak{p}}^k;a)=\{\begin{array}{ll}1+\left(\frac{a}{\mathfrak{p}}\right),& \mathfrak{p}\nmid 2a,\\ 1,& k\le v_{\mathfrak{p}}(a),\\ 0,& k>v_{\mathfrak{p}}(a),\text{vp(a) odd},\\ 1+\left(\frac{a/{\pi }_{\mathfrak{p}}^{v_{\mathfrak{p}}(a)}}{\mathfrak{p}}\right),& k>v_{\mathfrak{p}}(a),\text{vp(a)>0 even},\mathfrak{p}\nmid 2,\\ \le \mathfrak{N}{\mathfrak{p}}^{k-v_{\mathfrak{p}}(a)}-\mathfrak{N}{\mathfrak{p}}^{k-v_{\mathfrak{p}}(a)-1},& v_{\mathfrak{p}}(a)<k\le v_{\mathfrak{p}}(4a)+1,\text{vp(a) even},\mathfrak{p}\mid 2,\\ \eta ({\mathfrak{p}}^{v_{\mathfrak{p}}(4a)+1};a),& k>v_{\mathfrak{p}}(4a)+1,\text{vp(a) even},\mathfrak{p}\mid 2.\end{array}$$

Full text

1. 1 Introduction
2. 2 Integral models of split torsors
3. 3 Arithmetic functions
4. 4 The surfaces and their torsors
5. 5 Parameterization
6. 6 The first summation
7. 7 The remaining summations
8. 8 The leading constant

1. Introduction

1.1. Manin’s conjecture

Manin’s conjecture makes a precise prediction for the number of rational points of bounded anticanonical height on (possibly singular) Fano varieties over number fields [FMT89, Pey95, BT98, Pey03]. For simplicity, we state it for an anticanonically embedded del Pezzo surface $S\subset\mathds{P}^{n}_{K}$ of degree $\geq 3$ with at most $\mathbf{ADE}$-singularities over a number field $K$: Let $U$ be the complement of the lines on $S$, and let $H$ be the height function on $S(K)$ induced by the exponential Weil height on $\mathds{P}^{n}(K)$. Then we expect that

[TABLE]

as $B\to\infty$, where $\rho({\widetilde{S}})$ is the Picard number of a minimal desingularization ${\widetilde{S}}$ of $S$ over $K$, and $c_{S,H}$ is a positive constant that can be written as an explicit prescribed product of invariants of $K$ and $S$, including local densities on ${\widetilde{S}}$.

Two major approaches to Manin’s conjecture are harmonic analysis on adelic points and the universal torsor method. The harmonic analysis approach can only be applied to equivariant compactifications of algebraic groups, which include just a few types of del Pezzo surfaces, mostly of high degree; see [DL10, DL15] for their classification. For these, it can give very general results, covering all varieties in a given class over arbitrary number fields; see [FMT89, BT98, CLT02], for example. The universal torsor method can in principle be applied to all Fano varieties, but in each instance so far, it treats only a single example or a smaller family of varieties, with new difficulties arising in every case. It has been applied mostly to split varieties (that is, varieties for which the Picard group and the geometric Picard group coincide; for del Pezzo surfaces of degree $\leq 7$, this is equivalent to the lines being defined over the ground field) over $\mathds{Q}$. The universal torsor method over arbitrary number fields has been developed in [FP16]. Due to the possible presence of nonsplit tori, it does not seem to be the right way to treat nonsplit varieties (that is, varieties with a nontrivial Galois action on their geometric Picard group), as we explain below.

Here, we propose a new variation, the split torsor method for nonsplit varieties, and we prove Manin’s conjecture for all the nonsplit forms of the variety considered in [FP16] over arbitrary number fields.

1.2. Torsors

Let $X$ be a projective variety over a number field $K$ with algebraic closure ${\overline{K}}$. Colliot-Thélène and Sansuc studied torsors $\pi\colon Y\to X$ under (quasi)tori $T$ over $X$ [CTS87]. Such a torsor has a type, namely the homomorphism of $\operatorname{Gal}({\overline{K}}/K)$-modules

[TABLE]

that sends a character $\chi\colon T_{\overline{K}}\to\mathds{G}_{\mathrm{m},{\overline{K}}}$ to the class of the extension $\chi_{*}Y$. If $\operatorname{Pic}(X_{\overline{K}})$ is free and finitely generated, then one may consider torsors under the Néron–Severi torus $T$ with $\widehat{T}=\operatorname{Pic}(X_{\overline{K}})$ whose type is the identity. These are called universal torsors and are particularly useful, for example, to study the Hasse principle and weak approximation on geometrically rational varieties.

1.3. Universal torsor method

The basic idea of the universal torsor method for Manin’s conjecture consists in constructing a model $\mathcal{T}$ of the Néron–Severi torus $T$ over the ring of integers $\mathcal{O}_{K}$ of $K$ and a torsor $\pi\colon\mathcal{Y}\to\mathcal{X}$ under $\mathcal{T}$ that is an $\mathcal{O}_{K}$-model of a universal torsor $\pi\colon Y\to X$, and using the induced map

[TABLE]

to parameterize rational points on $X$ by the $\mathcal{T}(\mathcal{O}_{K})$-orbits on $\mathcal{Y}(\mathcal{O}_{K})$. After lifting the height function, one can use techniques of analytic number theory to count these orbits of bounded height.

If $X$ is split and $K=\mathds{Q}$, the map (1.1) is surjective and has finite fibers.

If $X$ is nonsplit, the map (1.1) does not need to be surjective even if $K=\mathds{Q}$, see for example [BBP12, §4]. Nonetheless, torsor parameterizations have been used to verify Manin’s conjecture for some nonsplit varieties over $\mathds{Q}$ that split over $\mathds{Q}(\sqrt{-1})$. See [BF04] for a smooth quintic del Pezzo surface, [BB11] for a smooth quartic del Pezzo surface, [BB07] for a quartic del Pezzo surface of type ${\mathbf{D}}_{4}$ and [BBP12, BB12, BT13, Des16] for quartic del Pezzo surfaces of type $2{\mathbf{A}}_{1}$ (Châtelet surfaces). For example in [BBP12, Corollary 5.25], several (but finitely many) twists of a universal torsor are used to parameterize the rational points on the underlying variety. For all such varieties, the Néron–Severi torus $T$ is not split, but $\mathcal{T}(\mathds{Z})$ is still finite.

Finiteness of $\mathcal{T}(\mathds{Z})$ also holds for varieties over $\mathds{Q}$ that split over imaginary quadratic fields $\mathds{Q}(\sqrt{a})$, but here one may have to deal with bad reduction at primes dividing $a$; apparently no examples with $a<-1$ have been worked out.

Already in the case of nonsplit varieties over $\mathds{Q}$ that split over a real quadratic field $\mathds{Q}(\sqrt{a})$, the set $\mathcal{T}(\mathds{Z})$ is infinite. A typical example is $\mathcal{T}\cong\mathcal{T}_{a}\times\mathds{G}_{\mathrm{m},\mathds{Z}}^{\rho-1}$, where $\mathcal{T}_{a}$ is a model of the norm-$1$-torus for $\mathds{Q}(\sqrt{a})/\mathds{Q}$ with

[TABLE]

This holds for example for the surfaces treated in the present paper if one takes $a>0$ in Theorem 1.1 below.

The main difficulty in the application of the universal torsor method for nonsplit varieties that split over real quadratic fields would be the construction of a fundamental domain for the action of $\mathcal{T}(\mathds{Z})$ on a model $\mathcal{Y}\to\mathcal{X}$ of a universal torsor $Y\to X$. An additional difficulty could be the need for twists of these torsors to cover $X(\mathds{Q})$. Apparently this strategy has never been implemented.

1.4. Split torsor method

For nonsplit varieties, we propose to work with the split torsor instead of universal torsors. The split torsor $\pi^{\prime}\colon Y^{\prime}\to X$ is a torsor under the split torus $T^{\prime}$ with character group $\widehat{T}^{\prime}=\operatorname{Pic}(X)$ whose type is the natural map $\operatorname{Pic}(X)\to\operatorname{Pic}(X_{\overline{K}})$; it is unique up to isomorphism and the map $Y^{\prime}(K)\to X(K)$ is surjective.

To make the split torsor $\pi^{\prime}\colon Y^{\prime}\to X$ explicit, we have developed a theory and constructions of Cox rings of arbitrary type in [DP19]. We can construct an $\mathcal{O}_{K}$-model $\pi^{\prime}\colon\mathcal{Y}^{\prime}\to\mathcal{X}$ using the theory that we develop in Section 2. Assuming $K=\mathds{Q}$ and $\operatorname{Pic}(X)\cong\mathds{Z}^{\rho^{\prime}}$, the map

[TABLE]

is surjective, and it is a $(2^{\rho^{\prime}}:1)$-map.

In [BB07], a parameterization of the rational points on a quartic del Pezzo surface of type ${\mathbf{D}}_{4}$ over $\mathds{Q}$ that splits over $\mathds{Q}(i)$ is produced by elementary manipulations of the defining equations, without a geometric interpretation. Using the techniques from [DP19] and Section 2, one can show that a model of the split torsor gives the same parameterization; see [DP19, Example 5.1].

The split torsor method not only avoids the difficulties due to the splitting fields of quasitori, but it produces a parameterization that enables the application in the nonsplit setting of known counting techniques typically used on split varieties.

The difficulty of the resulting counting problem depends on the variety in question. We expect that the primes of $\mathcal{O}_{K}$ that are ramified in the splitting field of $X$ will typically be places of bad reduction for the model $\mathcal{X}$ and cause considerable additional complications, as it happens for the varieties considered in this paper. In our case, the first new difficulty appears at the Möbius inversion step, introducing an extra summation over congruence classes. This is eventually encoded in a fairly complicated arithmetic function that we study via related Hecke $L$-functions.

If $K$ is an arbitrary number field, the maps (1.1) and (1.2) do not need to be surjective nor have finite fibers. For split varieties, both problems have been addressed and solved in [FP16] via twists of torsors and fundamental domains for the action of $\mathcal{T}(\mathcal{O}_{K})$. We show that a similar approach works also for nonsplit varieties.

1.5. Our application

We investigate Manin’s conjecture for quartic del Pezzo surfaces with singularities of type ${\mathbf{A}}_{3}+{\mathbf{A}}_{1}$ over an arbitrary number field $K$. In Section 4.1, we show that each such surface is isomorphic over $K$ to precisely one of the surfaces $S_{a}\subset\mathds{P}^{4}_{K}$ defined by

[TABLE]

where $a$ runs through a set of representatives of $K^{\times}/K^{\times 2}$.

We assume that $a$ is an algebraic integer. We could further assume that $a$ is squarefree (that is, there is no $c\in\mathcal{O}_{K}\smallsetminus\mathcal{O}_{K}^{\times}$ with $c^{2}\mid a$), but this assumption would not simplify our work since there still could be prime ideals ${\mathfrak{p}}$ in $\mathcal{O}_{K}$ with ${\mathfrak{p}}^{2}\mid a$.

The two singularities on $S_{a}$ are $(1:0:0:0:0)$ of type ${\mathbf{A}}_{3}$ and $(0:0:1:0:0)$ of type ${\mathbf{A}}_{1}$. Let $\sqrt{a}\in{\overline{K}}$ be a square root of $a$. The surface $S_{a}$ contains three lines: $\{x_{1}=x_{3}=x_{4}=0\}$ defined over $K$, and $\{x_{1}+\sqrt{a}x_{3}=x_{2}=x_{4}=0\}$ and $\{x_{1}-\sqrt{a}x_{3}=x_{2}=x_{4}=0\}$ defined over $K(\sqrt{a})$.

If $a$ is a square, the surface $S_{a}\cong S_{1}$ is split over $K$. Manin’s conjecture for $S_{1}$ is proved in [Der09, §8] over $\mathds{Q}$, in [DF14b, §2] over imaginary quadratic fields and in [FP16] over arbitrary number fields. If $a$ is not a square, the surface $S_{a}$ is nonsplit over $K$ and splits over $K(\sqrt{a})$; our main result is a proof of Manin’s conjecture for all these surfaces over an arbitrary number field $K$.

The following arithmetic function plays an important role in our analysis and appears in the nonarchimedean local densities in the constant $c_{S_{a},H}$. For every ideal ${\mathfrak{q}}$ of $K$, let

[TABLE]

As we see in Section 3, it is closely related to the Legendre symbol [Neu99, §V.3], which is defined for prime ideals ${\mathfrak{p}}\nmid 2a$ as

[TABLE]

We denote by $\pi_{\mathfrak{p}}$ a uniformizer for the completion $K_{\mathfrak{p}}$ and observe that the symbol

[TABLE]

is well defined when $v_{\mathfrak{p}}(a)$ is even and ${\mathfrak{p}}\nmid 2$, as $a/\pi_{\mathfrak{p}}^{v_{\mathfrak{p}}(a)}$ is a unit in $K_{\mathfrak{p}}$. We refer to Section 1.7 for the standard notation that we use in the description of the leading constant $c_{S_{a},H}$ in the theorem below.

Theorem 1.1**.**

Let $K$ be a number field and let $H$ be the exponential Weil height on $\mathds{P}^{4}_{K}$. Assume that $a\in\mathcal{O}_{K}$ is not a square. Let $S_{a}\subset\mathds{P}^{4}_{K}$ be the singular quartic del Pezzo surface defined by (1.3). Let $U$ be the complement of the lines on $S_{a}$. Let $\epsilon>0$. For $B\to\infty$, we have

[TABLE]

Here,

[TABLE]

where

[TABLE]

with

[TABLE]

[TABLE]

and, for $v\mid\infty$ and $(y_{5},y_{6},y_{7})\in K_{v}^{3}$,

[TABLE]

This agrees with the conjectures of Manin and Peyre [Pey03, Formule empirique 5.1].

Remark 1.2*.*

In the case $K=\mathds{Q}$, we may assume that $a\neq 1$ is a squarefree integer. Then

[TABLE]

Considering the ${\mathfrak{p}}$-adic densities in our leading constant, it is not surprising that the primes ${\mathfrak{p}}\mid 2a$, and in particular those with even $v_{\mathfrak{p}}(a)$, cause considerable technical difficulties. These appear at many places of the proof.

1.6. Strategy of proof

The proof implements the split torsor method that we have proposed above. Let ${\widetilde{S}}_{a}$ be the minimal desingularization of $S_{a}$. Here, the type of the torsor is

[TABLE]

In Section 4, we construct a split torsor $Y^{\prime}\to{\widetilde{S}}_{a}$ over $K$ and an $\mathcal{O}_{K}$-model $\mathcal{Y}^{\prime}\to\widetilde{\mathcal{S}}_{a}$ using the techniques developed in Section 2 and in [DP19]. In Section 5.1, we use $\mathcal{Y}^{\prime}\to\widetilde{\mathcal{S}}_{a}$ and its $h_{K}^{5}$ twists (where $h_{K}$ is the class number of $K$) to produce an explicit parameterization of $U(K)$ in terms of coordinates $(a_{1},\dots,a_{8})$ satisfying the torsor equation

[TABLE]

coprimality conditions and height conditions up to the action of $\mathcal{T}^{\prime}(\mathcal{O}_{K})\cong(\mathcal{O}_{K}^{\times})^{5}$.

As in [FP16], the five degrees of freedom coming from this action allow us to choose a fundamental domain such that all conjugates of $a_{1},\dots,a_{4}$ and of the height function have roughly the same size; see Section 5.2. For fixed $a_{1},\dots,a_{4}$, the volume of the fundamental domain cut by the height conditions for the remaining variables is roughly $\prod_{v\mid\infty}\omega_{v}(S_{a})\cdot B/|N(a_{2}a_{3}a_{4})|$, where $N$ denotes the norm of $K/\mathds{Q}$. In Section 5.3, we remove the coprimality conditions on $a_{5},a_{6},a_{7}$ by Möbius inversions and interpret the torsor equation (1.6) as a congruence modulo $a_{1}$, eliminating $a_{8}$. This is much more subtle than in the split case and isolates a summation over $\rho$ as in (1.4).

For fixed $a_{1},\dots,a_{4},B$, the first summation in Section 6 estimates the number of $a_{5},a_{6},a_{7}$ satisfying our congruence. It has many steps. In Section 6.1, we prove a general summation lemma that will be applied to all error terms in the first summation. In Section 6.2, we remove the “spikes” in our counting domain where $a_{6}$ or $a_{7}$ have small conjugates. As in [FP16], we show that the counting problem for $(a_{5},a_{6},a_{7})$ consists in estimating the number of lattice points lying in a set definable in Wilkie’s o-minimal structure $\mathds{R}_{\exp}$ [Wil96]. Therefore, we can apply [BW14]; the error terms depend on the volumes of the projections of this set to coordinate hyperplanes, which we estimate in Section 6.4. In Section 6.5, we estimate the error term coming from the “spikes”. This completes the first summation, which shows roughly that after turning the summation over $a_{1},\dots,a_{4}$ and the $h_{K}^{5}$ twists into a summation over ideals ${\mathfrak{a}}_{1},\dots,{\mathfrak{a}}_{4}$, we have

[TABLE]

see Proposition 6.9. Here, $\vartheta_{1}$ is an arithmetic function coming from the coprimality conditions and involving the function $\eta(\cdot;a)$ defined in (1.4).

In the split case, the remaining summations in [FP16, §12] are a straightforward application of the theory developed in [DF14a, §7]. In our nonsplit case, we must study averages of arithmetic functions involving the Legendre symbol (1.5); this is done in Section 7.1 using the corresponding Hecke $L$-function, which we discuss in Section 3, combined with Perron’s formula. In Section 7.2, we estimate the sum over ${\mathfrak{a}}_{1},\dots,{\mathfrak{a}}_{4}$ as

[TABLE]

which completes the proof of the asymptotic formula.

In Section 8, we show that our leading constant agrees with Peyre’s prediction in [Pey03]. We study the convergence factors in Section 8.1 and compute the local densities in Section 8.2. Again, the most difficult places are ${\mathfrak{p}}\mid 2a$ with even $v_{\mathfrak{p}}(a)$.

1.7. Notation

Additionally to the notation introduced in the course of this introduction, we will use the following: For the number field $K$, the residue of the Dedekind zeta function at $s=1$ is

[TABLE]

where $r_{1}$ is its number of real embeddings, $r_{2}$ its number of complex embeddings, $R_{K}$ its regulator, $h_{K}$ its class number, $\mu_{K}$ its group of roots of unity, and $\Delta_{K}$ its discriminant. Let $d=[K:\mathds{Q}]$ be its degree.

Let $\Omega_{K}$ be its set of places. For $v\in\Omega_{K}$, let $K_{v}$ be the corresponding completion of $K$. Let $\Omega_{\infty}$ be the set of archimedean places; we also write $v\mid\infty$ for $v\in\Omega_{\infty}$. Let $\Omega_{f}$ be the set of nonarchimedean places. Let $U_{K}$ be a free part of the unit group (that is, $\mathcal{O}_{K}^{\times}=U_{K}\times\mu_{K}$).

Let $\mathscr{I}_{K}$ be the monoid of nonzero ideals in $\mathcal{O}_{K}$ and let $\mathscr{C}\subset\mathscr{I}_{K}$ be a system of representatives of the class group of $\mathcal{O}_{K}$. Let $\mathfrak{N}{\mathfrak{q}}$ be the absolute norm of a (fractional) ideal ${\mathfrak{q}}$, and let $N(a)=N_{K/\mathds{Q}}(a)$ be the norm of $a\in K$ over $\mathds{Q}$. The symbol ${\mathfrak{p}}$ always denotes a prime ideal of $\mathcal{O}_{K}$. Let $\mu_{K}\colon\mathscr{I}_{K}\to\{-1,0,1\}$ be the Möbius function. For ${\mathfrak{q}}\in\mathscr{I}_{K}$, let $\omega_{K}({\mathfrak{q}})$ be the number of distinct prime ideals dividing ${\mathfrak{q}}$.

For $v\in\Omega_{K}$ lying above $w\in\Omega_{\mathds{Q}}$, let $d_{v}\coloneqq[K_{v}:\mathds{Q}_{w}]$ be the local degree, and let $|\cdot|_{v}=|N_{K_{v}/\mathds{Q}_{w}}(\cdot)|_{w}$ be the $v$-adic absolute value, where $|\cdot|_{w}$ is the usual $p$-adic or real absolute value on $\mathds{Q}_{w}$. Also for $v\in\Omega_{K}$, let $\sigma_{v}\colon K\to K_{v}$ be the $v$-adic embedding, we use the same symbol for the component-wise map $\sigma_{v}\colon K^{n}\to K_{v}^{n}$, and denote by $\sigma\colon K\to\prod_{v\in\Omega_{K}}K_{v}$ the diagonal map. For $a\in K$, we often write $|a|_{v}$ for $|\sigma_{v}(a)|_{v}$. The exponential Weil height of a rational point $\mathbf{x}=(x_{0}:\dots:x_{n})\in\mathds{P}^{n}(K)$ is

[TABLE]

For each $a\in K$, we fix a square root of $a$ in ${\overline{K}}$ and we denote it by $\sqrt{a}$.

We call $x\in K$ defined (or invertible) modulo an ideal ${\mathfrak{q}}$ if $v_{\mathfrak{p}}(x)\geq 0$ (or $v_{\mathfrak{p}}(x)=0$) for all ${\mathfrak{p}}\mid{\mathfrak{q}}$. If $x,y\in K$ are defined modulo ${\mathfrak{q}}$, the notation ${x}\equiv_{{\mathfrak{q}}}{y}$ means that $v_{\mathfrak{p}}(x-y)\geq v_{\mathfrak{p}}({\mathfrak{q}})$ for all ${\mathfrak{p}}\mid{\mathfrak{q}}$.

When we use Landau’s $O$-notation and Vinogradov’s $\ll$-notation, we mean that the corresponding inequalities hold for all values in the relevant range. The implied constants may always depend on $K$ and $a$; additional dependencies are denoted by subscripts.

All volumes of subsets of $\mathds{R}^{n}$ are computed with respect to the usual Lebesgue measure, unless stated otherwise.

Acknowledgements

The first author was partly supported by grant DE 1646/4-2 of the Deutsche Forschungsgemeinschaft. Some of this work was done while he was on sabbatical leave at the University of Oxford. The second author was partly supported by grant ES 60/10-1 of the Deutsche Forschungsgemeinschaft. The authors are grateful to D. R. Heath-Brown, E. Sofos and the anonymous referee for their helpful remarks.

2. Integral models of split torsors

We extend the construction of integral models of split universal torsors developed in [FP16, §3] to (not necessarily universal) torsors under split tori; see also [Pie15, §3]. We refer to [CTS87] for the theory of torsors and to [DP19] for the theory of Cox rings.

Construction 2.1**.**

Let $A$ be a noetherian integral domain. We fix a separable closure $\overline{k}$ of the fraction field of $A$. Let $\overline{X}$ be an integral projective $\overline{k}$-variety with a finitely generated Cox ring $\overline{R}$ of type $\lambda\colon\mathds{Z}^{r}\to\operatorname{Pic}(\overline{X})$ such that $\lambda(\mathds{Z}^{r})$ contains an ample divisor class. Let $\eta_{1},\dots,\eta_{N}$ be $\mathds{Z}^{r}$-homogeneous elements of $\overline{R}$ such that

[TABLE]

for a homogeneous ideal $I$ as in [DP19, Proposition 4.4]. Assume that $I$ is generated by polynomials $g_{1},\dots,g_{s}\in A[\eta_{1},\dots,\eta_{N}]$. Let

[TABLE]

be monic monomials such that the complement $\overline{Y}$ of the closed subset of $\operatorname{Spec}\overline{R}$ defined by $f_{1},\dots,f_{t}$ is a torsor of $\overline{X}$ of type $\lambda$ (cf. [DP19, Corollary 4.3]).

Let

[TABLE]

with the $\mathds{Z}^{r}$-grading induced by the $\mathds{Z}^{r}$-grading on $\overline{R}$. Assume that $(R;f_{1},\dots,f_{t})$ satisfies the following condition:

[TABLE]

Let $Y$ be the complement of the closed subset of $\operatorname{Spec}R$ defined by $f_{1},\dots,f_{t}$. For $i\in\{1,\dots,t\}$, let $U_{i}\coloneqq\operatorname{Spec}R[f_{i}^{-1}]$, let $R_{i}$ be the degree-[math]-part of the ring $R[f_{i}^{-1}]$ and let $V_{i}\coloneqq\operatorname{Spec}(R_{i})$. Let $X$ be the $A$-scheme obtained by gluing $\{V_{i}\}_{1\leq i\leq t}$, and let $\pi\colon Y\to X$ be the morphism induced by the inclusions $R_{i}\to R[f_{i}^{-1}]$ for $i\in\{1,\dots,t\}$.

Remark 2.2*.*

As in [FP16, Construction 3.1], $Y_{\overline{k}}\cong\overline{Y}$, $X_{\overline{k}}\cong\overline{X}$ and $\pi$ is an $A$-model of the torsor morphism $\overline{Y}\to\overline{X}$. The $\mathds{Z}^{r}$-grading on $R$ induces an action of $\mathds{G}_{m,A}^{r}$ on $Y$. The morphism $\pi$ is surjective and of finite presentation, and it gives an $X$-torsor under $\mathds{G}_{m,A}^{r}$ (compatible with the structure of $\overline{X}$-torsor of type $\lambda$ on $\overline{Y}$) if and only if $\pi$ is flat and the morphism of schemes $\psi\colon\mathds{G}^{r}_{m,A}\times_{\operatorname{Spec}A}Y\to Y\times_{X}Y$ defined by $(\underline{u},\underline{a})\mapsto(\underline{u}*\underline{a},\underline{a})$ is an isomorphism. The same proof as in [FP16, Theorem 3.3] shows that $\pi$ is an $X$-torsor under $\mathds{G}_{m,A}^{r}$ if $(R;f_{1},\dots,f_{t})$ satisfies the following condition:

[TABLE]

Remark 2.3*.*

If $A$ is a field, then $\pi\colon Y\to X$ is a torsor of type $\lambda\colon\mathds{Z}^{r}\to\operatorname{Pic}(\overline{X})$ (with trivial $\operatorname{Gal}(\overline{k}/A)$-action on $\mathds{Z}^{r}$) regardless of condition (2.2) by fpqc descent.

Remark 2.4*.*

Let $k$ be a finite extension of the fraction field of $A$ contained in $\overline{k}$. If $\widetilde{X}$ is a $k$-variety such that ${\widetilde{X}}_{\overline{k}}\cong\overline{X}$ and $\lambda$ factors through $\operatorname{Pic}(\widetilde{X})\to\operatorname{Pic}(\overline{X})$, the Cox ring $\overline{R}$ has a natural structure of $\operatorname{Gal}(\overline{k}/k)$-equivariant Cox ring of ${\widetilde{X}}_{\overline{k}}$ in the sense of [DP19, Definition 3.1], and we can choose $\eta_{1},\dots,\eta_{N}\in\overline{R}^{\operatorname{Gal}(\overline{k}/k)}$. Then $R_{k}$ is a Cox ring of $\widetilde{X}$ of type $\lambda\colon\mathds{Z}^{r}\to\operatorname{Pic}({\widetilde{X}}_{\overline{k}})$ and $Y_{k}\to\widetilde{X}$ is a torsor of $\widetilde{X}$ of type $\lambda\colon\mathds{Z}^{r}\to\operatorname{Pic}({\widetilde{X}}_{\overline{k}})$. In particular, $X_{k}\cong\widetilde{X}$.

Remark 2.5*.*

Straightforward generalizations of [FP16, Lemma 3.5, Proposition 3.6] hold in the setting of Construction 2.1 with the very same statement and the very same proof.

For the sake of precision we report the results analogous to [FP16, Propositions 3.7, 3.8].

Proposition 2.6**.**

Assume the setting of Construction 2.1.

1. (i)

Let $C_{\overline{k}}$ and $C_{A}$ be the ideals of $\overline{k}[\eta_{1},\dots,\eta_{N}]$ and $A[\eta_{1},\dots,\eta_{N}]$, respectively, generated by $f_{1},\dots,f_{t},g_{1},\dots,g_{s}$. If $f_{1},\dots,f_{t}$ have all the same degree $m\in\mathds{Z}^{r}$ such that $\lambda(m)$ is very ample, and

[TABLE]

then $X$ is projective over $A$. 2. (ii)

Assume that $A$ is a Dedekind domain, $R$ is an integral domain, $\operatorname{Spec}(R)\to\operatorname{Spec}(A)$ has geometrically integral fibers, and $\pi$ is flat (the last holds, for example, if (2.2) is satisfied). If the Jacobian matrix

[TABLE]

has rank $N-\dim\overline{X}-r$ for all $\underline{a}\in Y(\overline{k({\mathfrak{p}})})$ and ${\mathfrak{p}}\in\operatorname{Spec}(A)$, where $\overline{k({\mathfrak{p}})}$ is an algebraic closure of the residue field $k({\mathfrak{p}})$, then $X$ is smooth over $A$.

Proof.

For projectivity it suffices to replace [ADHL15, Corollary 1.6.3.6] by [DP19, Corollary 4.3] and [FP16, Lemma 3.5] by its straightforward analog (cf. Remark 2.5) in the proof of [FP16, Proposition 3.8]. Smoothness is proved by the same argument used in [FP16, Proposition 3.7]. ∎

Remark 2.7*.*

Assume that $\overline{k}$ has characteristic 0, that $\overline{X}$ is normal and that $\operatorname{Pic}(\overline{X})$ is finitely generated. By [ADHL15, Theorem 1.5.1.1], a Cox ring of $\overline{X}$ of identity type is an integral domain, hence also $\overline{R}$ is an integral domain, as the grading group of $\overline{R}$ is free. Then $R$ is an integral domain if and only if $I\cap A[\eta_{1},\dots,\eta_{N}]=(g_{1},\dots,g_{s})$.

3. Arithmetic functions

In this section, we study the function $\eta(\cdot;a)$ introduced in (1.4), and the Hecke $L$-function of the Legendre symbol, which plays an important role in Section 7.1 and in the study of the Tamagawa number in Section 8.1.

Let $K$ be a number field and fix an $a\in\mathcal{O}_{K}$ that is not a square. Recall the Legendre symbol from (1.5).

Lemma 3.1**.**

The function $\eta({\mathfrak{q}};a)$ defined in (1.4) is multiplicative in ${\mathfrak{q}}$. For a prime ideal ${\mathfrak{p}}$ and $k\in\mathds{Z}_{>0}$, we have

[TABLE]

Furthermore,

[TABLE]

Proof.

The multiplicativity of $\eta({\mathfrak{q}};a)$ follows from the Chinese remainder theorem. For ${\mathfrak{p}}\nmid 2a$, we have $\eta({\mathfrak{p}}^{k};a)=1+\left(\frac{a}{{\mathfrak{p}}}\right)$ for $k=1$ by the definition of $\eta$, and for $k>1$ by Hensel’s lemma. For the rest of the proof, we assume ${\mathfrak{p}}\mid 2a$ and write $n=v_{\mathfrak{p}}(a)$.

For $n>0$ and $k\leq n$, we have

[TABLE]

For odd $n\geq 1$ and $k>n$, we have

[TABLE]

because the first condition implies $v_{\mathfrak{p}}(\rho)=\left\lceil{n/2}\right\rceil$, hence $v_{\mathfrak{p}}(\rho^{2})=n+1$, which contradicts the second condition.

For even $n\geq 0$ and $k>n$, we have

[TABLE]

For ${\mathfrak{p}}\nmid 2$, we observe that for any uniformizer $\pi_{\mathfrak{p}}$ in $\mathcal{O}_{{\mathfrak{p}}}$,

[TABLE]

For ${\mathfrak{p}}\mid 2$, we have $\eta({\mathfrak{p}}^{n+1};a)\leq\mathfrak{N}{\mathfrak{p}}-1$ since there are only $\mathfrak{N}{\mathfrak{p}}-1$ possibilites for $\rho\pmod{{\mathfrak{p}}^{n/2+1}}$ satisfying $v_{\mathfrak{p}}(\rho)=n/2$. Every such $\rho$ has at most $\mathfrak{N}{\mathfrak{p}}^{k-n-1}$ lifts $\rho^{\prime}\pmod{{\mathfrak{p}}^{k-n/2}}$, hence $\eta({\mathfrak{p}}^{k};a)\leq\mathfrak{N}{\mathfrak{p}}^{k-n-1}\eta({\mathfrak{p}}^{n+1};a)$ for $k>n+1$.

By Hensel’s lemma, each $\rho_{0}\pmod{{\mathfrak{p}}^{k_{0}-n/2}}$ satisfying the congruence ${\rho^{2}}\equiv{a}\pmod{{\mathfrak{p}}^{k_{0}}}$ lifts to a unique solution $\rho\pmod{{\mathfrak{p}}^{k-n/2}}$ satisfying ${\rho^{2}}\equiv{a}\pmod{{\mathfrak{p}}^{k}}$ for every $k>k_{0}$ as soon as $k_{0}=2(v_{\mathfrak{p}}(2\rho))+1=v_{\mathfrak{p}}(4a)+1$.

The upper bound for $\eta({\mathfrak{q}};a)$ follows from the multiplicativity of $\eta$ and the fact that, for $k>0$, we have $\eta({\mathfrak{p}}^{k};a)\leq 2$ for all ${\mathfrak{p}}\nmid 2$, while $\eta({\mathfrak{p}}^{k};a)\leq\mathfrak{N}{\mathfrak{p}}^{v_{\mathfrak{p}}(4)+1}$ for ${\mathfrak{p}}\mid 2$. ∎

Example 3.2**.**

For $K=\mathds{Q}$ and squarefree $a\in\mathds{Z}$, we have

[TABLE]

For an ideal ${\mathfrak{q}}=\prod_{i=1}^{r}{\mathfrak{p}}_{i}^{e_{i}}$ (with prime ideals ${\mathfrak{p}}_{i}$ and $e_{i}\in\mathds{Z}_{>0}$) that is coprime to $2a\mathcal{O}_{K}$, we have as in [Neu99, Definition VI.8.2]

[TABLE]

For ${\mathfrak{q}}\in\mathscr{I}_{K}$, let

[TABLE]

Recall the definition [Lan18, Definition X] of characters modulo an ideal ${\mathfrak{f}}$. In the terminology of [Hei67, §1], [Neu99, VII.6.8], these are are (generalized) Dirichlet characters on the narrow ray class group modulo ${\mathfrak{f}}$. The following result must be well-known; over $\mathds{Q}$, see [Ser73, §VI.1.3, Proposition 5].

Lemma 3.3**.**

The function $\chi$ as in (3.1) is a nonprincipal character modulo $8a\mathcal{O}_{K}$.

Proof.

Let ${\mathfrak{f}}\coloneqq 8a\mathcal{O}_{K}$. By definition, $\chi$ is a multiplicative function on the monoid of all ideals in $\mathscr{I}_{K}$ that are coprime to ${\mathfrak{f}}$.

Let $x\in\mathcal{O}_{K}$ be totally positive with ${x}\equiv{1}\pmod{{\mathfrak{f}}}$. By reciprocity [Neu99, Theorem VI.8.3],

[TABLE]

where $\left(\frac{\cdot,\cdot}{{\mathfrak{p}}}\right)$ is the Hilbert symbol [Neu99, §V.3]. Since $x$ is totally positive, $\left(\frac{x,a}{{\mathfrak{p}}}\right)=1$ for all ${\mathfrak{p}}\mid\infty$. By Hensel’s lemma, ${x}\equiv{1}\pmod{8}$ implies that $x$ is a square in $K_{\mathfrak{p}}$ for every ${\mathfrak{p}}\mid 2$, hence $\left(\frac{x,a}{{\mathfrak{p}}}\right)=1$ for all ${\mathfrak{p}}\mid 2$. Writing $a\mathcal{O}_{K}=\prod_{{\mathfrak{p}}\mid a}{\mathfrak{p}}^{e_{\mathfrak{p}}}$, we have

[TABLE]

as ${\left(\frac{x}{{\mathfrak{p}}}\right)}\equiv{x^{(\mathfrak{N}{\mathfrak{p}}-1)/2}}\pmod{{\mathfrak{p}}}$ and ${x}\equiv{1}\pmod{{\mathfrak{p}}}$ for all ${\mathfrak{p}}\mid a$. In total, this proves $\chi(x\mathcal{O}_{K})=1$.

Next, we claim that $\chi({\mathfrak{a}})=\chi({\mathfrak{b}})$ for all ideals ${\mathfrak{a}}\sim{\mathfrak{b}}\pmod{{\mathfrak{f}}}$ as in [Lan18, Definition VIII] (that is, ${\mathfrak{a}}+{\mathfrak{f}}={\mathfrak{b}}+{\mathfrak{f}}=\mathcal{O}_{K}$ and there are totally positive $\alpha,\beta\in\mathcal{O}_{K}$ with $\alpha{\mathfrak{a}}=\beta{\mathfrak{b}}$ and $\alpha\equiv\beta\equiv 1\pmod{{\mathfrak{f}}}$). By multiplicativity

[TABLE]

Since $\chi(\alpha\mathcal{O}_{K})=\chi(\beta\mathcal{O}_{K})=1$ as above, our claim follows.

By the Grunwald–Wang Theorem [AT09, §X], the fact that $a$ is not a square in $K$ implies that it is not a square in $K_{\mathfrak{p}}$ for infinitely many ${\mathfrak{p}}$, in particular for some ${\mathfrak{p}}\nmid 2a$, and for these $\chi({\mathfrak{p}})=\left(\frac{a}{{\mathfrak{p}}}\right)=-1$. Therefore, $\chi$ is not the principal character modulo ${\mathfrak{f}}$. ∎

Lemma 3.4**.**

Assume that $a\in\mathcal{O}_{K}$ is not a square. Let $C\geq 0$ and $\epsilon>0$. For $t\geq 0$, we have

[TABLE]

Proof.

Let $\Sigma$ be the sum that we want to estimate. We may assume $t\geq 1$. As in [DF14a, Lemma 2.9], we deduce

[TABLE]

Since $\eta({\mathfrak{p}};a)=1+\left(\frac{a}{{\mathfrak{p}}}\right)$ for ${\mathfrak{p}}\nmid 2a$, an application of [Lan18, Satz LXXXIV] to the principal character and to $\chi$ as in (3.1) shows that the sum over ${\mathfrak{p}}$ is

[TABLE]

where the $O_{C,a}(1)$-term accounts for the summands with ${\mathfrak{p}}\mid 2a$. Estimating $o_{a}(\log\log t)$ by $\frac{\epsilon}{1+C}\log\log t$ for $t\gg_{C,a,\epsilon}1$ completes the proof. ∎

Lemma 3.5**.**

Let $\chi$ be as in (3.1).

1. (i)

For $s\in\mathds{C}$ with $\Re(s)>1$, we have

[TABLE]

where the series and the Euler product converge absolutely. It can be extended to an entire function on $\mathds{C}$. 2. (ii)

The series $L(s,\chi)$ converges conditionally for $\Re(s)>1-\frac{2}{d+1}$. 3. (iii)

We have

[TABLE]

where the product converges conditionally. 4. (iv)

For $\epsilon>0$ and $|\Im(s)|\geq 2$ and $\Re(s)\geq-\epsilon$, we have

[TABLE]

Proof.

For (i), see [Lan18, Definition XVII, Satz LVIII]. Since $\chi$ is nonprincipal, the second statement is [Lan18, Satz LXIII].

For (ii), let $\delta\coloneqq\frac{2}{d+1}$. By [Lan18, Satz XCV],

[TABLE]

Let $\Re(s)>1-\delta$. By partial summation, for $x>1$,

[TABLE]

Therefore,

[TABLE]

As $x\to\infty$, this shows that $L(s,\chi)$ converges.

For (iii), [Ros99, Theorem 4] and [Lan18, Satz LXXXIII] imply that

[TABLE]

where $L(1,\chi)\neq 0$ by [Lan18, Satz LXIV].

For (iv), see [Lan18, Satz LXX]. ∎

4. The surfaces and their torsors

In this section, we study the nonsplit quartic del Pezzo surfaces with singularities ${\mathbf{A}}_{3}+{\mathbf{A}}_{1}$, their Cox rings and their split torsors.

4.1. The surfaces

Proposition 4.1**.**

Every quartic del Pezzo surface with singularities ${\mathbf{A}}_{3}+{\mathbf{A}}_{1}$ over $K$ is isomorphic to precisely one of the surfaces $S_{a}\subset\mathds{P}^{4}_{K}$ defined by (1.3), where $a$ runs through a system of representatives of $K^{\times}/K^{\times 2}$.

Proof.

Let $X$ be such a singular surface with minimal desingularization ${\widetilde{X}}$. By [CT88, Proposition 6.1], ${\widetilde{X}}_{\overline{K}}$ contains seven negative curves: four $(-2)$-curves $D_{1},\dots,D_{4}$ and three $(-1)$-curves $D_{5},D_{6},D_{7}$ whose intersections are encoded in the Dynkin diagram:

[TABLE]

We use the same name for a curve and its strict transforms under blow-ups. As discussed in the proof of [CT88, Lemma 7.4], the simultaneous contraction of $D_{6},D_{7}$ and the interated contraction of $D_{5},D_{4},D_{3}$ defines a morphism $\rho\colon{\widetilde{X}}\to\mathds{P}^{2}_{K}$ over $K$. We observe that $\rho(D_{1})$, $\rho(D_{2})$ are lines in $\mathds{P}^{2}_{K}$ defined over $K$, that $\rho(D_{3})$, $\rho(D_{4})$, $\rho(D_{5})$ are the intersection point $\rho(D_{1})\cap\rho(D_{2})$, and that $\rho(D_{6})$, $\rho(D_{7})$ are distinct points on $\rho(D_{2})$ over ${\overline{K}}$ that are either conjugate under $\operatorname{Gal}({\overline{K}}/K)$ (and therefore defined over a quadratic extension of $K$) or each defined over $K$. We note that ${\widetilde{X}}$ can be recovered from the line $\rho(D_{1})$ and the (possibly conjugate) points $\rho(D_{6}),\rho(D_{7})$ by the simultaneous blow-up of $\rho(D_{6})$, $\rho(D_{7})$ and three interated blow-ups that are uniquely determined by $\rho(D_{1})$, $\rho(D_{2})$.

Let $a\in K$ be such that $\rho(D_{6})$, $\rho(D_{7})$ are defined over $K(\sqrt{a})$. Up to a linear change of coordinates $(z_{0}:z_{1}:z_{2})$ on $\mathds{P}^{2}_{K}$ over $K$, we may assume that $\rho(D_{1})=\{z_{0}=0\}$, $\rho(D_{6})=(1:0:\sqrt{a})$, and $\rho(D_{7})=(1:0:-\sqrt{a})$. This shows that ${\widetilde{X}}$ is uniquely determined by the class of $a$ in $K^{\times}/K^{\times 2}$. Applying the anticanonical map, the same holds for $X$.

On the other hand, $S_{a}$ defined by (1.3) is a quartic del Pezzo surface of type ${\mathbf{A}}_{3}+{\mathbf{A}}_{1}$ that splits over $K(\sqrt{a})$ because it is isomorphic via a linear change of coordinates over $K(\sqrt{a})$ to the split quartic del Pezzo surface of type ${\mathbf{A}}_{3}+{\mathbf{A}}_{1}$ described in [Der14, §3.4]. Therefore, $X\cong S_{a}$. ∎

From now on, we fix an element $a\in\mathcal{O}_{K}$ that is not a square, and we denote $S_{a}$ by $S$. Let $\phi\colon{\widetilde{S}}\to S$ be a minimal desingularization. The projection

[TABLE]

from the line $\{x_{1}=x_{3}=x_{4}=0\}$ maps the other two lines $\{x_{1}\pm\sqrt{a}x_{3}=x_{2}=x_{4}=0\}$ to $(1:0:\mp\sqrt{a})$ and induces a morphism $\rho\colon{\widetilde{S}}\to\mathds{P}^{2}_{K}$ that is precisely the sequence of blow-ups in the proof of Proposition 4.1.

4.2. Cox rings

By [Der14, §3.4], there is a basis $\ell_{0},\dots,\ell_{5}$ of $\operatorname{Pic}({\widetilde{S}}_{{\overline{K}}})$ such that the $(-2)$-curves in the Dynkin diagram (4.1) have classes

[TABLE]

and the $(-1)$-curves have classes

[TABLE]

in $\operatorname{Pic}({\widetilde{S}}_{{\overline{K}}})$. Let $D_{8}$ and $D_{9}$ be the strict transforms of the lines $\{z_{2}+\sqrt{a}z_{0}=0\}$ and $\{z_{2}-\sqrt{a}z_{0}=0\}$ under $\rho$, respectively. Then

[TABLE]

Let $\Lambda\coloneqq\bigoplus_{i=1}^{9}\mathds{Z}D_{i}$. Since the divisors classes $[D_{2}],\dots,[D_{7}]$ form a basis of $\operatorname{Pic}({\widetilde{S}}_{{\overline{K}}})$, the kernel $\Lambda_{0}$ of the group homomorphism $\Lambda\to\operatorname{Pic}({\widetilde{S}}_{{\overline{K}}})$ that sends each $D_{i}$ to its divisor class is a free group of rank 3 with basis

[TABLE]

Let $\Lambda_{0}\to{\overline{K}}({\widetilde{S}}_{{\overline{K}}})^{\times}$ be the group homomorphism defined by

[TABLE]

Then the ${\overline{K}}$-algebra

[TABLE]

endowed with the $\operatorname{Pic}({\widetilde{S}}_{{\overline{K}}})$-grading induced by assigning degree $[D_{i}]$ to $\eta_{i}$ for $i\in\{1,\dots,9\}$, together with the character (4.2) gives a Cox ring of ${\widetilde{S}}_{{\overline{K}}}$ of type $\operatorname{id}_{\operatorname{Pic}({\widetilde{S}}_{{\overline{K}}})}$ in the sense of [DP19, Proposition 4.4].

We observe that the Galois action of $\operatorname{Gal}({\overline{K}}/K)$ on ${\widetilde{S}}_{{\overline{K}}}$ exchanges $D_{6}$ with $D_{7}$ and $[D_{8}]$ with $[D_{9}]$, while $D_{1},\dots,D_{5}$ are fixed. Hence $\operatorname{Pic}({\widetilde{S}})=\operatorname{Pic}({\widetilde{S}}_{{\overline{K}}})^{\operatorname{Gal}(\overline{K}/K)}\cong\mathds{Z}^{5}$ with basis $\ell_{0},\dots,\ell_{3},\ell_{4}+\ell_{5}$.

We compute a Cox ring of ${\widetilde{S}}$ of type $\operatorname{Pic}({\widetilde{S}})\hookrightarrow\operatorname{Pic}({\widetilde{S}}_{{\overline{K}}})$ in the sense of [DP19].

Lemma 4.2**.**

The ${\overline{K}}$-algebra

[TABLE]

where $\zeta_{i}$ have degrees $[E_{i}]$ for

[TABLE]

is a Cox ring of ${\widetilde{S}}_{{\overline{K}}}$ of type $\operatorname{Pic}({\widetilde{S}})\hookrightarrow\operatorname{Pic}({\widetilde{S}}_{{\overline{K}}})$.

Proof.

According to [DP19, Corollary 2.10, Definition 3.12], every Cox ring of ${\widetilde{S}}_{{\overline{K}}}$ of type $\operatorname{Pic}({\widetilde{S}})\hookrightarrow\operatorname{Pic}({\widetilde{S}}_{{\overline{K}}})$ is isomorphic to the sub-${\overline{K}}$-algebra of (4.3) generated by the monomials $\prod_{j=1}^{9}\eta_{j}^{e_{j}}$ with $e_{1},\dots,e_{9}\in\mathds{Z}_{\geq 0}$ such that $\sum_{i=1}^{9}e_{j}[D_{j}]\in\operatorname{Pic}({\widetilde{S}})$, that is,

[TABLE]

The relation in (4.3) implies $\zeta_{9}=-\zeta_{7}-\zeta_{2}^{2}\zeta_{3}\zeta_{4}^{3}\zeta_{6}$ and $\zeta_{1}\zeta_{8}+\zeta_{7}(\zeta_{7}+\zeta_{2}^{2}\zeta_{3}\zeta_{4}^{3}\zeta_{6})=0$. The definition of the $\zeta_{i}$ gives a surjective map from $\overline{R}$ onto a Cox ring of ${\widetilde{S}}_{{\overline{K}}}$ of type $\operatorname{Pic}({\widetilde{S}})\hookrightarrow\operatorname{Pic}({\widetilde{S}}_{{\overline{K}}})$. Since the relation defining $\overline{R}$ is irreducible and $\dim\overline{R}=\dim S_{{\overline{K}}}+\operatorname{rk}\operatorname{Pic}({\widetilde{S}})$, we conclude that $\overline{R}$ is a Cox ring of ${\widetilde{S}}_{{\overline{K}}}$ of the desired type. ∎

Lemma 4.3**.**

Every Cox ring of ${\widetilde{S}}$ of type $\operatorname{Pic}({\widetilde{S}})\hookrightarrow\operatorname{Pic}({\widetilde{S}}_{{\overline{K}}})$ is isomorphic to the $K$-algebra

[TABLE]

with $\xi_{i}$ homogeneous of degree $E_{i}$ for all $i\in\{1,\dots,8\}$.

Proof.

We observe that the action of $\operatorname{Gal}(K(\sqrt{a})/K)$ on (4.3) that fixes $\eta_{1},\dots,\eta_{5}$, exchanges $\eta_{6}$ with $\eta_{7}$ and $\eta_{8}$ with $\eta_{9}$ turns (4.3) and $\overline{R}$ into $\operatorname{Gal}({\overline{K}}/K)$-equivariant Cox rings of ${\widetilde{S}}_{{\overline{K}}}$ as in [DP19, Definition 3.1]. Let $\zeta_{9}\coloneqq-\zeta_{7}-\zeta_{2}^{2}\zeta_{3}\zeta_{4}^{3}\zeta_{6}$ as element of $\overline{R}$. We observe that the elements $\zeta_{7}$ and $\zeta_{9}$ are exchanged by the Galois action, while all the other $\zeta_{i}$ are fixed. Let $\xi_{i}\coloneqq\zeta_{i}$ for $i\in\{1,\dots,6\}$ and consider the following change of variables:

[TABLE]

Then

[TABLE]

and $\xi_{9}=-\xi_{2}^{2}\xi_{3}\xi_{4}^{3}\xi_{6}$. So $4a\zeta_{7}\zeta_{9}=a\xi_{9}^{2}-\xi_{7}^{2}=a\xi_{2}^{4}\xi_{3}^{2}\xi_{4}^{6}\xi_{6}^{2}-\xi_{7}^{2}$.

Thus,

[TABLE]

has $\operatorname{Gal}({\overline{K}}/K)$-invariant generators and relation. Hence it descend to a Cox ring of ${\widetilde{S}}$ of type $\operatorname{Pic}({\widetilde{S}})\hookrightarrow\operatorname{Pic}({\widetilde{S}}_{{\overline{K}}})$. Up to isomorphism, there is only one Cox ring of ${\widetilde{S}}$ of type $\operatorname{Pic}({\widetilde{S}})\hookrightarrow\operatorname{Pic}({\widetilde{S}}_{{\overline{K}}})$ because the Galois action on $\operatorname{Pic}({\widetilde{S}})$ is trivial (cf. the comment after [DP19, Proposition 3.7]). ∎

4.3. A torsor over ${\overline{K}}$

The next lemma extends [Bou11, Remark 6] to the Cox rings introduced in [DP19].

Lemma 4.4**.**

Let $X$ be a smooth surface over an algebraically closed field $k$. Let $\widetilde{R}$ be a finitely generated Cox ring of $X$ of type $\lambda\colon M\to\operatorname{Pic}(X)$ such that $\lambda(M)$ contains an ample divisor class. Let $\eta_{1},\dots,\eta_{N}$ be homogeneous generators of $\widetilde{R}$. For every $i\in\{1,\dots,N\}$, let $D_{i}$ be the divisor on $X$ defined by $\eta_{i}$. Then the open subset of $\operatorname{Spec}\widetilde{R}$ defined by $\prod_{j\notin J}\eta_{j}\neq 0$ for all $J\subseteq\{1,\dots,N\}$ such that $\bigcap_{j\in J}D_{j}\neq\emptyset$ is a torsor of $X$ of type $\lambda$.

Proof.

Let $D$ be a very ample divisor on $X$ such that $[D]\in\lambda(M)$. Let $Y=\operatorname{Spec}\widetilde{R}\smallsetminus V(\widetilde{R}_{[D]})$, where $\widetilde{R}_{[D]}$ is the degree-$[D]$-part of $\widetilde{R}$. Then $Y$ is a torsor of $X$ of type $\lambda$ by [DP19, Corollary 4.3]. Let $\mathfrak{J}$ be the set of subsets $J\subseteq\{1,\dots,N\}$ such that $\bigcap_{j\in J}D_{j}=\emptyset$. Since for every $i\in\{1,\dots,N\}$ the preimage of $D_{i}$ under $Y\to X$ is $Y\cap V(\eta_{i})$, a subset $J\subseteq\{1,\dots,N\}$ belongs to $\mathfrak{J}$ if and only if $\widetilde{R}_{[D]}$ is contained in the ideal $(\eta_{j}:j\in J)$. Hence, $\widetilde{R}_{[D]}\subseteq\bigcap_{J\in\mathfrak{J}}(\eta_{j}:j\in J)$. We observe that $V(\prod_{j\notin J}\eta_{j}:J\notin\mathfrak{J})\subseteq V(\bigcap_{J\in\mathfrak{J}}(\eta_{j}:j\in J))$, because for every point $x\in V(\prod_{j\notin J}\eta_{j}:J\notin\mathfrak{J})=\bigcap_{J\notin\mathfrak{J}}\bigcup_{j\notin J}V(\eta_{j})$, the set $J_{x}\coloneqq\{j\in\{1,\dots,N\}:x\in V(\eta_{j})\}$ belongs to $\mathfrak{J}$. Hence, $Y\subseteq\operatorname{Spec}\widetilde{R}\smallsetminus V(\prod_{j\notin J}\eta_{j}:J\notin\mathfrak{J})$. For the other inclusion let $x\in\operatorname{Spec}\widetilde{R}\smallsetminus V(\prod_{j\notin J}\eta_{j})$ for some $J\subseteq\{1,\dots,N\}$, $J\notin\mathfrak{J}$. Since $\widetilde{R}_{[D]}\nsubseteq(\eta_{j}:j\in J)$, and $\widetilde{R}_{[D]}$ is generated by monomials, there exists $s\in\widetilde{R}_{[D]}$ such that $s(x)\neq 0$. Hence $x\in Y$. ∎

By Lemma 4.4 and the description of the negative curves on ${\widetilde{S}}_{{\overline{K}}}$ from [Der14, §3.4], the closed subset of $\operatorname{Spec}R$ defined by the monomials

[TABLE]

where $A_{i_{1},\dots,i_{s}}\coloneqq\prod_{1\leq j\leq 7,j\notin\{i_{1},\dots,i_{s}\}}\xi_{j}$, is the complement of a torsor of ${\widetilde{S}}$ of type $\operatorname{Pic}({\widetilde{S}})\hookrightarrow\operatorname{Pic}({\widetilde{S}}_{{\overline{K}}})$.

4.4. A torsor over $\mathcal{O}_{K}$

Let $\pi\colon\mathcal{Y}\to\widetilde{\mathcal{S}}$ be the $\mathcal{O}_{K}$-model of the torsor $Y\to{\widetilde{S}}$ defined by

[TABLE]

and the monomials (4.4) as in Construction 2.1.

Proposition 4.5**.**

The $\mathcal{O}_{K}$-morphism $\pi\colon\mathcal{Y}\to\widetilde{\mathcal{S}}$ is a torsor under $\mathds{G}_{\mathrm{m},\mathcal{O}_{K}}^{5}$, and the $\mathcal{O}_{K}$-scheme $\widetilde{\mathcal{S}}$ is projective.

Proof.

The first statement follows from Remark 2.2 because for each monomial in (4.4) the degrees of the variables appearing in the monomial generate $\operatorname{Pic}({\widetilde{S}})$.

From [FP16, Remark 4.2], the divisor class $A\coloneqq 9\ell_{0}-3\ell_{1}-2\ell_{2}-\ell_{3}-(\ell_{4}+\ell_{5})$ is ample. Let $C_{K}$ and $C_{\mathcal{O}_{K}}$ be the ideals generated by $g\coloneqq\xi_{1}\xi_{8}+\xi_{7}^{2}-a\xi_{2}^{4}\xi_{3}^{2}\xi_{4}^{6}\xi_{6}^{2}$ and the monomials of degree $A$ in $K[\xi_{1},\dots,\xi_{8}]$ and in $\mathcal{O}_{K}[\xi_{1},\dots,\xi_{8}]$, respectively. Then the radicals of $C_{K}$ and $C_{\mathcal{O}_{K}}$ are the radicals of the ideal generated by $g$ and the following monomials

[TABLE]

in $K[\xi_{1},\dots,\xi_{8}]$ and in $\mathcal{O}_{K}[\xi_{1},\dots,\xi_{8}]$, respectively.

The relations

[TABLE]

show that $\sqrt{C_{K}}=\sqrt{C^{\prime}_{K}}$ and $\sqrt{C_{\mathcal{O}_{K}}}=\sqrt{C^{\prime}_{\mathcal{O}_{K}}}$, where $C^{\prime}_{K}$ and $C^{\prime}_{\mathcal{O}_{K}}$ are the ideals generated by $g$ and the monomials (4.4) in $K[\xi_{1},\dots,\xi_{8}]$ and in $\mathcal{O}_{K}[\xi_{1},\dots,\xi_{8}]$, respectively.

The polynomials

[TABLE]

generate the ideal $C^{\prime}_{\mathcal{O}_{K}}$, as

[TABLE]

and can be easily checked to form a Gröbner basis of $C^{\prime}_{\mathcal{O}_{K}}$ for the lexicographic monomial ordering with respect to $\xi_{8}>\dots>\xi_{1}$ via [AL94, Algorithm 4.2.2].

Since the leading coefficient of each polynomial in the Gröbner basis is invertible in $\mathcal{O}_{K}$, $C^{\prime}_{K}\cap\mathcal{O}_{K}[\xi_{1},\dots,\xi_{8}]=C^{\prime}_{\mathcal{O}_{K}}$ by [AL94, Proposition 4.4.4]. Hence $\sqrt{C_{K}}\cap\mathcal{O}_{K}[\xi_{1},\dots,\xi_{8}]=\sqrt{C_{\mathcal{O}_{K}}}$, and the $\mathcal{O}_{K}$-model $\widetilde{\mathcal{S}}$ that we constructed above is projective by Proposition 2.6(ii). ∎

5. Parameterization

In this section, we parameterize the rational points on ${\widetilde{S}}$ by integral points on its torsors, we construct a good fundamental domain for the action of $\mathds{G}_{\mathrm{m}}^{5}(\mathcal{O}_{K})$ and we perform a Möbius inversion.

5.1. Passage to the split torsor

Under the identification $\operatorname{Pic}({\widetilde{S}})\cong\mathds{Z}^{5}$ given by the basis $\ell_{0},\dots,\ell_{3},\ell_{4}+\ell_{5}$, the action of $\mathds{G}_{\mathrm{m}}^{5}(\mathcal{O}_{K})$ on $Y(K)$ is given by

[TABLE]

where

[TABLE]

and $\underline{u}^{(m_{0},\dots,m_{4})}=\prod_{i=0}^{4}u_{i}^{m_{i}}$ for all $\underline{u}=(u_{0},\dots,u_{4})\in(\mathcal{O}_{K}^{\times})^{5}$.

Lemma 5.1**.**

For the surface $S$,

[TABLE]

where $M_{\underline{\mathfrak{c}}}(B)$ is the set of all

[TABLE]

with height condition

[TABLE]

torsor equation

[TABLE]

coprimality conditions

[TABLE]

Here, $\mathcal{F}$ is a fundamental domain for the action of $U_{K}\times(\mathcal{O}_{K}^{\times})^{4}$ on $(K^{\times})^{6}\times K^{2}$ described by (5.1), $\mathcal{O}_{j*}\coloneqq\mathcal{O}_{j}^{\neq 0}$ for $j\in\{1,\dots,6\}$ and $\mathcal{O}_{j*}\coloneqq\mathcal{O}_{j}$ for $j\in\{7,8\}$, where $\mathcal{O}_{j}\coloneqq{\underline{\mathfrak{c}}}^{m^{(j)}}=\prod_{i=0}^{4}\mathfrak{c}_{i}^{m^{(j)}_{i}}$ for all $j\in\{1,\dots,8\}$. For $v\in\Omega_{K}$ and $(x_{1},\dots,x_{7})\in K_{v}^{7}$ with $x_{1}\neq 0$,

[TABLE]

Finally,

[TABLE]

and ${\mathfrak{a}}_{j}\coloneqq a_{j}\mathcal{O}_{j}^{-1}$ for all $j\in\{1,\dots,8\}$.

Proof.

The map $\psi\colon Y\to{\widetilde{S}}\to S\subseteq\mathds{P}^{4}_{K}$ that sends $(\xi_{1},\dots,\xi_{8})$ to

[TABLE]

is the composition of the chosen anticanonical embedding of $S$, the minimal desingularization of $S$ and the torsor over ${\widetilde{S}}$ from Section 4.1. The complement of the lines of $\mathds{P}^{4}_{{\overline{K}}}$ contained in $S$ is

[TABLE]

so $\psi^{-1}(U)=\{\xi_{1}\xi_{2}\xi_{3}\xi_{4}\xi_{5}\xi_{6}\neq 0\}\subseteq\operatorname{Spec}R$. Since $U$ is contained in the smooth locus of $S$, Proposition 4.5 and [FP16, Theorem 2.7] give

[TABLE]

where ${}_{{\underline{\mathfrak{c}}}}\pi\colon{}_{{\underline{\mathfrak{c}}}}\mathcal{Y}\to\widetilde{\mathcal{S}}$ is a ${\underline{\mathfrak{c}}}$-twist of $\pi$ and ${}_{{\underline{\mathfrak{c}}}}\mathcal{Y}(\mathcal{O}_{K})\cap\psi^{-1}(U(K))$ is the set of $\underline{a}=(a_{1},\dots,a_{8})\in(\mathcal{O}_{1*}\times\dots\times\mathcal{O}_{8*})$ that satisfy (5.3) and (5.4) because the radical of the ideal of $\mathcal{R}$ generated by (4.4) equals the radical of the ideal

[TABLE]

Since the monomials (4.4) belong to the radical of the ideal generated by the monomials defining $\psi$,

[TABLE]

(where we eliminate $a_{8}$ using (5.3)) for all $(a_{1},\dots,a_{8})\in{}_{{\underline{\mathfrak{c}}}}\mathcal{Y}(\mathcal{O}_{K})$. ∎

5.2. The fundamental domain

Here we choose an explicit fundamental domain for Lemma 5.1. Define $\Sigma$, $\delta$, $l$, $F$, $F(\infty)$, $F(B)$, $\mathscr{F}_{1}$ as in [FP16, §5]. We recall that $\mathscr{F}_{1}\subseteq K^{\times}$ is a fundamental domain for the multiplicative action of $\mathcal{O}_{K}^{\times}$ such that

[TABLE]

for every $a^{\prime}\in\mathscr{F}_{1}$ and every $v\mid\infty$.

For $\underline{a}=(a_{1},\dots,a_{4})\in(K^{\times})^{4}$ and $(x_{5v},x_{6v},x_{7v})\in(K_{v}^{\times})^{2}\times K_{v}$, we define

[TABLE]

For $B\in\mathds{R}_{>0}\cup\{\infty\}$, let $S_{F}(\underline{a};B)$ be the set

[TABLE]

and

[TABLE]

As in [FP16, §5], the set

[TABLE]

is a fundamental domain for the action of $U_{K}\times(\mathcal{O}_{K}^{\times})^{4}$ on $(K^{\times})^{6}\times K^{2}$.

Since $F(B)=F+(-\infty,\log B]\delta$, where $\delta=(d_{v})_{v\in\Omega_{\infty}}$ and $F\subseteq\mathds{R}^{\Omega_{\infty}}$ is contained in the hyperplane where the sum of the coordinates vanish, $(\sigma_{v}(\underline{a}),x_{5v},x_{6v},x_{7v})_{v}$ satisfies the height condition (5.2) for all $(x_{5v},x_{6v},x_{7v})_{v}\in S_{F}(\underline{a},u_{\underline{\mathfrak{c}}}B)$.

As we will see in Section 6.5, the contribution of the first summation to the main term consists of the volume of $S_{F}(\underline{a};B)$, which we compute here.

For the next result, recall the definition of $\omega_{v}(S)$ in Theorem 1.1.

Lemma 5.2**.**

With respect to the usual Lebesgue measure on $\prod_{v\mid\infty}K_{v}^{3}=\mathds{R}^{3d}$, for $B>0$, we have

[TABLE]

Proof.

This is analogous to [FP16, Lemma 5.1]. ∎

5.3. Möbius inversion

In this section, we deal with the coprimality conditions in (5.4) that involve $a_{5},\dots,a_{8}$. The remaining ones are encoded in the arithmetic function $\vartheta_{0}$ defined below. We also replace the torsor equation (5.3) by a sum over congruence classes for a parameter $\rho$ as in (1.4) and eliminate the variable $a_{8}$.

Recall the notation introduced in Lemma 5.1. We write

[TABLE]

Let

[TABLE]

Let ${\underline{\mathfrak{d}}}=({\mathfrak{d}}_{56},{\mathfrak{d}}_{58},{\mathfrak{d}}_{5},{\mathfrak{d}}_{6},{\mathfrak{d}}_{7})$ be a 5-tuple of nonzero ideals. Let

[TABLE]

For fixed ${\underline{\mathfrak{c}}}\in\mathscr{C}^{5}$, $\underline{a}$, ${\underline{\mathfrak{d}}}$ as above, let ${\mathfrak{g}}\coloneqq{\mathfrak{d}}_{58}{\mathfrak{a}}_{1}+a\mathcal{O}_{K}$, and let ${\mathfrak{g}}^{\prime}$ be the unique ideal with $v_{\mathfrak{p}}({\mathfrak{g}}^{\prime})=\lceil v_{\mathfrak{p}}({\mathfrak{g}})/2\rceil$ for all prime ideals ${\mathfrak{p}}$. Define the fractional ideals

[TABLE]

For $\rho\in\mathcal{O}_{K}$, let $\mathscr{G}({\underline{\mathfrak{c}}},\underline{a},{\underline{\mathfrak{d}}},\rho)$ be the additive subgroup of $K^{3}$ consisting of all $(a_{5},a_{6},a_{7})$ with $a_{5}\in{\mathfrak{b}}_{5}$ and $a_{6}\in{\mathfrak{b}}_{6}$ and $a_{7}\in\gamma_{7}\cdot a_{6}+{\mathfrak{b}}_{7}$ with $\gamma_{7}\in\mathcal{O}_{K}$ satisfying

[TABLE]

Note that such a $\gamma_{7}$ exists if ${\mathfrak{d}}_{7}+{\mathfrak{d}}_{58}{\mathfrak{a}}_{1}=\mathcal{O}_{K}$, as ${\mathfrak{d}}_{7}\mathfrak{c}_{1}\mathfrak{c}_{2}\mathfrak{c}_{3}+{\mathfrak{d}}_{58}{\mathfrak{a}}_{1}\mathfrak{c}_{1}\mathfrak{c}_{2}\mathfrak{c}_{3}=\mathfrak{c}_{1}\mathfrak{c}_{2}\mathfrak{c}_{3}$ divides $a_{2}^{2}a_{3}a_{4}^{3}$.

The following technical lemma is similar to [DF14a, Lemma 5.6].

Lemma 5.3**.**

Let ${\mathfrak{a}}$ be an ideal and ${\mathfrak{f}}$ a nonzero fractional ideal of $\mathcal{O}_{K}$. Let $y_{1},y_{2}\in{\mathfrak{f}}$ with $y_{1}{\mathfrak{f}}^{-1}+{\mathfrak{a}}=\mathcal{O}_{K}$. Then $y_{2}/y_{1}$ is defined modulo ${\mathfrak{a}}$ and, for $x\in\mathcal{O}_{K}$, we have

[TABLE]

Proof.

The quotient $y_{2}/y_{1}$ is invertible modulo ${\mathfrak{a}}$ since $v_{\mathfrak{p}}(y_{1})=v_{\mathfrak{p}}({\mathfrak{f}})\leq v_{\mathfrak{p}}(y_{2})$ for every ${\mathfrak{p}}\mid{\mathfrak{a}}$. Furthermore, ${xy_{1}}\equiv{y_{2}}\pmod{{\mathfrak{a}}{\mathfrak{f}}}$ if and only if $v_{\mathfrak{p}}(x-y_{2}/y_{1})\geq v_{\mathfrak{p}}({\mathfrak{a}})-v_{\mathfrak{p}}(y_{1}{\mathfrak{f}}^{-1})$, which is equivalent to ${x}\equiv_{{\mathfrak{a}}}{y_{2}/y_{1}}$ by our assumptions. ∎

Lemma 5.4**.**

Let ${\underline{\mathfrak{c}}}\in\mathscr{C}^{5}$ and $B\geq 0$. With $\mathscr{F}_{1},\mathscr{F}_{0}(\underline{a};u_{\underline{\mathfrak{c}}}B)$ as in Section 5.2 and $\mathscr{G}({\underline{\mathfrak{c}}},\underline{a},{\underline{\mathfrak{d}}},\rho)$ defined above, we have

[TABLE]

where the second sum runs over 5-tuples of ideals ${\underline{\mathfrak{d}}}$ satisfying

[TABLE]

[TABLE]

and the third sum runs over

[TABLE]

Proof.

For fixed $\underline{a}$ with $\vartheta_{0}(\underline{{\mathfrak{a}}})=1$ and $B>0$, let $\mathscr{F}_{0}\coloneqq\mathscr{F}_{0}(\underline{a};u_{\underline{\mathfrak{c}}}B)$, and

[TABLE]

We apply Möbius inversions: For ${\mathfrak{a}}_{5}+{\mathfrak{a}}_{8}=\mathcal{O}_{K}$, we introduce ${\mathfrak{d}}_{58}$. This gives

[TABLE]

There is a one-to-one correspondence between such $(a_{5},\dots,a_{8})$ satisfying the torsor equation and triples $(a_{5},a_{6},a_{7})\in({\mathfrak{d}}_{58}\mathcal{O}_{5}\times\mathcal{O}_{6}\times\mathcal{O}_{7})\cap\mathscr{F}_{0}$ satisfying

[TABLE]

Furthermore, this congruence and the other coprimality conditions imply that ${\mathfrak{d}}_{58}+{\mathfrak{a}}_{2}{\mathfrak{a}}_{3}{\mathfrak{a}}_{4}=\mathcal{O}_{K}$ and that every ${\mathfrak{p}}$ with odd $v_{\mathfrak{p}}(a)$ satisfies $v_{\mathfrak{p}}({\mathfrak{d}}_{58}{\mathfrak{a}}_{1})\leq v_{\mathfrak{p}}(a)$ (otherwise, we would have ${\mathfrak{p}}\mid{\mathfrak{a}}_{2}{\mathfrak{a}}_{3}{\mathfrak{a}}_{4}{\mathfrak{a}}_{6}$, contradicting ${\mathfrak{a}}_{1}+{\mathfrak{a}}_{2}{\mathfrak{a}}_{3}{\mathfrak{a}}_{4}{\mathfrak{a}}_{6}=\mathcal{O}_{K}$). Hence $\tilde{M}$ equals

[TABLE]

We observe that ${\mathfrak{d}}_{58}{\mathfrak{a}}_{1}+{\mathfrak{a}}_{2}{\mathfrak{a}}_{3}{\mathfrak{a}}_{4}{\mathfrak{a}}_{6}=\mathcal{O}_{K}$, but ${\mathfrak{g}}\coloneqq{\mathfrak{d}}_{58}{\mathfrak{a}}_{1}+a\mathcal{O}_{K}$ is not necessarily $\mathcal{O}_{K}$. As above, let ${\mathfrak{g}}^{\prime}$ be the unique ideal with $v_{\mathfrak{p}}({\mathfrak{g}}^{\prime})=\lceil v_{\mathfrak{p}}({\mathfrak{g}})/2\rceil$ for all prime ideals ${\mathfrak{p}}$. Then $a_{7}\in{\mathfrak{g}}^{\prime}\mathcal{O}_{7}$. Let ${\mathfrak{q}}\coloneqq{\mathfrak{d}}_{58}{\mathfrak{a}}_{1}{\mathfrak{g}}^{-1}$.

We claim: For every $a_{7}\in\mathcal{O}_{7}$ with (5.13) there is a unique $\rho\pmod{{\mathfrak{q}}{\mathfrak{g}}^{\prime}}$ with

[TABLE]

Conversely, if $a_{7}\in\mathcal{O}_{7}$ satisfies the first congruence in (5.14) for some $\rho$ with the second congruence in (5.14), then $a_{7}$ satisfies (5.13).

For the proof of this claim, our first step is to observe: Every $\rho\in\mathcal{O}_{K}$ satisfying the second congruence in (5.14) also satisfies the last condition in (5.14). To prove this observation, we must show that $\min\{v_{\mathfrak{p}}(\rho),v_{\mathfrak{p}}({\mathfrak{q}}{\mathfrak{g}}^{\prime})\}=v_{\mathfrak{p}}({\mathfrak{g}}^{\prime})$ for every prime ideal ${\mathfrak{p}}$. Since ${\mathfrak{g}}={\mathfrak{q}}{\mathfrak{g}}+a\mathcal{O}_{K}$, we have $v_{\mathfrak{p}}({\mathfrak{g}})=\min\{v_{\mathfrak{p}}({\mathfrak{q}}{\mathfrak{g}}),v_{\mathfrak{p}}(a)\}$. In the case $v_{\mathfrak{p}}({\mathfrak{g}})=v_{\mathfrak{p}}({\mathfrak{q}}{\mathfrak{g}})\leq v_{\mathfrak{p}}(a)$, we have $v_{\mathfrak{p}}({\mathfrak{q}})=0$, our congruence implies $v_{\mathfrak{p}}(\rho)\geq v_{\mathfrak{p}}({\mathfrak{g}})/2$, and therefore $v_{\mathfrak{p}}(\rho)\geq v_{\mathfrak{p}}({\mathfrak{g}}^{\prime})$, which proves the observation. In the case $v_{\mathfrak{p}}({\mathfrak{g}})=v_{\mathfrak{p}}(a)<v_{\mathfrak{p}}({\mathfrak{q}}{\mathfrak{g}})$, our congruence gives $2\mid v_{\mathfrak{p}}(a)$ and $v_{\mathfrak{p}}(\rho)=v_{\mathfrak{p}}(a)/2=v_{\mathfrak{p}}({\mathfrak{g}}^{\prime})$, which proves the observation.

We write $A=a_{2}^{2}a_{3}a_{4}^{3}a_{6}\in\mathcal{O}_{2}^{2}\mathcal{O}_{3}\mathcal{O}_{4}^{3}\mathcal{O}_{6}=\mathfrak{c}_{0}$ and ${\mathfrak{A}}=A\mathfrak{c}_{0}^{-1}={\mathfrak{a}}_{2}^{2}{\mathfrak{a}}_{3}{\mathfrak{a}}_{4}^{3}{\mathfrak{a}}_{6}$ for simplicity. Now we prove one direction of our claim: Since $a_{7}\in\mathcal{O}_{7}=\mathfrak{c}_{0}$ and $A\in\mathfrak{c}_{0}$ with $A\mathfrak{c}_{0}^{-1}+{\mathfrak{q}}{\mathfrak{g}}^{\prime}={\mathfrak{A}}+{\mathfrak{q}}{\mathfrak{g}}^{\prime}=\mathcal{O}_{K}$, we can apply Lemma 5.3 (with ${\mathfrak{a}}={\mathfrak{q}}{\mathfrak{g}}^{\prime}$, ${\mathfrak{f}}=\mathfrak{c}_{0}$, $y_{1}=A$, $y_{2}=a_{7}$, $x=\rho$) to see that there is a $\rho\in\mathcal{O}_{K}$ satisfying the first part of (5.14), and that such a $\rho$ is unique modulo ${\mathfrak{q}}{\mathfrak{g}}^{\prime}$. This congruence implies ${\mathfrak{g}}^{\prime}\mathfrak{c}_{0}\mid a_{7}-\rho A$; together with ${\mathfrak{g}}^{\prime}\mathfrak{c}_{0}\mid a_{7}$ and ${\mathfrak{g}}^{\prime}+{\mathfrak{A}}=\mathcal{O}_{K}$, this gives ${\mathfrak{g}}^{\prime}\mid\rho$. By [DF14a, Lemma 5.7] (with ${\mathfrak{a}}_{1}={\mathfrak{q}}$, ${\mathfrak{a}}_{2}={\mathfrak{g}}^{\prime}\mathfrak{c}_{0}$), ${a_{7}^{2}}\equiv{\rho^{2}A^{2}}\pmod{{\mathfrak{q}}{\mathfrak{g}}^{\prime 2}\mathfrak{c}_{0}^{2}}$; since ${\mathfrak{g}}\mid{\mathfrak{g}}^{\prime 2}$, this implies ${a_{7}^{2}}\equiv{\rho^{2}A^{2}}\pmod{{\mathfrak{q}}{\mathfrak{g}}\mathfrak{c}_{0}^{2}}$. With (5.13), we get ${\rho^{2}A^{2}}\equiv{aA^{2}}\pmod{{\mathfrak{q}}{\mathfrak{g}}\mathfrak{c}_{0}^{2}}$. By Lemma 5.3 (with ${\mathfrak{a}}={\mathfrak{q}}{\mathfrak{g}}$, ${\mathfrak{f}}=\mathfrak{c}_{0}^{2}$, $y_{1}=A^{2}$, $y_{2}=aA^{2}$, $x=\rho^{2}$), we have ${\rho^{2}}\equiv{a}\pmod{{\mathfrak{q}}{\mathfrak{g}}}$, the second congruence in (5.14).

Now we prove the converse direction of the claim: Given such $a_{7}$ and $\rho$, the first congruence in (5.14) together with [DF14a, Lemma 5.7] (which we may apply with ${\mathfrak{a}}_{1}={\mathfrak{q}}$ and ${\mathfrak{a}}_{2}={\mathfrak{g}}^{\prime}\mathfrak{c}_{0}$ since the third part of (5.14) gives $\rho A\in{\mathfrak{g}}^{\prime}\mathfrak{c}_{0}$ and the first congruence in (5.14) then gives $a_{7}\in{\mathfrak{g}}^{\prime}\mathfrak{c}_{0}$) implies ${a_{7}^{2}}\equiv{\rho^{2}A}\pmod{{\mathfrak{q}}{\mathfrak{g}}^{\prime 2}\mathfrak{c}_{0}}$ and hence ${a_{7}^{2}}\equiv{\rho^{2}A}\pmod{{\mathfrak{q}}{\mathfrak{g}}\mathfrak{c}_{0}}$. The second congruence in (5.14) and Lemma 5.3 (with ${\mathfrak{a}}={\mathfrak{q}}{\mathfrak{g}}$, ${\mathfrak{f}}=\mathfrak{c}_{0}^{2}$, $y_{1}=A^{2}$, $y_{2}=aA^{2}$, $x=\rho^{2}$) give ${\rho^{2}A^{2}}\equiv{aA^{2}}\pmod{{\mathfrak{q}}{\mathfrak{g}}\mathfrak{c}_{0}^{2}}$. Therefore, $a_{7}$ satisfies (5.13).

Our claim implies that

[TABLE]

Next, we remove ${\mathfrak{a}}_{6}+{\mathfrak{a}}_{5}=\mathcal{O}_{K}$ by Möbius inversion:

[TABLE]

Here, we could restrict ourselves to ${\mathfrak{d}}_{56}+{\mathfrak{a}}_{1}{\mathfrak{a}}_{2}{\mathfrak{a}}_{3}{\mathfrak{a}}_{4}=\mathcal{O}_{K}$.

Three more Möbius inversions give

[TABLE]

Note that $a_{7}\in{\mathfrak{d}}_{7}{\mathfrak{g}}^{\prime}\mathcal{O}_{7}$ and ${a_{7}}\equiv{\rho a_{2}^{2}a_{3}a_{4}^{3}a_{6}}\pmod{{\mathfrak{q}}{\mathfrak{g}}^{\prime}\mathfrak{c}_{0}}$ are equivalent to

[TABLE]

By our construction of $\gamma_{7}$, if $a_{7}=\gamma_{7}a_{6}$, then this system of equivalences is satisfied, and this value for $a_{7}$ is unique modulo the lcm of the moduli. Hence it is equivalent to

[TABLE]

6. The first summation

In this section, we perform the summation over the variables $a_{5},a_{6},a_{7}$. We first show that the region where $a_{5},a_{6}$ are small does not contribute to the main term; see Section 6.2. This yields a new counting problem to which [BW14] applies. We estimate the error term in Section 6.4 and the main term in Section 6.5, where the summation over the remaining variables $a_{1},\dots,a_{4}$ is translated into a summation over ideals ${\mathfrak{a}}_{1},\dots,{\mathfrak{a}}_{4}$.

6.1. Summation of error terms

Lemma 6.2 will be used to bound all error terms in the first summation when summed over the remaining variables $a_{1},\dots,a_{4}$ and the auxiliary variables ${\mathfrak{d}}$ and $\rho$.

Lemma 6.1**.**

Let ${\mathfrak{a}}$ be a nonzero fractional ideal of $K$ and $b\in K$. Let $y_{v}\in K_{v}$ and $c_{v}>0$ for all $v\mid\infty$. Let

[TABLE]

Then

[TABLE]

Proof.

For real $v$, the condition $|\sigma_{v}(a^{\prime})^{2}-y_{v}|_{v}\leq c_{v}$ describes one or two intervals in $\mathds{R}$ of length $\ll c_{v}^{1/2}$ (see [Der09, Lemma 5.1(a)]). For complex $v$, this condition describes a subset of $\mathds{C}$ contained in one or two balls of radius $\ll c_{v}^{1/2}$ as in [DF14a, Lemma 3.2]. Arguing as in [DF14a, Lemma 3.3] and [FP16, Lemma 7.1] proves the result. ∎

Lemma 6.2**.**

Let ${\underline{\mathfrak{c}}}\in\mathscr{C}^{5}$, $\epsilon,\alpha_{5},\alpha_{6},\alpha_{7}>0$, $\alpha\in[0,1]$ with $\alpha\alpha_{5}+\alpha_{6}>1$ and $(1-\alpha)\alpha_{5}+\alpha_{7}>1$, $(e_{1},\dots,e_{4})\in\mathds{Z}^{4}$ with $e_{i}>0$ for some $i\in\{1,\dots,4\}$ and $\beta\in\mathds{R}_{\neq 0}$. Consider the conditions

[TABLE]

Then, for $B\geq 3$, we have

[TABLE]

where the implicit constant also depends on $K,\alpha_{5},\alpha_{6},\alpha_{7},\alpha,e_{1},\dots,e_{4},\beta$ and the implicit constants in (6.1).

Proof.

The proof is similar to [FP16, Lemma 7.3]. We have

[TABLE]

The sum over $\rho$ gives the factor

[TABLE]

using Lemma 3.1. The sums over ${\mathfrak{d}}_{56},{\mathfrak{d}}_{58}$ converge. The sums over ${\mathfrak{d}}_{5},{\mathfrak{d}}_{6},{\mathfrak{d}}_{7}$ are $\ll(1+\epsilon/5)^{\omega_{K}({\mathfrak{a}}_{1}{\mathfrak{a}}_{2}{\mathfrak{a}}_{3}{\mathfrak{a}}_{4})}$. We may replace the summation over $\underline{a}$ by a summation over ideals $\underline{{\mathfrak{a}}}$. Therefore, it is enough to show that

[TABLE]

where

[TABLE]

Indeed, if $e_{i}>0$, we sum over ${\mathfrak{a}}_{i}$ first using the first bound from (6.2) and then over the remaining ${\mathfrak{a}}_{j}$ using the remaining bounds from (6.2). Here, we combine [DF14a, Lemma 2.4] with Lemma 3.4 for the summation over ${\mathfrak{a}}_{1}$ and with [DF14a, Lemma 2.9] for the summation over ${\mathfrak{a}}_{2},{\mathfrak{a}}_{3},{\mathfrak{a}}_{4}$. ∎

6.2. Small conjugates

In Lemma 5.4, we have $|N(a_{i})|\geq\mathfrak{N}{\mathfrak{b}}_{i}$, for $i=5,6$, since $a_{i}\neq 0$ and $a_{i}\in{\mathfrak{b}}_{i}$ by definition of $\mathscr{G}({\underline{\mathfrak{c}}},\underline{a},{\underline{\mathfrak{d}}},\rho)$. We would like to strengthen this to the condition $|a_{i}|_{v}\geq\mathfrak{N}{\mathfrak{b}}_{i}^{d_{v}/d}$ for all $v\mid\infty$. We set

[TABLE]

and

[TABLE]

We show that we can replace $\mathscr{F}_{0}(\underline{a};u_{\underline{\mathfrak{c}}}B)$ by $\mathscr{F}_{0}^{*}({\underline{\mathfrak{c}}},\underline{a},{\underline{\mathfrak{d}}};u_{\underline{\mathfrak{c}}}B)$ in Lemma 5.4 and that we may restrict the summation over $\underline{a}$ to those satisfying the new height condition

[TABLE]

Lemma 6.3**.**

Let ${\underline{\mathfrak{c}}}\in\mathscr{C}^{5}$ and $\epsilon>0$. Then, for $B\geq 3$,

[TABLE]

Proof.

We show that we can replace $\mathscr{F}_{0}(\underline{a};u_{\underline{\mathfrak{c}}}B)$ by $\mathscr{F}_{0}^{*}({\underline{\mathfrak{c}}},\underline{a},{\underline{\mathfrak{d}}};u_{\underline{\mathfrak{c}}}B)$ as in [FP16, Lemmas 7.4, 7.5], using Lemma 6.1 instead of [FP16, Lemma 7.1] to prove the analogue of [FP16, (7.13)], and Lemma 6.2 instead of [FP16, Lemma 7.3] to show that the error terms are sufficiently small.

We show that we can introduce the condition (6.3) as in [FP16, Lemma 7.6], again using Lemma 6.2 instead of [FP16, Lemma 7.3]. ∎

6.3. Definability in an o-minimal structure

In order to apply [BW14] as in [FP16], we interpret $|\mathscr{G}({\underline{\mathfrak{c}}},\underline{a},{\underline{\mathfrak{d}}},\rho)\cap\mathscr{F}_{0}^{*}({\underline{\mathfrak{c}}},\underline{a},{\underline{\mathfrak{d}}};u_{\underline{\mathfrak{c}}}B)|$ in Lemma 6.3 as the number of lattice points in a set that is definable in an o-minimal structure. Here and in the following, we write for simplicity

[TABLE]

We define the $\mathds{R}$-linear map

[TABLE]

by

[TABLE]

Let $\Lambda=\Lambda({\underline{\mathfrak{c}}},\underline{a},{\underline{\mathfrak{d}}},\rho)\coloneqq\tau(\sigma(\mathscr{G}))$. Since $\sigma\colon K^{3}\to\prod_{v\mid\infty}K_{v}^{3}\cong\mathds{R}^{3d}$ is an embedding, we have

[TABLE]

Because of (6.4), we want to estimate $|\Lambda\cap\tau(S_{F}^{*})|$. For this, we would like to use [BW14, Theorem 1.3].

Let $Z$ be the set of all

[TABLE]

satisfying

[TABLE]

where $\exp\colon\mathds{R}^{\Omega_{\infty}}\to\mathds{R}_{>0}^{\Omega_{\infty}}$ is the coordinate-wise exponential function.

For

[TABLE]

we define the fiber

[TABLE]

Then $\tau(S_{F}^{*})=Z_{T}$ for $T\coloneqq(u_{\underline{\mathfrak{c}}}B,\mathfrak{N}{\mathfrak{b}}_{5}^{1/d},\mathfrak{N}{\mathfrak{b}}_{6}^{1/d},\mathfrak{N}{\mathfrak{b}}_{7}^{1/d},(\sigma_{v}(a_{1}),\dots,\sigma_{v}(a_{4}))_{v})$.

Lemma 6.4**.**

The set $Z\subset\mathds{R}^{4+7d}$ is definable in the o-minimal structure $\mathds{R}_{\exp}=\langle\mathds{R};<,+,\cdot,-,\exp\rangle$. The fibers $Z_{T}$ are bounded.

Proof.

This analogous to [FP16, Lemma 9.1]. ∎

6.4. Volumes of projections

For every coordinate subspace $W$ of $\prod_{v\mid\infty}K_{v}^{3}=\mathds{R}^{3d}$, let $V_{W}=V_{W}({\underline{\mathfrak{c}}},\underline{a},{\underline{\mathfrak{d}}};u_{\underline{\mathfrak{c}}}B)$ be the $(\dim W)$-dimensional volume of the orthogonal projection of $\tau(S_{F}^{*})$ to $W$.

Lemma 6.5**.**

For every ${\underline{\mathfrak{c}}},\underline{a},{\underline{\mathfrak{d}}},\rho$ as in Lemma 6.3, we have

[TABLE]

where $W$ runs over all proper coordinate subspaces of $\mathds{R}^{3d}$.

Proof.

As in [FP16, Lemma 8.2], the set $\Lambda$ is a lattice of rank $3d$ and determinant $(2^{-r_{2}}\sqrt{|\Delta_{K}|})^{3}$, and Minkowski’s first successive minimum $\lambda_{1}$ of this lattice with respect to the unit ball in $\mathds{R}^{3d}$ is at least $1$.

Now we argue as in [FP16, Lemma 10.1]. We start with (6.4) and apply [BW14, Theorem 1.3], which shows that

[TABLE]

By the definition of $\tau$, we have $\operatorname{vol}(\tau(S_{F}^{*}))=\operatorname{vol}(S_{F}^{*})/\mathfrak{N}({\mathfrak{b}}_{5}{\mathfrak{b}}_{6}{\mathfrak{b}}_{7})$. ∎

We want to estimate the $V_{W}$. By the definition of $F(\infty)$ and of $S_{F}^{*}$, every $(x_{5v},x_{6v},x_{7v})_{v}\in S_{F}^{*}$ satisfies

[TABLE]

for every $v\mid\infty$. In particular, with the notation

[TABLE]

and with (5.8), it satisfies

[TABLE]

Here, (6.5), (6.6) and (6.7) follow from the condition

[TABLE]

In particular, (6.7) follows from the condition

[TABLE]

using [Der09, Lemma 5.1(1)] and [DF14a, Lemma 3.5(1)]. For (6.8), we compute

[TABLE]

using (5.9), (5.11), (6.5), (5.8), and (6.9).

Define $\tau_{v}\colon K_{v}^{3}\to K_{v}^{3}$ by $(x_{5v},x_{6v},x_{7v})\mapsto(\mathfrak{N}{\mathfrak{b}}_{5}^{-1/d}x_{5v},\mathfrak{N}{\mathfrak{b}}_{6}^{-1/d}x_{6v},\mathfrak{N}{\mathfrak{b}}_{7}^{-1/d}x_{7v})$. Let

[TABLE]

Then

[TABLE]

Let

[TABLE]

Lemma 6.6**.**

Let $v\mid\infty$. For $P=(p_{5},p_{6},p_{7})\in\{0,\dots,d_{v}\}^{3}$, let $V_{P}$ be the volume of the orthogonal projection of $\tau_{v}(S_{F}^{(v)})$ to

[TABLE]

where

[TABLE]

Then

[TABLE]

Proof.

The computations are exactly the same as [FP16, Lemmas 10.2, 10.3], except that the indices $5,6,7$ are shifted by one and that we have condition (6.7) instead of $|x_{7v}|\ll c_{7}^{d_{v}/d}|x_{6v}|_{v}^{-1/2}$. But either condition is sufficient to show that

[TABLE]

and they are not used elsewhere. ∎

Lemma 6.7**.**

Let ${\underline{\mathfrak{c}}}\in\mathscr{C}^{5}$. Let $W$ be a proper coordinate subspace of $\prod_{v\mid\infty}K_{v}^{3}=\mathds{R}^{3d}$. Let $V_{W}$ be the $\dim(W)$-dimensional volume of the orthogonal projection of $\tau(S_{F}^{*})$ to $W$. Then, for $\epsilon>0$, we have

[TABLE]

Proof.

For $j=0,5,6,7$, we estimate

[TABLE]

We have $\Sigma_{0}\ll B(\log B)^{4+\epsilon}$. Indeed, we argue similarly as in Lemma 6.2, using Lemma 3.4 for the summation over ${\mathfrak{a}}_{1}$. For all other $j=5,6,7$, we have $\Sigma_{P}\ll B(\log B)^{3+\epsilon}$ because of Lemma 6.2 with the third height condition for $j=5,7$ and with (6.3) for $j=6$.

Finally, we argue as in [FP16, Lemma 10.4], applying Hölder’s inequality. ∎

6.5. Completion of the first summation

We show that $S_{F}(\underline{a};u_{\underline{\mathfrak{c}}}B)$ and $S_{F}^{*}$ have roughly the same volume. Here, we use the condition

[TABLE]

Lemma 6.8**.**

Let ${\underline{\mathfrak{c}}}\in\mathscr{C}^{5}$ and $\epsilon>0$. Then

[TABLE]

Proof.

This is analogous to [FP16, Lemma 11.1], using Lemma 6.2 instead of [FP16, Lemma 7.3]. ∎

Proposition 6.9**.**

For $\epsilon>0$ and $B\geq 3$, we have

[TABLE]

with the height condition

[TABLE]

and the notation

[TABLE]

with

[TABLE]

(using the notation $\operatorname{supp}(\mathbf{v})\coloneqq\{i:v_{i}\neq 0\}$ from [Der09, Definition 7.1]) and

[TABLE]

Proof.

By Lemmas 5.1, 6.3, 6.5, 6.7, 6.8, 5.2,

[TABLE]

By definition (see (5.6), (5.9)),

[TABLE]

In the summation over ${\underline{\mathfrak{c}}}=(\mathfrak{c}_{0},\mathfrak{c}_{1},\mathfrak{c}_{2},\mathfrak{c}_{3},\mathfrak{c}_{4})\in\mathscr{C}^{5}$, the summands are independent of $\mathfrak{c}_{0}$. Hence we can replace this sum by the factor $h_{K}$ and a sum over $(\mathfrak{c}_{1},\dots,\mathfrak{c}_{4})\in\mathscr{C}^{4}$; we also use the notation $\rho_{K}$ (1.7). As in the proof of [FP16, Lemma 11.3], we transform the summation over $(\mathfrak{c}_{1},\dots,\mathfrak{c}_{4})\in\mathscr{C}^{4}$ and $(a_{1},\dots,a_{4})\in\mathscr{F}_{1}^{4}\cap\mathcal{O}_{*}$ into a summation over $({\mathfrak{a}}_{1},\dots,{\mathfrak{a}}_{4})\in\mathscr{I}_{K}^{4}$. The conditions (6.3), (6.10) on $\underline{a}$ correspond to (6.11) on $\underline{{\mathfrak{a}}}$. It remains to show that

[TABLE]

This is a straightforward computation, as in [FP16, Lemma 11.2]. ∎

7. The remaining summations

The summation over the ideals ${\mathfrak{a}}_{1},\dots,{\mathfrak{a}}_{4}$ appearing in Proposition 6.9 is implemented in Section 7.2 by adapting the techniques developed in [DF14a, §7]. In order to do so, we first need to estimate the average size of the arithmetic function $\vartheta_{1}$.

7.1. Averages of arithmetic functions

Lemma 7.1**.**

Let $\vartheta_{1}$ be as in Proposition 6.9. Then, for all ${\mathfrak{a}}_{2},{\mathfrak{a}}_{3},{\mathfrak{a}}_{4}\in\mathscr{I}_{K}$, $x\geq 0$ and $\epsilon>0$, we have

[TABLE]

where

[TABLE]

and (with $r_{a}({\mathfrak{p}})$ as defined in Theorem 1.1)

[TABLE]

Proof.

We may assume ${\mathfrak{a}}_{3}+{\mathfrak{a}}_{4}=\mathcal{O}_{K}$ since the statement is trivial otherwise. The statement clearly holds for $x<2$; hence we can assume $x\geq 2$. For simplicity, we write

[TABLE]

Let $A_{\mathfrak{p}}(n)=\vartheta_{1,{\mathfrak{p}}}(n,v_{\mathfrak{p}}({\mathfrak{a}}_{2}),v_{\mathfrak{p}}({\mathfrak{a}}_{3}),v_{\mathfrak{p}}({\mathfrak{a}}_{4}))$. Then $\vartheta_{1}({\mathfrak{q}})=\prod_{\mathfrak{p}}A_{\mathfrak{p}}(v_{\mathfrak{p}}({\mathfrak{q}}))$. Let ${\mathfrak{b}}\coloneqq{\mathfrak{a}}_{2}{\mathfrak{a}}_{3}{\mathfrak{a}}_{4}$. We have

[TABLE]

By Lemma 3.1, $1-\frac{3}{\mathfrak{N}{\mathfrak{p}}^{2}}<A_{\mathfrak{p}}(0)<1$ for all ${\mathfrak{p}}\nmid 2a$ and $0<A_{\mathfrak{p}}(0)\leq 1$ for all ${\mathfrak{p}}\mid 2a$. In particular, $c_{0}\coloneqq\prod_{\mathfrak{p}}A_{\mathfrak{p}}(0)$ is a positive real number $\leq 1$. Furthermore, for $n>0$, we have

[TABLE]

For every ${\mathfrak{p}}$ and $n>0$, we define the multiplicative functions $B_{0},B_{1}\colon\mathscr{I}_{K}\to\mathds{R}$ by

[TABLE]

(using Lemma 3.1) and

[TABLE]

For ${\mathfrak{p}}\nmid 2a$, we observe that

[TABLE]

We consider the formal Dirichlet series

[TABLE]

with $\chi$ as in (3.1). We will discuss their convergence below. Since $\vartheta_{1}=(\vartheta_{1}*\mu_{K})*1$, we have

[TABLE]

where $\zeta_{K}(s)$ is the Dedekind zeta function.

A computation as in [DF14a, Lemma 2.2(1)] and [Der09, Proposition 6.8(3)] shows that

[TABLE]

hence

[TABLE]

Since $B_{0},B_{1}$ are multiplicative, we may compare the Euler products of $G_{0}(s)$ and $G_{1}(s)$. Their Euler factors differ at most at the primes ${\mathfrak{p}}\mid 2a{\mathfrak{b}}$. More precisely, we have $G_{0}(s)=G_{1}(s)H_{1}(s)$ with

[TABLE]

Similarly, we have $G_{1}(s)=H_{2}(s)L(s,\chi)$ with

[TABLE]

by (7.1). In total,

[TABLE]

Now we discuss the analytic properties of these functions. By [Lan18, Satz LXI, LXX], $\zeta_{K}(s)$ has a simple pole with residue $\rho_{K}$ at $s=1$ and is otherwise analytic; its Dirichlet series converges absolutely for $\Re(s)>1$. For $L(s,\chi)$, we refer to Lemma 3.5. By (7.1), we have $1+\frac{B_{1}({\mathfrak{p}})}{\mathfrak{N}{\mathfrak{p}}^{s}}\neq 0$ for $\Re(s)>0$, hence $H_{1}(s)$ is analytic with absolutely convergent Dirichlet series for $\Re(s)>0$. For $\Re(s)>1/2$, because of (7.2), the function $H_{2}(s)$ is analytic and its Dirichlet series converges absolutely.

In total, this shows that $G(s)$ is analytic for $\Re(s)>1/2$. Its Dirichlet series and its Euler product converge absolutely for $\Re(s)>1$. Since the Euler products of $L(s,\chi)$, $H_{1}(s)$, $H_{2}(s)$ converge for $s=1$ to $L(1,\chi)$, $H_{1}(1)$, $H_{2}(1)$, respectively (by Lemma 3.5(ii) and absolute convergence, respectively), the same is true for $G_{0}(s)$. Multiplying by the convergent product $c_{0}$ shows that

[TABLE]

where the right hand side must be convergent. A computation reveals that each ${\mathfrak{p}}$-adic factor of (7.3) agrees with $\vartheta_{2,{\mathfrak{p}}}$. Therefore, $G(1)=\vartheta_{2}$. Using Lemma 3.1, we check that

[TABLE]

for all ${\mathfrak{p}}$. Therefore, $\vartheta_{2,{\mathfrak{p}}}\neq 0$ for all ${\mathfrak{p}}$, and hence $G(1)\neq 0$.

Therefore, $F(s)$ has a simple pole with residue $\rho_{K}\vartheta_{2}$ at $s=1$ and is otherwise analytic for $\Re(s)>1/2$. Its Dirichlet series is absolutely convergent for $\Re(s)>1$.

Next, we discuss the vertical growth of our functions. We define $C\coloneqq 1+\epsilon$. By [Lan18, Satz LXX], $\zeta_{K}(s)\ll_{\epsilon}|\Im(s)|^{(\frac{1}{2}+2\epsilon)d}$ for $\Re(s)\geq-2\epsilon$ and $|\Im(s)|\geq 2$; by Lemma 3.5(iv), the same bound holds for $L(s,\chi)$. By absolute convergence, $|\zeta_{K}(s)|$ and $|L(s,\chi)|$ are $\ll_{\epsilon}1$ for $\Re(s)\geq 1+2\epsilon$. Define $k_{\epsilon}(\sigma)\coloneqq(1+2\epsilon)d-d\sigma$. By Phragmen-Lindelöf (as in [Ten15, Remark to Theorem II.1.20]), in the domain $-2\epsilon\leq\Re(s)\leq 1+2\epsilon$ with $|\Im(s)|\geq 2$, we have $\zeta_{K}(s)L(s,\chi)\ll_{a,\Re(s),\epsilon}|\Im(s)|^{k_{\epsilon}(\Re(s))}$ for each $\Re(s)$ and $\zeta_{K}(s)L(s,\chi)\ll_{a,\epsilon}|\Im(s)|^{k_{\epsilon}(\Re(s))+\epsilon}$ uniformly. The Euler factors of $H_{1}(s)$ are $1$ for ${\mathfrak{p}}\nmid 2a{\mathfrak{b}}$ and $1+O(\frac{1}{\mathfrak{N}{\mathfrak{p}}^{s}})$ for ${\mathfrak{p}}\mid{\mathfrak{b}}$ by (7.1), hence $H_{1}(s)\ll_{a,\epsilon}(1+\epsilon)^{\omega_{K}({\mathfrak{b}})}$ for $\Re(s)\geq\epsilon$. By absolute convergence and since $H_{2}(s)$ is independent of ${\mathfrak{b}}$ by definition, $H_{2}(s)\ll_{a,\epsilon}1$ for $\Re(s)\geq 1/2+\epsilon$. In total, for $1/2+\epsilon\leq\Re(s)\leq 1+2\epsilon$ and $|\Im(s)|\geq 2$, we have $F(s)\ll_{a,\Re(s),\epsilon}C^{\omega_{K}({\mathfrak{b}})}|\Im(s)|^{k_{\epsilon}(\Re(s))}$ for each $\Re(s)$ and $F(s)\ll_{a,\epsilon}C^{\omega_{K}({\mathfrak{b}})}|\Im(s)|^{k_{\epsilon}(\Re(s))+\epsilon}$ uniformly.

We apply Perron’s formula as in [Brü95, Lemma 1.4.2] with $c=1+2\epsilon$ and $T=\max\{x^{1/(2d)},2\}$. We have

[TABLE]

since there are $\ll n^{\epsilon/2}$ ideals of norm $n$, and each satisfies $2^{\omega_{K}({\mathfrak{q}})}\ll_{\epsilon}\mathfrak{N}{\mathfrak{q}}^{\epsilon/2}$. Therefore,

[TABLE]

with the error term

[TABLE]

Let $b=1/2+\epsilon$. By Cauchy’s residue theorem for the rectangle with vertices $b\pm iT,c\pm iT$, the main term is

[TABLE]

Since $\operatorname{Res}_{s=1}F(s)=\rho_{K}\vartheta_{2}$, the residue is $\rho_{K}\vartheta_{2}x$. The first and third integral are (using $1/|s|\leq 1/T$)

[TABLE]

For the second integral, we use $|F(s)|\leq|\Im(s)|^{k(b)}$ and $|s|\geq|\Im(s)|$ (for $|\Im(s)|\geq 2$) and $|F(s)|\ll 1$ and $|s|\geq b$ (for $|\Im(s)|\leq 2$) to deduce

[TABLE]

In total, this proves the result. ∎

7.2. Completion of the proof

Proof of Theorem 1.1.

Our starting point is Proposition 6.9. We argue similarly as in [DF14a, §7], but $\vartheta_{1}$ is more complicated.

First, we sum over ${\mathfrak{a}}_{1}$. For $t_{1},\dots,t_{4}\in\mathds{R}_{\geq 1}$, let

[TABLE]

Since the two inequalities in the definition of $V_{1}$ (which correspond to the height conditions (6.11)) imply $t_{1}t_{2}^{2}t_{3}^{2}t_{4}^{2}\leq B$, we have $V_{1}(t_{1},\dots,t_{4};B)=0$ unless $t_{1},\dots,t_{4}\leq B$. We claim that

[TABLE]

Indeed, by Lemma 7.1, we can apply [DF14a, Lemma 2.10] with $m=1$, $c_{1}=(1+\epsilon)^{\omega_{K}({\mathfrak{a}}_{2}{\mathfrak{a}}_{3}{\mathfrak{a}}_{4})}$, $b_{1}=1-1/(2d)+\epsilon$, $k_{1}=0$, $c_{g}=B/\mathfrak{N}({\mathfrak{a}}_{2}{\mathfrak{a}}_{3}{\mathfrak{a}}_{4})$, $a=-1$ for the summation over ${\mathfrak{a}}_{1}$. By [DF14a, Lemmas 2.9, 2.4, 2.10], the total error term is

[TABLE]

For the summations over ${\mathfrak{a}}_{2},{\mathfrak{a}}_{3},{\mathfrak{a}}_{4}$, let

[TABLE]

and

[TABLE]

hence

[TABLE]

Then $\theta_{2}\in\Theta_{3}^{\prime}(C)$ [DF14a, Definition 2.6] for $C=\max(\{\mathfrak{N}{\mathfrak{p}}:{\mathfrak{p}}\mid 2a\}\cup\{5\})$. Indeed, $\theta_{2}({\mathfrak{a}}_{2},{\mathfrak{a}}_{3},{\mathfrak{a}}_{4})\in[0,1]$ since

[TABLE]

is easily checked using (7.4). By definition, $\theta_{2}(0,0,0)=1$. If $|\operatorname{supp}(\mathbf{v})|=1$, then $\theta_{2,{\mathfrak{p}}}(\mathbf{v})\geq 1-\frac{C}{\mathfrak{N}{\mathfrak{p}}}$ holds for ${\mathfrak{p}}\nmid 2a$ using $|r_{a}({\mathfrak{p}})|=1$ and $C\geq 5$, and it holds for ${\mathfrak{p}}\mid 2a$ since $\theta_{2,{\mathfrak{p}}}(\mathbf{v})\geq 0$.

Therefore, we can apply [DF14a, Proposition 7.2] inductively (as in the proof of [DF14a, Proposition 7.3]; note that our definition (1.7) of $\rho_{K}$ differs from the one in [DF14a, §1.4] by the factor $h_{K}$) to deduce that

[TABLE]

where

[TABLE]

is the average value of $\vartheta_{2}({\mathfrak{a}}_{2},{\mathfrak{a}}_{3},{\mathfrak{a}}_{4})$ when summed over ${\mathfrak{a}}_{2},{\mathfrak{a}}_{3},{\mathfrak{a}}_{4}$ (as defined before [DF14a, Proposition 2.3]).

Therefore,

[TABLE]

with

[TABLE]

The classes of $E_{1},\dots,E_{5}$ form a basis of $\operatorname{Pic}({\widetilde{S}})$. Together with $[E_{6}]=[E_{1}]-[E_{2}]-2[E_{4}]+[E_{5}]$, they generate the effective cone. We have $-K_{\widetilde{S}}=2[E_{1}]+[E_{2}]+2[E_{3}]+3[E_{5}]$. As in [DF14a, Lemma 8.1],

[TABLE]

Here, by [DJT08, Corollary 7.5],

[TABLE]

as in Theorem 1.1, where $S_{0}$ is a nonsplit smooth quartic del Pezzo surface with $\operatorname{Pic}(S_{0})$ of rank $5$ (with $\alpha(S_{0})=\frac{1}{36}$ by [DEJ14, Theorem 4.1]) and $|W(R)|=2!\cdot 4!$ is the order of the Weyl group of the root system ${\mathbf{A}}_{3}+{\mathbf{A}}_{1}$ (since all $(-2)$-curves on ${\widetilde{S}}_{\overline{K}}$ are already defined over $K$).

We check that $\vartheta_{3}=\prod_{\mathfrak{p}}\omega_{\mathfrak{p}}(S)$ with $\omega_{\mathfrak{p}}(S)$ as in Theorem 1.1 using [DF14a, Lemma 2.8, formula (2.2)], which we may apply to $\theta_{2}$ and therefore also directly to $\vartheta_{2}$.

By the discussion in the following Section 8, our asymptotic formula agrees with the conjectures of Manin and Peyre. ∎

8. The leading constant

In this section, we check that the leading constant in Theorem 1.1 agrees with [Pey03, Formule empirique 5.1]. Recall that $\phi\colon{\widetilde{S}}\to S$ is a minimal desingularization. Then ${\widetilde{S}}$ is an almost Fano variety in the sense of [Pey03, Definition 3.1] by the description in [Der14, §3.4] and an application of Kawamata–Viehweg vanishing [Vie82, Corollary 0.3]. Moreover, ${\widetilde{S}}$ satisfies [Pey03, Hypothèse 3.3] because it is rational.

8.1. Convergence factors

As in [FP16, §13], we construct an adelic metric on the anticanonical line bundle $\omega_{\widetilde{S}}^{-1}$ such that the corresponding height function on ${\widetilde{S}}(K)$ is $H\circ\phi$. We denote by $\omega_{H,v}$ the local measures, and by $\tau_{H}$ the Tamagawa measure corresponding to the adelic metric (see [Pey03, §4]). Then the leading constant is expected to be

[TABLE]

where $\alpha({\widetilde{S}})=\alpha_{S}$ as discussed in Section 7, $\beta({\widetilde{S}})=1$ as $\operatorname{Pic}({\widetilde{S}}_{{\overline{K}}})$ is a permutation $\operatorname{Gal}({\overline{K}}/K)$-module and

[TABLE]

for a finite set $S^{0}$ of finite places of $K$, as defined in [Pey03, Notations 4.5].

Lemma 8.1**.**

We have

[TABLE]

Proof.

We follow the strategy of [PT01, Proposition 5.1]. We use the notation $S^{c}\coloneqq\Omega_{f}\smallsetminus S^{0}$. With $\Lambda\coloneqq\operatorname{Pic}({\widetilde{S}}_{\overline{K}})$, by definition,

[TABLE]

with

[TABLE]

where $\operatorname{Fr}_{\mathfrak{p}}$ is the geometric Frobenius acting on $\operatorname{Pic}(\widetilde{\mathcal{S}}\times_{\mathds{Z}}\mathds{F}_{\mathfrak{p}})\otimes_{\mathds{Z}}\mathds{Q}$.

Enlarging $S^{0}$ does not change the expression that we want to compute. Therefore, we may assume that $S^{0}$ contains $\{{\mathfrak{p}}\in\Omega_{f}:{\mathfrak{p}}\mid 2a\}$ and that there is a model $\widetilde{\mathcal{S}}$ of ${\widetilde{S}}$ over the ring $\mathcal{O}_{S^{0}}$ of $S^{0}$-integers such that

[TABLE]

for all ${\mathfrak{p}}\in S^{c}$ by [Pey03, Remarque 4.4]. By a result of Weil [Man86, Theorem 23.1], for ${\mathfrak{p}}\in S^{c}$,

[TABLE]

In Lemma 8.5 we show that for finite places ${\mathfrak{p}}\nmid 2a$,

[TABLE]

Since $|\mathds{F}_{\mathfrak{p}}|=\mathfrak{N}{\mathfrak{p}}$, we deduce, for ${\mathfrak{p}}\in S^{c}$, that

[TABLE]

We have $\det(1-T\cdot f)=1-\operatorname{Tr}(f)\cdot T+\dots+(-1)^{n}\det(f)\cdot T^{n}$ in the polynomial ring $\mathds{Q}[T]$ for an endomorphism $f$ of an $n$-dimensional $\mathds{Q}$-vector space. Therefore,

[TABLE]

for some function

[TABLE]

With $\chi$ as in (3.1), let

[TABLE]

Here, $G(s)$ is analytic for $\Re(s)>1/2$, while $\zeta_{K,S^{0}}$ has a simple pole with residue

[TABLE]

at $s=1$ and $L_{S^{0}}(1,\chi)$ converges and is nonzero by Lemma 3.5(iii). Therefore,

[TABLE]

with

[TABLE]

We compute

[TABLE]

since $L_{S^{0}}(1,\chi)\neq 0$ by Lemma 3.5(iii). ∎

8.2. Local densities

We compute the local densities $\omega_{H,v}({\widetilde{S}}(K_{v}))$. The birational projection $S\dashrightarrow\mathds{P}^{2}$, $(x_{0}:\dots:x_{4})\mapsto(x_{1}:x_{3}:x_{4})$ induces the chart

[TABLE]

With $f_{1}=x_{0}x_{4}+x_{1}^{2}-ax_{3}^{2}$, $f_{2}=x_{2}x_{3}-x_{4}^{2}$, we have

[TABLE]

hence with $x_{4}=1$, we deduce analogously to [FP16, §13] for every $v\in\Omega_{K}$ that

[TABLE]

Lemma 8.2**.**

For every archimedean place $v$ of $K$, the constant $\omega_{v}(S)$ defined in Theorem 1.1 agrees with the $v$-adic density $\omega_{H,v}({\widetilde{S}}(K_{v}))$.

Proof.

This is analogous to the end of [FP16, §13]. ∎

For a prime ideal ${\mathfrak{p}}$ of $\mathcal{O}_{K}$, let $\mu_{\mathfrak{p}}$ be the Haar measure on $K_{\mathfrak{p}}$, normalized such that $\mu_{\mathfrak{p}}(\{x\in K_{\mathfrak{p}}:v_{\mathfrak{p}}(x)\geq 0\})=1$.

Lemma 8.3**.**

Let ${\mathfrak{p}}$ be a prime ideal in $\mathcal{O}_{K}$. Assume that $v_{\mathfrak{p}}(a)$ is even. Let $x_{3}\in K_{\mathfrak{p}}^{\times}$, and let $k\geq 0$. Then

[TABLE]

Proof.

Let $\gamma=2\gamma^{\prime}=v_{\mathfrak{p}}(a)\geq 0$. First, we consider the case $x_{3}=1$. The condition $v_{\mathfrak{p}}(a-x_{1}^{2})\geq 2\gamma^{\prime}+k$ holds if and only if $v_{\mathfrak{p}}(x_{1})=\gamma^{\prime}$ and ${x_{1}}\equiv{\rho}\pmod{{\mathfrak{p}}^{\gamma^{\prime}+k}}$ for some $\rho\in\mathcal{O}_{K}$ satisfying $\rho\mathcal{O}_{K}+{\mathfrak{p}}^{\gamma^{\prime}+k}={\mathfrak{p}}^{\gamma^{\prime}}$ and ${\rho^{2}}\equiv{a}\pmod{{\mathfrak{p}}^{2\gamma^{\prime}+k}}$. Since $\eta({\mathfrak{p}}^{2\gamma^{\prime}+k};a)$ is defined as the cardinality of the set of such $\rho\pmod{{\mathfrak{p}}^{\gamma^{\prime}+k}}$, our claim holds in the first case.

In the general case, the transformation $x_{1}\mapsto x_{1}/x_{3}$ reduces to the first case. ∎

Lemma 8.4**.**

For every $q\in\mathds{R}_{>1}$ and $n\in\mathds{Z}_{\geq 0}$, we have

[TABLE]

Proof.

We observe that

[TABLE]

Lemma 8.5**.**

For each finite place ${\mathfrak{p}}$ of $K$, we have

[TABLE]

with $\omega_{\mathfrak{p}}(S)$ as in Theorem 1.1.

Proof.

Let $q\coloneqq\mathfrak{N}{\mathfrak{p}}$. Let $\alpha=v_{\mathfrak{p}}(x_{1})$, $\beta=v_{\mathfrak{p}}(x_{3})$, and $\gamma=v_{\mathfrak{p}}(a)$. In the case $2\alpha=2\beta+\gamma$, let $\kappa\coloneqq v_{\mathfrak{p}}(a(x_{3}/x_{1})^{2}-1)\in\mathds{Z}_{\geq 0}$. Then

[TABLE]

Now it is straightforward to compute the maximum $M$ in (8.1) as

[TABLE]

The integration domain $K_{\mathfrak{p}}^{2}=\{(x_{1},x_{3})\in K_{\mathfrak{p}}^{2}:v_{\mathfrak{p}}(x_{1})=\alpha,\ v_{\mathfrak{p}}(x_{3})=\beta\}$ in (8.1) is the disjoint union of the sets

[TABLE]

Therefore,

[TABLE]

where $\omega_{{\mathfrak{p}},i}$ is (8.1) integrated over $V_{i}$ instead of $K_{\mathfrak{p}}^{2}$.

On $V_{1}$, we have $|x_{3}|_{\mathfrak{p}}\cdot M=q^{-\beta}\cdot q^{\beta}=1$. Writing $\beta=2\beta^{\prime}-1$ or $\beta=2\beta^{\prime}$ with $\beta^{\prime}\in\mathds{Z}_{>0}$, we have

[TABLE]

On $V_{2}$, we have $|x_{3}|_{\mathfrak{p}}\cdot M=q^{-\beta}\cdot q^{-2\alpha}$. Writing $\alpha^{\prime}=-\alpha$ and $\gamma^{\prime}=\left\lfloor{\gamma/2}\right\rfloor$ and using Lemma 8.4,

[TABLE]

On $V_{3}$, we have $|x_{3}|_{\mathfrak{p}}\cdot M=q^{-\beta}\cdot q^{-2\alpha}$. Writing $\alpha^{\prime}=-\alpha$ and using Lemma 8.4,

[TABLE]

On $V_{4}$, we have $|x_{3}|_{\mathfrak{p}}\cdot M=q^{-\beta}\cdot q^{-\beta}$. Writing $\beta^{\prime}=-\beta$, we have

[TABLE]

On $V_{5}$, we have $|x_{3}|_{\mathfrak{p}}\cdot M=q^{-\beta}\cdot q^{-(2\beta+\gamma)}$. Writing $\beta^{\prime}=-\beta$, we have

[TABLE]

So far, this shows that

[TABLE]

For odd $\gamma$, since $V_{6}=\emptyset$ and $\omega_{{\mathfrak{p}},6}=0$, this is $\omega_{H,{\mathfrak{p}}}({\widetilde{S}}(K_{\mathfrak{p}}))$, which agrees with Theorem 1.1.

Now assume that $\gamma=2\gamma^{\prime}=v_{\mathfrak{p}}(a)\geq 0$ is even. By the definition of $\kappa$,

[TABLE]

Therefore, using Lemma 8.3 in the second step, writing $\beta^{\prime}=-\beta$ in the third step, and exchanging orders of summation in the fourth step,

[TABLE]

For even $\gamma$, we have in total

[TABLE]

which also agrees with Theorem 1.1. ∎