# Paucity of rational points on fibrations with multiple fibres

Tim Browning  IST Austria
Am Campus 1
3400 Klosterneuburg
Austria<EMAIL_ADDRESS>,  Julian Lyczak  Department of Mathematical Sciences
University of Bath
Claverton Down
Bath BA2 7AY
UK<EMAIL_ADDRESS>and  Arne Smeets  Department of Mathematics
KU Leuven
Celestijnenlaan 200B
B-3001 Leuven
Belgium<EMAIL_ADDRESS>
###### Abstract.

Given a family of varieties over the projective line, we study the density of
fibres that are everywhere locally soluble in the case that components of
higher multiplicity are allowed. We use log geometry to formulate a new
sparsity criterion for the existence of everywhere locally soluble fibres and
formulate new conjectures that generalise previous work of Loughran–Smeets.
These conjectures involve geometric invariants of the associated multiplicity
orbifolds on the base of the fibration in the spirit of Campana. We give
evidence for the conjectures using Chebotarev’s theorem and sieve methods.

###### 2010 Mathematics Subject Classification:

14G05 (11G35, 11N36, 14A21, 14D10)

###### Contents

  1. 1 Introduction
  2. 2 Group-theoretic results
  3. 3 Pairs of integers with Frobenian conditions
  4. 4 Orbifolds and étale orbifold morphisms
  5. 5 A sparsity criterion
  6. 6 Multiple fibres via the large sieve
  7. 7 Examples: lower bounds and asymptotics

##  1\. Introduction

Let $X$ be a smooth, proper, geometrically irreducible variety over
$\mathbb{Q}$, which is equipped with a dominant morphism
$\pi:X\to\mathbb{P}^{1}$ with geometrically integral generic fibre. We shall
refer to such fibrations as standard. The focus of this article is on
situations where multiple fibres are present. Work of Colliot-Thélène,
Skorobogatov and Swinnerton-Dyer [7] shows that the set $X(\mathbb{Q})$ of
$\mathbb{Q}$-rational points on $X$ is not Zariski dense when there are at
least $5$ geometric double fibres. Our goal is to put this kind of result on a
quantitative footing by analysing the simpler question of solubility over the
ring of adèles $\mathbf{A}_{\mathbb{Q}}$. Let

$N_{\text{loc}}(\pi,H,B)=\\#\left\\{x\in\mathbb{P}^{1}(\mathbb{Q})\cap\pi(X(\mathbf{A}_{\mathbb{Q}})):H(x)\leqslant
B\right\\},$ (1.1)

where $H$ is a height function on $\mathbb{P}^{1}(\mathbb{Q})$. In general, we
will need to allow the height $H$ to be any adelic height on a line bundle
$\mathscr{O}(d)$. However, most of the time we shall use an
$\mathscr{O}(1)$-height. In this case we will simply write
$N_{\text{loc}}(\pi,H,B)=N_{\text{loc}}(\pi,B)$. Usually we will take the
naive height $H(x)=\max\\{|x_{0}|,|x_{1}|\\}$, if
$x\in\mathbb{P}^{1}(\mathbb{Q})$ is represented by a vector
$\mathbf{x}=(x_{0},x_{1})\in\mathbb{Z}_{\mathrm{prim}}^{2}$, in which case it
is easy to prove that $\\#\\{x\in\mathbb{P}^{1}(\mathbb{Q}):H(x)\leqslant
B\\}\sim\frac{2}{\zeta(2)}B^{2}$, as $B\to\infty$.

Loughran and Smeets [15] have shown that

$N_{\text{loc}}(\pi,B)\ll\frac{B^{2}}{(\log B)^{\Delta(\pi)}},$ (1.2)

for a certain exponent $\Delta(\pi)\geqslant 0$ that is defined in terms of
the data of the fibration. (Here, as throughout our work, all implied
constants are allowed to depend on the fibration $\pi$.) Roughly speaking, the
size of $\Delta(\pi)$ is determined by the number of non-split fibres of
$\pi$, thereby lending credence to a philosophy put forward by Serre [18] and
further developed by Loughran [12]. In [15, Conj. 1.6] it is conjectured that
the upper bound (1.2) is sharp provided that the fibre of $\pi$ over every
closed point of $\mathbb{P}^{1}$ has an irreducible component of multiplicity
one. (In fact, the work in [15] works over arbitrary number fields $k$ and
concerns fibrations $X\to\mathbb{P}^{n}$ over projective space of arbitrary
dimension, but we shall restrict to $k=\mathbb{Q}$ and $n=1$ in our work.) Our
goal is to explore what happens to $N_{\text{loc}}(\pi,B)$ when the assumption
about components of multiplicity one is violated.

There are relatively few examples in the number theory literature that feature
standard fibrations with multiple fibres. When the generic fibre of $\pi$ is
rationally connected, it follows from work of Graber, Harris and Starr [9]
that every fibre contains a geometrically integral component of multiplicity
one. In particular, when $\dim X=2$, we must look to fibrations over
$\mathbb{P}^{1}$ into curves of positive genus to find examples with multiple
fibres. Let $c,d,f\in\mathbb{Q}[t]$ be non-zero polynomials such that $f$ is
square-free of even degree and such that $f$ and $c-d$ are coprime. Let
$\pi\colon X\to\mathbb{P}^{1}$ be a smooth, proper model of the affine variety
cut out by the pair of equations

$x^{2}-c(t)=f(t)y^{2},\quad x^{2}-d(t)=f(t)z^{2}.$ (1.3)

Then it follows from [7, Prop. 4.1] that all the fibres of $\pi$ over the
zeros of $f$ are double fibres, and that the generic fibre is a geometrically
integral curve whose projective model is isomorphic to a curve of genus one.
When $\deg(f)\geqslant 6$, as pointed out by Loughran and Matthiesen [13, Thm.
1.4], the argument of [7, Cor. 2.2] implies that $N_{\text{loc}}(\pi,B)=O(1)$.
Further examples involving genus $2$ fibrations over $\mathbb{P}^{1}$ have
been worked out by Stoppino [20].

In the spirit of Campana [5], our approach to this problem comes from relating
the arithmetic of $\pi\colon X\to\mathbb{P}^{1}$ to the arithmetic of the
orbifold base $(\mathbb{P}^{1},\partial_{\pi})$, for a certain
$\mathbb{Q}$-divisor $\partial_{\pi}$, in the sense of Definition 4.6. For
each closed point $D\in(\mathbb{P}^{1})^{(1)}$, we let $m_{D}\geqslant 1$
denote the minimum multiplicity of the irreducible components of
$\pi^{-1}(D)$. Then we may define

$\partial_{\pi}=\sum_{D\in(\mathbb{P}^{1})^{(1)}}\left(1-\frac{1}{m_{D}}\right)[D].$
(1.4)

With this notation, we make the following conjecture.

######  Conjecture 1.1.

Let $\pi:X\to\mathbb{P}^{1}$ be a standard fibration such that the
$\mathbb{Q}$-divisor $-(K_{\mathbb{P}^{1}}+\partial_{\pi})$ is ample. Then

$N_{\text{loc}}(\pi,B)=O_{\varepsilon}\left(B^{2-\deg\partial_{\pi}+\varepsilon}\right),$

for any $\varepsilon>0$.

Note that $-\deg(K_{\mathbb{P}^{1}}+\partial_{\pi})=2-\deg\partial_{\pi}.$
Hence $-(K_{\mathbb{P}^{1}}+\partial_{\pi})$ is ample if and only if
$\deg\partial_{\pi}<2$. The main feature of Conjecture 1.1 is that we expect
$N_{\text{loc}}(\pi,B)$ to be much smaller in the presence of multiple fibres.
Our remaining results give evidence towards this, as well as a proposal about
the replacement of $B^{\varepsilon}$ by an explicit non-positive power of
$\log B$.

###  1.1. Upper bounds

For each closed point $D\in(\mathbb{P}^{1})^{(1)}$, let $S_{D}$ be the set of
geometrically irreducible components of $\pi^{-1}(D)$ of multiplicity $m_{D}$
and let $\kappa(D)$ be the residue field. For any number field $N/\mathbb{Q}$,
we write

$\delta_{D,N}(\pi)=\frac{\\#\\{\sigma\in\Gamma_{D,N}:\text{$\sigma$ acts with
a fixed point on $S_{D}$}\\}}{\\#\Gamma_{D,N}},$ (1.5)

where $\Gamma_{D,N}$ is a finite group through which the action of ${\rm
Gal}(\overline{N}/N)$ on $S_{D}$ factors. (We take $\delta_{D,N}(\pi)=0$ when
no such components exist.) Note that

$0\leqslant\delta_{D,N}(\pi)\leqslant 1.$ (1.6)

Moreover, we shall write $\delta_{D}(\pi)=\delta_{D,\kappa(D)}(\pi)$. When
$\pi^{-1}(D)$ has components of multiplicity one, this agrees with the
definition given by Loughran and Smeets [15, Eq. (1.4)]. A natural analogue of
the exponent appearing in [15, Thm. 1.2] is then

$\Delta(\pi)=\sum_{D\in(\mathbb{P}^{1})^{(1)}}\left(1-\delta_{D}(\pi)\right),$
(1.7)

which agrees with the exponent appearing in (1.2) whenever $\pi^{-1}(D)$
contains a multiplicity one component for every $D\in(\mathbb{P}^{1})^{(1)}$.

The following upper bound treats the case of one multiple fibre above a degree
$1$ point of $\mathbb{P}^{1}$, and is consistent with Conjecture 1.1.

######  Theorem 1.2.

Let $\pi:X\to\mathbb{P}^{1}$ be a standard fibration with a unique multiple
fibre at $0$. Then

$N_{\text{loc}}(\pi,B)\ll\frac{B^{2-\deg\partial_{\pi}}}{(\log
B)^{\Delta(\pi)}},$

where $\Delta(\pi)$ is given by (1.7).

It is tempting to suppose that the same estimate continues to hold when there
is more than one closed point of $\mathbb{P}^{1}$ above which multiple fibres
exist. However, in Theorem 7.1, we shall illustrate that a smaller exponent
than $\Delta(\pi)$ is sometimes necessary.

Let $\pi:X\to\mathbb{P}^{1}$ be a standard fibration and let
$D\in(\mathbb{P}^{1})^{(1)}$, which we suppose is defined by an irreducible
binary form $g\in\mathbb{Q}[x,y]$. Assume first that $g(1,0)\neq 0$. Then the
residue field is $\kappa(D)=\mathbb{Q}[x]/(g(x,1))$. Moreover, for any
$d\in\mathbb{N}$ and any $v\in\mathbb{Q}$, let
$N_{D,d,v}=\mathbb{Q}[x]/(h(x)),$ where

$h(x)=g(x^{d},v).$

For typical $v$ this forms a number field of degree $\deg(g)+d$, but in
general forms an étale algebra, since $h$ is not necessarily irreducible, with
a factorisation

$N_{D,d,v}=N_{D,d,v}^{(1)}\times\dots\times N_{D,d,v}^{(s_{D})}.$ (1.8)

It still remains to deal with the case $g(1,0)=0$. But then $D=\infty$ and it
readily follows that $\kappa(\infty)=\mathbb{Q}[y]/(g(1,y))=\mathbb{Q}$ and
$N_{\infty,d,v}=\mathbb{Q}[y]/(g(1,vy^{d}))=\mathbb{Q}$, for any
$v\in\mathbb{Q}$. We may now define

$\Theta_{v}(\pi)=\sum_{D\in\left(\mathbb{P}^{1}\right)^{(1)}}\sum_{k=1}^{s_{D}}\left(1-\delta_{D,N_{D,d,v}^{(k)}}(\pi)\right),$
(1.9)

in the notation of (1.5). Our main upper bound is as follows.

######  Theorem 1.3.

Let $\pi:X\to\mathbb{P}^{1}$ be a standard fibration with multiple fibres at
$0$ and $\infty$, and nowhere else. Let $d=\gcd(m_{0},m_{\infty})$. Then

$N_{\text{loc}}(\pi,B)\ll\frac{B^{2-\deg\partial_{\pi}}}{(\log
B)^{\min_{v\in\mathbb{Q}^{\times}/\mathbb{Q}^{\times,d}}\Theta_{v}(\pi)}}.$

It will be convenient to put

$\Theta(\pi)=\min_{v\in\mathbb{Q}^{\times}/\mathbb{Q}^{\times,d}}\Theta_{v}(\pi).$
(1.10)

Let us first note that $\Theta(\pi)\geqslant 0$, by (1.6). Secondly,
$\Delta(\pi)$ and $\Theta(\pi)$ can be different; in Theorem 7.1 we will see
an example with $\Theta(\pi)=0$, but $\Delta(\pi)=1$. However, we will see
that

$\Theta(\pi)=\Delta(\pi),\quad\text{if $\gcd(m_{0},m_{\infty})=1$}.$ (1.11)

The following result shows that there are only finitely many values that
$\Theta_{v}(\pi)$ can take.

######  Theorem 1.4.

Let $\pi:X\to\mathbb{P}^{1}$ be a standard fibration and let
$D\in(\mathbb{P}^{1})^{(1)}$. Let $E$ be the field of definition of the
elements of $S_{D}$ and let $N/\mathbb{Q}$ be a number field. Then
$\delta_{D,N}(\pi)=\delta_{D,N\cap E^{\text{normal}}}(\pi),$ where
$E^{\text{normal}}$ is the normal closure of $E$.

As we have seen, our understanding of $N_{\text{loc}}(\pi,B)$ is inexorably
linked to the arithmetic of the orbifold base
$(\mathbb{P}^{1},\partial_{\pi})$. The study of rational points on orbifolds
is the focus of work by Pieropan, Smeets, Tanimoto and Várilly-Alvarado [17],
which offers a far-reaching conjectural asymptotic formula for any orbifold
$(Y,\partial)$ with $\mathbb{Q}$-ample divisor $-(K_{Y}+\partial)$. Pieropan
and Schindler [16] have verified many cases of the conjecture when $Y$ is a
split toric variety over $\mathbb{Q}$. Their work covers the orbifolds that
arise in the proof of Theorem 1.2 and 1.3 and would yield the upper bound
$N_{\text{loc}}(\pi,B)=O(B^{2-\deg\partial_{\pi}}).$ In order to achieve the
desired non-positive powers of $\log B$, we need to incorporate extra
Chebotarev type conditions that arise when counting locally soluble fibres.

The proofs of Theorems 1.2 and 1.3 are based on the large sieve and will be
carried out in Section 6. A crucial ingredient will be a sparsity criterion,
which gives explicit control over which fibres are everywhere locallly
soluble. This criterion will be proved in Section 5 using log geometry, and
may be of independent interest.

Extending Theorem 1.3 to three multiple fibres represents a formidable
challenge. The easiest such case corresponds to the $\mathbb{Q}$-divisor

$\partial_{\pi}=\frac{1}{2}[0]+\frac{1}{2}[1]+\frac{1}{2}[\infty].$

Conjecture 1.1 would predict that
$N_{\text{loc}}(\pi,B)=O_{\varepsilon}(B^{1/2+\varepsilon})$, for any
$\varepsilon>0$. However, the best upper bound we have is due to Browning and
Van Valckenborgh [2], which only yields the exponent $3/5+\varepsilon$.

###  1.2. A new conjecture

We are now ready to reveal a new conjecture for the density of locally soluble
fibres for standard fibrations, in which multiple fibres are allowed. Let
$\pi\colon X\to\mathbb{P}^{1}$ be a standard fibration, and let
$\theta\colon\mathbb{P}^{1}\to(\mathbb{P}^{1},\partial_{\pi})$ be a finite
étale orbifiold morphism, as defined in Definition 4.2.

We assume that $(\mathbb{P}^{1},\partial_{\pi})$ does not admit a finite étale
orbifold morphism which factors through $\theta$, and $\theta$ is a $G$-torsor
under a finite étale group scheme $G$. Let
$\theta_{v}\colon\mathbb{P}^{1}\to\mathbb{P}^{1}$ denote the twist of $\theta$
by any $v\in\textup{H}^{1}({\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q}),G)$.
Finally, let $\pi_{v}\colon X_{v}\to\mathbb{P}^{1}$ denote the normalisation
of the pullback of $\pi$ along $\theta_{v}$.

######  Conjecture 1.5.

Let $\pi:X\to\mathbb{P}^{1}$ be a standard fibration such that the
$\mathbb{Q}$-divisor $-(K_{\mathbb{P}^{1}}+\partial_{\pi})$ is ample and
$X(\mathbf{A}_{\mathbb{Q}})\neq\emptyset$. Then there exists a constant
$c_{\pi}>0$ such that

$N_{\text{loc}}(\pi,B)\sim c_{\pi}\frac{B^{2-\deg\partial_{\pi}}}{(\log
B)^{\min_{v\in\textup{H}^{1}({\rm
Gal}(\overline{\mathbb{Q}}/\mathbb{Q}),G)}\Delta(\pi_{v})}}$

where $\Delta(\pi_{v})$ is given by (1.7).

Note that it follows from Theorem 1.4 that $\Delta(\pi_{v})$ takes only
finitely many values. In the special case that the orbifold base is simply
connected as an orbifold, which in the setting of Theorem 1.3 covers the case
$\gcd(m_{0},m_{\infty})=1$, we will have $\Theta(\pi)=\Delta(\pi)$. Thus
Conjecture 1.5 implies that

$N_{\text{loc}}(\pi,B)\sim c_{\pi}\frac{B^{2-\deg\partial_{\pi}}}{(\log
B)^{\Delta(\pi)}},$

in this case, which is consistent with the upper bound in Theorem 1.2. In
Corollary 4.9 we shall take $G=\mu_{d}$ and prove that
$\Theta_{v}(\pi)=\Delta(\pi_{v})$ in (1.9). Hence the upper bound in Theorem
7.1 is also consistent with Conjecture 1.5. In Section 7 we shall provide
further evidence for the conjecture, by establishing a range of estimates for
the variant $N_{\text{loc},S}(\pi,B)$ of $N_{\text{loc}}(\pi,B)$, in which
local solubility is only required away from a finite set $S$ of primes. In
Theorem 7.2, for example, we establish a precise lower bound for
$N_{\text{loc},S}(\pi,B)$ in the case that $\pi:X\to\mathbb{P}^{1}$ is a
standard fibration for which the only non-split fibres lie over $0$ and
$\infty$.

One further source of examples that can be used to illustrate our conjectures
is the class of Halphen surfaces. These were introduced by Halphen [10] in
1882 and correspond to standard fibrations admitting a unique multiple fibre.
In Theorems 7.3–7.8 we provide several estimates for $N_{\text{loc},S}(\pi,B)$
that are consistent with Conjecture 1.5, for appropriate surfaces of Halphen
type. In the proof of Theorem 7.8 we are led to a concrete problem in analytic
number theory that was solved by Friedlander and Iwaniec [8, Thm. 11.31].
Indeed, we need matching upper and lower bounds for the number of positive
integers $a,b$ satisfying $a^{6}+b^{2}\leqslant x$, as $x\to\infty$, such that
the only prime divisors of $a^{6}+b^{2}$ are those that split in a given cubic
Galois extension $K/\mathbb{Q}$. It would be useful to have a similar result
for non-Galois extensions, but this appears to be difficult.

######  Remark 1.6.

Returning to the example (1.3), we see that the associated
$\mathbb{Q}$-divisor $\partial_{\pi}$ has degree $\frac{1}{2}\deg(f)$. Since
$f$ is assumed to have even degree, it follows that
$-(K_{\mathbb{P}^{1}}+\partial_{\pi})$ is ample only when $\deg(f)=2$. When
$f$ is a quadratic polynomial, Conjecture 1.1 implies that
$N_{\text{loc}}(\pi,B)=O_{\varepsilon}(B^{1+\varepsilon})$ for any
$\varepsilon>0$. The orbifold base $(\mathbb{P}^{1},\partial_{\pi})$ admits
$\mu_{2}$-covers and it is possible to apply Conjecture 1.5 to predict an
explicit power of $\log B$. The outcome will depend on the Galois action on
the geometric components of the fibres.

###  1.3. Further questions

We expect similar conjectures to hold when looking at fibrations $\pi:X\to Y$
over other bases for which $-(K_{Y}+\partial_{\pi})$ is $\mathbb{Q}$-ample.
However, when $\dim(Y)>1$ the sparsity criterion we work out in Section 5 will
be significantly more complicated. Moreover, care also needs to be taken
around the effect of thin subsets of $Y(\mathbb{Q})$ on the counting problem.
A counter-example to the most naive expectation has recently been provided [3]
in the case that $Y$ is a split quadric in $\mathbb{P}^{3}$.

In a different direction, when $Y=\mathbb{P}^{1}$, we can extend the
definition (1.1) by defining $N_{\text{loc}}(\pi,B;Z)$ to be the number of
$x\in(\mathbb{P}^{1}(\mathbb{Q})\setminus
Z)\cap\pi(X(\mathbf{A}_{\mathbb{Q}}))$ for which $H(x)\leqslant B$, for any
thin subset $Z\subset\mathbb{P}^{1}(\mathbb{Q})$. It is then very natural to
ask whether or not we should expect a bound of the shape

$\displaystyle
N_{\text{loc}}(\pi,B;Z)\ll\frac{B^{\frac{1}{m_{0}}+\frac{1}{m_{\infty}}}}{(\log
B)^{\Delta(\pi)}},$

where $\Delta(\pi)$ is given by (1.7), if we have the freedom to remove any
thin set $Z$.

###  1.4. Summary of the paper

The main sparsity criterion for locally soluble fibres is Theorem 5.4. It is
proved using log geometry in Section 5 and leads to Chebotarev type conditions
about the splitting behaviour of primes. In Section 2 we shall collect
together some basic group-theoretic results that allow us to interpret the
output from Chebotarev’s theorem. Section 3 uses recent work of Arango-
Piñeros, Keliher and Keyes [1] to count pairs of power-full integers which lie
in the multiplicative span of Frobenian sets of primes. In Section 4 we shall
introduce the necessary background on orbifolds that is required to interpret
the exponent of $\log B$ in Conjecture 1.5. Section 6 contains the proof of
Theorems 1.2 and 1.3 and is based on an application of the large sieve.
Finally, Section 7 builds on the work in Section 3 and contains a range of
estimates for the modified counting function $N_{\text{loc},S}(\pi,B)$ in
specific examples.

### Acknowledgements

We are very grateful to Tim Santens for useful conversations. While working on
this paper the first author was supported by FWF grant P 36278.

##  2\. Group-theoretic results

We will need some preliminary results on the density of primes with a
prescribed splitting behaviour. Using Chebotarev’s theorem we will be able to
translate it into statements about groups and group actions. We begin by
proving some results in elementary group theory.

###  2.1. Group theory lemmas

Let $G$ be a finite group and let $H\subseteq G$ be a subgroup. For an element
$g\in G$ we will write $\operatorname{Fix}_{g}(G/H)$ for the set of fixed
points of $g$ under the natural action of $G$ on $G/H$.

######  Lemma 2.1.

Let $C\subseteq G$ be a conjugacy class. Then we have

$\sum_{g\in C}\\#\operatorname{Fix}_{g}(G/H)=\frac{\\#G}{\\#H}\\#(C\cap H).$

###### Proof.

First note that for conjugate elements $g,y\in C$ there is an element $u\in G$
such that $u^{-1}yu=g$. Hence

$\\{x\in G\colon x^{-1}gx=y\\}=\\{x\in
G\colon(ux)^{-1}y(ux)=y\\}=u^{-1}\operatorname{Stab}_{y},$

whose cardinality is $\\#G/\\#C$ by the orbit–stabiliser theorem, since $C$ is
the orbit of $y$ under conjugation. We now see that

$\displaystyle\sum_{g\in C}\\#\operatorname{Fix}_{g}(G/H)$
$\displaystyle=\\#\\{(g,xH)\in C\times G/H\colon gxH=xH\\}$
$\displaystyle=\\#\\{(g,xH)\in C\times G/H\colon x^{-1}gx\in H\\}.$

Hence

$\displaystyle\sum_{g\in C}\\#\operatorname{Fix}_{g}(G/H)$
$\displaystyle=\frac{1}{\\#H}\\#\\{(g,x)\in C\times G\colon x^{-1}gx\in H\cap
C\\}$ $\displaystyle=\frac{1}{\\#H}\\#\\{(g,x,y)\in C\times G\times(H\cap
C)\colon x^{-1}gx=y\\}$
$\displaystyle=\frac{1}{\\#H}\\#C\cdot\frac{\\#G}{\\#C}\cdot\\#(C\cap H),$

which proves the lemma. ∎

######  Lemma 2.2.

Let $S$ and $T$ be subgroups of $G$. Then

$\\#S\\#T=\\#(S\cap T)\\#(ST).$

###### Proof.

Consider the action $S\times T$ on $G$ by $(s,t)g=sgt^{-1}$. The stabiliser of
$e_{G}$ equals the image of diagonal map $S\cap T\hookrightarrow S\times T$
and the set $ST$ is the orbit of $e_{G}$. The result now follows from the
orbit–stabiliser formula. ∎

###  2.2. Density of primes

Let $F/\mathbb{Q}$ be a number field with ring of integers $\mathscr{O}_{F}$.
Define $\mathscr{P}_{F,m}$ to be the set of rational primes $p$ unramified in
$F$ which are divisible by exactly $m$ primes
$\mathfrak{p}_{i}\subseteq\mathscr{O}_{F}$ of degree $1$. Let

$\mathscr{P}_{F}=\bigcup_{m\geqslant 1}\mathscr{P}_{F,m}.$

We define

$\delta(E,K)=1-\sum_{m=1}^{d}m\operatorname{dens}\left(\mathscr{P}_{K,m}\cap\mathscr{P}_{E}\right),$

for any number fields $K,E\subseteq\overline{\mathbb{Q}}$ with
$d=[K\colon\mathbb{Q}]$. The main result of this section is the following
result.

######  Theorem 2.3.

Let $K,E\subseteq\overline{\mathbb{Q}}$ be two number fields with
$d=[K\colon\mathbb{Q}]$. Define

$\delta(E,K)=1-\sum_{m=1}^{d}m\operatorname{dens}\left(\mathscr{P}_{K,m}\cap\mathscr{P}_{E}\right).$

Let $L\subseteq\overline{\mathbb{Q}}$ be a Galois extension of $\mathbb{Q}$
which contains both $K$ and $E$. Then

$\delta(E,K)=1-\frac{\\#\\{\sigma\in{\rm Gal}(L/K)\colon\sigma\text{ fixes a
conjugate of }E\\}}{\\#{\rm Gal}(L/K)}.$

The quantity $\delta(E,K)$ generalises a quantity that is implicit in the work
of Loughran–Smeets [15, Eq. (1.4)]. Let $\pi:X\to\mathbb{P}^{1}$ be a standard
fibration and let $D$ be a closed point of $\mathbb{P}^{1}$ with residue field
$\kappa(D)$. Let $I_{D}(\pi)$ be the set of geometrically irreducible
components of $\pi^{-1}(D)$ of multiplicity one and let $E$ be the minimal
extension of $\kappa(D)$ over which the components of $I_{D}(\pi)$ are
defined. Then

$\delta_{D}(\pi)=1-\delta(E,\kappa(D))$

in [15, Eq. (1.4)]. Moreover, if we take $S_{D}$ to be the set of
geometrically irreducible components of $\pi^{-1}(D)$ of multiplicity $m_{D}$
and we let $E$ be the field of definition of the elements of $S_{D}$, then we
also have

$\delta_{D,N}(\pi)=1-\delta(E,N)$ (2.1)

in (1.5), for any number field $N/\mathbb{Q}$.

###### Proof of Theorem 2.3.

Write $G={\rm Gal}(L/\mathbb{Q})$ and let $K$ and $E$ be the fixed fields of
the subgroups $H_{1},H_{2}\subseteq G$. Then we have

$\mathscr{P}_{K,m}=\\{\text{primes }p\in\mathbb{Z}\text{ unramified in $L$ for
which }\\#\operatorname{Fix}_{\operatorname{Frob}_{p}}(G/H_{1})=m\\}$

and

$\mathscr{P}_{E}=\\{\text{primes }p\in\mathbb{Z}\text{ unramified in $L$ for
which }\\#\operatorname{Fix}_{\operatorname{Frob}_{p}}(G/H_{2})\geqslant
1\\}.$

Note that

$C_{m}=\\{g\in G\colon\\#\operatorname{Fix}_{g}(G/H_{1})=m\text{ and
}\\#\operatorname{Fix}_{g}(G/H_{2})\geqslant 1\\}$

is closed under conjugation, since conjugate elements have the same number of
fixed points. By Chebotarev’s theorem, in the form presented by Serre [19,
Thm. 3.4], for example, we therefore obtain

$\operatorname{dens}(\mathscr{P}_{K,m}\cap\mathscr{P}_{E})=\frac{\\#C_{m}}{\\#G}.$

Let $T=\bigcup_{t\in G}tH_{2}t^{-1}$, which we note is closed under
conjugation. Since $g\in G$ has at least a fixed point on $G/H_{2}$ if and
only if $g\in T$, we arrive at

$\displaystyle\sum_{m=1}^{d}m\operatorname{dens}\left(\mathscr{P}_{K,m}\cap\mathscr{P}_{E}\right)$
$\displaystyle=\frac{1}{\\#G}\sum_{m=1}^{d}m\\#C_{m}$
$\displaystyle=\frac{1}{\\#G}\sum_{g\in T}\\#\operatorname{Fix}_{g}(G/H_{1}).$

We may now conclude from Lemma 2.1 that

$\sum_{m=1}^{d}m\operatorname{dens}\left(\mathscr{P}_{K,m}\cap\mathscr{P}_{E}\right)=\frac{\\#(T\cap
H_{1})}{\\#H_{1}}.$ (2.2)

The statement of the theorem follows on noting that $H_{1}={\rm Gal}(L/K)$ and
$T=\\{\sigma\in G\colon\sigma\text{ fixes a conjugate of }E\\}$. ∎

Note that we could not have applied the Chebotarev Theorem to $\\#(T\cap
H_{1})$, since $T\cap H_{1}$ is not necessarily fixed under conjugation in
$G$. It is however closed under conjugation in $H_{1}$.

###  2.3. Computation of $\delta$ in specific cases

Theorem 2.3 allows us to compute the density $\delta(E,K)$ in the common
Galois closure $L$ of both $K$ and $E$. The following theorem says that this
can be reduced to a computation in a Galois closure of $E$.

######  Proposition 2.4.

Let $E^{\text{normal}}$ be the normal closure of $E$ in
$\overline{\mathbb{Q}}$. Then

$\delta(E,K)=\delta(E,E^{\text{normal}}\cap K).$

###### Proof.

We adopt the notation from the proof of Theorem 2.3. Let $H_{2}^{\\{j\\}}$ be
the conjugates of $H_{2}$ indexed by a set $J$. For a set $I\subseteq J$ we
write $H_{2}^{I}=\bigcap_{i\in I}H^{\\{i\\}}$. The field
$E^{\text{normal}}\cap K$ corresponds to the subgroup $\langle
H_{1},H_{2}^{J}\rangle\subseteq G$ generated by $H_{1}$ and $H_{2}^{J}$.
(Since $H_{2}^{J}$ is normal one can actually show that $\langle
H_{1},H_{2}^{J}\rangle=H_{1}H_{2}^{J}$.) It follows from Lemma 2.2 that

$\frac{\\#(S\cap H_{1})}{\\#H_{1}}=\frac{\\#(S\cap\langle
H_{1},H_{2}^{J}\rangle)}{\\#\langle H_{1},H_{2}^{J}\rangle},$

when $S$ is equal to $H_{2}^{I}$ for any $I\subseteq J$. Since both sides are
additive in $S$, the statement extends to $S=T=\bigcup_{j\in
J}H_{2}^{\\{j\\}}$ by the principle of inclusion and exclusion. ∎

###### Proof of Theorem 1.4.

Combine Proposition 2.4 with (2.1). ∎

Our remaining results summarise some special situations in which we can use
Theorem 2.3 and Proposition 2.4 to calculate the densities $\delta(E,K)$
easily.

######  Lemma 2.5.

If $E/\mathbb{Q}$ is Galois, then $\delta(E,K)=1-\frac{\deg(E\cap K)}{\deg
E}$.

###### Proof.

Since $E/\mathbb{Q}$ is Galois, $E$ is also Galois over $E^{\text{normal}}\cap
K=E\cap K$. Thus we conclude $\delta(E,K)=\delta(E,E\cap
K)=1-\frac{1}{[E\colon E\cap K]}$. ∎

######  Lemma 2.6.

If $E\subseteq K$ then $\delta(E,K)=0$.

###### Proof.

Since $K,E$ are the fixed fields of the subgroups $H_{1},H_{2}\subseteq{\rm
Gal}(L/\mathbb{Q})$, we have $E\subseteq K$ if and only if $H_{2}\supseteq
H_{1}$. But then $H_{1}\subseteq H_{2}\subseteq T=\bigcup_{t\in
G}tH_{2}t^{-1}$, whence $\tfrac{\\#(T\cap H_{1})}{\\#H_{1}}=1$ in (2.2). ∎

######  Lemma 2.7.

If $K/\mathbb{Q}$ is Galois and $KE=E^{\text{normal}}$, then
$\delta(E,K)=1-\frac{\deg(E\cap K)}{\deg E}.$

###### Proof.

Since $KE=E^{\text{normal}}$ and $K/\mathbb{Q}$ is Galois we have $H_{1}\cap
H_{2}^{\\{j\\}}=H_{2}^{J}$ for all $j\in J$. Thus

$\displaystyle\frac{\\#(T\cap
H_{1})}{\\#H_{1}}=\frac{\\#H_{2}^{J}}{\\#H_{1}}=\frac{\deg K}{\deg
E^{\text{normal}}}=\frac{\deg K}{\deg KE}$

in (2.2). Since $K$ is Galois we have $[KE\colon K]=[E\colon E\cap K]$, from
which the lemma follows. ∎

##  3\. Pairs of integers with Frobenian conditions

We say that a set $\mathscr{P}$ of rational primes is Frobenian if there is a
finite Galois extension $K/\mathbb{Q}$ and a union of conjugacy classes $H$ in
${\rm Gal}(K/\mathbb{Q})$ such that $\mathscr{P}$ is equal to the set of
primes $p$ that are unramified in $K$ and for which the Frobenius conjugacy
class of $p$ in ${\rm Gal}(K/\mathbb{Q})$ lies in $H$. In this section we
produce an asymptotic formula for the density of coprime integers
$a_{0},a_{1}$ which are both power-full and lie in the multiplicative span of
a Frobenian set of primes.

It will be convenient to introduce the notation

$c_{S}(\alpha)=\prod_{p\in S}\left(1-\frac{1}{p^{\alpha}}\right),$ (3.1)

for any $\alpha>0$ and any finite set of primes $S$. We shall prove the
following result.

######  Proposition 3.1.

For $i\in\\{0,1\\}$ let $m_{i}\in\mathbb{N}$ and let $\mathscr{P}_{i}$ be a
Frobenian set of rational primes of density $\partial_{i}$. Then, for any
finite set of primes $S$, we have

$\displaystyle\\#$
$\displaystyle\left\\{(a_{0},a_{1})\in\mathbb{Z}_{\mathrm{prim}}^{2}:|a_{i}|\leqslant
B,~{}p\not\in S\Rightarrow\big{[}m_{i}\mid v_{p}(a_{i})\text{ and }(p\mid
a_{i}\Rightarrow p\in\mathscr{P}_{i})\big{]}\right\\}$ $\displaystyle\hskip
113.81102pt\sim c_{m_{i},\mathscr{P}_{i},S}\frac{B^{1/m_{0}+1/m_{1}}}{(\log
B)^{2-\partial_{0}-\partial_{1}}},$

as $B\to\infty$, where

$\displaystyle c_{m_{i},\mathscr{P}_{i},S}=~{}$
$\displaystyle\frac{4m_{0}^{1-\partial_{0}}m_{1}^{1-\partial_{1}}}{\Gamma(\partial_{0})\Gamma(\partial_{1})}\cdot\frac{c_{S}(\frac{1}{m_{0}}+\frac{1}{m_{1}})}{c_{S}(\frac{1}{m_{0}})c_{S}(\frac{1}{m_{1}})}\prod_{\begin{subarray}{c}p\in\mathscr{P}_{0}\cap\mathscr{P}_{1}\\\
p\not\in S\end{subarray}}\left(1-\frac{1}{p^{2}}\right)$
$\displaystyle\quad\times\prod_{p\in\mathscr{P}_{0}\cap
S}\left(1-\frac{1}{p}\right)\prod_{p\in\mathscr{P}_{0}}\left(1-\frac{1}{p}\right)^{-1+\partial_{0}}\prod_{p\not\in\mathscr{P}_{0}}\left(1-\frac{1}{p}\right)^{\partial_{0}}$
$\displaystyle\quad\times\prod_{p\in\mathscr{P}_{1}\cap
S}\left(1-\frac{1}{p}\right)\prod_{p\in\mathscr{P}_{1}}\left(1-\frac{1}{p}\right)^{-1+\partial_{1}}\prod_{p\not\in\mathscr{P}_{1}}\left(1-\frac{1}{p}\right)^{\partial_{1}}.$

There are only $O(1)$ elements with $a_{0}a_{1}=0$ that contribute to the
counting function. Let $M(B)=M(m_{i},\mathscr{P}_{i},B,S)$ denote the overall
contribution with $a_{0}a_{1}\neq 0$. Hence, on accounting for signs, we have

$\displaystyle
M(B)=4\\#\left\\{(a_{0},a_{1})\in\mathbb{N}^{2}:\begin{array}[]{l}a_{0},a_{1}\leqslant
B,~{}\gcd(a_{0},a_{1})=1\\\ p\not\in S\Rightarrow\big{[}m_{i}\mid
v_{p}(a_{i})\text{ and }(p\mid a_{i}\Rightarrow
p\in\mathscr{P}_{i})\big{]}\end{array}\right\\}.$

For $(a_{0},a_{1})$ appearing in the counting function, we may clearly write

$a_{0}=b_{0}u_{0}^{m_{0}}\quad\text{ and }\quad a_{1}=b_{1}u_{1}^{m_{1}},$

where $p\mid b_{0}b_{1}\Rightarrow p\in S$, $\gcd(u_{0}u_{1},\prod_{p\in
S}p)=1$, and $p\mid u_{i}\Rightarrow p\in\mathscr{P}_{i}$. Moreover, we have
$\gcd(b_{0},b_{1})=\gcd(u_{0},u_{1})=1$. Let
$\mathscr{Q}=\mathscr{P}_{0}\cap\mathscr{P}_{1}$.

We proceed by introducing the counting functions

$M_{i}(x)=\\#\left\\{v\leqslant x:p\mid v\Rightarrow
p\in\mathscr{P}_{i,S}\right\\},$

for $i=0,1$, where
$\mathscr{P}_{i,S}=\mathscr{P}_{i}\setminus(S\cap\mathscr{P}_{i}).$ On using
the Möbius function to detect the condition $\gcd(u_{0},u_{1})=1$, we may now
write

$\displaystyle M(B)$
$\displaystyle=4\sum_{\begin{subarray}{c}b_{0},b_{1}\in\mathbb{N}\\\
\gcd(b_{0},b_{1})=1\\\ p\mid b_{0}b_{1}\Rightarrow p\in
S\end{subarray}}\sum_{\begin{subarray}{c}k\in\mathbb{N}\\\ p\mid k\Rightarrow
p\in\mathscr{Q}_{S}\end{subarray}}\mu(k)M_{0}\left(k^{-1}(B/b_{0})^{1/m_{0}}\right)M_{1}\left(k^{-1}(B/b_{1})^{1/m_{1}}\right),$

where $\mathscr{Q}_{S}=\mathscr{Q}\setminus(S\cap\mathscr{Q}).$ The treatment
of $M_{i}(x)$ is handled by the following result.

######  Lemma 3.2.

Let $i\in\\{0,1\\}$. Then

$M_{i}(x)\sim\frac{\kappa_{i,S}}{\Gamma(\partial_{i})}\frac{x}{(\log
x)^{1-\partial_{i}}},$

as $x\to\infty$, where

$\kappa_{i,S}=\prod_{p\in\mathscr{P}_{i}\cap
S}\left(1-\frac{1}{p}\right)\prod_{p\in\mathscr{P}_{i}}\left(1-\frac{1}{p}\right)^{-1+\partial_{i}}\prod_{p\not\in\mathscr{P}_{i}}\left(1-\frac{1}{p}\right)^{\partial_{i}}.$
(3.2)

###### Proof.

Let $i\in\\{0,1\\}$. There are several approaches to estimating $M_{i}(x)$,
but the one we shall adopt is via a general result of Wirsing [21] on mean
values of multiplicative arithmetic functions $g:\mathbb{N}\to[0,1]$. (In
fact, this result applies to general non-negative multiplicative arithmetic
functions under further assumptions on the behaviour of $g$ at prime powers.)
Suppose that

$\sum_{p\leqslant x}g(p)\log p\sim\tau x,$

for some $\tau>0$. Then it follows that

$\sum_{n\leqslant x}g(n)\sim\frac{e^{-\gamma\tau}}{\Gamma(\tau)}\frac{x}{\log
x}\prod_{p\leqslant
x}\left(1+\frac{g(p)}{p}+\frac{g(p^{2})}{p^{2}}+\dots\right),$

where $\gamma$ is Euler’s constant.

In our case we take

$g(n)=\begin{cases}1&\text{ if $p\mid n\Rightarrow p\in\mathscr{P}_{i,S}$,}\\\
0&\text{ otherwise.}\end{cases}$

Then, since $\mathscr{P}_{i}$ is a Frobenian set of primes of density
$\partial_{i}$, it follows from the Chebotarev density theorem that

$\displaystyle\sum_{p\leqslant x}g(p)\log p$
$\displaystyle=\sum_{\begin{subarray}{c}p\leqslant x\\\
p\in\mathscr{P}_{i,S}\end{subarray}}\log p\sim\partial_{i}\log x,$

as $x\to\infty$. Hence $\tau=\partial_{i}$ and we obtain

$M(x)\sim\frac{e^{-\gamma\partial_{i}}}{\Gamma(\partial_{i})}\frac{x}{\log
x}\prod_{\begin{subarray}{c}p\leqslant x\\\
p\in\mathscr{P}_{i,S}\end{subarray}}\left(1-\frac{1}{p}\right)^{-1},$

as $x\to\infty$. It remains to study

$\prod_{\begin{subarray}{c}p\leqslant x\\\
p\in\mathscr{P}_{i,S}\end{subarray}}\left(1-\frac{1}{p}\right)^{-1}=\prod_{p\in\mathscr{P}_{i}\cap
S}\left(1-\frac{1}{p}\right)\prod_{\begin{subarray}{c}p\leqslant x\\\
p\in\mathscr{P}_{i}\end{subarray}}\left(1-\frac{1}{p}\right)^{-1}.$

However, on appealing to recent work of Arango-Piñeros, Keliher and Keyes [1,
Thm. A], we quickly arrive at the expression

$\displaystyle\prod_{\begin{subarray}{c}p\leqslant x\\\
p\in\mathscr{P}_{i}\end{subarray}}\left(1-\frac{1}{p}\right)^{-1}$
$\displaystyle\sim\left(\frac{\log x}{e^{-\gamma_{K}}}\right)^{\partial_{i}},$

as $x\to\infty$, where

$e^{-\gamma_{K}}=e^{-\gamma}\prod_{p\in\mathscr{P}_{i}}\left(1-\frac{1}{p}\right)^{\partial_{i}^{-1}-1}\prod_{p\not\in\mathscr{P}_{i}}\left(1-\frac{1}{p}\right)^{-1}.$

It now follows that

$\prod_{\begin{subarray}{c}p\leqslant x\\\
p\in\mathscr{P}_{i,S}\end{subarray}}\left(1-\frac{1}{p}\right)^{-1}\sim\kappa_{i,S}(\log
x)^{\partial_{i}}e^{\gamma\partial_{i}},$

in the notation of lemma. Inserting this into our previous asymptotic formula
for $M_{i}(x)$, we finally arrive at the statement of the lemma. ∎

We clearly have

$\displaystyle\left(\log\left(k^{-1}(B/b_{i})^{1/m_{i}}\right)\right)^{-(1-\partial_{i})}$
$\displaystyle=m_{i}^{1-\partial_{i}}(\log
B)^{-(1-\partial_{i})}\left(1+O\left(\frac{\log kb_{i}}{\log
B}\right)\right),$

for $i=0,1$. Hence, on substituting Lemma 3.2 into our previous expression for
$M(B)$, we thereby obtain

$\displaystyle M(B)$
$\displaystyle=~{}4\sum_{\begin{subarray}{c}b_{0},b_{1}\in\mathbb{N}\\\
\gcd(b_{0},b_{1})=1\\\ p\mid b_{0}b_{1}\Rightarrow p\in
S\end{subarray}}\sum_{\begin{subarray}{c}k\in\mathbb{N}\\\ p\mid k\Rightarrow
p\in\mathscr{Q}_{S}\end{subarray}}A_{b_{0},b_{1},k}(B)+o\left(\frac{B^{1/m_{0}+1/m_{1}}}{(\log
B)^{2-\partial_{0}-\partial_{1}}}\right),$

with

$\displaystyle A_{b_{0},b_{1},k}(B)$
$\displaystyle=\frac{\kappa_{0,S}\kappa_{1,S}}{\Gamma(\partial_{0})\Gamma(\partial_{1})}\cdot\frac{\mu(k)m_{0}^{1-\partial_{0}}m_{1}^{1-\partial_{1}}\left(k^{-1}(B/b_{0})^{1/m_{0}}\right)\left(k^{-1}(B/b_{1})^{1/m_{1}}\right)}{(\log
B)^{2-\partial_{0}-\partial_{1}}}$
$\displaystyle=\frac{\kappa_{0,S}\kappa_{1,S}}{\Gamma(\partial_{0})\Gamma(\partial_{1})}\cdot
m_{0}^{1-\partial_{0}}m_{1}^{1-\partial_{1}}\cdot\frac{B^{1/m_{0}+1/m_{1}}}{(\log
B)^{2-\partial_{0}-\partial_{1}}}\cdot\frac{\mu(k)}{k^{2}}\cdot\frac{1}{b_{0}^{1/m_{0}}b_{1}^{1/m_{1}}},$

and where $\kappa_{0,S},\kappa_{1,S}$ are given by (3.2)

Next, on recalling the notation of (3.1), a simple calculation furnishes the
identities

$\sum_{\begin{subarray}{c}b_{0},b_{1}\in\mathbb{N}\\\ \gcd(b_{0},b_{1})=1\\\
p\mid b_{0}b_{1}\Rightarrow p\in
S\end{subarray}}\frac{1}{b_{0}^{1/m_{0}}b_{1}^{1/m_{1}}}=\frac{c_{S}(\frac{1}{m_{0}}+\frac{1}{m_{1}})}{c_{S}(\frac{1}{m_{0}})c_{S}(\frac{1}{m_{1}})}$

and

$\sum_{\begin{subarray}{c}k\in\mathbb{N}\\\ p\mid k\Rightarrow
p\in\mathscr{Q}_{S}\end{subarray}}\frac{\mu(k)}{k^{2}}=\prod_{\begin{subarray}{c}p\in\mathscr{P}_{0}\cap\mathscr{P}_{1}\\\
p\not\in S\end{subarray}}\left(1-\frac{1}{p^{2}}\right).$

Hence, it follows that the asymptotic formula in Proposition 3.1 holds with
the leading constant

$\displaystyle c_{m_{i},\mathscr{P}_{i},S}$
$\displaystyle=4\cdot\frac{\kappa_{0,S}\kappa_{1,S}}{\Gamma(\partial_{0})\Gamma(\partial_{1})}\cdot
m_{0}^{1-\partial_{0}}m_{1}^{1-\partial_{1}}\cdot\frac{c_{S}(\frac{1}{m_{0}}+\frac{1}{m_{1}})}{c_{S}(\frac{1}{m_{0}})c_{S}(\frac{1}{m_{1}})}\cdot\prod_{\begin{subarray}{c}p\in\mathscr{P}_{0}\cap\mathscr{P}_{1}\\\
p\not\in S\end{subarray}}\left(1-\frac{1}{p^{2}}\right),$

where $\kappa_{0,S},\kappa_{1,S}$ are given by (3.2). This therefore completes
the proof of Proposition 3.1.

##  4\. Orbifolds and étale orbifold morphisms

Campana related the study of fibrations $\pi\colon X\to Y$ of varieties over a
fixed field $k$ to orbifolds on the base [4]. He studied multiplicity
orbifolds, but since these are the only orbifolds in this paper we will simply
call them orbifolds. In this section we summarise the construction of the most
important invariant of orbifolds.

###  4.1. Orbifold pairs

Throughout this section let $k$ be an arbitrary field of characteristic $0$.

######  Definition 4.1.

An orbifold is a pair $(B,\Delta)$, where $B$ is a normal, proper $k$-scheme
and $\Delta$ is a $\mathbb{Q}$-divisor

$\Delta=\sum_{D}\left(1-\frac{1}{m_{D}}\right)[D]$

for positive integers $m_{D}$ associated to prime divisors $D$ on $B$. We call
$m_{D}$ the multiplicity of the orbifold over $D$.

######  Definition 4.2.

Let $(B,\Delta)$ be an orbifold on a normal and proper $k$-variety $B$. A
finite étale (orbifold) morphism is a morphism $\theta\colon C\to B$, with $C$
normal, which is

  1. (i)

finite,

  2. (ii)

étale away from $\Delta$,

  3. (iii)

has the property $e(D^{\prime}/D)\mid m_{D}$, for any prime divisor
$D^{\prime}\mid D$ (meaning any prime divisor $D^{\prime}\subset C$ above
$D\subset B$), where $e(D^{\prime}/D)$ is the ramification index.

Let us explain the use of the word étale. Consider a finite dominant morphism
$\theta\colon C\to B$ between integral, normal, proper $k$-varieties. Then we
can always endow $B$ with an orbifold structure such that $\theta$ becomes a
finite étale orbifold morphism, by assigning
$m_{D}=\mathrm{lcm}\\{e(D^{\prime}/D)\colon D^{\prime}\mid D\\}$. If $B$ has
an orbifold divisor $\Delta$ under which $\theta$ is a finite étale orbifold
morphism, then we can endow $C$ with the $\mathbb{Q}$-divisor

$\Delta_{C}=\sum_{D^{\prime}}\left(1-\frac{1}{m_{D^{\prime}}}\right)[D^{\prime}],\quad\text{where
}m_{D^{\prime}}=\frac{m_{D}}{e(D^{\prime}/D)}.$

This is the unique orbifold structure on $C$ such that the orbifold morphism
$(C,\Delta_{C})\to(B,\Delta)$ is étale in codimension 1, in the sense of [6,
Definition 2.21]. In the latter case, the Riemann–Hurwitz formula yields

$K_{C,\Delta_{C}}=\theta^{*}K_{B,\Delta},$

where $K_{B,\Delta}=K_{B}+\Delta$ is the canonical divisor class on an
orbifold $(B,\Delta)$. (This statement can be proven along similar lines to
the proof of Proposition 4.7(c).)

######  Proposition 4.3.

Let $C_{1},C_{2}\to C$ be morphisms of normal $k$-varieties. Let
$V=\widetilde{C_{1}\times_{C}C_{2}}$ be the normalisation of the product
$C_{1}\times_{C}C_{2}$.

$C_{1}\times_{C}C_{2}$$V$$C_{2}$$C_{1}$$C$

Let $D_{V}\subset V$ be a prime divisor lying above prime divisors
$D_{i}\subset C_{i}$ and $D\subset C$. Then

$e(D_{V}/D_{1})=\frac{e_{2}}{\gcd(e_{1},e_{2})},$

where $e_{i}=e(D_{i}/D)$ for $i=1,2$.

###### Proof.

Replacing the prime divisors with their generic points we can compute the
normalisation étale locally over $D$. Hence we consider the normalisation of
the tensor product of the two homomorphisms $\varrho_{i}\colon k[\\![t]\\!]\to
k[\\![t_{i}]\\!]$ given by $t\mapsto t_{i}^{e_{i}}$. The tensor product is
$R=k[\\![t_{1},t_{2}]\\!]/(t_{1}^{e_{1}}-t_{2}^{e_{2}})$ generated by the
images of the $t_{i}$. Let us define $d=\gcd(e_{1},e_{2})$. We can write $R$
as the product

$R=\prod_{\stackrel{{\scriptstyle p\mid
X^{d}-1}}{{\text{irr.}}}}k[\\![t_{1},t_{2},\zeta]\\!]/(p(\zeta),t_{1}^{e_{1}/d}-\zeta
t_{2}^{e_{2}/d}).$

All factors are principal ideal domains, since polynomials $X^{a}-\lambda
Y^{b}$ with $\lambda\in k^{\times}$ and $\gcd(a,b)=1$, are even irreducible
over an algebraic closure $\overline{k}$. We will compute the integral closure
of each component separately. Let us write $\alpha_{1}e_{1}+\alpha_{2}e_{2}=d$
for $\alpha_{i}\in\mathbb{Z}$. Then $T=t_{1}^{\alpha_{2}}t_{2}^{\alpha_{1}}$
is integral in each factor, since
$T^{e_{1}/d}=t_{2}\cdot(t_{1}^{e_{1}/d}/t_{2}^{e_{2}/d})^{\alpha_{2}}$ and
$T^{e_{2}/d}=t_{1}\cdot(t_{2}^{e_{2}/d}/t_{1}^{e_{1}/d})^{\alpha_{1}}$. It
follows that

$k[\\![t_{1},t_{2},\zeta]\\!]/(p(\zeta),t_{1}^{e_{1}/d}-\zeta
t_{2}^{e_{2}/d})\hookrightarrow k[\\![T,\zeta]\\!]/(p(\zeta))$

is the integral closure. Finally, to compute $e(D_{V}/D_{1})$ we look at the
image of $t_{1}$ under the map

$k[\\![t_{1}]\\!]\to k[\\![T,\alpha]\\!]/(p(\alpha)),$

which has valuation $e_{2}/d$. ∎

######  Remark 4.4.

In [6, Definition 11.1], Campana defines the orbifold fundamental group
$\pi_{1}(X|\Delta)$ for a complex orbifold $(X|\Delta)$ and relates it to
covers unramified away from $\Delta$. Likewise, we can define the (algebraic)
orbifold fundamental group and relate it to the structure of all finite étale
orbifold morphisms over a fixed base $(B,\Delta)$ of dimension $1$. (Note that
we could do this in arbitrary dimension, if we allow finite étale morphisms to
be defined away from a codimension $2$ locus.) Consider the category
$\operatorname{\mathbf{FEt}}_{(B,\Delta)}$ of all finite étale orbifold
morphisms to $(B,\Delta)$, where the morphisms are given by $B$-morphisms.
Given a point $\overline{x}\in
B(\overline{k})\setminus\operatorname{supp}(\Delta)$ we have the fibre functor

$F\colon\operatorname{\mathbf{FEt}}_{(B,\Delta)}\to\operatorname{\mathbf{Sets}}$

given by $C\mapsto C_{\overline{x}}$, and one can show that
$(\operatorname{\mathbf{FEt}}_{(B,\Delta)},F)$ is a Galois category. The only
non-trivial part is to show that $\operatorname{\mathbf{FEt}}_{(B,\Delta)}$
has products, but this follows from Proposition 4.3. In particular, this
implies that for any two finite étale covers of $(C,\partial)$, there is
another cover mapping to both. We define the (algebraic) orbifold fundamental
group $\pi^{\text{orb}}_{1}(B,\Delta)$ to be the automorphism group of the
fibre functor $F$. Many relations between the topological and algebraic
fundamental group can be directly translated to fundamental groups of
orbifolds. For example, if $k\subset\mathbb{C}$ then

$\pi^{\text{orb}}_{1}(B,\Delta)=\widehat{\pi_{1}(B(\mathbb{C})|\Delta)}.$

Campana studied the complex orbifold fundamental group in [6, Sections 11 and
12] and has several results and conjectures about their structure.

For our application we will need the following definition.

######  Definition 4.5.

Let $G/k$ be a finite étale group and $(B,\Delta)$ an orbifold. Let
$\theta\colon C\to B$ be a finite étale orbifold morphism endowed with a
$G$-action on $C$, which is compatible with $\theta$. We say that $\theta$ is
a $G$-torsor (of orbifolds) if the restriction of $\theta$ away from the
support of $\Delta$ is a $G$-torsor.

Since we are dealing with curves, it makes sense to talk about torsors. The
natural morphism $G\times_{B}C\to C\times_{B}C$ is not necessarily an
isomorphism over $B$, but it is so over $B\setminus\Delta$ by definition.
Since $G\times_{B}C$ is a smooth curve over $k$, this morphism factors through
the normalisation $G\times_{B}C\to\widetilde{C\times_{B}C}\to C\times_{B}C$.
Now $G\times_{B}C\to\widetilde{C\times_{B}C}$ is morphism between normal
curves, which is an isomorphism on a dense open subset. Note that this agrees
with the observation that $\widetilde{C\times_{B}C}\to C$ is unramified by
Proposition 4.3; $\widetilde{C\times_{B}C}$ is just a union of copies of $C$.

Categorically, the product of two normal covers of $B$ is the normalisation of
the usual fibre product, which means that a $G$-torsor of orbifolds is indeed
a torsor.

###  4.2. Orbifold base of a fibration

As we saw in Section 1, we can associate a natural orbifold to any fibration.
In this section we discuss this further, before passing to our reasoning
behind Conjecture 1.5.

######  Definition 4.6.

Consider a fibration $\pi\colon X\to Y$, which we assume is a morphism between
integral, normal, proper $k$-schemes such that the generic fibre is
geometrically irreducible. For a prime divisor $D\subset Y$ with generic point
$\eta_{D}$ we define $m_{D}$ as the minimum multiplicity of the components of
$X_{\eta_{D}}$ as a divisor on $X$. The orbifold base of $\pi$ is
$(Y,\partial_{\pi})$ where

$\partial_{\pi}=\sum_{D}\left(1-\frac{1}{m_{D}}\right)[D].$

Possibly up to thin sets, we expect the geometry of the base orbifold
$(Y,\partial_{\pi})$ to govern the arithmetic properties of the fibration. We
henceforth focus our attention on standard fibrations $\pi:X\to\mathbb{P}^{1}$
defined over $\mathbb{Q}$, with the aim of interpreting the growth of the
counting function $N_{\text{loc}}(\pi,B)$ that was defined in (1.1).
Occasionally we will write $N^{\circ}_{\text{loc}}(\pi,B)$ for the same
counting function, but excluding the finitely many points in the orbifold
divisors $\partial_{\pi}$.

Let us begin by discussing the conjectured power of $B$ in Conjecture 1.5,
which is equal to

$2-\deg\partial_{\pi}=-\deg(K_{\mathbb{P}^{1},\partial_{\pi}}),$ (4.1)

where $K_{\mathbb{P}^{1},\partial_{\pi}}=K_{\mathbb{P}^{1}}+\partial_{\pi}$.
The following result relates the geometry of $\pi$ to the geometry of a
normalisation of the fibre product of $\pi$ with a finite cover.

######  Proposition 4.7.

Let $\pi:X\to\mathbb{P}^{1}$ be a standard fibration and let

$\theta\colon\mathbb{P}^{1}\to\mathbb{P}^{1}$

be a (possibly ramified) finite cover of degree $d$. We define
$\pi_{\theta}:X_{\theta}\to\mathbb{P}^{1}$ to be the normalisation of the
fibre product of $\theta$ and $\pi$. Then we have the following properties.

  1. (a)

$\pi_{\theta}\colon X_{\theta}\to\mathbb{P}^{1}$ is a standard fibration.

  2. (b)

The orbifold multiplicities $m_{P^{\prime}}$ for $\pi_{\theta}$ satisfy

$m_{P^{\prime}}\geqslant\frac{m_{P}}{e(P^{\prime}/P)}$

for any prime divisor $P^{\prime}$ of $\mathbb{P}^{1}$, where
$P=\theta(P^{\prime})$. We have equality precisely when condition (iii) in
Definition 4.2 is satisfied at $P^{\prime}$.

  3. (c)

We have

$\deg(K_{\mathbb{P}^{1},\partial_{\pi_{\theta}}})\geqslant
d\deg(K_{\mathbb{P}^{1},\partial_{\pi}}),$

with equality precisely when $\theta$ is a finite étale orbifold morphism.

###### Proof.

  1. (a)

This is clear from the definition.

  2. (b)

Consider a component $Z^{\prime}$ of the fibre of $\pi_{\theta}$ over a prime
divisor $P^{\prime}$ of $\mathbb{P}^{1}$. Suppose that $Z^{\prime}$ lies over
$Z\subseteq X$ and $P^{\prime}$ lies over $P$. Let
$m_{P^{\prime}}(Z^{\prime})$ and $m_{P}(Z)$ denote the multiplicities of these
components in their resepective fibres. We wish to apply Proposition 4.3 with
$C_{1}\to C$ being the morphism $\theta:\mathbb{P}^{1}\to\mathbb{P}^{1}$ and
$C_{2}\to C$ being the morphism $\pi:X\to\mathbb{P}^{1}$. Then $V\to C_{1}$ is
the morphism $\pi_{\theta}:X_{\theta}\to\mathbb{P}^{1}$. It follows that

$e(Z^{\prime}/P^{\prime})=\frac{e(Z/P)}{\gcd\left(e(Z/P),e(P^{\prime}/P)\right)}.$

Hence, since the ramification indices over a codimension one point are
precisely the multiplicities of the different components of the fibre, we
obtain

$m_{P^{\prime}}(Z^{\prime})=\frac{m_{P}(Z)}{\gcd\\!\left(m_{P}(Z),e(P^{\prime}/P)\right)}.$

Since $m_{P}(Z)\geqslant m_{P}$ and
$\gcd\\!\left(m_{P}(Z),e(P^{\prime}/P)\right)\leqslant e(P^{\prime}/P)$ we
conclude $m_{P^{\prime}}(Z^{\prime})\geqslant\frac{m_{P}}{e(P^{\prime}/P)}$
for all components $Z^{\prime}$ in the fibre over $P^{\prime}$.

Clearly, if $m_{P^{\prime}}=\frac{m_{P}}{e(P^{\prime}/P)}$ we have
$e(P^{\prime}/P)\mid m_{P}$. Now suppose that $e(P^{\prime}/P)\mid m_{P}$. To
prove the statement we must show that there is a component $Z^{\prime}$ over
$P^{\prime}$ with $m_{P}(Z^{\prime})=\frac{m_{P}}{e(P^{\prime}/P)}$. By the
definition of $m_{P}$ there exists a component $Z$ over $P$ with
$m_{P}=m_{P}(Z)$. Now let $Z^{\prime}$ be any component over $P^{\prime}$
which lies over $P$. Then

$m_{P^{\prime}}(Z^{\prime})=\frac{m_{P}(Z)}{\gcd\\!\left(m_{P}(Z),e(P^{\prime}/P)\right)}=\frac{m_{P}}{\gcd\\!\left(m_{P},e(P^{\prime}/P)\right)}=\frac{m_{P}}{e(P^{\prime}/P)}.$

This concludes the proof of part (b).

  3. (c)

We will prove the result for orbifolds equipped with a degree $d$ morphism
$(C^{\prime},\partial^{\prime})\to(C,\partial)$, for general smooth curves $C$
and $C^{\prime}$, in order to distinguish between the two copies of
$\mathbb{P}^{1}$. The statement is invariant under base change, so we can
assume we are working over an algebraically closed field $k=\overline{k}$. We
begin by noting that

$\deg K_{C,\partial}=2g(C)-2+\sum_{P\in
C^{(1)}}\left(1-\frac{1}{m_{P}}\right)$

and

$\deg
K_{C^{\prime},\partial^{\prime}}=2g(C^{\prime})-2+\sum_{P^{\prime}\in{C^{\prime}}^{(1)}}\left(1-\frac{1}{m_{P^{\prime}}}\right).$

The Riemann–Hurwitz formula yields

$2g(C^{\prime})-2=d(2g(C)-2)+\sum_{P^{\prime}\in
C^{\prime(1)}}\left(e(P^{\prime}/P)-1\right),$

where $P=\theta(P^{\prime})$. Hence

$\deg K_{C^{\prime},\partial^{\prime}}=d(2g(C)-2)+\sum_{P^{\prime}\in
C^{\prime(1)}}\left(e(P^{\prime}/P)-\frac{1}{m_{P^{\prime}}}\right).$

It now follows that

$\displaystyle\deg K_{C^{\prime},\partial^{\prime}}$ $\displaystyle-d\deg
K_{C,\partial}$
$\displaystyle=\sum_{P^{\prime}\in{C^{\prime}}^{(1)}}\left(e(P^{\prime}/P)-\frac{1}{m_{P^{\prime}}}\right)-d\sum_{P\in
C^{(1)}}\left(1-\frac{1}{m_{P}}\right)$ $\displaystyle=\sum_{P\in
C^{(1)}}\left[\left(\sum_{P^{\prime}\mid
P}e(P^{\prime}/P)-d\right)+\left(\frac{d}{m_{P}}-\sum_{P^{\prime}\mid
P}\frac{1}{m_{P^{\prime}}}\right)\right].$

Using $\sum_{P^{\prime}\mid P}e(P^{\prime}/P)=d$ we see that the first terms
all vanish and so

$\deg K_{C^{\prime},\partial^{\prime}}-d\deg K_{C,\partial}=\sum_{P\in
C^{(1)}}\sum_{P^{\prime}\mid
P}\left(\frac{e(P^{\prime}/P)}{m_{P}}-\frac{1}{m_{P^{\prime}}}\right).$

This is clearly non-negative by (b), and we have equality if and only if
condition (iii) of Definition 4.2 is satisfied at all $P^{\prime}$. ∎

In the setting of this result, it follows that the points in
$N_{\text{loc}}(\pi,B)$ that are counted by
$N_{\text{loc}}(\pi_{\theta},H_{\theta},B)$ are expected to contribute at most
to the same order of $B$, where $H_{\theta}$ is the pullback height along
$\theta$. Indeed, in Conjecture 1.1 we have

$N_{\text{loc}}(\pi_{\theta},H_{\theta},B)=O_{\varepsilon}\left((B^{\frac{1}{d}})^{\deg(-K_{\mathbb{P}^{1},\partial_{\pi_{\theta}}})+\varepsilon}\right),$

for any $\varepsilon>0$, where we use $B^{1/d}$ since $H_{\theta}$ is an
$\mathscr{O}(d)$-height on $\mathbb{P}^{1}$. Hence, in the light of
Proposition 4.7(c), we should expect no higher order contribution from
$N_{\text{loc}}(\pi_{\theta},H_{\theta},B)$ to $N_{\text{loc}}(\pi,B)$.
Moreover, we should obtain the same exponent of $B$ when $\theta$ is a finite
étale orbifold morphism.

We are now ready to address the possible power of $\log B$. Let $\pi\colon
X\to\mathbb{P}^{1}$ be a standard fibration and suppose that
$\theta\colon\mathbb{P}^{1}\to\mathbb{P}^{1}$ is a $G$-torsor of orbifolds
under a finite étale group scheme $G$ of degree $d$, as presented in
Definition 4.5. We write $\theta_{v}\colon\mathbb{P}^{1}\to\mathbb{P}^{1}$ for
the twists of $\theta$ by $v\in\textup{H}^{1}({\rm
Gal}(\overline{\mathbb{Q}}/\mathbb{Q}),G)$. Finally, we shall write
$\pi_{v}\colon X_{v}\to\mathbb{P}^{1}$ for the normalisation of the pullback
of $\pi$ along $\theta_{v}$. We are now ready to compare the counting function
$N_{\text{loc}}^{\circ}(\pi,B)$ with the counting functions
$N_{\text{loc}}^{\circ}(\pi_{v},H_{v},B)$, for various
$v\in\textup{H}^{1}({\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q}),G)$, where
$H_{v}$ is the pullback height along $\theta_{v}$. (Note that this is a
$\mathscr{O}(d)$-height on the codomain $\mathbb{P}^{1}$ of $\theta_{v}$.)

######  Proposition 4.8.

In the setting above we have the following.

  1. (a)

A point $x\in\mathbb{P}^{1}(\mathbb{Q})$ is counted by
$N^{\circ}_{\text{loc}}(\pi,B)$ if and only if there exists
$v\in\textup{H}^{1}({\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q}),G)$ and
$y\in\mathbb{P}^{1}(\mathbb{Q})$ such that $\theta_{v}(y)=x$, and such that
$y$ is counted by $N^{\circ}_{\text{loc}}(\pi_{v},H_{v},B)$.

  2. (b)

We have

$N^{\circ}_{\text{loc}}(\pi,B)=\frac{1}{\\#G(\mathbb{Q})}\sum_{v\in\textup{H}^{1}({\rm
Gal}(\overline{\mathbb{Q}}/\mathbb{Q}),G)}N^{\circ}_{\text{loc}}(\pi_{v},H_{v},B).$

  3. (c)

Let $\theta_{v}^{-1}(D)=\bigcup_{1\leqslant i\leqslant s_{D}}E^{(i)}_{v}$ be a
decomposition into irreducible components, and write
$N^{(i)}_{v}=\kappa(E^{(i)}_{v})$ for their function fields. Then

$\Delta(\pi_{v})=\sum_{D\in(\mathbb{P}^{1})^{(1)}}\sum_{i=1}^{s_{D}}\left(1-\delta_{D,N_{D,v}^{(i)}}(\pi)\right),$

where $\delta_{D,N_{D,v}^{(i)}}$ is given by (1.5).

  4. (d)

The expression $\Delta(\pi_{v})$ only assumes finitely many values.

###### Proof.

  1. (a)

Let $U\subseteq\mathbb{P}^{1}$ be the image of the étale locus of $\theta$.
The restrictions $\theta_{v}\colon U_{v}\to U$ are $G$-torsors, and so we have
a partition

$U(\mathbb{Q})=\bigsqcup_{v\in\textup{H}^{1}({\rm
Gal}(\overline{\mathbb{Q}}/\mathbb{Q}),G)}\theta_{v}\left(U_{v}(\mathbb{Q})\right).$

Furthermore, the fibre of $\pi_{v}$ over $y\in U_{v}(\mathbb{Q})$ is
isomorphic to the fibre of $\pi$ over $x=\theta_{v}(y)$. Hence one of these
fibres is locally soluble precisely when the other is. Finally, since
$\theta\colon\mathbb{P}^{1}\to\mathbb{P}^{1}$ has degree $d$, the pullback of
the $\mathscr{O}(1)$-height pulls back to a $\mathscr{O}(d)$-height.

  2. (b)

This follows from the partition in (a), and the fact that each fibre has
$\\#G(\mathbb{Q})$ points.

  3. (c)

This directly follows from the definition of $\delta_{D,N}$ and $\pi_{v}$.

  4. (d)

This follows from Theorem 1.4. ∎

In the setting of Theorem 1.3 we consider $\mu_{d}$-covers parametrised by
$\mathbb{Q}^{\times}/\mathbb{Q}^{\times,d}$. The following result therefore
follows from part (c) of Proposition 4.8.

######  Corollary 4.9.

We have $\Delta(\pi_{v})=\Theta_{v}(\pi)$ in (1.9).

In principle there might be infinitely many twists $\pi_{v}$ for which
$\Delta(\pi_{v})$ differs from the expected exponent $\Delta(\pi)$ defined in
(1.7). The following example illustrates an instance where the points counted
by the covers for which $\Delta(\pi_{v})=\Delta(\pi)$ can form a non-trivial
cothin set in $\mathbb{P}^{1}(\mathbb{Q})$.

######  Example 4.10.

Consider the fibration $\pi\colon X\to\mathbb{P}^{1}$ with three double fibres
over $0$, $-1$ and $\infty$, together with precisely one other non-split fibre
over $1$, which has multiplicity one and is split by a quadratic extension
$K/\mathbb{Q}$. Let $C_{v}$ be the conic

$v_{1}x_{1}^{2}+v_{2}x_{2}^{2}=x_{0}^{2}$

in $\mathbb{P}^{2}$ defined by
$v=(v_{1},v_{2})\in\mathbb{Q}^{\times}/\mathbb{Q}^{\times,2}\times\mathbb{Q}^{\times}/\mathbb{Q}^{\times,2}$.
We apply the partition in part (b) of Proposition 4.8 with the full family of
twists

$\theta_{v}:C_{v}\to\mathbb{P}^{1},\quad[x_{0}\colon x_{1}\colon
x_{2}]\mapsto[v_{1}x_{0}^{2}\colon v_{2}x_{1}^{2}].$

This is the finest partition in the sense of Remark 4.4, since we have
$\pi_{1}^{\text{orb}}(\mathbb{P}^{1},\partial_{\pi})=\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$,
and any $\theta_{v}$ is geometrically a universal orbifold cover. Consider the
fibres $\theta_{v}^{-1}(1)$ as $v$ varies, which on algebras are biquadratic
étale $\mathbb{Q}$-algebras $\prod_{i}N^{(i)}_{1,v}$. Infinitely many of these
contain the splitting field $K$ of the fibre and for such $v$ we have

$1-\delta_{1,\mathbb{Q}}<1=1-\delta_{1,N_{1,v}^{(\alpha)}}(\pi)<\sum_{i}\left(1-\delta_{1,N_{1,v}^{(i)}}(\pi)\right),$

where $\alpha$ is such that $K\subseteq N^{(\alpha)}_{1,v}$. However, each of
these infinitely many
$(\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z})$-covers factors through
only two $\mathbb{Z}/2\mathbb{Z}$-covers. Hence, the set of points counted
through the $v$ for which

$1-\delta_{1,\mathbb{Q}}\neq\sum_{i}\left(1-\delta_{1,N_{1,v}^{(i)}}(\pi)\right),$

is a thin set. In the case of a non-trivial Galois action on the components of
the multiple fibres, we will need to deal with them in a similar manner to
conclude that the points counted in the covers $\theta_{v}$ for
$\Delta(\pi_{v})\neq\Delta(\pi)$ form a thin set.

##  5\. A sparsity criterion

Let $k$ be a number field. Let $X$ and $Y$ be smooth, proper varieties over
$k$, and let $D$ and $E$ be strict normal crossings divisors on $X$ and $Y$
respectively, where $f^{-1}(E)\subseteq D$. Assume that the induced morphism
$f:(X,D)\to(Y,E)$ is a _toroidal_ morphism; i.e., a toroidal morphism between
toroidal embeddings, or equivalently, a log smooth morphism of (Zariski) log
regular schemes. Fix $Q\in Y(k)$. We want to understand when $f^{-1}(Q)$ is
everywhere locally soluble.

Let $S$ be a finite set of places including all places of bad reduction for
$f$. This means that we have a good model
$\overline{f}:(\mathscr{X},\mathscr{D})\to(\mathscr{Y},\mathscr{E})$ for $f$
over $\mathscr{O}_{k,S}$ with the property that
$\overline{f}^{-1}(\mathscr{E})\subseteq\mathscr{D}$, such that
$(\mathscr{X},\mathscr{D})$ and $(\mathscr{Y},\mathscr{E})$ are still log
regular, and such that $\overline{f}$ is still log smooth with respect to the
divisorial log structures induced by $\mathscr{D}$ and $\mathscr{E}$.

Let $v\not\in S$ be a finite place of $k$. Let
$\mathscr{Q}_{v}\in\mathscr{Y}(\mathscr{O}_{v})$ be the unique lift of $Q\in
Y(k)$ to an $\mathscr{O}_{v}$-point. We will give necessary and sufficient
conditions for the existence of an $\mathscr{O}_{v}$-point $\mathscr{P}_{v}$
on $\mathscr{X}$ such that $f(\mathscr{P})_{v}=\mathscr{Q}_{v}$, for $v\gg 0$.

If $\mathscr{Q}_{v}\not\subseteq\mathscr{E}$, then the $\mathscr{O}_{v}$-point
$\mathscr{Q}_{v}$ can be seen as a morphism

$\mathscr{Q}_{v}\colon(\mathrm{Spec}\,\mathscr{O}_{v})^{\dagger}\to(\mathscr{Y},\mathscr{E})$

of log schemes, where $(\mathrm{Spec}\,\mathscr{O}_{v})^{\dagger}$ is the
scheme $\mathrm{Spec}\,\mathscr{O}_{v}$ equipped with the divisorial log
structure induced by the closed point. This morphism induces a morphism of
associated _Kato fans_

$F(\mathscr{Q}_{v})\colon\mathrm{Spec}\,\mathbb{N}\cong
F((\mathrm{Spec}\,\mathscr{O}_{v})^{\dagger})\to F(\mathscr{Y},\mathscr{E}).$

In other words, we get an $\mathbb{N}$-valued point $F(\mathscr{Q}_{v})\in
F(\mathscr{Y},\mathscr{E})(\mathbb{N})$.

If $\mathscr{Q}_{v}$ is the image of
$\mathscr{P}_{v}\in\mathscr{X}(\mathscr{O}_{v})$, then clearly
$F(\mathscr{Q}_{v})$ cannot lie anywhere in
$F(\mathscr{Y},\mathscr{E})(\mathbb{N})$; it needs to be an element of the
potentially smaller set

$\mathrm{image}\left(F(\mathscr{X},\mathscr{D})(\mathbb{N})\to
F(\mathscr{Y},\mathscr{E})(\mathbb{N})\right).$

This means that if $F(\mathscr{Q}_{v})$ does _not_ lie in the image of
$F(\mathscr{X},\mathscr{D})(\mathbb{N})$, then surely $\mathscr{Q}_{v}$ cannot
lift to a $\mathscr{O}_{v}$-point on $\mathscr{X}$. This is a sparsity
criterion in the sense of [15, §2], but still a rather naïve one, since it
does not take important arithmetic information into account.

######  Definition 5.1.

Let $\overline{\mathscr{P}}_{v}$ be an $\mathbb{F}_{p}$-point on
$\mathscr{X}$. With the notation above, we define
$F(\mathscr{X},\mathscr{D})(\mathbb{N})_{\overline{\mathscr{P}}_{v}}$ as the
subset of $F(\mathscr{X},\mathscr{D})(\mathbb{N})$ with the property that
$\overline{\mathscr{P}}_{v}$ lies is the logarithmic stratum associated to the
image of the closed point $\mathbb{N}_{>0}$ of $\mathrm{Spec}\,\mathbb{N}$.

######  Proposition 5.2.

With notation as above, let $\overline{\mathscr{P}}_{v}$ be an
$\mathbb{F}_{p}$-point on $\mathscr{X}_{\mathscr{Q}_{v}}$ and assume that
$F(\mathscr{Q}_{v})$ does _not_ lie in

$\mathrm{image}\left(F(\mathscr{X},\mathscr{D})(\mathbb{N})_{\overline{\mathscr{P}}_{v}}\to
F(\mathscr{Y},\mathscr{E})(\mathbb{N})\right).$

Then
$\overline{\mathscr{P}}_{v}\in\mathscr{X}_{\mathscr{Q}_{v}}(\mathbb{F}_{p})$
does not lift to
$\mathscr{P}_{v}\in\mathscr{X}_{\mathscr{Q}_{v}}(\mathscr{O}_{v})$.

###### Proof.

Assume that $\overline{\mathscr{P}}_{v}$ lifts, i.e.,
$\mathscr{Q}_{v}=\overline{f}(\mathscr{P}_{v})$ for some
$\mathscr{P}_{v}\in\mathscr{X}(\mathscr{O}_{v})$ with
$\overline{\mathscr{P}}_{v}=\mathscr{P}_{v}\mod{v}$ (which is the image of
$\mathrm{Spec}\,\mathbb{F}_{v}$ under $\mathscr{P}_{v}$). Therefore the image
of $F(\mathscr{P}_{v})\in F(\mathscr{X},\mathscr{D})(\mathbb{N})$ under the
map $F(\mathscr{X},\mathscr{D})(\mathbb{N})\to
F(\mathscr{Y},\mathscr{E})(\mathbb{N})$ comes from
$F(\mathscr{X},\mathscr{D})(\mathbb{N})_{\overline{\mathscr{P}}_{v}}$, as
desired. ∎

In fact, the logarithmic Hensel lemma [14, Proposition 5.13] yields more, as
in the following result.

######  Proposition 5.3.

If $\overline{\mathscr{P}}_{v}$ is an $\mathbb{F}_{v}$-point on
$\mathscr{X}_{\mathscr{Q}_{v}}$, the following are equivalent:

  * (a)

$\overline{\mathscr{P}}_{v}$ lifts to an $\mathscr{O}_{v}$-point on
$\mathscr{X}_{\mathscr{Q}_{v}}$;

  * (b)

$F(\mathscr{Q}_{v})\in\mathrm{image}\left(F(\mathscr{X},\mathscr{D})(\mathbb{N})_{\overline{\mathscr{P}}_{v}}\to
F(\mathscr{Y},\mathscr{E})(\mathbb{N})\right)$.

###### Proof.

Since we have already shown that (a) implies (b), it remains to prove the
reverse implication. This is an application of [14, Proposition 5.13]. Indeed,
let $s^{\dagger}=\mathrm{Spec}\,\mathbb{F}_{v}$ with the standard log
structure of rank $1$, and $S^{\dagger}=\mathrm{Spec}\,\mathscr{O}_{v}$. Let
$j:s^{\dagger}\to S^{\dagger}$ be the canonical closed immersion.

By assumption there is an element $p_{v}\in
F(\mathscr{X},\mathscr{D})(\mathbb{N})_{\overline{\mathscr{P}}_{v}}$ which
maps to $F(\mathscr{Q}_{v})\in F(\mathscr{Y},\mathscr{E})(\mathbb{N})$, and
there is an $\mathbb{F}_{v}$-point $u\colon\mathrm{Spec}\ \mathbb{F}_{v}\to X$
on the associated stratum of $(\mathscr{X},\mathscr{D})$. We can uniquely make
$u$ into a morphism of log schemes $s^{\dagger}\to(\mathscr{X},\mathscr{D})$
such that $F(u)=p_{v}$ under the identification
$F(\mathbb{N})=F(s^{\dagger})$, similar to the proof of Proposition 6.1 in
[14].

Since $F(f)$ maps $F(u)$ to $F(\mathscr{Q}_{v})$ we have a commutative
diagram:

$s^{\dagger}$$S^{\dagger}$$(\mathscr{X},\mathscr{D})$$(\mathscr{Y},\mathscr{E})$$u$$\mathscr{Q}_{v}$$j$$\overline{f}$

Now [14, Proposition 5.13] provides a lift
$S^{\dagger}\to(\mathscr{X},\mathscr{D})$ of $\mathscr{Q}_{v}$. The morphism
of schemes which underlies this lift is the $\mathscr{O}_{v}$-point
$\mathscr{P}_{v}$ we are looking for. ∎

Using this statement we can give precise conditions for locally solubility. We
allow ourselves to work over a general number field $k/\mathbb{Q}$ and so
define a standard fibration to be a a dominant morphism
$\pi:X\to\mathbb{P}^{1}$ with geometrically integral generic fibre, such that
$X$ is a smooth, proper, geometrically irreducible $k$-variety.

Let $E$ be the reduced divisor of $\mathbb{P}^{1}$ of the non-split fibres of
$\pi$. Let $D$ be the reduced divisor underlying $\pi^{-1}(E)$. By embedded
resolutions of singularities, there exists a birational morphism
$X^{\prime}\to X$ such that the pullback $D^{\prime}$ of $D$ has strict normal
crossings. Since $X\setminus D\cong X^{\prime}\setminus D^{\prime}$ over
$\mathbb{P}^{1}$ we see that $N_{\text{loc}}(\pi^{\prime},B)$ differs by a
constant from $N_{\text{loc}}(\pi,B)$, where $\pi^{\prime}\colon X^{\prime}\to
X\to Y$ is the composition. Thus, for the purposes of upper and lower bounds,
we can assume without loss of generality that the reduced subschemes of the
non-split fibres of $\pi$ have strict normal crossings.

######  Theorem 5.4.

Let $X\to\mathbb{P}^{1}$ be a standard fibration whose non-split fibres in
their reduced subscheme structure are sncd. There exists a finite set of
primes $S$ and a model $\mathscr{X}\to\mathbb{P}^{1}_{\mathscr{O}_{S}}$ such
that the following holds for $v\not\in S$. Fix a point
$\mathscr{Q}\in\mathbb{P}^{1}(\mathscr{O}_{S})$ for which the fibre $X_{Q}$ is
split. Then any $\mathbb{F}_{v}$-point
$\overline{\mathscr{P}}_{v}\in\mathscr{X}_{\mathscr{Q}}(\mathbb{F}_{v})$ lifts
to a point $\mathscr{P}_{v}\in\mathscr{X}_{\mathscr{Q}}(\mathscr{O}_{v})$
precisely if for every closed point $V(h)\in(\mathbb{P}^{1})^{(1)}$ we have
that $v(h(\mathscr{Q}))$ lies in the positive linear span of the
multiplicities $m_{i}$ of the components of $\mathscr{X}_{V(h)}$ that contain
$\overline{\mathscr{P}}_{v}$.

Note that the last condition is trivially satisfied for all closed points
$V(h)$ for which $v(h(\mathscr{Q}))=0$, and also for those for which if
$X_{V(h)}$ is split. By restricting $S$ further, we can assume that there is
at most one non-split fibre $X_{V(h)}$ for which we have to check this
condition.

###### Proof of Theorem 5.4.

By the definition of $D$ on $X$ and $E$ on $\mathbb{P}^{1}$, we see that
$(X,D)\to(\mathbb{P}^{1},E)$ is log smooth. For a suitable finite set of
primes $S$ this extends to $\mathscr{O}_{S}$-schemes and divisors, such that
$\mathscr{D}\subseteq\mathscr{X}$ and
$\mathscr{E}\subseteq\mathbb{P}^{1}_{\mathscr{O}_{S}}$ still have strict
normal crossings and
$(\mathscr{X},\mathscr{D})\to(\mathbb{P}^{1}_{\mathscr{O}_{S}},\mathscr{E})$
is also log smooth. We will check that this model satisfies the condition.

Consider
$\overline{\mathscr{P}}_{v}\in\mathscr{X}_{\mathscr{Q}}(\mathbb{F}_{v})$ and
let $V(h)\subset\mathbb{P}^{1}_{\mathscr{O}_{S}}$ be the unique non-split
fibre containing $\overline{\mathscr{Q}}_{v}=\pi(\overline{\mathscr{P}}_{v})$.
Suppose that we can write $v(h(\mathscr{Q}))=\sum_{i}a_{i}m_{i}$ with
$a_{i}>0$ integers, and $m_{i}$ the multiplicities of the $r$ components of
$X_{v(h)}$ which contain $\overline{\mathscr{P}}_{v}$. Around
$\overline{\mathscr{P}}_{v}$ and $\overline{\mathscr{Q}}_{v}$ the Kato fans
have affine charts $\mathbb{N}^{r}$ and $\mathbb{N}$. Under this
identification we have $F(\mathscr{Q}_{v})=v(h(\mathscr{Q}))\in\mathbb{N}$ and
$F(\mathscr{X},\mathscr{D})(\mathbb{N})\to
F(\mathbb{P}^{1}_{\mathscr{O}_{S}},\mathscr{E})(\mathbb{N})$ is given by
$(u_{i})\mapsto\sum m_{i}u_{i}$. Hence the result follows from Proposition
5.3. ∎

######  Remark 5.5.

In [15, § 2] the following was proven: if $v(h(\mathscr{Q}))=1$ then
$\mathscr{X}_{\mathscr{Q}}$ is a regular scheme. This implies that any
$\mathbb{F}_{v}$-point on $\mathscr{X}_{\mathscr{Q}}$ which lies on the
intersection of at least two components of the reduction
$\mathscr{X}_{\mathscr{Q},v}$ does not lift to a $\mathbb{Q}_{v}$-point on
$\mathscr{X}_{\mathscr{Q}}$. This last statement directly follows from our
criterion above, since then the valuation $v(h(\mathscr{Q}))=1$ cannot
possibly lie in the positive linear span of two positive integers.

The above conditions make it easy to check if an $\mathbb{F}_{v}$-point lifts.
However, one cannot deduce the existence of $\mathbb{F}_{v}$-points purely
from valuations and multiplicities, as explained by Loughran and Matthiesen
[13, Lemma 6.2]. In general, this only allows us to give necessary conditions
for local solubility.

######  Corollary 5.6.

Let $X\to\mathbb{P}^{1}$ be a standard fibration and let
$Q\in\mathbb{P}^{1}(k)$. Suppose that $X_{Q}(k_{v})\neq\emptyset$ for
$v\not\in S$. Then for every closed point $D=V(h)\in(\mathbb{P}^{1})^{(1)}$,
we have either $v(h(Q))>m_{D}$, or else $v(h(Q))=m_{D}$ and $v$ belongs to

$T_{D}=\left\\{v\not\in S:\text{$\operatorname{Frob}_{v}$ fixes an element of
$S_{D}$}\right\\}.$

(Recall that $S_{D}$ is the set of geometric components of $X_{D}$ of minimum
multiplicity $m_{D}$.)

In the special case that the non-split fibres all lie about $k$-rational
points in $\mathbb{P}^{1}$, we can (after possibly extending the set $S$
again) make this even more precise, as follows.

######  Corollary 5.7.

Let $X\to\mathbb{P}^{1}$ be a standard fibration and let
$Q\in\mathbb{P}^{1}(k)$. Assume that the non-split fibres of
$X\to\mathbb{P}^{1}$ all lie above $k$-rational points. Then
$X_{Q}(k_{v})\neq\emptyset$ precisely if for every
$V(h)\in(\mathbb{P}^{1})^{(1)}$ the fibre $X_{V(h)}$ has intersecting
geometric components of multiplicity $m_{i}$ which are fixed by
$\operatorname{Frob}_{v}$, such that $v(h(Q))$ lies in the positive linear
span of the $m_{i}$.

###### Proof.

We will start with $S$ and $\mathscr{X}\to\mathbb{P}^{1}_{\mathscr{O}_{S}}$ as
above. By the results above we have that $P_{v}\in X_{Q}(k_{v})$ reduces to an
$\mathbb{F}_{v}$-point on $\mathscr{X}$. Since this $\mathbb{F}_{v}$-points
lifts we get the result.

For the inverse implication we will need to enlarge $S$, as follows. Firstly
we do so to assume that all fibres of
$\mathscr{X}\setminus\mathscr{D}\to\mathbb{P}^{1}_{\mathscr{O}_{S}}\setminus\mathscr{E}$
are geometrically integral. Hence by Lang–Weil we find a smooth
$\mathbb{F}_{v}$-point all those fibres except for finitely many $v$. Now let
$W$ be a geometric component of a non-open stratum of $(X,D)$, which is
defined over $k^{\prime}/k$. The closure $\mathscr{W}$ of $W$ will have
geometrically irreducible fibres over all but finitely many places of
$k^{\prime}$. Hence after enlarging $S$ we see that $\mathscr{W}$ has an
$\mathbb{F}_{v^{\prime}}$-point for all $v^{\prime}\mid v$, for $v\not\in S$.
Since there are only finitely many strata and each has again finitely many
components we can enlarge $S$ to make this true for all possible $W$.

Suppose now that $\operatorname{Frob}_{v}$ fixes the components of $D$ which
define the stratum containing $W$. Then for any $v^{\prime}\mid v$ we see that
$W$ contains an $\mathbb{F}_{v^{\prime}}=\mathbb{F}_{v}$-point. We can lift
this point under the conditions in Theorem 5.4. ∎

##  6\. Multiple fibres via the large sieve

We place ourselves in the setting of Theorems 1.2 and 1.3. Let $\pi\colon
X\to\mathbb{P}^{1}$ be a standard fibration with orbifold divisor

$\partial_{\pi}=\left(1-\frac{1}{m_{0}}\right)[0]+\left(1-\frac{1}{m_{\infty}}\right)[\infty],$

in the notation of (1.4), for $m_{0},m_{\infty}\in\mathbb{N}$. Note that
$2-\deg\partial_{\pi}=\frac{1}{m_{0}}+\frac{1}{m_{\infty}}$. We define
$d=\gcd(m_{0},m_{\infty})$. We shall apply the theory from Section 4 to the
family of $\mu_{d}$-torsors

$\theta_{v}\colon\mathbb{P}^{1}\to\mathbb{P}^{1},\quad[x_{0}\colon
x_{1}]\to[v_{0}x_{0}^{d}\colon v_{1}x_{1}^{d}],$

which are parametrised by
$v=v_{1}/v_{0}\in\mathbb{Q}^{\times}/\mathbb{Q}^{\times,d}=\text{H}^{1}({\rm
Gal}(\overline{\mathbb{Q}}/\mathbb{Q}),\mu_{d})$. Let $\pi_{v}\colon
X_{v}\to\mathbb{P}^{1}$ be the normalisation of the pullback of $\pi$ along
$\theta_{v}$.

The main result of this section is the following, which pertains to the
density of locally soluble fibres on the standard fibration
$\pi_{v}:X_{v}\to\mathbb{P}^{1}$, relative to the pullback height $H_{v}$
along $\theta_{v}$. We denote by $\operatorname{rad}(n)=\prod_{p\mid n}p$, the
square-free radical of any $n\in\mathbb{N}$.

######  Proposition 6.1.

Let $\varepsilon>0$ and let
$v=v_{1}/v_{0}\in\mathbb{Q}^{\times}/\mathbb{Q}^{\times,d}$. Then

$N_{\text{loc}}(\pi_{v},H_{v},B)\ll_{\varepsilon}c_{v,\varepsilon}B^{\frac{1}{m_{0}}+\frac{1}{m_{\infty}}},$

where

$c_{v,\varepsilon}=\frac{|v_{0}v_{1}|^{\varepsilon}}{\operatorname{rad}(v_{0})|v_{0}|^{1/m_{0}}\operatorname{rad}(v_{1})|v_{1}|^{1/m_{\infty}}}.$
(6.1)

Furthermore, if $|v_{0}v_{1}|\leqslant B^{\varepsilon}$, then

$\displaystyle
N_{\text{loc}}(\pi_{v},H_{v},B)\ll_{\varepsilon}c_{v,\varepsilon}\frac{B^{\frac{1}{m_{0}}+\frac{1}{m_{\infty}}}}{(\log
B)^{\Theta_{v}(\pi)}},$

where $\Theta_{v}(\pi)$ is given by (1.9).

We shall begin the proof of this result in Section 6.2. Our argument is based
on the large sieve, which is recalled in Section 6.1. Taking the result on
faith for the moment, we proceed to show how it can be used to establish
Theorems 1.2 and 1.3.

######  Remark 6.2.

Proposition 6.1 is consistent with Conjecture 1.5 for a fixed choice of
$v\in\mathbb{Q}^{\times}/\mathbb{Q}^{\times,d}$. Indeed, we have
$H_{v}(x)=H(x)^{d}$ where $H(x)$ is an $\mathscr{O}(1)$-height on
$\mathbb{P}^{1}$. It follows that

$N_{\text{loc}}(\pi_{v},B)=N_{\text{loc}}(\pi_{v},H,B)=N_{\text{loc}}(\pi_{v},H_{v},B^{d})\ll_{v}\frac{B^{\frac{d}{m_{0}}+\frac{d}{m_{\infty}}}}{(\log
B)^{\Theta_{v}(\pi)}}.$

The orbifold base of $\pi_{v}$ is $(X_{v},\partial_{\pi_{v}})$, with

$\partial_{\pi_{v}}=\left(1-\frac{d}{m_{0}}\right)[0]+\left(1-\frac{d}{m_{\infty}}\right)[\infty],$

by part (b) of Proposition 4.7. It follows from (4.1) and part (c) of
Proposition 4.7 that that $\frac{d}{m_{0}}+\frac{d}{m_{\infty}}=-d\deg
K_{\pi,\partial_{\pi}}=2-\deg\partial_{\pi_{v}}$. Moreover,
$\Theta_{v}(\pi)=\Delta(\pi_{v})$, by part (c) of Proposition 4.8.

###### Proof of Theorem 1.2.

In this case there is only one multiple fibre above $0$ and so $m_{\infty}=1$
and $d=1$. Thus $\text{H}^{1}({\rm
Gal}(\overline{\mathbb{Q}}/\mathbb{Q}),\mu_{d})$ is the trivial group and it
follows directly from Proposition 6.1 that

$N_{\text{loc}}(\pi,B)\ll\frac{B^{\frac{1}{m_{0}}+1}}{(\log
B)^{\Theta_{1}(\pi)}}.$

We have already seen that $\frac{1}{m_{0}}+1=2-\deg\partial_{\pi}$. Moreover,
we saw that $\Theta_{1}(\pi)=\Delta(\pi)$ in (1.11). ∎

###### Proof of Theorem 1.3.

We appeal to the decomposition in part (b) of Proposition 4.8. This gives

$N_{\text{loc}}(\pi,B)\ll\sum_{v=v_{1}/v_{0}\in\mathbb{Q}^{\times}/\mathbb{Q}^{\times,d}}N_{\text{loc}}(\pi_{v},H_{v},B).$

For any $\delta>0$ we clearly have

$\displaystyle\sum_{n>x}\frac{1}{\operatorname{rad}(n)n^{\delta}}<\sum_{n=1}^{\infty}\frac{(n/x)^{\delta/2}}{\operatorname{rad}(n)n^{\delta}}$
$\displaystyle=x^{-\delta/2}\prod_{p}\left(1+\sum_{k=1}^{\infty}\frac{1}{p^{1+k\delta/2}}\right)$
$\displaystyle\ll_{\delta}x^{-\delta/2}.$

Let $\varepsilon>0$. In the light of the latter bound, it follows from the
first part of Proposition 6.1 that there exists $\delta(\varepsilon)>0$, such
that the terms with $|v_{0}v_{1}|>B^{\varepsilon}$ make an overall
contribution $O_{\varepsilon}(B^{1/m_{0}+1/m_{\infty}-\delta(\varepsilon)})$
to $N_{\text{loc}}(\pi,B)$. For the terms with $|v_{0}v_{1}|\leqslant
B^{\varepsilon}$, we apply the second part of Proposition 6.1.

This easily leads to the conclusion that

$N_{\text{loc}}(\pi,B)\ll_{\varepsilon}B^{\frac{1}{m_{0}}+\frac{1}{m_{\infty}}-\delta(\varepsilon)}+\sum_{|v_{0}v_{1}|\leqslant
B^{\varepsilon}}c_{v,\varepsilon}\frac{B^{\frac{1}{m_{0}}+\frac{1}{m_{\infty}}}}{(\log
B)^{\Theta_{v}(\pi)}}\ll_{\varepsilon}\frac{B^{\frac{1}{m_{0}}+\frac{1}{m_{\infty}}}}{(\log
B)^{\Theta(\pi)}},$

where $\Theta(\pi)$ is given by (1.10). The statement of the theorem follows,
since we have already remarked that
$\frac{1}{m_{0}}+\frac{1}{m_{\infty}}=2-\deg\partial_{\pi}$. ∎

###  6.1. The large sieve

We begin by stating the version of the large sieve that we shall use in this
paper.

######  Lemma 6.3.

Let $m\in\mathbb{N}$, let $B_{0},B_{1}\geqslant 1$ and let
$\Omega\subset\mathbb{Z}^{2}$. For each prime $p$ assume that there exists
$\overline{\omega}(p)\in[0,1)$ such that the reduction modulo $p^{m}$ of
$\Omega$ has cardinality at most $(1-\overline{\omega}(p))p^{2m}$. Then

$\\#\left\\{\mathbf{x}\in\Omega:|x_{i}|\leqslant B_{i},\text{ for
$i=0,1$}\right\\}\ll\frac{(B_{0}+Q^{2m})(B_{1}+Q^{2m})}{L(Q)},$

for any $Q\geqslant 1$, where

$L(Q)=\sum_{q\leqslant Q}\mu^{2}(q)\prod_{p\mid
q}\frac{\overline{\omega}(p)}{1-\overline{\omega}(p)}.$

###### Proof.

When $m=1$ this is a straightforward rephrasing of the multidimensional large
sieve worked out by Kowalski [11, Thm. 4.1]. The extension to $m>1$ is routine
and will not be explained here. ∎

###  6.2. Preliminary steps

Recall that $d=\gcd(m_{0},m_{\infty})$. Henceforth, we usually write
$\mathbf{v}=(v_{0},v_{1})\in\mathbb{Z}_{\mathrm{prim}}^{2}$ for the point
$v=v_{1}/v_{0}\in\mathbb{Q}^{\times}/\mathbb{Q}^{\times,d}$. We may clearly
proceed under the assumption that $v_{0},v_{1}$ are both free of $d$th powers.

Let $S$ be a large enough finite set of primes, as required for the arguments
in Section 5 to go through. Suppose that
$E_{1},\dots,E_{r}\in(\mathbb{P}^{1})^{(1)}$ are the closed points distinct
from $0$ and $\infty$, where $\pi_{v}$ is not smooth. For each $1\leqslant
j\leqslant r$, assume that $E_{j}=V(h_{j})$ for a square-free binary form
$h_{j}\in\mathbb{Z}_{S}[x_{0},x_{1}]$. We may further assume that $h_{j}$ is
irreducible over $\mathbb{Q}$ and coprime to the monomial $x_{0}x_{1}$, and
that the coefficients of $h_{j}$ are relatively coprime.

We proceed by defining the sets

$\displaystyle T_{0}$ $\displaystyle=\left\\{p\not\in
S:\text{$\operatorname{Frob}_{p}$ fixes an element of $S_{0}$}\right\\},$
$\displaystyle T_{\infty}$ $\displaystyle=\left\\{p\not\in
S:\text{$\operatorname{Frob}_{p}$ fixes an element of $S_{\infty}$}\right\\},$
$\displaystyle U_{j}$ $\displaystyle=\left\\{p\not\in
S:\text{$\operatorname{Frob}_{p}$ fixes an element of $S_{E_{j}}$}\right\\},$

for $1\leqslant j\leqslant r$. The fibre $X_{v,y}$ of the fibration
$\pi_{v}\colon X_{v}\to\mathbb{P}^{1}$ has a $\mathbb{Q}_{p}$-point precisely
if $X_{\pi_{v}(y)}$ does and thus we can apply the sparsity conditions
Corollary 5.6. This yields the upper bound $N_{\text{loc}}(\pi_{v},B)\leqslant
M_{\mathbf{v}}(B)$, where $M_{\mathbf{v}}(B)$ is defined to be the number of
$\mathbf{y}=(y_{0},y_{1})\in\mathbb{Z}^{2}$ such that
$\gcd(v_{0}y_{0},v_{1}y_{1})=1$ and
$\max\\{|v_{0}y_{0}^{d}|,|v_{1}y_{1}^{d}|\\}\leqslant B$, with

$p\not\in S\Rightarrow\begin{cases}\left(v_{p}(x_{0})=m_{0}\text{ and }p\in
T_{0}\right),\text{ or }v_{p}(x_{0})>m_{0},\\\
\left(v_{p}(x_{1})=m_{\infty}\text{ and }p\in T_{\infty}\right),\text{ or
}v_{p}(x_{1})>m_{\infty},\\\ p\|h_{j}(\mathbf{x})\Rightarrow p\in
U_{j},\end{cases}$

where $(x_{0},x_{1})=(v_{0}y_{0}^{d},v_{1}y_{1}^{d})$. Write
$v_{0}=a_{0}w_{0}^{\prime}$ and $y_{0}=b_{0}z_{0}$, where
$w_{0}^{\prime}z_{0}$ is coprime to all the primes in $S$ and $p\mid
a_{0}b_{0}\Rightarrow p\in S$. Let $p\not\in S$. Then
$v_{p}(w_{0}^{\prime}z_{0}^{d})=m_{0}$ if and only if
$v_{p}(w_{0}^{\prime})=0$ and $v_{p}(z_{0})=m_{0}/d$, since $w_{0}^{\prime}$
is free of $d$th powers. Similarly, if $v_{p}(w_{0}^{\prime}z_{0}^{d})>m_{0}$
then either $v_{p}(z_{0})>m_{0}/d$, or else $v_{p}(z_{0})=m_{0}/d$ and $p\mid
w_{0}^{\prime}$. This suggests that we may write

$v_{0}=a_{0}w_{0},\quad y_{0}=b_{0}s_{0}^{m_{0}/d}t_{0}^{m_{0}/d}u_{0},$

where

  * •

$p\mid a_{0}b_{0}\Rightarrow p\in S$;

  * •

$p\mid s_{0}w_{0}u_{0}\Rightarrow p\not\in S$;

  * •

$s_{0},t_{0}$ square-free;

  * •

$p\mid w_{0}\Rightarrow p\mid s_{0}$;

  * •

$p\mid t_{0}\Rightarrow p\in T_{0}$; and

  * •

$u_{0}$ is $(m_{0}/d+1)$-full.

Similarly, we have a factorisation

$v_{1}=a_{1}w_{1},\quad
y_{1}=b_{1}s_{1}^{m_{\infty}/d}t_{1}^{m_{\infty}/d}u_{1},$

where

  * •

$p\mid a_{1}b_{1}\Rightarrow p\in S$;

  * •

$p\mid s_{1}w_{1}u_{1}\Rightarrow p\not\in S$;

  * •

$s_{1},t_{1}$ square-free;

  * •

$p\mid w_{1}\Rightarrow p\mid s_{1}$;

  * •

$p\mid t_{1}\Rightarrow p\in T_{\infty}$; and

  * •

$u_{1}$ is $(m_{\infty}/d+1)$-full.

There are $O_{\varepsilon}(|v_{0}v_{1}|^{\varepsilon})$ choices for
$a_{i},s_{i},w_{i}\in\mathbb{Z}$ for $i=0,1$, by the standard estimate for the
divisor function. We fix a choice of $b_{0},b_{1},u_{0},u_{1}$ and write

$A_{0}=a_{0}b_{0}^{d}s_{0}^{m_{0}}u_{0}^{d}w_{0}\quad\text{ and }\quad
A_{1}=a_{1}b_{1}^{d}s_{1}^{m_{\infty}}u_{1}^{d}w_{1}.$ (6.2)

Note that we have $\gcd(A_{0},A_{1})=1$. Moreover, let

$R_{0}=\left(\frac{B}{|A_{0}|}\right)^{1/m_{0}},\qquad
R_{1}=\left(\frac{B}{|A_{1}|}\right)^{1/m_{\infty}},$

and

$g_{j}(\mathbf{t})=h_{j}(A_{0}t_{0}^{m_{0}},A_{1}t_{1}^{m_{\infty}}),\text{
for $1\leqslant j\leqslant r$}.$ (6.3)

The binary form $g_{j}(t_{0},t_{1})$ is square-free and coprime to the
monomial $t_{0}t_{1}$, since $h_{j}(t_{0},t_{1})$ satisfies these properties.
For a (possibly infinite) set $T$ of primes, let

$\mathbf{1}_{T}(n)=\begin{cases}1&\text{ if $p\mid n\Rightarrow p\in T$,}\\\
0&\text{ otherwise.}\end{cases}$

In what follows, we will also write $T^{c}$ to denote the complement of $T$ in
the full set of primes.

Then, with all this notation in mind, we have

$M_{\mathbf{v}}(B)\ll\sum_{v_{0}=a_{0}w_{0}}\sum_{v_{1}=a_{1}w_{1}}\sum_{\begin{subarray}{c}b_{0},b_{1}\\\
p\mid b_{0}b_{1}\Rightarrow p\in
S\end{subarray}}\sum_{u_{0},u_{1}\in\mathbb{Z}}L(R_{0},R_{1}),$

where

$L(R_{0},R_{1})=\sum_{\begin{subarray}{c}(t_{0},t_{1})\in\mathbb{Z}^{2}\\\
|t_{0}|\leqslant R_{0},~{}|t_{1}|\leqslant
R_{1}\end{subarray}}\mu^{2}(t_{0}t_{1})\mathbf{1}_{T_{0}}(t_{0})\mathbf{1}_{T_{\infty}}(t_{1})\prod_{j=1}^{r}\mathbf{1}^{\sharp}_{U_{j}}\left(t_{0},t_{1}\right),$
(6.4)

and where

$\mathbf{1}_{U_{j}}^{\sharp}(t_{0},t_{1})=\begin{cases}1&\text{ if
$p\|g_{j}(\mathbf{t})\Rightarrow p\in U_{j}$},\\\ 0&\text{
otherwise.}\end{cases}$

The trivial bound for $L(R_{0},R_{1})$ is

$\displaystyle L(R_{0},R_{1})$
$\displaystyle\ll\frac{B^{1/m_{0}+1/m_{\infty}}}{|A_{0}|^{1/m_{0}}|A_{1}|^{1/m_{\infty}}}$
$\displaystyle\ll\frac{B^{1/m_{0}+1/m_{\infty}}}{|s_{0}||v_{0}|^{1/m_{0}}|s_{1}||v_{1}|^{1/m_{\infty}}|b_{0}u_{0}|^{d/m_{0}}|b_{1}u_{1}|^{d/m_{\infty}}},$

by (6.2). Clearly

$|s_{i}|\gg\operatorname{rad}(v_{i}),\quad\text{for $i=0,1$},$ (6.5)

for a suitable implied constant depending only on $S$. Note that

$\sum_{\begin{subarray}{c}|b_{0}|>J\\\ p\mid b_{0}\Rightarrow p\in
S\end{subarray}}|b_{0}|^{-d/m_{0}}\ll\frac{1}{J^{d/m_{0}}},$

for any $J\geqslant 1$. Similarly,

$\sum_{\begin{subarray}{c}|u_{0}|>J\\\ \text{$u_{0}$ is
$(m_{0}/d+1)$-full}\end{subarray}}|u_{0}|^{-d/m_{0}}\ll\frac{1}{J^{d^{2}/(m_{0}(m_{0}+d))}},$

Let $\varepsilon>0$. In what follows it will be convenient to recall the
notation (6.1) for $c_{v,\varepsilon}$ in the statement of Proposition 6.1. It
now follows that the overall contribution to $M_{\mathbf{v}}(B)$ from
parameters $b_{0},u_{0}$ in the range $\min(|b_{0}|,|u_{0}|)>B^{\varepsilon}$,
or parameters $b_{1},u_{1}$ in the range
$\min(|b_{1}|,|u_{1}|)>B^{\varepsilon}$ is clearly

$\ll_{\varepsilon}c_{v,\varepsilon}B^{1/m_{0}+1/m_{\infty}-\varepsilon/(m_{0}^{2}m_{\infty}^{2})},$

since we have seen that there are
$O_{\varepsilon}(|v_{0}v_{1}|^{\varepsilon})$ choices for
$a_{i},s_{i},w_{i}\in\mathbb{Z}$ associated to a particular choice of
$\mathbf{v}$. Thus we deduce that

$\begin{split}M_{\mathbf{v}}(B)\ll_{\varepsilon}~{}&\sum_{v_{0}=a_{0}w_{0}}\sum_{v_{1}=a_{0}w_{1}}\sum_{\begin{subarray}{c}|b_{0}|,|b_{1}|\leqslant
B^{\varepsilon}\\\ p\mid b_{0}b_{1}\Rightarrow p\in
S\end{subarray}}\sum_{|u_{0}|,|u_{1}|\leqslant
B^{\varepsilon}}L(R_{0},R_{1})\\\
&\quad+c_{v,\varepsilon}B^{1/m_{0}+1/m_{\infty}-\varepsilon/(m_{0}^{2}m_{\infty}^{2})}.\end{split}$
(6.6)

###  6.3. Application of the large sieve

We shall now apply Lemma 6.3 to estimate (6.4), which we shall apply with
$m=2$. Let $\Omega\subset\mathbb{Z}^{2}$ be the set of vectors
$\mathbf{t}\in\mathbb{N}^{2}$ such that
$\mathbf{1}_{T_{0}}(t_{0})\mathbf{1}_{T_{\infty}}(t_{1})=1$ and for which
$p\in U_{j}$ whenever there exists an index $j$ such that
$p\|g_{j}(\mathbf{t})$. For any prime $p\not\in S$, let

$A_{0}(p)=\\{\mathbf{t}\in(\mathbb{Z}/p^{2}\mathbb{Z})^{2}:\text{$p\mid t_{0}$
and $p\not\in T_{0}$}\\}$

and

$A_{\infty}(p)=\\{\mathbf{t}\in(\mathbb{Z}/p^{2}\mathbb{Z})^{2}:\text{$p\mid
t_{1}$ and $p\not\in T_{\infty}$}\\}.$

Similarly, let

$B_{j}(p)=\\{\mathbf{t}\in(\mathbb{Z}/p^{2}\mathbb{Z})^{2}:\text{
$p\|g_{j}(\mathbf{t})$ and $p\not\in U_{j}$}\\}$

for $1\leqslant j\leqslant r$. Then
$\\#\Omega\bmod{p^{2}}\leqslant(1-\overline{\omega}(p))p^{4}$, where

$\overline{\omega}(p)=\frac{\\#\left(A_{0}(p)\cup A_{\infty}(p)\cup
B_{1}(p)\cup\dots\cup B_{r}(p)\right)}{p^{4}}.$

In particular, we have $\overline{\omega}(p)\in[0,1)$. The following result is
concerned with estimating this quantity.

######  Lemma 6.4.

Let $p\not\in S$ and let $d=\gcd(m_{0},m_{\infty})$. Then

$\overline{\omega}(p)=\frac{\mathbf{1}_{T_{0}^{c}}(p)}{p}+\frac{\mathbf{1}_{T_{\infty}^{c}}(p)}{p}+\sum_{j=1}^{r}\frac{\mathbf{1}_{U_{j}^{c}}(p)\nu_{j}(p;\mathbf{v})}{p^{2}}+O\left(\frac{\gcd(p,A_{0}A_{1})}{p^{2}}\right),$

where

$\nu_{j}(p;\mathbf{v})=\\#\\{\mathbf{t}\in\mathbb{F}_{p}^{2}:h_{j}(v_{0}t_{0}^{d},v_{1}t_{1}^{d})=0\\}.$

###### Proof.

Recall that $\gcd(A_{0},A_{1})=1$, that $g_{j}(t_{0},t_{1})$ is defined in
(6.3), and that $g_{j}(t_{0},t_{1})$ is square-free and coprime to the
monomial $t_{0}t_{1}$. If $p\mid A_{0}A_{1}$ we take the trivial upper bound

$\\#\left(A_{0}(p)\cup A_{\infty}(p)\cup B_{1}(p)\cup\dots\cup
B_{r}(p)\right)=O(p^{3}),$

whence $\overline{\omega}(p)=O(1/p),$ which is satisfactory.

Suppose henceforth that $p\nmid A_{0}A_{1}$. We proceed by noting that the
intersection of any two sets in the union $A_{0}(p)\cup A_{\infty}(p)\cup
B_{1}(p)\cup\dots\cup B_{r}(p)$ contains $O(p^{2})$ elements of
$(\mathbb{Z}/p^{2}\mathbb{Z})^{2}$. Thus

$\overline{\omega}(p)=\frac{\mathbf{1}_{T_{0}^{c}}(p)}{p}+\frac{\mathbf{1}_{T_{\infty}^{c}}(p)}{p}+\sum_{j=1}^{r}\frac{\\#B_{j}(p)}{p^{4}}+O\left(\frac{1}{p^{2}}\right).$

Turning to $\\#B_{j}(p)$ for $j\in\\{1,\dots,r\\}$, we write
$\mathbf{u}=\mathbf{x}+p\mathbf{y}$ for
$\mathbf{x},\mathbf{y}\in\mathbb{F}_{p}^{2}$. Thus

$\\#\\{\mathbf{t}\in(\mathbb{Z}/p^{2}\mathbb{Z})^{2}:p^{2}\mid
g_{j}(\mathbf{t})\\}=\sum_{\begin{subarray}{c}\mathbf{x}\in\mathbb{F}_{p}^{2}\\\
g_{j}(\mathbf{x})=0\end{subarray}}\\#\left\\{\mathbf{y}\in\mathbb{F}_{p}^{2}:\mathbf{y}.\nabla
g_{j}(\mathbf{x})=-g_{j}(\mathbf{x})/p\right\\}.$

On enlarging $S$, we can assume that $\nabla g_{j}(\mathbf{x})\neq\mathbf{0}$
for any $\mathbf{x}$ in the sum. Thus each of the $O(p)$ values of
$\mathbf{x}$ produces $O(p)$ choices of $\mathbf{y}$, giving

$\\#\\{\mathbf{t}\in(\mathbb{Z}/p^{2}\mathbb{Z})^{2}:p^{2}\mid
g_{j}(\mathbf{t})\\}=O(p^{2}).$

Hence

$\\#B_{j}(p)=\mathbf{1}_{U_{j}^{c}}(p)p^{2}\\#\\{\mathbf{t}\in\mathbb{F}_{p}^{2}:g_{j}(\mathbf{t})=0\\}+O(p^{2}).$

Putting this together we have shown that

$\overline{\omega}(p)=\frac{\mathbf{1}_{T_{0}^{c}}(p)}{p}+\frac{\mathbf{1}_{T_{\infty}^{c}}(p)}{p}+\sum_{j=1}^{r}\frac{\mathbf{1}_{U_{j}^{c}}(p)\lambda_{j}(p;A_{0},A_{1})}{p^{2}}+O\left(\frac{1}{p^{2}}\right),$

where

$\displaystyle\lambda_{j}(p;A_{0},A_{1})$
$\displaystyle=\\#\\{\mathbf{t}\in\mathbb{F}_{p}^{2}:h_{j}(A_{0}t_{0}^{m_{0}},A_{1}t_{1}^{m_{\infty}})=0\\}.$

for $1\leqslant j\leqslant r$. In order to complete the proof of the lemma, it
will suffice to prove that

$\lambda_{j}(p;A_{0},A_{1})=\nu_{j}(p;\mathbf{v})+O(1),$ (6.7)

for $1\leqslant j\leqslant r$, in the notation of the lemma.

To see this, let $e$ be the least common multiple of $m_{0}$ and $m_{\infty}$,
so that $e=m_{0}m_{\infty}/d$. We pick a generator
$\alpha\in\mathbb{F}_{p}^{*}$ of
$\mathbb{F}_{p}^{*}/(\mathbb{F}_{p}^{*})^{e}$. Then it is easily confirmed
that

$\langle\alpha^{de/m_{0}}\rangle=(\mathbb{F}_{p}^{*})^{d}/(\mathbb{F}_{p}^{*})^{m_{0}}\quad\text{
and
}\quad\langle\alpha^{de/m_{\infty}}\rangle=(\mathbb{F}_{p}^{*})^{d}/(\mathbb{F}_{p}^{*})^{m_{\infty}},$

on noting that $(\mathbb{F}_{p}^{*})^{m_{0}}$ and
$(\mathbb{F}_{p}^{*})^{m_{\infty}}$ are subgroups of
$(\mathbb{F}_{p}^{*})^{d}$. (Indeed, to check the first equality, for example,
it suffices to confirm that $\alpha^{de/m_{0}}$ has order $m_{0}/d$ in
$\mathbb{F}_{p}^{*}$.) The group
$(\mathbb{F}_{p}^{*})^{d}/(\mathbb{F}_{p}^{*})^{m_{0}}$ has order
$N_{0}=\gcd(m_{0},p-1)$ and, likewise,
$(\mathbb{F}_{p}^{*})^{d}/(\mathbb{F}_{p}^{*})^{m_{\infty}}$ has order
$N_{\infty}=\gcd(m_{\infty},p-1)$. It follows from this that any non-zero
$d$th power in $\mathbb{F}_{p}$ can be represented uniquely as
$u^{m_{0}}\alpha^{edk/m_{0}}$ for some $k\in\mathbb{Z}/N_{0}\mathbb{Z}$, or as
$u^{m_{\infty}}\alpha^{ed\ell/m_{\infty}}$ for some
$\ell\in\mathbb{Z}/N_{\infty}\mathbb{Z}$.

Define

$\lambda_{j}(p;A_{0},A_{1};k,\ell)=\\#\\{\mathbf{t}\in\mathbb{F}_{p}^{2}:h_{j}(A_{0}t_{0}^{m_{0}}\alpha^{edk/m_{0}},A_{1}t_{1}^{m_{\infty}}\alpha^{ed\ell/m_{\infty}})=0\\},$

for any $k\in\mathbb{Z}/N_{0}\mathbb{Z}$ and
$\ell\in\mathbb{Z}/N_{\infty}\mathbb{Z}$. Let
$\beta=\alpha^{-edk/m_{0}-ed\ell/m_{\infty}}$. On multiplying through by
$\beta^{\deg(h_{j})}$ and recalling that $h_{j}$ is homogeneous, we obtain

$\displaystyle\lambda_{j}(p;A_{0},A_{1};k,\ell)$
$\displaystyle=\\#\\{\mathbf{t}\in\mathbb{F}_{p}^{2}:h_{j}(A_{0}t_{0}^{m_{0}}\alpha^{edk/m_{0}}\beta,A_{1}t_{1}^{m_{\infty}}\alpha^{ed\ell/m_{\infty}}\beta)=0\\}$
$\displaystyle=\\#\\{\mathbf{t}\in\mathbb{F}_{p}^{2}:h_{j}(A_{0}t_{0}^{m_{0}}\alpha^{-ed\ell/m_{\infty}},A_{1}t_{1}^{m_{\infty}}\alpha^{-edk/m_{0}})=0\\}.$

But $ed/m_{\infty}=m_{0}$ and $ed/m_{0}=m_{\infty}$. Hence a simple change of
variables yields

$\lambda_{j}(p;A_{0},A_{1};k,\ell)=\lambda_{j}(p;A_{0},A_{1};0,0).$ (6.8)

Let $\nu_{j}^{*}(p;A_{0},A_{1})$ denote the contribution to
$\nu_{j}(p;A_{0},A_{1})$ from $t_{0}t_{1}\neq 0$, and similarly for
$\lambda_{j}^{*}(p;A_{0},A_{1};k,\ell)$. Then we may write

$\displaystyle\nu_{j}(p;A_{0},A_{1})$
$\displaystyle=\nu_{j}^{*}(p;A_{0},A_{1})+O(1)$
$\displaystyle=\frac{1}{N_{0}N_{\infty}}\sum_{k\in\mathbb{Z}/N_{0}\mathbb{Z}}\sum_{\ell\in\mathbb{Z}/N_{\infty}\mathbb{Z}}\lambda_{j}^{*}(p;A_{0},A_{1};k,\ell)+O(1)$
$\displaystyle=\lambda_{j}^{*}(p;A_{0},A_{1};0,0)+O(1),$

by (6.8). Noting that
$\lambda_{j}^{*}(p;A_{0},A_{1};0,0)=\lambda_{j}(p;A_{0},A_{1})+O(1)$, we have
therefore shown that

$\lambda_{j}(p;A_{0},A_{1})=\nu_{j}(p;A_{0},A_{1})+O(1).$

At this point we recall the factorisation (6.2), together with the fact that
$v_{i}=a_{i}s_{i}w_{i}$, for $i=0,1$. Hence, since $p\nmid A_{0}A_{1}$, a
simple change of variables shows that

$\displaystyle\nu_{j}(p;A_{0},A_{1})$
$\displaystyle=\\#\\{\mathbf{t}\in\mathbb{F}_{p}^{2}:h_{j}(v_{0}(b_{0}s_{0}^{m_{0}/d}t_{0})^{d},v_{1}(b_{1}s_{1}^{m_{\infty}/d}t_{1})^{d})=0\\}$
$\displaystyle=\nu_{j}(p;\mathbf{v}),$

from which the claim (6.7) follows. ∎

We will need to study the average size of $\overline{\omega}(p)$ as $p$
varies. We break this into the following results.

######  Lemma 6.5.

We have

$\sum_{\begin{subarray}{c}p\leqslant x\\\ p\not\in
T_{0}\end{subarray}}\frac{1}{p}=(1-\delta_{0,\mathbb{Q}}(\pi))\log\log x+O(1)$

and

$\sum_{\begin{subarray}{c}p\leqslant x\\\ p\not\in
T_{\infty}\end{subarray}}\frac{1}{p}=(1-\delta_{\infty,\mathbb{Q}}(\pi))\log\log
x+O(1),$

in the notation of (1.5).

###### Proof.

This is a straightforward consequence of the Chebotarev density theorem, in
the form presented by Serre [19, Thm. 3.4], for example. ∎

Our next result concerns the average behaviour of the function
$\nu_{j}(p;\mathbf{v})$ in Lemma 6.4, as we average over primes $p\not\in
U_{j}$. This is more difficult and requires the use of notation introduced at
the start of Section 2.2, which we recall here. For a number field
$F/\mathbb{Q}$, let $\mathscr{P}_{F}$ denote the set of primes
$p\in\mathbb{Z}$ that are unramified in $F$ and for which there exists a prime
ideal $\mathfrak{p}\mid p\mathfrak{o}_{F}$ of residue degree $1$. For any
positive integer $m\leqslant[F:\mathbb{Q}]$ we write $\mathscr{P}_{F,m}$ for
the subset of $p\in\mathscr{P}_{F}$ for which there are precisely $m$ prime
ideals above $p$ of residue degree $1$.

For each $j\in\\{1,\dots,r\\}$, define the étale algebra

$N_{E_{j},d,v_{1}/v_{0}}=\mathbb{Q}[x]/(r_{j}(x)),$

where $r_{j}(x)=h_{j}(x^{d},v_{1}/v_{0}).$ As in (1.8), this has a
factorisation into number fields

$N_{E_{j},d,v_{1}/v_{0}}=N^{(1)}\times\dots\times N^{(s)},$

where $N^{(k)}=N_{E_{j},d,v}^{(k)}$, for $1\leqslant k\leqslant s$, where the
dependency of $s$ on $j$ is suppressed for legibility.

######  Lemma 6.6.

For each $j\in\\{1,\dots,r\\}$, we have

$\sum_{\begin{subarray}{c}p\leqslant x\\\ p\not\in
U_{j}\end{subarray}}\frac{\nu_{j}(p;\mathbf{v})}{p^{2}}=\sum_{k=1}^{s}(1-\delta_{D,N^{(k)}}(\pi))\log\log
x+O\left(1+\omega(v_{0}v_{1})\right),$

in the notation of (1.5), where $\omega(n)$ denotes the number of distinct
prime factors of $n\in\mathbb{Z}$.

###### Proof.

We have

$\sum_{\begin{subarray}{c}p\leqslant x\\\ p\not\in
U_{j}\end{subarray}}\frac{\nu_{j}(p;\mathbf{v})}{p^{2}}=\sum_{\begin{subarray}{c}p\leqslant
x\\\ p\not\in U_{j}\\\ p\nmid
v_{0}v_{1}\end{subarray}}\frac{\nu_{j}(p;\mathbf{v})}{p^{2}}+\sum_{\begin{subarray}{c}p\leqslant
x\\\ p\not\in U_{j}\\\ p\mid
v_{0}v_{1}\end{subarray}}\frac{\nu_{j}(p;\mathbf{v})}{p^{2}}$

Since $\gcd(v_{0},v_{1})=1$ the second term is seen to be

$\ll\sum_{\begin{subarray}{c}p\leqslant x\\\ p\mid
v_{0}v_{1}\end{subarray}}\frac{1}{p}\ll\omega(v_{0}v_{1}).$

Next, we see that

$\sum_{\begin{subarray}{c}p\leqslant x\\\ p\not\in U_{j}\\\ p\nmid
v_{0}v_{1}\end{subarray}}\frac{\nu_{j}(p;\mathbf{v})}{p^{2}}=\sum_{\begin{subarray}{c}p\leqslant
x\\\ p\not\in U_{j}\\\ p\nmid
v_{0}v_{1}\end{subarray}}\frac{\\#\\{t\in\mathbb{F}_{p}:h_{j}(t^{d},v_{1}/v_{0})=0\\}}{p}+O\left(1\right).$

Write $r_{j}(t)=r_{j}(t^{d},v_{1}/v_{0})$ and let
$r_{j}(t)=r_{j}^{(1)}(t)\dots r_{j}^{(s)}(t)$ be its factorisation into
irreducible factors over $\mathbb{Q}$. Then $N^{(k)}$ is the number field
$\mathbb{Q}[t]/(r_{j}^{(k)})$, for $1\leqslant k\leqslant s$. We have

$\sum_{\begin{subarray}{c}p\leqslant x\\\ p\not\in U_{j}\\\ p\nmid
v_{0}v_{1}\end{subarray}}\frac{\nu_{j}(p;\mathbf{v})}{p^{2}}=\sum_{k=1}^{s}\sum_{\begin{subarray}{c}p\leqslant
x\\\ p\not\in U_{j}\\\ p\nmid
v_{0}v_{1}\end{subarray}}\frac{\\#\\{t\in\mathbb{F}_{p}:r_{j}^{(k)}(t)=0\\}}{p}+O(1).$

To begin with, it follows from the prime ideal theorem that

$\sum_{\begin{subarray}{c}p\leqslant
x\end{subarray}}\frac{\\#\\{t\in\mathbb{F}_{p}:r_{j}^{(k)}(t)=0\\}}{p}=\log\log
x+O\left(1+\omega(v_{0}v_{1})\right).$

Next, we note that $p\in U_{j}$ if and only if $\operatorname{Frob}_{p}$ fixes
a component of $S_{E_{j}}$. Let $F_{j}$ denote the field of definition of the
elements of $S_{E_{j}}$. Then, for any $p\not\in S$, the condition $p\in
U_{j}$ is equivalent to the condition $p\in\mathscr{P}_{F_{j}}$. Likewise, for
any positive integer $m\leqslant[N^{(k)}:\mathbb{Q}]$, we will have
$\\#\\{t\in\mathbb{F}_{p}:r_{j}^{(k)}(t)=0\\}=m$ if and only if
$p\in\mathscr{P}_{N^{(k)},m}$. Hence

$\displaystyle\sum_{\begin{subarray}{c}p\leqslant x\\\ p\not\in
U_{j}\end{subarray}}\frac{\nu_{j}(p;\mathbf{v})}{p^{2}}=~{}$
$\displaystyle\sum_{k=1}^{s}\left(\log\log
x-\sum_{m=1}^{[N^{(k)}:\mathbb{Q}]}m\sum_{\begin{subarray}{c}p\leqslant x\\\
p\in\mathscr{P}_{N^{(k)},m}\cap\mathscr{P}_{F_{j}}\end{subarray}}\frac{1}{p}\right)+O\left(1+\omega(v_{0}v_{1})\right).$

The remaining sum over primes is susceptible to a further application of the
Chebotarev density theorem. Once coupled with Theorem 2.3 and (2.1), this
leads to the statement of the lemma. ∎

We may combine the previous two results to produce a lower bound for the
quantity $L(Q)$ in Lemma 6.3, with the choice of $\overline{\omega}(p)$ from
Lemma 6.4.

######  Lemma 6.7.

For any $\varepsilon>0$, we have the lower bound

$L(Q)\gg_{\varepsilon}\frac{(\log
Q)^{\Theta_{v}(\pi)}}{|A_{0}A_{1}|^{\varepsilon}},$

where $\Theta_{v}(\pi)$ is given by (1.9).

###### Proof.

Since $1-\overline{\omega}(p)\leqslant 1$, we have

$L(Q)\geqslant\sum_{q\leqslant Q}\mu^{2}(q)\prod_{p\mid
q}\overline{\omega}(p).$

There are many results in the literature concerning mean values of non-
negative arithmetic functions. However, we can get by with the relatively
crude lower bound found in [8, Thm. A.3], which is based on an application of
Rankin’s trick. Let $\gamma:\mathbb{N}\to\mathbb{R}_{\geqslant 0}$ be a
multiplicative arithmetic function that is supported on square-free integers
and which satisfies

$\sum_{y<p\leqslant x}\gamma(p)\log p\leqslant a\log(x/y)+b,$ (6.9)

for any $x>y>2$, for appropriate constants $a,b>0$. Then it follows from [8,
Thm. A.3] that

$\sum_{n\leqslant x}\gamma(n)\gg\prod_{p\leqslant x}\left(1+\gamma(p)\right),$
(6.10)

where the implied constant is allowed to depend on $a$ and $b$. We seek to
apply this with

$\gamma(n)=\mu^{2}(n)\prod_{p\mid n}\overline{\omega}(p).$

It is clear from Lemma 6.4 that $\overline{\omega}(p)=O(1/p)$. Hence

$\sum_{y<p\leqslant x}\gamma(p)\log p\ll 1+\sum_{y<p\leqslant x}\frac{\log
p}{p}\ll 1+\log(x/y),$

uniformly in $v_{0}$ and $v_{1}$. Hence (6.9) holds for $a,b=O(1)$ and it
follows from (6.10) that

$L(Q)\gg\prod_{p\leqslant Q}\left(1+\overline{\omega}(p)\right),$

for an absolute implied constant. On appealing once more to Lemma 6.4, we find
that

$\displaystyle\log\left(\prod_{p\leqslant
Q}\left(1+\overline{\omega}(p)\right)\right)=~{}$
$\displaystyle\sum_{\begin{subarray}{c}p\leqslant Q\\\ p\not\in
T_{0}\end{subarray}}\frac{1}{p}+\sum_{\begin{subarray}{c}p\leqslant Q\\\
p\not\in
T_{\infty}\end{subarray}}\frac{1}{p}+\sum_{j=1}^{r}\sum_{\begin{subarray}{c}p\leqslant
Q\\\ p\not\in
U_{j}\end{subarray}}\frac{\nu_{j}(p;\mathbf{v})}{p^{2}}+O\left(1+\omega(A_{0}A_{1})\right).$

These sums are estimated using Lemmas 6.5 and 6.6, leading to the conclusion
that

$\log\left(\prod_{p\leqslant
Q}\left(1+\overline{\omega}(p)\right)\right)=\widetilde{\Theta}(\pi,v_{1}/v_{0})\log\log
Q+O\left(1+\omega(A_{0}A_{1})\right),$

where

$\widetilde{\Theta}(\pi,v_{1}/v_{0})=2-\delta_{0,\mathbb{Q}}(\pi)-\delta_{\infty,\mathbb{Q}}(\pi)+\sum_{j=1}^{r}\sum_{k=1}^{s}\left(1-\delta_{E_{j},N_{E_{j},d,v_{1}/v_{0}}^{(k)}}(\pi)\right),$

in the notation of (1.5). Clearly
$\widetilde{\Theta}(\pi,v_{1}/v_{0})=\Theta_{v}(\pi)$, the latter being
defined in (1.9). Hence, the statement of the lemma follows on exponentiating
and using the fact that $\omega(n)\ll(\log|n|)/(\log\log|n|)$ for any non-zero
$n\in\mathbb{Z}$. ∎

###  6.4. Completion of the proof of Proposition 6.1

We begin by focusing on the estimation of the quantity $L(R_{0},R_{1})$ that
was defined in (6.4). In view of (6.2), we see that
$A_{0}=v_{0}(b_{0}s_{0}^{m_{0}/d}u_{0})^{d}$ and
$A_{1}=v_{1}(b_{1}s_{1}^{m_{\infty}/d}u_{1})^{d}$. Recall that $s_{i}\mid
v_{i}$ for $i=0,1$. Taking $Q=B^{\varepsilon}$, we note that

$R_{0}^{m_{0}}=\frac{B}{|A_{0}|}\geqslant\frac{B}{|v_{0}(s_{0}b_{0}u_{0})^{m_{0}}|}\geqslant
B^{1-(1+3m_{0})\varepsilon}\geqslant Q^{4m_{0}},$

provided that $\varepsilon\leqslant 1/(1+7m_{0})$. Similarly, we can assume
that $R_{1}\geqslant Q^{4}$ if $\varepsilon>0$ is chosen to be sufficiently
small. Hence, with these choices, we’ll have

$(R_{0}+Q^{4})(R_{1}+Q^{4})\ll
R_{0}R_{1}\ll\frac{B^{1/m_{1}+1/m_{\infty}}}{|A_{0}|^{1/m_{0}}|A_{1}|^{1/m_{\infty}}}.$

We may now apply Lemma 6.7 in Lemma 6.3 to deduce that

$\displaystyle L(R_{0},R_{1})\ll_{\varepsilon}~{}$
$\displaystyle\frac{B^{1/m_{0}+1/m_{\infty}}}{|A_{0}|^{1/m_{0}}|A_{1}|^{1/m_{\infty}}}\cdot\frac{|A_{0}A_{1}|^{\varepsilon}}{(\log
B)^{\Theta_{v}(\pi)}}.$

Substituting this into (6.6), recalling (6.5) and summing over
$b_{0},b_{1},u_{0},u_{1}$, the statement of Proposition 6.1 easily follows.

##  7\. Examples: lower bounds and asymptotics

Let $\pi:X\to\mathbb{P}^{1}$ be a standard fibration. It is clear from the
constructions in Section 5 that we only be able to interpret local solubility
conditions outside a given finite set $S$ of primes. With more work one might
be able to incorporate local solubility at places in $S$, but this should not
change the order of growth, which is the main interest in this paper.
Accordingly, for any finite set $S$ of primes, we introduce the counting
function

$N_{\text{loc},S}(\pi,B)=\\#\left\\{x\in\mathbb{P}^{1}(\mathbb{Q})\cap\pi(X(\mathbf{A}_{\mathbb{Q}}^{S})):H(x)\leqslant
B\right\\},$

where $H$ is the usual height function on $\mathbb{P}^{1}(\mathbb{Q})$ and
$\mathbf{A}_{\mathbb{Q}}^{S}$ is the set of adèles away from $S$. We clearly
have $N_{\text{loc},S}(\pi,B)\geqslant N_{\text{loc}}(\pi,B)$ and we expect
these two counting functions to have the same order of magnitude.

We shall prove several results about Halphen surfaces. Let $m>1$ be an
integer. A Halphen pencil is a geometrically irreducible pencil of plane
curves of degree $3m$ with multiplicity $m$ at $9$ base points
$P_{1},\dots,P_{9}$. We let $X$ be the Halphen surface of order $m$ obtained
by blowing up $\mathbb{P}^{2}$ at these nine points, as introduced by Halphen
[10] in 1882. We shall assume that $P_{1},\dots,P_{9}$ are globally defined
over $\mathbb{Q}$, so that $X$ is a smooth, proper, geometrically integral
surface defined over $\mathbb{Q}$. In fact, $X$ is a rational elliptic surface
and we obtain a standard morphism $\pi:X\to\mathbb{P}^{1}$, such that there
exists a unique fibre of multiplicity $m$. In particular, $\pi$ does not admit
a section.

###  7.1. Lower bounds

In this section we establish some lower bounds for $N_{\text{loc,S}}(\pi,B)$.
The following result demonstrates that Conjecture 1.5 would be false with the
exponent $\Delta(\pi)$ and that it is indeed sometimes necessary to take a
smaller exponent.

######  Theorem 7.1.

Let $\pi:X\to\mathbb{P}^{1}$ be a standard fibration. Assume it only has non-
split fibres above $0$, $1$ and $\infty$, comprising geometrically irreducible
double fibres over $0$ and $\infty$, and a non-split fibre of multiplicity one
above $1$ that is split by a quadratic extension. Then

$B\ll N_{\text{loc},S}(\pi,B)\ll B.$

###### Proof.

Suppose that $F=\mathbb{Q}(\sqrt{d})$ is the quadratic extension that splits
the fibre above $1$, for square-free $d\in\mathbb{Z}$. Then it is clear that

$\displaystyle 0\leqslant\Theta(\pi)$
$\displaystyle=\min_{\text{$K/\mathbb{Q}$
quadratic}}\left(1-\delta_{1,K}(\pi)\right)\leqslant 1-\delta_{1,F}(\pi)=0.$

Hence the upper bound is a direct consequence of Theorem 1.3.

For the lower bound we compose the exact counting problem, using Corollary
5.7. Thus there exists a finite set of places $S$, containing the prime
divisors of $2d$, such that

$N_{\text{loc},S}(\pi,B)=\frac{1}{2}\\#\left\\{(a,b)\in\mathbb{Z}_{\mathrm{prim}}^{2}\colon\begin{array}[]{l}|a|,|b|\leqslant
B\\\ p\not\in S\Rightarrow 2\mid v_{p}(a)\text{ and }2\mid v_{p}(b)\\\
p\not\in S,p\mid a-b\Rightarrow p\in\mathscr{P}_{F}\end{array}\right\\}.$

The lower bound is provided by taking pairs $(a,b)$ of the form
$(u^{2},dv^{2})$. ∎

In this result we have $2-\deg\partial_{\pi}=1$, so that the exponent of $B$
matches the predicted exponent of $B$ in Conjectures 1.1 and 1.5. We also have
$\delta_{0,\mathbb{Q}}(\pi)=\delta_{\infty,\mathbb{Q}}(\pi)=1$ and
$\delta_{1,\mathbb{Q}}(\pi)=\frac{1}{2}$, so that $\Delta(\pi)=\frac{1}{2}$.
However, we saw in the proof that $\Theta(\pi)=0$. Thus Theorem 7.1 is in
agreement with Conjecture 1.5.

Let us describe what is going on geometrically. Consider the finite étale
orbifold $\mu_{2}$-cover $\theta_{v}\colon\mathbb{P}^{1}\to\mathbb{P}^{1}$
given by $(x\colon y)\mapsto(x^{2}\colon vy^{2})$, and the pullback fibrations
$\pi_{v}\colon X_{v}\to\mathbb{P}^{1}$ obtained from normalisation of the
pullback of $\pi$ along $\theta_{v}$. By Proposition 4.7 we see that the two
double fibres of $\pi$ pull back to components of mutiplicity one on
$\pi_{v}$. Also, all fibres which do not lie over $1$ in the composition
$X_{v}\xrightarrow{\pi_{v}}\mathbb{P}^{1}\xrightarrow{\theta_{v}}\mathbb{P}^{1}$
are split. We proceed by studying the fibres over $1$.

First we study the fibre of $1$ in $\theta_{v}$. For
$v\in\mathbb{Q}^{\times}/\mathbb{Q}^{\times,2}$, we have
$\theta^{-1}(1)=\operatorname{Spec}A$, where $A$ is the degree $2$ étale
algebra $\mathbb{Q}(\sqrt{v})$ if $v\not\in\mathbb{Q}^{\times,2}$, and
$\mathbb{Q}\times\mathbb{Q}$ for $v\in\mathbb{Q}^{\times,2}$. This gives

$\Delta(\pi_{v})=\sum_{D^{\prime}\mid
D}\left(1-\delta_{D^{\prime}}(\pi_{v})\right)=\begin{cases}0&\text{ if
}v\equiv d,\\\ \frac{1}{2}+\frac{1}{2}=1&\text{ if }v\equiv 1,\\\
\frac{1}{2}&\text{ otherwise},\end{cases}$

where the sum range over all points $D^{\prime}$ lying above
$D=1\in(\mathbb{P}^{1})^{(1)}$. In the first case, the fibre over $1$ (which
is split by $F$) pulls back to a $F$-point, and becomes split. In the second,
case the fibre pulls back to two $\mathbb{Q}$-points. In the last case, the
fibre is irreducible and its residue field is linearly disjoint from the
splitting field, and we obtain $\Delta(\pi_{v})=\Delta(\pi)$, in general.

Theorem 7.1 indicates that the main contribution to the point count comes from
the single cover $\pi_{d}$. If we were to exclude the thin set of points
coming from this cover, we are left with infinitely many covers $\pi_{v}$,
with $\Delta(\pi_{v})=\Delta(\pi)$ for $v\neq 1$. Proposition 4.7(b) implies
that the covers have no multiple fibres, since it gives

$m_{P^{\prime}}=\frac{m_{P}}{\gcd\\!\left(m_{P},e(P^{\prime}/P)\right)}=\frac{2}{\gcd(2,2)}=1,$

for each $P^{\prime}\mid 0,\infty$. Hence, in the light of the original
Loughran–Smeets conjecture [15, Conj. 1.6], we expect the remaining covers to
contribute order $B/\sqrt{\log B}$ to the counting function, apart from the
cover corresponding to $1$, which should contribute order $B/\log B$.

Our second lower bound deals with the case of precisely two non-split fibres
and is consistent with Conjecture 1.5, since
$\deg\partial_{\pi}=2-\frac{1}{m_{0}}-\frac{1}{m_{\infty}}$.

######  Theorem 7.2.

Let $\pi:X\to\mathbb{P}^{1}$ be a standard fibration for which the only non-
split fibres lie over $0$ and $\infty$. Then

$N_{\text{loc,S}}(\pi,B)\gg\frac{B^{\frac{1}{m_{0}}+\frac{1}{m_{\infty}}}}{(\log
B)^{\Delta(\pi)}}.$

###### Proof.

We begin by using Corollary 5.7 to give explicit conditions for local
solubility away from $S$, after passing to a sncd model
$X^{\prime}\to\mathbb{P}^{1}$. This leads to the conclusion that
$N_{\text{loc},S}(\pi,B)$ is equal to the number of $x=(x_{0}\colon
x_{1})\in\mathbb{P}^{1}(\mathbb{Q})$ with $H(x)\leqslant B$, such that for
each $i\in\\{0,1\\}$ and every $p\not\in S$, $\operatorname{Frob}_{p}$ fixes a
collection of intersecting components $Z_{j}$ of $X^{\prime}_{D_{i}}$ such
that $v_{p}(x_{i})\in\langle m(Z_{j})\rangle_{\mathbb{N}}$, where
$D_{i}=V(x_{i})$. The following is clearly a sufficient condition for the
fibre over $x$ to have $\mathbb{Q}_{p}$-point: for all $i$, the Frobenius
$\operatorname{Frob}_{p}$ fixes a component of $Z$ of minimimal multiplicity
in $X^{\prime}_{D_{i}}$, and $m(Z)\mid v_{p}(x_{i})$. The density
$\partial_{i}$ of rational primes $p$ for which $\operatorname{Frob}_{p}$
fixes an element of $S_{D_{i}}$ is equal to
$\delta_{D_{i}}(\pi)=\delta_{D_{i},\kappa(D_{i})}(\pi)$, in the notation of
(1.5). Hence the statement of the theorem now follows from Proposition 3.1 and
(1.7). ∎

###  7.2. Halphen surfaces with one non-split fibre

Generically, a Halphen surface has no other non-split fibre apart from the
multiple one. Even in these cases the counting problem still depends on the
Galois action on the components of the multiple fibres, and how these
components intersect. We proceed to record some results which illustrate this
phenomenon, in the course of which it will be convenient to keep in mind the
notation (3.1).

We begin with the following result, which agrees with Conjecture 1.5, since
$\deg\partial_{\pi}=1-\frac{1}{m}$ and $\Delta(\pi)=0$.

######  Theorem 7.3.

Let $X\to\mathbb{P}^{1}$ be a Halphen surface with a single non-split fibre
over $0$, that is the fibre of multiplicity $m$. Suppose that this fibre has a
geometric component fixed by ${\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$.
Then there exists a finite set $S$ such that

$N_{\text{loc},S}(\pi,B)\sim c_{\pi,S}B^{1+\frac{1}{m}},$

where

$c_{\pi,S}=\frac{12c_{S}(1+\frac{1}{m})}{\pi^{2}c_{S}(\frac{1}{m})}\prod_{p\in
S}\left(1+\frac{1}{p}\right)^{-1}.$

###### Proof.

By Corollary 5.7 we see that there is a finite set of places $S$ such that

$N_{\text{loc},S}(\pi,B)=\frac{1}{2}\\#\\{(a,b)\in\mathbb{Z}_{\mathrm{prim}}^{2}\colon|a|,|b|\leqslant
B,~{}p\not\in S\Rightarrow m\mid v_{p}(a)\\}.$

We may apply Proposition 3.1 with $m_{0}=m$ and $m_{1}=1$, and with
$\mathscr{P}_{0}=\mathscr{P}_{1}$ equal to the full set of rational primes. In
particular $\partial_{0}=\partial_{1}=1$ and it follows that
$N_{\text{loc},S}(\pi,B)\sim c_{\pi,S}B^{1+\frac{1}{m}}$, as $B\to\infty$,
where

$\displaystyle c_{\pi,S}=~{}$
$\displaystyle\frac{2c_{S}(1+\frac{1}{m})}{c_{S}(1)c_{S}(\frac{1}{m})}\prod_{\begin{subarray}{c}p\not\in
S\end{subarray}}\left(1-\frac{1}{p^{2}}\right)\prod_{p\in
S}\left(1-\frac{1}{p}\right)^{2},$

in the notation of (3.1). The statement easily follows on simplifying the
expression for the constant. ∎

The following two result agree with Conjecture 1.5, since in both cases we
have $\deg\partial_{\pi}=1-\frac{1}{m}$ and $\Delta(\pi)=\frac{2}{3}$.
Moreover, in these two examples, we have multiple fibres which do not have a
geometrically integral component. This demonstrates the need to define (1.5)
in terms of $S_{D}$, for each divisor $D$, which allows us to work with the
Galois action on the components of a fibre of minimum multiplicity.

######  Theorem 7.4.

Let $X\to\mathbb{P}^{1}$ be a Halphen surface with a single non-split fibre
over $0$, that is the fibre of multiplicity $m$. Suppose that this fibre
consists of three conjugate lines split by a cubic Galois extension
$K/\mathbb{Q}$ that do not all meet in a point. Then there exists a finite set
$S$ such that

$N_{\text{loc},S}(\pi,B)\sim c_{\pi,S}\frac{B^{1+\frac{1}{m}}}{(\log
B)^{\frac{2}{3}}},$

where

$\displaystyle c_{\pi,S}=~{}$
$\displaystyle\frac{2m^{\frac{2}{3}}c_{S}(1)^{\frac{1}{3}}c_{S}(1+\frac{1}{m})}{\Gamma(\frac{1}{3})c_{S}(\frac{1}{m})}\prod_{\begin{subarray}{c}p\in\mathscr{P}_{K}\\\
p\not\in
S\end{subarray}}\left(1+\frac{1}{p}\right)\left(1-\frac{1}{p}\right)^{\frac{1}{3}}\prod_{\begin{subarray}{c}p\not\in\mathscr{P}_{K}\\\
p\not\in S\end{subarray}}\left(1-\frac{1}{p}\right)^{\frac{1}{3}}.$

###### Proof.

Suppose that the three conjugate lines are split by the cubic Galois extension
$K/\mathbb{Q}$. By Corollary 5.7 we see that there is a finite set of places
$S$ such that $N_{\text{loc},S}(\pi,B)$ is equal to

$\frac{1}{2}\\#\left\\{(a,b)\in\mathbb{Z}_{\mathrm{prim}}^{2}\colon\begin{array}[]{l}|a|,|b|\leqslant
B\\\ \big{[}p\not\in S\text{ and }p\mid a\big{]}\Rightarrow\big{[}m\mid
v_{p}(a)\text{ and }p\in\mathscr{P}_{K}\big{]}\end{array}\right\\},$

where $\mathscr{P}_{K}$ is the set of rational primes $p$ that are unramified
in $K$ and split completely. We may apply Proposition 3.1 with $m_{0}=m$ and
$m_{1}=1$, and with $\mathscr{P}_{0}=\mathscr{P}_{K}$ and $\mathscr{P}_{1}$
equal to the full set of rational primes. In particular $\partial_{0}=1/3$ and
$\partial_{1}=1$. It follows that

$N_{\text{loc},S}(\pi,B)\sim c_{\pi,S}\frac{B^{1+\frac{1}{m}}}{(\log
B)^{\frac{2}{3}}}$

where

$\displaystyle c_{\pi,S}=~{}$
$\displaystyle\frac{2m^{\frac{2}{3}}}{\Gamma(\frac{1}{3})}\cdot\frac{c_{S}(1+\frac{1}{m})}{c_{S}(\frac{1}{m})}\prod_{\begin{subarray}{c}p\in\mathscr{P}_{K}\\\
p\not\in S\end{subarray}}\left(1-\frac{1}{p^{2}}\right)$
$\displaystyle\quad\times\prod_{p\in\mathscr{P}_{K}\cap
S}\left(1-\frac{1}{p}\right)\prod_{p\in\mathscr{P}_{K}}\left(1-\frac{1}{p}\right)^{-\frac{2}{3}}\prod_{p\not\in\mathscr{P}_{K}}\left(1-\frac{1}{p}\right)^{\frac{1}{3}}.$

The statement of the proposition follows on simplifying this expression. ∎

The next result agrees with Conjecture 1.5, since
$\deg\partial_{\pi}=1-\frac{1}{m}$ and $\Delta(\pi)=\frac{2}{3}$.

######  Theorem 7.5.

Let $X\to\mathbb{P}^{1}$ be a Halphen surface with a single non-split fibre
over $0$, that is the fibre of multiplicity $m$. Suppose that this fibre
consists of three conjugate lines split by a cubic Galois extension
$K/\mathbb{Q}$ that do meet in a point. Then there exists a finite set $S$
such that

$\frac{B^{1+\frac{1}{m}}}{(\log B)^{\frac{2}{3}}}\ll
N_{\text{loc},S}(\pi,B)\ll\frac{B^{1+\frac{1}{m}}}{(\log B)^{\frac{2}{3}}}.$

###### Proof.

The upper bound follows from Theorem 1.2. The lower bound was proven in
Theorem 7.2. ∎

Theorem 7.5 illustrates the need for the non-split fibres to be sncd; the
counting problem for this setting is

$p\not\in S,p\mid a\Rightarrow\Big{[}\big{(}3m\mid v_{p}(a)\big{)}\text{ or
}\big{(}m\mid v_{p}(a)\text{ and }p\in\mathscr{P}_{K}\big{)}\Big{]}.$

The condition $3m\mid v_{p}(a)$ comes from a Galois fixed component of
multiplicity $3m$ on the multiple fibre of the sncd-model of $X$. However, no
such component exists on the multiple fibre of $X$ itself.

###  7.3. Halphen surfaces with two non-split fibres

In practice, it can be difficult to construct Halphen surfaces with more than
one non-split fibre. We present two such examples, both of which verify
Conjecture 1.5.

######  Theorem 7.6.

There exists a Halphen surface $X\to\mathbb{P}^{1}$ of degree $2$ with two
non-split fibres: the multiple fibre is geometrically irreducible and has
multiplicity $2$, and the other is a sncd divisor of Kodaira classification
$I_{6}$ split by a cubic Galois extension $K/\mathbb{Q}$. Moreover, there
exists a finite set of places $S$, and an explicit constant $c_{\pi,S}>0$ such
that

$N_{\text{loc},S}(\pi,B)\sim c_{\pi,S}\frac{B^{1+\frac{1}{2}}}{(\log
B)^{\frac{2}{3}}}.$

###### Proof.

Let us first fix the cyclic cubic number field $K/\mathbb{Q}$. Now choose two
sets of three conjugate points $P_{i},Q_{i}\in\mathbb{P}^{2}(K)$, indexed by
$i\in\mathbb{Z}/3\mathbb{Z}$. We let $R_{i}$ be the intersecting point of the
lines $P_{i+1}P_{i+2}$ and $Q_{i+1}Q_{i+2}$. For generic choices of $P_{i}$
and $Q_{i}$, the $R_{i}$ are well-defined and there is a unique smooth cubic
through the nine points $P_{i}$, $Q_{i}$ and $R_{i}$.

We will consider $X=\operatorname{Bl}_{P_{i},Q_{i},R_{i}}\mathbb{P}^{2}$. The
two non-split fibres of $X$ come from the double cubic passing through these
nine points, and the sextic curve which is geometrically the union of the six
lines $P_{i+1}P_{i+2}$ and $Q_{i+1}Q_{i+2}$. Under blowup the first curve
turns into a geometrically integral fibre of multiplicity $2$, and the other
into six lines meeting in a cycle. The three lines $P_{1}P_{2}$, $P_{2}P_{3}$
and $P_{3}P_{1}$ are permuted by ${\rm Gal}(K/\mathbb{Q})$ and no longer meet
on $X$. For a generic choice of $P_{i}$ and $Q_{i}$ there will be no other
non-split fibres.

Let us assume the multiple fibre lies above $0$ and the other non-split fibre
over $\infty$. The fibres of $X\to\mathbb{P}^{1}$ are all sncd, so we can
directly compose the counting problem to find that

$N_{\text{loc},S}(\pi,B)=\frac{1}{2}\\#\left\\{(a,b)\in\mathbb{Z}_{\mathrm{prim}}^{2}\colon\begin{array}[]{l}|a|,|b|\leqslant
B\\\ p\not\in S\Rightarrow 2\mid v_{p}(a)\\\ \big{[}p\not\in S\text{ and
}p\mid b\big{]}\Rightarrow p\in\mathscr{P}_{K}\end{array}\right\\}.$

Such a counting problem is dealt with by Proposition 3.1. ∎

######  Theorem 7.7.

There exists a Halphen surface $X\to\mathbb{P}^{1}$ of degree $3$ with two
non-split fibres: the multiple fibre is geometrically irreducible and has
multiplicity $3$, and the other is a non-sncd divisor of Kodaira
classification $I_{3}$ split by a cubic Galois extension $K/\mathbb{Q}$.
Moreover, there exists a finite set of places $S$ such that

$\frac{B^{1+\frac{1}{3}}}{(\log B)^{\frac{2}{3}}}\ll
N_{\text{loc},S}(\pi,B)\ll\frac{B^{1+\frac{1}{3}}}{(\log B)^{\frac{2}{3}}}.$

We will return to this surface in Section 7.4 to create another interesting
example. There we will assume that the multiple fibre lies over $0$ and the
remaining non-split fibre lies over $\infty$.

###### Proof of Theorem 7.7.

Let $E/\mathbb{Q}$ be an elliptic curve with
$E(\mathbb{Q})_{\text{tors}}=\mathbb{Z}/9\mathbb{Z}$. Let $K/\mathbb{Q}$ be a
cyclic cubic number field $K/\mathbb{Q}$, such that
$\operatorname{rank}E(\mathbb{Q})<\operatorname{rank}E(K)$. We will fix

  * (i)

a generator $\sigma\in{\rm Gal}(K/\mathbb{Q})$,

  * (ii)

a generator $A\in E(\mathbb{Q})_{\text{tors}}$,

  * (iii)

$B\in E(K)\setminus E(\mathbb{Q})$ such that $B+\sigma(B)+\sigma^{2}(B)=O\in
E(\mathbb{Q})$, and any

  * (iv)

$C\in E(K)\setminus E(\mathbb{Q})$.

With this notation in mind, consider the nine points

$P_{i}=\sigma^{i}(C),\quad Q_{i}=\sigma^{i}(-2C+B+A)\quad\text{ and
}R_{i}=\sigma^{i}(C+2A).$

For general choices, we find that
$\operatorname{Bl}_{P_{i},Q_{i},R_{i}}\mathbb{P}^{2}$ is a Halphen surface of
degree $3$. In particular, there is a smooth cubic through the nine points,
which becomes the geometrically irreducible triple fibre on $X$. Moreover, we
have

$\sum_{i}(P_{i}+Q_{i}+R_{i})-P_{j}+R_{j}=O,$

so that there is a cubic curve which passes through all nine points except
$P_{j}$ and has a singularity at $R_{j}$. The union of these three curves
becomes the the non-split $I_{3}$-fibre, split by $K$.

For the lower bound we may apply Theorem 7.2 and the upper bound follows from
Theorem 1.2. ∎

###  7.4. A non-split fibre over a point of higher degree

Our final result concerns a surface of Halphen type, with a fibration over
$\mathbb{P}^{1}$ that has one multiple fibre and a non-split fibre over a
degree $2$ point. Our local solubility criteria do not apply to this case in
general, but we are nonetheless able to deduce explicit criteria.

Consider the Halphen surface $\pi:X\to\mathbb{P}^{1}$ from Theorem 7.7 with
$m=3$, with a multiple fibre over $0$ and a non-split fibre over $\infty$
split by a Galois cubic extension $K/\mathbb{Q}$. Let
$\pi^{\prime}:X^{\prime}\to\mathbb{P}^{1}$ be the normalisation of the
pullback of $\pi$ along the morphism
$\theta\colon\mathbb{P}^{1}\to\mathbb{P}^{1}$ given by $[u\colon
v]\mapsto[u^{2}\colon u^{2}+v^{2}]$. We claim that the surface $X^{\prime}$
has a unique multiple fibre over $u=0$, whose multiplicity is $3$, and that
the only other non-split fibre lies over the degree $2$ point $u^{2}+v^{2}=0$,
and is split by $K$. To see this we note that the fibres of the pullback of
$X$ are precisely the fibres of $X^{\prime}$, and normalisation only changes
the fibres over $0$ and $\infty$. The multiplicities of the new fibres can
then be computed using Proposition 4.3. Note that
$\partial_{\pi^{\prime}}=\frac{2}{3}[0]$ and
$\Delta(\pi^{\prime})=1-\delta_{u^{2}+v^{2}}(\pi^{\prime})=\frac{2}{3}.$ We
shall now prove the following result, which is easily seen to agree with the
prediction in Conjecture 1.5.

######  Theorem 7.8.

For the surface $\pi^{\prime}:X^{\prime}\to\mathbb{P}^{1}$ as above, there
exists a finite set $S$ such that

$\frac{B^{\frac{4}{3}}}{(\log B)^{\frac{2}{3}}}\ll
N_{\text{loc},S}(\pi^{\prime},B)\ll\frac{B^{\frac{4}{3}}}{(\log
B)^{\frac{2}{3}}}.$

###### Proof.

The upper bound follows directly from Theorem 1.2. To prove the lower bound,
we note that for all but finitely many points
$x\in\mathbb{P}^{1}(\mathbb{Q})$, the fibre of $X^{\prime}\to\mathbb{P}^{1}$
is isomorphic to the fibre of $X\to\mathbb{P}^{1}$ over
$\theta(x)\in\mathbb{P}^{1}(\mathbb{Q})$. Hence we can apply the criterion in
Corollary 5.7 to determine local solubility for $X$. Noting that
$v_{p}(u^{2})$ is divisible by $3$ precisely if this is true for $v_{p}(u)$,
we find that $N_{\text{loc},S}(\pi^{\prime},B)$ is

$\displaystyle\frac{1}{2}\\#\left\\{(u,v)\in\mathbb{Z}_{\mathrm{prim}}^{2}:\begin{array}[]{l}|u|,|v|\leqslant
B\\\ p\not\in S\Rightarrow 3\mid v_{p}(u)\\\ \big{[}p\not\in S\text{ and
}p\mid u^{2}+v^{2}\big{]}\Rightarrow
p\in\mathscr{P}_{K}\end{array}\right\\}+O(1).$

On restricting to positive coprime $u,v$ and demanding that $u$ is cube, we
arrive at the lower bound

$N_{\text{loc},S}(\pi^{\prime},B)\geqslant\frac{1}{2}M(B)+O(1),$

where

$\displaystyle
M(B)=\\#\left\\{(u,v)\in\mathbb{Z}_{\mathrm{prim}}^{2}:\begin{array}[]{l}0\leqslant
u^{3},v\leqslant B\\\ p\mid u^{6}+v^{2}\Rightarrow
p\in\mathscr{P}_{K}\end{array}\right\\}.$

Note that $u^{3},v\leqslant B$ whenever $u^{6}+v^{2}\leqslant B^{2}$. Hence

$M(B)\geqslant\\#\left\\{(u,v)\in\mathbb{Z}_{\geqslant
0}^{2}:\begin{array}[]{l}\gcd(u,v)=1,~{}u^{6}+v^{2}\leqslant B^{2}\\\ p\mid
u^{6}+v^{2}\Rightarrow p\in\mathscr{P}_{K}\end{array}\right\\}.$

The right hand side is exactly the quantity estimated via the $\beta$-sieve by
Friedlander and Iwaniec [8, Thm. 11.31], with the outcome that

$M(B)\gg\left(\frac{B^{2}}{\log(B^{2})}\right)^{\frac{2}{3}}.$

The statement of the theorem now follows. ∎

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