POLYNOMIALS REPRESENTED BY NORM FORMS VIA THE BETA SIEVE

Abstract

A central question in Arithmetic geometry is to determine for which polynomials $f \in \mathbb {Z}[t]$ and which number fields K the Hasse principle holds for the affine equation $f(t) = \mathbf {N}_{K/\mathbb {Q}}(\mathbf {x}) \neq 0$. Whilst extensively studied in the literature, current results are largely limited to polynomials and number fields of low degree. In this paper, we establish the Hasse principle for a wide family of polynomials and number fields, including polynomials that are products of arbitrarily many linear, quadratic or cubic factors. The proof generalises an argument of Irving [27], which makes use of the beta sieve of Rosser and Iwaniec. As a further application of our sieve results, we prove new cases of a conjecture of Harpaz and Wittenberg on locally split values of polynomials over number fields, and discuss consequences for rational points in fibrations.

1 Introduction

Let K be a number field of degree n, and let $f \in \mathbb {Z}[t]$ be a polynomial. A central problem in arithmetic geometry is to determine under what conditions f can take values equal to a norm of an element of K. In order to address this question, we take an integral basis $\omega _1, \ldots , \omega _n$ for K, viewed as a vector space over $\mathbb {Q}$ , and define the norm form as $\mathbf {N}(\mathbf {x}) = N_{K/\mathbb {Q}}(\omega _1x_1+ \cdots +\omega _nx_n)$ , where $N_{K/\mathbb {Q}}(\cdot )$ is the field norm. We then seek to understand when the equation

(1.1) $$ \begin{align} f(t) = \mathbf{N}(\mathbf{x}) \neq 0 \end{align} $$

has a solution with $(t,x_1, \ldots , x_n) \in \mathbb {Q}^{n+1}$ . A necessary condition for solubility over $\mathbb {Q}$ is that (1.1) must have solutions in $\mathbb {R}^{n+1}$ and in $\mathbb {Q}_p^{n+1}$ for every prime p. We say that the Hasse principle holds if this condition alone is sufficient to guarantee existence of a solution to (1.1) over $\mathbb {Q}$ .

Local to global questions for (1.1) have received much attention over the years. The first case to consider is when f is a nonzero constant polynomial. Here, the Hasse principle for (1.1) is known as the Hasse norm principle. More precisely, we say that the Hasse norm principle holds for the extension $K/\mathbb {Q}$ if $\mathbb {Q}^{\times }\cap N_{K/\mathbb {Q}}(I_K) = N_{K/\mathbb {Q}}(K^{\times })$ , where $I_K = \mathbb {A}_K^{\times }$ is the group of ideles of K. The Hasse norm principle has been extensively studied, beginning with the work of Hasse himself, who established that it holds for cyclic extensions $K/\mathbb {Q}$ (a result known as the Hasse norm theorem), but does not hold for certain biquadratic extensions, such as $K = \mathbb {Q}(\sqrt {13},\sqrt {-3})$ . The Hasse norm principle is also known to hold if the degree of K is prime (Bartels [1]) or if the normal closure of K has Galois group $S_n$ (Kunyavskiı̆ and Voskresenskiı̆ [32]) or $A_n$ for $n \neq 4$ (Macedo [35]).

When $[K:\mathbb {Q}]=2$ and f is irreducible of degree $3$ or $4$ , (1.1) defines a Châtelet surface. There are now many known counterexamples to the Hasse principle for Châtelet surfaces, including one by Iskovskikh [28], which we discuss in more detail in Example 5.4. However, Colliot-Thélène, Sansuc and Swinnerton-Dyer [14] prove that the Brauer–Manin obstruction accounts for all failures of the Hasse principle. A similar result holds when f is an irreducible polynomial of degree at most $3$ and $[K:\mathbb {Q}]=3$ , as proved by Colliot-Thélène and Salberger [8]. Both of these results make use of fibration and descent methods.

In the case when f is an irreducible quadratic and K is a quartic extension containing a root of f, the Hasse principle and weak approximation are known to hold for (1.1) thanks to the work of Browning and Heath-Brown [4]. This result was generalised by Derenthal, Smeets and Wei [16, Theorem 2] to prove that the Brauer–Manin obstruction is the only obstruction to the Hasse principle and weak approximation for irreducible quadratics f and arbitrary number fields K. Moreover, in [16, Theorem 4], they give an explicit description of the Brauer groups that can be obtained in this family.

Results when f is not irreducible have so far been limited to products of linear polynomials. Suppose that f takes the form

(1.2) $$ \begin{align} f(t) = c\prod_{i=1}^r (t-e_i)^{m_i}, \end{align} $$

for some $c\in \mathbb {Q}^*, e_1, \ldots , e_r \in \mathbb {Q}$ and $m_1, \ldots , m_r \in \mathbb {N}$ . When $r=1$ , the Brauer–Manin obstruction is the only obstruction to the Hasse principle and weak approximation for any smooth projective model of (1.1). This is a special case of the work of Colliot-Thélène and Sansuc [9] on principal homogeneous spaces under algebraic tori. Heath-Brown and Skorobogatov [26] treat the case $r=2$ by combining descent methods with the Hardy–Littlewood circle method, under the assumption that $\gcd (m_1,m_2,\deg K)=1$ . This assumption was later removed by Colliot-Thélène, Harari and Skorobogatov [13]. Thanks to the work of Browning and Matthiesen [5], it is now settled that for any number field K and polynomial f of the form (1.2) (for arbitrary $r \geqslant 1$ ), the Brauer–Manin obstruction is the only obstruction to the Hasse principle and weak approximation for any smooth projective model of (1.1). Their result is inspired by additive combinatorics results of Green, Tao and Ziegler [20], [21], combined with vertical torsors introduced by Schindler and Skorobogatov [40].

In general, it has been conjectured by Colliot-Thélène [7] that all failures of the Hasse principle for any smooth projective model of (1.1) are explained by the Brauer–Manin obstruction. Assuming Schinzel’s hypothesis, this holds true for f an arbitrary polynomial and $K/\mathbb {Q}$ a cyclic extension, as demonstrated by work of Colliot-Thélène and Swinnerton-Dyer on pencils of Severi–Brauer varieties [12]. Recently, Skorobogatov and Sofos also establish unconditionally that when $K/\mathbb {Q}$ is cyclic, (1.1) satisfies the Hasse principle for a positive proportion of polynomials f of degree d, when their coefficients are ordered by height [42, Theorem 1.3].

In [27], Irving introduces an entirely new approach to studying the Hasse principle for (1.1), which rests on sieve methods. Irving’s main result [27, Theorem 1.1] states that if $f\in \mathbb {Z}[t]$ is an irreducible cubic, then the Hasse principle holds for (1.1) under the following assumptions:

1. (1) K satisfies the Hasse norm principle.

2. (2) There exists a prime $q\geqslant 7$ and a finite set of primes S, such that for all $p\notin S$ , either $p\equiv 1 \ (\mathrm {mod}\ q)$ or the inertia degrees of p in $K/\mathbb {Q}$ are coprime.

3. (3) The number field generated by f is not contained in the cyclotomic field $\mathbb {Q}(\zeta _q)$ .

An example provided by Irving in [27] is the number field $\mathbb {Q}(\alpha )$ , where $\alpha $ is a root of $x^q-2$ and $q \geqslant 7$ is prime. We shall comment on this further in Example 5.9.

In this paper, we generalize Irving’s arguments to establish the Hasse principle for a wide new family of polynomials and number fields. Our results cover for the first time polynomials of arbitrarily large degree which are not a product of linear factors. In fact, under suitable assumptions on K, we can deal with polynomials that are products of arbitrarily many linear, quadratic and cubic factors.

Throughout this paper, we let $\widehat {K}$ denote the Galois closure of K, and we let $G = \mathrm {Gal}(\widehat {K}/\mathbb {Q})$ , viewed as a permutation group on n letters. We define

(1.3) $$ \begin{align} T(G) = \frac{1}{\#G}\#\{\sigma \in G: \textrm{ the cycle lengths of }\sigma\textrm{ are not coprime}\}. \end{align} $$

We now state our main results.

Theorem 1.1. Let K be a number field satisfying the Hasse norm principle. Let $f\in \mathbb {Z}[t]$ be a polynomial, all of whose irreducible factors have degree at most 2. Let k denote the number of distinct irreducible factors of f, and let j denote the number of distinct irreducible quadratic factors of f which generate a quadratic field contained in $\widehat {K}$ . Suppose that $T(G) \leqslant \frac {0.39}{k+j+1}$ . Then the Hasse principle holds for (1.1).

In practice, the constant $0.39$ can be improved slightly, particularly when the majority of the factors of f are linear, although it will always be less than $1/2$ . The precise optimal constant is obtained by finding the maximal value of $\kappa $ such that (4.55) holds.

When $G=S_n$ , we shall see in Lemma 5.10 that $T(S_n) \rightarrow 0$ as $n \rightarrow \infty $ , so in the setting of Theorem 1.1, we can establish the Hasse principle provided n is sufficiently large in terms of the degree of f. We illustrate this by treating the case when f is a product of two irreducible quadratics.

Corollary 1.2. Let $f \in \mathbb {Z}[t]$ be a product of two quadratic polynomials such that the number field L generated by f is a biquadratic extension of $\mathbb {Q}$ . Let K be a number field of degree n with $G=S_n$ . Suppose that $L\cap \widehat {K} = \mathbb {Q}$ . Then the Hasse principle holds for (1.1), provided that

$$ \begin{align*}n\not\in\{2,3,\ldots, 10, 12,14,15,16,18,20,22,24,26,28,30,36,42,48\}.\end{align*} $$

We remark that without the assumption $L\cap \widehat {K} = \mathbb {Q}$ , a similar result to Corollary 1.2 still holds, although a larger list of degrees n would need to be excluded. For example, if $L\cap \widehat {K}$ is quadratic, then the Hasse principle holds for (1.1) for all primes $n\geqslant 11$ and all integers $n>90$ , whilst if $L\cap \widehat {K} = L$ , then the Hasse prinicple holds for all primes $n\geqslant 13$ and all integers $n>150$ .

We cannot hope to deal with all small values of n in Corollary 1.2. For example, the work of Iskovskikh [28] shows that the Hasse principle can fail when $n=2$ (see Example 5.4). However, as we shall discuss in Appendix A, in the case $n\geqslant 3$ , there is no Brauer–Manin obstruction to the Hasse principle, and so according to the conjecture of Colliot-Thélène mentioned above, we should expect the Hasse principle to hold.

Our second main result allows f to contain irreducible cubic factors but requires more restrictive assumptions on the number field K, more similar to Irving’s setup in [27].

Theorem 1.3. Let $f\in \mathbb {Z}[t]$ be a polynomial, all of whose irreducible factors have degree at most $3$ . Then the Hasse principle holds for (1.1) under the following assumptions for K.

1. (1) K satisfies the Hasse norm principle.

2. (2) The set $\mathcal {P}$ of primes p for which the inertia degrees of p in $K/\mathbb {Q}$ are not coprime satisfies Assumption 2.2.

As an example, Assumption 2.2 is satisfied if there exists a prime q such that $\frac {\deg f +1}{q-1}\leqslant 0.32380$ , and such that for all but finitely many primes $p\not \equiv 1 \ (\mathrm {mod}\ q)$ , the inertia degrees of p in $K/\mathbb {Q}$ are coprime. The constant $0.32380$ appearing in Assumption 2.2 could likely be improved with more work, and in specific examples, the required bounds could be computed more precisely using (4.59). We remark that we have also dropped the assumption made in [27] that the number field generated by f is not contained in $\mathbb {Q}(\zeta _q)$ . This assumption is not essential to Irving’s argument, but allows for the treatment of smaller values of q. Reinserting this assumption and optimising (4.59), we could recover Irving’s result from our work.

We prove Theorems 1.1 and 1.3 by applying the beta sieve of Rosser and Iwaniec [18, Theorem 11.13]. The main sieve results we obtain are stated in Theorems 2.1 and 2.3. These results in fact prove the existence of a solution to (1.1) with t arbitrarily close to a given adelic solution. Consequently, the above results could be extended to prove weak approximation for (1.1), provided that weak approximation holds for the norm one torus $\mathbf {N}(\mathbf {x})=1$ . For example, the work of Kunyavskiı̆ and Voskresenskiı̆ [32] and Macedo [35] demonstrates that weak approximation for the norm one torus holds when $G=S_n$ or $G=A_n$ and $n \neq 4$ , and so weak approximation holds in the setting of Corollary 1.2.

In Section 6, we find a second application of Theorem 2.1 to a conjecture of Harpaz and Wittenberg [23, Conjecture 9.1], which we restate in Conjecture 6.1 and henceforth refer to as the Harpaz–Wittenberg conjecture. The conjecture concerns a collection of number field extensions $L_i/k_i/k$ , $i\in \{1, \ldots , n\}$ , where $k_i \cong k[t]/(P_i(t))$ for monic irreducible polynomials $P_i \in k[t]$ . Roughly speaking, the conjecture predicts, under certain hypotheses, the existence of an element $t_0\in k$ such that $P_1(t_0), \ldots ,P_n(t_0)$ are locally split (i.e., each place in $k_i$ dividing $P_i(t_0)$ has a degree $1$ place of $L_i$ above it).

A major motivation for the conjecture is the development of the theory of rational points in fibrations. Given a fibration $\pi :X\rightarrow \mathbb {P}^1_{k}$ , a natural question is to what extent we can deduce arithmetic information about X from arithmetic information about the fibres of $\pi $ . A famous conjecture of Colliot-Thélène [7, p.174] predicts that for any smooth, proper, geometrically irreducible, rationally connected variety X over a number field k, the rational points $X(k)$ are dense in the Brauer–Manin set $X(\mathbb {A}_k)^{\operatorname {Br}}$ . (In other words, the Brauer–Manin obstruction is the only obstruction to weak approximation.) Applied to this conjecture, the above question becomes whether density of $X(k)$ in $X(\mathbb {A}_k)^{\operatorname {Br}}$ follows from density of $X_c(k)$ in $X_c(\mathbb {A}_k)^{\operatorname {Br}}$ for a general fibre of $\pi $ (see [24, Question 1.2]). Applications of the Harpaz–Wittenberg conjecture to this question are studied in [23] and [24].

Harpaz and Wittenberg [23, Section 9.2] demonstrate that their conjecture follows from the homogeneous version of Schinzel’s hypothesis (commonly reffered to as $(\textrm {HH}_1)$ ) in the case of abelian extensions $L_i/k_i$ , or more generally, almost abelian extensions (see [23, Definition 9.4]). Examples of almost abelian extensions include cubic extensions, and extensions of the form $k(c^{1/p})/k$ for $c\in k$ and p prime. The work of Heath-Brown and Moroz [25] establishes $(\textrm {HH}_1)$ for primes represented by binary cubic forms, from which the Harpaz–Wittenberg conjecture can be deduced in the case $k=\mathbb {Q}, n=1$ and $\deg P_1 = 3$ . Using a geometric reformulation of [23, Conjecture 9.1], the authors establish their conjecture in low-degree cases – namely, when $\sum _{i=1}^n [k_i:k] \leqslant 2$ or $\sum _{i=1}^n [k_i:k]=3$ and $[L_i:k_i]=2$ for all i.

The Harpaz–Wittenberg conjecture is related to the study of polynomials represented by norm forms. As a consequence of the work of Matthiesen [36] on norms as products of linear polynomials, the Harpaz–Wittenberg conjecture holds in the case $k_1=\cdots = k_n = k =\mathbb {Q}$ [23, Theorem 9.14]. Similarly, we can deduce from [27, Theorem 1.1] that the Harpaz–Wittenberg conjecture holds in the case $n=2, k=\mathbb {Q}, k_1=K,k_2=\mathbb {Q}, L_1 = K(2^{1/q})$ and $L_2 = \mathbb {Q}(2^{1/q})$ , where $q\geqslant 7$ is a prime such that $K\not \subseteq \mathbb {Q}(\zeta _q)$ [23, Theorem 9.15].

Besides the work of Matthiesen [36] for $k_1 = \cdots = k_n = k = \mathbb {Q}$ , the aforementioned results apply only to the case $n\leqslant 2$ . In Section 6, we prove the following theorem, which establishes the Harpaz–Wittenberg conjecture in a new family of extensions $k_1/\mathbb {Q}, \ldots , k_n/\mathbb {Q}$ , where n may be arbitrarily large, and each extension $k_i/\mathbb {Q}$ may have degree up to $3$ .

Theorem 1.4. Let $n\geqslant 1$ . Let $k=\mathbb {Q}$ , and for $i\in \{1, \ldots , n\}$ , let $k_i, M_i$ be linearly disjoint number fields over $\mathbb {Q}$ . Let $L_i = M_ik_i$ be the compositum of $k_i$ and $M_i$ . Define

(1.4) $$ \begin{align}T_i = \frac{1}{\#\mathrm{Gal}(\widehat{M_i}/\mathbb{Q})}\#\{\sigma \in \mathrm{Gal}(\widehat{M}_i/\mathbb{Q}): \sigma \textrm{ has no fixed point}\}. \end{align} $$

Let $d = \sum _{i=1}^n [k_i:\mathbb {Q}]$ . Then the Harpaz–Wittenberg conjecture holds in the following cases.

1. (1) $[k_i:\mathbb {Q}]\leqslant 2$ for all $i\in \{1, \ldots , n\}$ and $\sum _{i=1}^n T_i \leqslant 0.39/d$ .

2. (2) $[k_i:\mathbb {Q}]\leqslant 3$ for all $i\in \{1, \ldots , n\}$ , and there exist primes $q_i$ satisfying $\sum _{i=1}^n 1/(q_i-1)\leqslant 0.32380/d$ , and integers $t_i$ coprime to $q_i$ , such that for all but finitely many primes $p \not \equiv t_i\ (\mathrm {mod}\ q_i)$ , there is a place of degree $1$ in $M_i$ above p.

Corollary 1.5. Let $q_1, \ldots , q_n$ be distinct primes, and let $r_1, \ldots , r_n \in \mathbb {N}$ be such that $g_i(x) = x^{q_i}-r_i$ is irreducible for all i. Let $M_i = \mathbb {Q}[x]/(g_i)$ and let $k_i,L_i$ and d be as in Theorem 1.4. Suppose that one of the following holds:

1. (1) $[k_i:\mathbb {Q}] \leqslant 2$ for all $i\in \{1, \ldots , n\}$ and $\sum _{i=1}^n 1/q_i \leqslant 0.39/d$ ,

2. (2) $[k_i:\mathbb {Q}] \leqslant 3$ for all $i \in \{1,\ldots ,n\}$ and $\sum _{i=1}^n 1/(q_i-1) \leqslant 0.32380/d$ .

Then the Harpaz–Wittenberg conjecture holds for $k=\mathbb {Q}$ and for such choices of $k_i$ and $L_i$ .

We remark that when applied to the setting of [23, Theorem 9.15], the above result requires a stronger bound on q. However, with a more careful optimisation of (4.58), it should be possible to recover [23, Theorem 9.15] from our approach.

By combining Theorem 1.4 with [23, Theorem 9.17] (with the choice $B=0, M" = \emptyset $ and $M' = \mathbb {P}^1_k\backslash U$ ), we obtain the following result about rational points in fibrations.

Theorem 1.6. Let X be a smooth, proper, geometrically irreducible variety over $\mathbb {Q}$ . Let $\pi :X \rightarrow \mathbb {P}^1_{\mathbb {Q}}$ be a dominant morphism whose general fibre is rationally connected. Let $k_1, \ldots , k_n$ denote the residue fields of the closed points of $\mathbb {P}^1_{\mathbb {Q}}$ above which $\pi $ has nonsplit fibres, and let $L_i/k_i$ be finite extensions which split these nonsplit fibres. Assume that

1. (1) The smooth fibres of $\pi $ satisfy the Hasse principle and weak approximation.

2. (2) The hypotheses of Theorem 1.4 hold.

Then $X(\mathbb {Q})$ is dense in $X(\mathbb {A}_{\mathbb {Q}})^{\operatorname {\mathrm {Br}}(X)}$ .

It would be interesting to investigate whether Condition $(1)$ in Theorem 1.6 could be relaxed to the assumption that the smooth fibres $X_c(\mathbb {Q})$ are dense in $X_c(\mathbb {A}_{\mathbb {Q}})^{\operatorname {\mathrm {Br}}(X_c)}$ , as in the setting of [24, Question 1.2] discussed above. This would require an extension of Theorem 1.4 to cover a stronger version of the Harpaz–Wittenberg conjecture, involving strong approximation of an auxiliary variety W off a finite set of places [24, Proposition 6.1]. Strong approximation of W was studied by Browning and Schindler [6], for example, who established [24, Question 1.2] in the case when the rank of $\pi $ is at most $3$ , and at least one of its nonsplit fibres lies above a rational of $\mathbb {P}^1_{\mathbb {Q}}$ .

2 Main sieve results for binary forms

Let $f \in \mathbb {Z}[x,y]$ be a non-constant binary form with nonzero discriminant. We write $f(x,y)$ as a product of distinct irreducible factors

(2.1) $$ \begin{align} f(x,y) = \prod_{i=0}^m f_i(x,y)\prod_{i=m+1}^k f_i(x,y), \end{align} $$

where $f_i(x,y)$ are linear forms for $1\leqslant i\leqslant m$ , and forms of degree $k_i\geqslant 2$ for $m+1\leqslant i \leqslant k$ . If $y\mid f(x,y)$ , then we define $f_0(x,y)=y$ , and otherwise, we let $f_0(x,y)=1$ . Hence, we have $y\nmid f_i(x,y)$ for all $i\geqslant 1$ .

For $i\in \{0,\ldots , k\}$ , we define

(2.2) $$ \begin{align} \nu_i(p) &= \#\{[x:y]\in \mathbb{P}^1(\mathbb{F}_p): f_i(x,y) \equiv 0 \ (\mathrm{mod}\ p)\}, \end{align} $$

(2.3) $$ \begin{align} \nu(p) &=\#\{[x:y]\in \mathbb{P}^1(\mathbb{F}_p): f(x,y) \equiv 0 \ (\mathrm{mod}\ p)\}. \end{align} $$

Let $\mathcal {P}$ be a set of primes, and let $\mathcal {P}_{\leqslant x} = \{p\in \mathcal {P}:p\leqslant x\}$ . We denote by $\pi (x)$ the number of primes less than x. For all $i\in \{0,\ldots , k\}$ , we need to assume $\mathcal {P}$ has the following properties, for some $\alpha , \theta _i>0$ and any $A\geqslant 1$ :

(2.4) $$ \begin{align} \sum_{p \in \mathcal{P}_{\leqslant x}} 1 &= \alpha \pi(x)\left(1+O_A\left((\log x)^{-A}\right)\right), \end{align} $$

(2.5) $$ \begin{align} \sum_{p \in \mathcal{P}_{\leqslant x}} \nu_i(p) &= \alpha\theta_i \pi(x)\left(1+O_A\left((\log x)^{-A}\right)\right). \end{align} $$

The reason we require explicit error terms in (2.4) and (2.5) is so that the sieve dimensions, introduced in Section 4.2, exist. We note that for $i=0$ , we have $\theta _0=1$ if $f_0(x,y) = y$ , and $\theta _0=0$ if $f_0(x,y)=1$ . Additionally, from (2.5), it follows that

(2.6) $$ \begin{align} \sum_{p \in \mathcal{P}_{\leqslant x}} \nu(p) = \alpha\theta \pi(x)\left(1+O_A\left((\log x)^{-A}\right)\right), \end{align} $$

where $\theta = \theta _0+ \cdots + \theta _k$ .

Let $\mathcal {B}\subseteq [-1,1]^2$ denote a bounded region whose boundary is a piecewise continuous simple closed curve of finite length. The perimeter of $\mathcal {B}$ will always be assumed to be bounded by some absolute constant C. In the applications in Sections 5 and 6, we shall make the choice

(2.7) $$ \begin{align}\mathcal{B} = \left\{(x,y) \in (0,1]^2: \left|\frac{x}{y}-r\right|<\xi\right\},\end{align} $$

for a fixed real number $r>0$ and a small parameter $\xi>0$ , and so we may choose $C=4$ , for example. We also define $\mathcal {B}N=\{(Nx,Ny): (x,y) \in \mathcal {B}\}.$

Let $\Delta $ be an integer and let $a_0,b_0 \in \mathbb {Z}/\Delta \mathbb {Z}$ . We now state the main sieve results which will be used in the proof of Theorem 1.1 and Theorem 1.3. They concern the sifting function

(2.8) $$ \begin{align} S(\mathcal{P}, \mathcal{B},N) = \#\left\{(a,b)\in \mathcal{B}N\cap \mathbb{Z}^2: \begin{aligned} &a\equiv a_0, b\equiv b_0 \ (\mathrm{mod}\ \Delta)\\ &p\mid f(a,b) \implies p\notin \mathcal{P} \end{aligned} \right\}. \end{align} $$

Theorem 2.1. Let $f(x,y)$ be a binary form consisting of distinct irreducible factors, all of degree at most $2$ . Then there exists a finite set of primes $S_0$ , depending on f, such that the following holds:

Let S be a finite set of primes containing $S_0$ . Let $\Delta $ be an integer with only prime factors in S, and let $a_0, b_0 \in \mathbb {Z}/\Delta \mathbb {Z}$ . Let $\mathcal {P}$ be a set of primes disjoint from S and satisfying (2.4) and (2.5) for some $\alpha , \theta _i>0$ . Assume that $\alpha \theta \leqslant 0.39$ . Then $S(\mathcal {P},\mathcal {B},N)>0$ for N sufficiently large.

We also have a similar sieve result when f may contain irreducible factors of degree up to $3$ , but with a less general sifting set $\mathcal {P}$ , satisfying the following assumption.

Assumption 2.2. There exists a positive integer n, a finite set of primes $S_0$ , primes $q_1, \ldots , q_n$ , and integers $t_1, \ldots , t_n$ with $q_j \nmid t_j$ for all $j \in \{1, \ldots , n\}$ , such that

(2.9) $$ \begin{align} \mathcal{P}\backslash S_0 \subseteq \bigcup_{j=1}^n \{p \equiv t_j \ (\mathrm{mod}\ q_j)\} \end{align} $$

and

$$\begin{align*}\deg f \sum_{j=1}^n \frac{1}{q_j-1} \leqslant 0.32380. \end{align*}$$

Theorem 2.3. Let $f(x,y)$ be a binary form consisting of distinct irreducible factors, all of degree at most $3$ . Then there exists a finite set of primes $S_0$ , depending on f, such that the following holds:

Let $\mathcal {P}$ be a set of primes satisfying Assumption 2.2 with the above choice of $S_0$ . Let S be a finite set of primes containing $S_0$ . Let $\Delta $ be an integer with only prime factors in S, and let $a_0, b_0 \in \mathbb {Z}/\Delta \mathbb {Z}$ . Then $S(\mathcal {P}, \mathcal {B},N)>0$ for N sufficiently large.

For brevity, in the remainder of the paper, we shall denote the condition $a\equiv a_0 \ (\mathrm {mod}\ \Delta ), b\equiv b_0 \ (\mathrm {mod}\ \Delta )$ by $C(a,b)$ .

3 Levels of distribution

Crucial to the success of the beta sieve in proving Theorems 2.1 and 2.3 is a good level of distribution result, which provides an approximation of the quantities

$$ \begin{align*}\#\{(a,b) \in \mathcal{B}N\cap \mathbb{Z}^2: p\mid f_i(a,b), d \mid f(a,b)\}\end{align*} $$

by multiplicative functions, at least on average over p and d. (Here, and throughout this section, we keep the notation from Section 2.) In this section, we provide such an estimate, following similar arguments developed by Daniel [15, Lemma 3.3]. We slightly generalise the setup as follows:

Let $g_1,g_2$ be binary forms with nonzero discriminants. Throughout this section, we fix $S, \Delta $ , and $C(a,b)$ , and assume that S contains all primes dividing the discriminants of $g_1$ and $g_2$ . We allow all implied constants to depend only the degrees of $g_1$ and $g_2$ and a small positive constant $\epsilon $ , which for convenience, we allow to take different values at different points in the argument.

Let $\mathcal {R}$ be a bounded region of $\mathbb {R}^2$ whose boundary is a piecewise continuous simple closed curve of finite length. We denote by $\operatorname {Vol}(\mathcal {R})$ and $P(\mathcal {R})$ the volume and perimeter of $\mathcal {R}$ , respectively. Let

(3.1) $$ \begin{align} A(d_1,d_2) &= \#\{(a,b)\in \mathcal{R}\cap \mathbb{Z}^2:C(a,b), d_1\mid g_1(a,b), d_2\mid g_2(a,b)\}, \end{align} $$

(3.2) $$ \begin{align} \varrho (d_1,d_2)&=\#\{(a,b) \ (\mathrm{mod}\ d_1d_2): d_1\mid g_1(a,b), d_2\mid g_2(a,b)\}. \quad\end{align} $$

We define

(3.3) $$ \begin{align} r(d_1,d_2) = A(d_1,d_2)-\frac{\varrho (d_1,d_2)\operatorname{Vol}(\mathcal{R})}{d^2\Delta^2}. \end{align} $$

In what follows, we let $d=d_1d_2$ , and we assume that $\gcd (d_1,d_2)=\gcd (d,\Delta )=1$ . The main aim of this section is to prove the following proposition.

Proposition 3.1. Suppose that $g_1$ does not contain any linear factors. Then for any $D_1,D_2>0$ and any $\epsilon>0$ , we have

$$ \begin{align*} \sum_{\substack{d_1\leqslant D_1, d_2\leqslant D_2\\ \gcd(d_1,d_2)=\gcd(d,\Delta)=1}}\sup_{P(\mathcal{R})\leqslant N}\left|r(d_1, d_2)\right|\ll (D_1D_2)^{\epsilon}(D_1D_2+N(D_1D_2)^{1/2}+ND_2). \end{align*} $$

As a corollary, we obtain the following level of distribution result.

Corollary 3.2. Suppose that $g_1$ does not contain any linear factors. Let $\mathcal {B} \subseteq [-1,1]^2$ be as in Section 2, and let $\mathcal {R} = \mathcal {B}N$ . Then for any $\epsilon>0$ , there exists $\delta>0$ such that for any $D_1,D_2>0$ with $D_2 \ll N^{1-\epsilon }$ and $D_1D_2 \ll N^{2-\epsilon }$ , we have

(3.4) $$ \begin{align} \sum_{\substack{d_1\leqslant D_1, d_2\leqslant D_2\\ \gcd(d_1,d_2)=\gcd(d,\Delta)=1}}\left|A(d_1,d_2)-\frac{N^2\varrho (d_1,d_2)\operatorname{Vol}(\mathcal{B})}{d^2\Delta^2}\right| \ll N^{2-\delta}. \end{align} $$

Proposition 3.1 and Corollary 3.2 are generalisations of Irving’s results from [27, Section 3], which can be recovered by taking $g_1(x,y) = f(x,y)$ to be the cubic form Irving considered, and $g_2(x,y)=yf(x,y)$ . The method of proof is inspired by the pioneering work of Daniel on the divisor-sum problem for binary forms, which requires a similar level of distribution result (see [15, Lemma 3.3]). Daniel’s argument is more delicate, keeping track of powers of $\log N$ in place of factors of $N^{\epsilon }$ , and Corollary 3.2 could be similarly refined, but this yields no advantage for our applications.

Before proceeding with the proof of Proposition 3.1, we recall the following standard lattice point counting result.

Lemma 3.3. Let $\Lambda \subseteq \mathbb {R}^2$ be a full-rank lattice, and let $\mathcal {R}\subseteq \mathbb {R}^2$ be as defined before Proposition 3.1. Then

$$\begin{align*}\#(\mathcal{R}\cap \Lambda) = \frac{\operatorname{Vol}(\mathcal{R})}{\det \Lambda} + O\left(\frac{P(\mathcal{R})}{\lambda_1} + 1\right), \end{align*}$$

where $\lambda _1$ is the length of a shortest nonzero vector in $\Lambda $ .

Proof. Let $\mathcal {F}$ be a fundamental domain of $\Lambda $ . The translates $v + \mathcal {F}$ for $v \in \Lambda $ tile $\mathbb {R}^2$ . Define sets

$$\begin{align*}S^- = \{v \in \Lambda: (v + \mathcal{F}) \subseteq \mathcal{R}\}, \qquad S^+ = \{v \in \Lambda: (v + \mathcal{F})\cap \mathcal{R} \neq \emptyset\}. \end{align*}$$

Then

$$\begin{align*}\frac{\operatorname{Vol}(S^-+ \Lambda)}{\det(\Lambda)} = \#S^- \leqslant \#(\mathcal{R}\cap \Lambda) \leqslant \#S^+ = \frac{\operatorname{Vol}(S^++ \Lambda)}{\det(\Lambda)}. \end{align*}$$

Moreover, $S^- + \Lambda \subseteq \mathcal {R} \subseteq S^+ + \Lambda $ , so $\operatorname {Vol}(S^- + \Lambda ) \leqslant \operatorname {Vol}(\mathcal {R}) \leqslant \operatorname {Vol}( S^+ + \Lambda )$ . Therefore,

$$\begin{align*}\left|\#(\mathcal{R}\cap \Lambda) - \frac{\operatorname{Vol}(\mathcal{R})}{\det(\Lambda)}\right| \leqslant \#S^+ - \#S^-. \end{align*}$$

However, $S^+ - S^- = \{v \in \Lambda : (v + \mathcal {F}) \cap \partial \mathcal {R} \neq \emptyset \},$ where $\partial \mathcal {R}$ denotes the boundary of $\mathcal {R}$ . Each segment of $\partial \mathcal {R}$ of length $\lambda _1$ can intersect at most four translates of $\mathcal {F}$ . Therefore, $S^+ - S^- \ll P(\mathcal {R})/\lambda _1 + 1$ , as required.

We now commence with the proof of Proposition 3.1. We introduce the quantities $R^*(d_1,d_2), \varrho ^*(d_1,d_2)$ and $r^*(d_1,d_2)$ which are defined similarly to $A(d_1,d_2), \varrho (d_1,d_2)$ and $r(d_1,d_2)$ but with the added condition $\gcd (a,b,d)=1$ . We write $(a_1,b_1) \sim (a_2,b_2)$ if there exists an integer $\lambda $ such that $(a_1,b_1)\equiv \lambda (a_2,b_2)\ (\mathrm {mod}\ d)$ . This forms an equivalence relation on points $(a,b)\in \mathbb {Z}^2$ with $\gcd (a,b,d)=1$ . Moreover, the properties $g_1(a,b) \equiv 0 \ (\mathrm {mod}\ d_1)$ and $g_2(a,b)\equiv 0 \ (\mathrm {mod}\ d_2)$ are preserved under this equivalence. We may therefore define

$$ \begin{align*}\mathcal{U}(d_1,d_2) = \left.\left\{a,b \ (\mathrm{mod}\ d): \begin{array}{l l} &\displaystyle \gcd(a,b,d)=1\\ &\displaystyle d_1\mid g_1(a,b), d_2\mid g_2(a,b) \end{array}\right\}\middle/\sim.\right.\end{align*} $$

For $\mathcal {C}\in \mathcal {U}(d_1,d_2)$ , we define

$$ \begin{align*}\Lambda(\mathcal{C}) = \{y \in \mathbb{Z}^2: y \equiv \lambda(a,b) \ (\mathrm{mod}\ d)\textrm{ for some }(a,b)\in \mathcal{C} \textrm{ and some } \lambda \in \mathbb{Z}\}.\end{align*} $$

It is easy to check that $\Lambda (\mathcal {C})$ is a lattice in $\mathbb {Z}^2$ , and its set of primitive points is $\mathcal {C}$ . For $e \in \mathbb {Z}$ , we define

$$ \begin{align*}\Lambda(\mathcal{C},e) = \{(a,b) \in \Lambda(\mathcal{C}): e\mid \gcd(a,b)\}.\end{align*} $$

By Möbius inversion, we have

$$ \begin{align*}R^*(d_1,d_2) = \sum_{\mathcal{C} \in \mathcal{U}(d_1,d_2)}\sum_{e\mid d}\mu(e)\#\{(a,b) \in \mathcal{R}\cap \Lambda(\mathcal{C},e): C(a,b)\}.\end{align*} $$

Since $\gcd (d,\Delta )=1$ , the set $\{(a,b) \in \Lambda (\mathcal {C},e):C(a,b)\}$ is a coset of the lattice $\Lambda (\mathcal {C},e\Delta )$ , which has determinant $de\Delta ^2$ . Therefore, by Lemma 3.3,

(3.5) $$ \begin{align} R^*(d_1,d_2) = \sum_{\mathcal{C} \in \mathcal{U}(d_1,d_2)}\sum_{e\mid d}\mu(e) \left(\frac{\operatorname{Vol}(\mathcal{R})}{de\Delta^2}+O\left(1+\frac{P(\mathcal{R})}{\lambda_1(\mathcal{C})}\right)\right), \end{align} $$

where $\lambda _1(\mathcal {C})$ denotes the length of the shortest nonzero vector in $\Lambda (\mathcal {C})$ .

Each equivalence class $\mathcal {C} \in \mathcal {U}(d_1,d_2)$ consists of $\varphi (d)$ elements, and so

$$ \begin{align*}\sum_{\mathcal{C} \in \mathcal{U}(d_1,d_2)}\sum_{e\mid d}\frac{\mu(e)}{e} = \sum_{\mathcal{C} \in \mathcal{U}(d_1,d_2)}\frac{\varphi(d)}{d} = \frac{\varrho ^*(d_1,d_2)}{d}.\end{align*} $$

Moreover, we have $\#\mathcal {U}(d_1,d_2)\ll d^{\epsilon }$ , as we now explain. We observe that $\#\mathcal {U}(d_1,d_2) =\varrho ^*(d_1,d_2)/\varphi (d)$ , and $\varrho ^*$ is multiplicative by the Chinese remainder theorem. For primes $p\notin S$ , we may apply Hensel’s lemma to show that $\varrho ^*(p^e,1),\varrho ^*(1,p^e) = O(p^e)$ for any integer $e \geqslant 1$ . Therefore, by the trivial bound for the divisor function [22, Section 18.1], we conclude that

(3.6) $$ \begin{align} \#\mathcal{U}(d_1,d_2) = \frac{\varrho ^*(d_1,d_2)}{\varphi(d)}\ll \frac{d^{1+\epsilon}}{\varphi(d)}\ll d^{\epsilon}. \end{align} $$

Applying (3.5), and (3.6), we obtain

(3.7) $$ \begin{align} \begin{aligned} &\sum_{\substack{d_1\leqslant D_1, d_2\leqslant D_2\\ \gcd(d_1,d_2)=\gcd(d,\Delta)=1}}\sup_{P(\mathcal{R})\leqslant N}\left|r^*(d_1,d_2)\right|\\ &\ll_{\epsilon}(D_1D_2)^{\epsilon}\left(D_1D_2+N\sum_{\substack{d_1\leqslant D_1, d_2\leqslant D_2\\ \gcd(d_1,d_2)= \gcd(d,\Delta)=1}}\sum_{\mathcal{C} \in \mathcal{U}(d_1,d_2)}\lambda_1(\mathcal{C})^{-1}\right). \end{aligned} \end{align} $$

Let $v_1(\mathcal {C})$ denote a shortest nonzero vector of $\Lambda (\mathcal {C})$ , and let $\|\cdot \|$ be the usual Euclidean norm. Then $\|v_1(\mathcal {C})\|^2 \ll |\det \Lambda (\mathcal {C})|=d \leqslant D_1D_2$ . Therefore,

(3.8) $$ \begin{align} \sum_{\substack{d_1\leqslant D_1, d_2\leqslant D_2\\ \gcd(d_1,d_2)=\gcd(d,\Delta)=1}}\sum_{\mathcal{C} \in \mathcal{U}(d_1,d_2)}\lambda_1(\mathcal{C})^{-1} \ll \sum_{0<a^2+b^2\ll D_1D_2}\frac{M(a,b)}{\sqrt{a^2+b^2}}, \end{align} $$

where

For any $d_1,d_2$ enumerated by $M(a,b)$ , we have $d_1\mid g_1(a,b)$ and $d_2\mid g_2(a,b)$ , so

$$ \begin{align*}M(a,b) \leqslant \#\{d_1\leqslant D_1, d_2\leqslant D_2: d_1\mid g_1(a,b), d_2\mid g_2(a,b)\}.\end{align*} $$

Since $g_1$ contains no linear factors, we know that $g_1(a,b)\neq 0$ whenever $(a,b) \neq (0,0)$ . Suppose in addition that $g_2(a,b) \neq 0$ . Then by the trivial bound for the divisor function, we have $M(a,b) \ll (D_1D_2)^{\epsilon }$ . We deduce that

$$ \begin{align*} \sum_{\substack{0<a^2+b^2\ll D_1D_2\\g_2(a,b)\neq 0}}\frac{M(a,b)}{\sqrt{a^2+b^2}}&\ll (D_1D_2)^{\epsilon}\sum_{0<a^2+b^2\ll D_1D_2}\frac{1}{\sqrt{a^2+b^2}}\\ &\ll (D_1D_2)^{1/2 + \epsilon}. \end{align*} $$

Now suppose that $g_2(a,b)=0$ . Then as above, we have $O(D_1^{\epsilon })$ choices for $d_1$ , but now $D_2$ choices for $d_2$ , so that $M(a,b) \ll D_1^{\epsilon }D_2$ . We obtain

$$ \begin{align*} \sum_{\substack{0<a^2+b^2\ll D_1D_2\\g_2(a,b)= 0}}\frac{M(a,b)}{\sqrt{a^2+b^2}}&\ll D_1^{\epsilon}D_2\sum_{\substack{0<a^2+b^2\ll D_1D_2\\ g_2(a,b)=0}}\frac{1}{\sqrt{a^2+b^2}}. \end{align*} $$

For a fixed $b\neq 0$ , $g_2(a,b)$ is a nonzero polynomial in a, and so has $O(1)$ roots. Therefore,

$$ \begin{align*} \sum_{\substack{0<a^2+b^2\ll D_1D_2\\ g_2(a,b)=0}}\frac{1}{\sqrt{a^2+b^2}}&= \sum_{ \substack{0<a^2+b^2\ll D_1D_2\\ b \neq 0\\g_2(a,b)= 0}}\frac{1}{\sqrt{a^2+b^2}}+\sum_{\substack{0<a^2 \ll D_1D_2\\ g_2(a,0)=0}}\frac{1}{a}\\ &\ll \sum_{b \ll \sqrt{D_1D_2}}\frac{1}{b}+\sum_{a \ll \sqrt{D_1D_2}}\frac{1}{a}\\ &\ll (D_1D_2)^{\epsilon}. \end{align*} $$

To summarize, we have established the following generalisation of [27, Lemma 3.2].

Lemma 3.4. Suppose that $g_1$ does not contain any linear factors. Then for any $D_1,D_2>0$ and any $\epsilon>0$ , we have

$$ \begin{align*} \sum_{\substack{d_1\leqslant D_1, d_2\leqslant D_2\\ \gcd(d_1,d_2)= \gcd(d,\Delta)=1}}\sup_{P(\mathcal{R})\leqslant N}\left|r^*(d_1,d_2)\right|\ll (D_1D_2)^{\epsilon}(D_1D_2+N(D_1D_2)^{1/2}+ND_2). \end{align*} $$

Now we remove the restriction $\gcd (a,b,d)=1$ . Below, we write $A(d_1,d_2)=A(\mathcal {R},d_1,d_2; C(a,b))$ in order to make the dependence on $\mathcal {R}$ and $C(a,b)$ clear. Let $k_1=\deg g_1$ and $k_2=\deg g_2$ . We work with multiplicative functions $\psi _k$ for $k=k_1$ and $k=k_2$ , which map prime powers $p^r$ to $p^{{\lceil r/k \rceil }}$ . We follow the same argument as Irving, but with $\psi _{k_1},\psi _{k_2}$ in place of $\psi _3,\psi _4$ . The motivation for this definition of $\psi _k$ comes from the fact that for any integers $d,e,k \geqslant 1$ with $e\mid \psi _k(d)$ , and for any prime p, we have

(3.9) $$ \begin{align} p\mid \frac{\psi_k(d)}{e} \iff p \mid \frac{d}{\gcd(d,e^k)}. \end{align} $$

Since $\gcd (d_1,d_2)=1$ , we have

(3.10) $$ \begin{align} A(\mathcal{R},d_1,d_2;C(a,b)) = \sum_{\substack{e_1\mid \psi_{k_1}(d_1)\\e_2\mid \psi_{k_2}(d_2)}}N(d_1,d_2,e_1,e_2), \end{align} $$

where

(3.11) $$ \begin{align} N(d_1,d_2,e_1,e_2) = \#\left\{(a,b) \in \mathcal{R}\cap \mathbb{Z}^2: \begin{array}{l l} &\displaystyle C(a,b), d_1\mid g_1(a,b), d_2\mid g_2(a,b),\\ &\displaystyle \gcd(a,b,\psi_{k_1}(d_1)\psi_{k_2}(d_2)) = e_1e_2 \end{array} \right\}. \end{align} $$

We make a change of variables $a' = a/e_1e_2, b' = b/e_1e_2$ in (3.11). Let $\overline {e_1e_2}$ denote the multiplicative inverse of $e_1e_2$ modulo $\Delta $ , which exists due to the assumption $\gcd (d_1d_2, \Delta )=1$ . The congruence condition $C(a,b)$ is equivalent to the congruence condition $a' \equiv \overline {e_1e_2}a_0 \ (\mathrm {mod}\ \Delta )$ and $b' \equiv \overline {e_1e_2}b_0\ (\mathrm {mod}\ \Delta )$ , which we denote by $C'(a',b')$ . We have

$$ \begin{align*} d_1\mid g_1(a,b) &\iff d_1 \mid (e_1e_2)^{k_1} g_1(a',b') \\ &\iff d_1 \mid e_1^{k_1}g_1(a',b') \\ &\iff \frac{d_1}{\gcd(d_1,e_1^{k_1})}\mid g_1(a',b'), \end{align*} $$

and similarly for $d_2 \mid g_2(a,b)$ . For convenience, we define

$$ \begin{align*}f_1 = \frac{d_1}{\gcd(d_1,e_1^{k_1})}, \quad f_2 =\frac{d_2}{\gcd(d_2,e_2^{k_2})}.\end{align*} $$

Changing notation from $a',b'$ back to $a,b$ , we deduce that $N(d_1,d_2,e_1,e_2)$ can be rewritten as

(3.12) $$ \begin{align} &\#\left\{(a,b) \in \mathcal{R}/(e_1e_2)\cap \mathbb{Z}^2: \begin{array}{l l} &\displaystyle C'(a,b), f_1\mid g_1(a,b), f_2\mid g_2(a,b), \nonumber\\ &\displaystyle \gcd(a,b,\psi_{k_1}(d_1)\psi_{k_2}(d_2)/e_1e_2) = 1 \end{array} \right\}\\ &=\#\left\{(a,b) \in \mathcal{R}/(e_1e_2)\cap \mathbb{Z}^2: \begin{array}{l l} &\displaystyle C'(a,b), f_1\mid g_1(a,b), f_2\mid g_2(a,b),\\ &\displaystyle \gcd(a,b,f_1f_2)= 1 \end{array} \right\}.\nonumber\\ &=R^*\left(\mathcal{R}/(e_1e_2), f_1,f_2;C'(a,b)\right). \end{align} $$

The above arguments, but with the congruence conditions removed, and with the specific choice $\mathcal {R} = [0,d_1d_2]^2$ also demonstrate that

(3.13) $$ \begin{align} \varrho (d_1,d_2) &=\sum_{\substack{e_1\mid \psi_{k_1}(d_1)\\e_2\mid \psi_{k_2}(d_2)}}\#\left\{(a,b) \in \mathcal{R}/(e_1e_2)\cap \mathbb{Z}^2:\begin{array}{l l} &\displaystyle f_1\mid g_1(a,b), f_2\mid g_2(a,b),\nonumber\\ &\displaystyle \gcd(a,b,\psi_{k_1}(d_1)\psi_{k_2}(d_2)/e_1e_2) = 1 \end{array} \right\}\\ &= \sum_{\substack{e_1\mid \psi_{k_1}(d_1)\\e_2\mid \psi_{k_2}(d_2)}}\left(\frac{d_1d_2}{e_1e_2f_1f_2}\right)^2\varrho ^*(f_1,f_2). \end{align} $$

We denote the quantity

$$ \begin{align*}R^*(\mathcal{R}/(e_1e_2), f_1,f_2; C'(a,b)) - \frac{\operatorname{Vol}(\mathcal{R}/(e_1e_2))\varrho ^*(f_1,f_2)}{(f_1f_2\Delta)^2}\end{align*} $$

by $E(e_1,e_2,f_1,f_2)$ . Combining (3.10), (3.12) and (3.13), we have

(3.14) $$ \begin{align} &\sum_{\substack{d_1\leqslant D_1,d_2\leqslant D_2\\ (d_1,d_2)=(d_1d_2,\Delta)=1}}\sup_{P(\mathcal{R})\leqslant N}\left|r(d_1,d_2)\right|=\sum_{\substack{d_1\leqslant D_1,d_2\leqslant D_2\\ (d_1,d_2)=(d_1d_2,\Delta)=1}}\sup_{P(\mathcal{R})\leqslant N}\sum_{\substack{e_1\mid \psi_{k_1}(d_1)\\e_2\mid \psi_{k_2}(d_2)}}|E(e_1,e_2,f_1,f_2)| \end{align} $$

(3.15) $$ \begin{align} &\leqslant \sum_{e_1\leqslant D_1,e_2\leqslant D_2}\sum_{\substack{f_1\leqslant D_1/e_1, f_2 \leqslant D_2/e_2\\ (f_1,f_2) = (f_1f_2,\Delta) =1}}\delta(e_1,f_1)\delta(e_2,f_2)\sup_{P(\mathcal{R})\leqslant N}\left|E(e_1,e_2,f_1,f_2)\right|, \end{align} $$

where for integers $e,f,k,D \geqslant 1$ , we have defined

$$ \begin{align*}\delta(e,f) = \#\left\{d\leqslant D: e\mid \psi_{k}(d), f = \frac{d}{\gcd(d,e^{k})}\right\}.\end{align*} $$

We claim that $\delta (e,f) \ll e^{\epsilon }$ . To see this, suppose that p is a prime and let $r=\nu _p(d), s=\nu _p(e)$ and $t = \nu _p(f)$ . There is a unique choice of r for a given $k,s$ and t provided that $t>0$ – namely, $r=ks+t$ . If $t=0$ , then we deduce from $f = d/\gcd (d,e^{k})$ that $r\leqslant ks$ . Taking a product over primes, we conclude that each d enumerated by $\delta (e,f)$ is a divisor of $e^k$ multiplied by a quantity that is uniquely determined by e and f. The claim follows since the number of divisors of $e^k$ is $O(e^{\epsilon })$ . In our situation, where $e_1\leqslant D_1$ and $e_2 \leqslant D_2$ , we obtain $\delta (e_1,f_1)\delta (e_2,f_2) \ll (D_1D_2)^{\epsilon }.$ Therefore, applying Lemma 3.4 for each choice of $e_1,e_2$ in (3.15), we conclude that

$$ \begin{align*} &\sum_{\substack{d_1\leqslant D_1,d_2\leqslant D_2\\ (d_1,d_2)=(d_1d_2,\Delta)=1}}\sup_{P(\mathcal{R}\leqslant N)}\left|r(d_1,d_2)\right|\ll (D_1D_2)^{\epsilon}\sum_{\substack{e_1\leqslant D_1\\e_2\leqslant D_2}}\left(\frac{D_1D_2}{e_1e_2}+N\left(\frac{D_1D_2}{e_1e_2}\right)^{1/2}+\frac{ND_2}{e_2}\right)\\ &\ll (D_1D_2)^{\epsilon}\left(D_1D_2 + N(D_1D_2)^{1/2}+ND_2\right), \end{align*} $$

which completes the proof of Proposition 3.1.

Remark 3.5. If $g_2(a,b)\neq 0$ for all $(a,b) \neq (0,0)$ , then we do not need to consider the case $g_2(a,b)=0$ in the analysis of the sum in (3.8), and so in our final level of distribution result, we do not require the assumption $D_2 \ll N^{1-\epsilon }$ .

When $g_1(a,b)$ does contain linear factors, we can still obtain a basic level of distribution result from the above argument using the trivial estimate $\lambda _1(\mathcal {C})^{-1}\leqslant 1$ in (3.7). This establishes the following lemma.

Lemma 3.6. Let $g_1,g_2$ be arbitrary binary forms with nonzero discriminant. Then for any $\epsilon>0$ , there exists $\delta>0$ such that for any $D_1,D_2>0$ with $D_1D_2 \ll N^{1-\epsilon }$ , we have

$$ \begin{align*}\sum_{\substack{d_1\leqslant D_1, d_2\leqslant D_2\\ \gcd(d_1,d_2)=\gcd(d,\Delta)=1}}\sup_{P(\mathcal{R})\leqslant N}\left|r(d_1,d_2)\right| \ll N^{2-\delta}.\end{align*} $$

4 Application of the beta sieve

In this section, we prove Theorem 2.1 and Theorem 2.3 by combining the level of distribution results from Section 3 with the beta sieve of Rosser and Iwaniec [18, Theorem 11.12]. We state the precise version of this theorem we need in Theorem 4.1.

We recall some of the notation from Section 2. We fix a region $\mathcal {R}=\mathcal {B}N$ for some $\mathcal {B}\subseteq [-1,1]^2$ as in Section 2. Then $\mathcal {R}$ has volume $\gg N^2$ and perimeter $\ll N$ . Let d denote the largest degree among the irreducible factors of f. (We specialise to the cases $d=2$ and $d=3$ later. The reason our methods are unable to deal with larger values of d is explained in Remark 4.5.) Then there exists $x \ll N^d$ such that the largest prime factor of $f(a,b)$ for $(a,b)\in \mathcal {R}\cap \mathbb {Z}^2$ is strictly less than x. Let S be a finite set of primes, including all primes dividing the discriminant of $f(x,y)$ . In Section 4.6, we also append to S all primes bounded by some constant $P_1$ . Let $\Delta $ be an integer with only prime factors in S. Without loss of generality, we may assume every prime in S divides $\Delta $ because taking a multiple of $\Delta $ can only decrease the sifting function $S(\mathcal {P}, \mathcal {B}, N)$ from (4.2). Additionally, by taking an appropriate multiple of $\Delta $ and appropriate lifts of $a_0, b_0$ , we may assume that $\nu _p(f(a_0, b_0)) < \nu _p(\Delta )$ for all $p\in S$ .

All implied constants in this section are allowed to depend on $\Delta $ . Let $\mathcal {P}$ be a set of primes disjoint from S satisfying (2.4) and (2.5). We also define $\mathcal {P'}$ to be the set of primes not in $\mathcal {P}\cup S$ . Let $P(x)$ denote the product of primes in $\mathcal {P}_{< x}$ , and similarly for $P'(x)$ . We also define $X = \operatorname {Vol}(\mathcal {R})/\Delta ^2$ .

For a sequence of non-negative real numbers $\mathcal {A} = (a_n)$ , and a parameter $z\geqslant 1$ , we define the sifting function

$$ \begin{align*}S(\mathcal{A}, \mathcal{P}, z) = \sum_{\gcd(n,P(z))=1}a_n.\end{align*} $$

We make the choice

(4.1) $$ \begin{align} a_n = \#\{(a,b) \in \mathcal{R}\cap \mathbb{Z}^2: C(a,b), f(a,b) = n\}, \end{align} $$

so that

(4.2) $$ \begin{align} \begin{aligned} S(\mathcal{A},\mathcal{P},x)&=\#\{(a,b)\in \mathcal{R}\cap \mathbb{Z}^2: C(a,b), \gcd(f(a,b), P(x))=1\}\\ &=S(\mathcal{B},\mathcal{P},N), \end{aligned} \end{align} $$

where $S(\mathcal {B},\mathcal {P},N)$ is as defined in (2.8). Our aim is to prove that $S(\mathcal {A}, \mathcal {P}, x)>0$ for sufficiently large N (which may depend on $\Delta $ ). For a prime $p\in \mathcal {P}$ and for any $i\in \{0,\ldots , k\}$ , we also consider the sequences $\mathcal {A}_p, \mathcal {A}_p^{(i)}$ defined similarly to (4.1) but with the additional conditions $p\mid f(a,b), p\mid f_i(a,b)$ , respectively, so that

$$ \begin{align*} S(\mathcal{A}_p, \mathcal{P}, p) &=\#\{(a,b)\in \mathcal{R}\cap \mathbb{Z}^2: C(a,b), p\mid f(a,b), \gcd(f(a,b), P(p))=1\},\\ S(\mathcal{A}_p^{(i)}, \mathcal{P},p) &=\#\{(a,b)\in \mathcal{R}\cap \mathbb{Z}^2: C(a,b), p\mid f_i(a,b), \gcd(f(a,b), P(p))=1\}. \end{align*} $$

Using the Buchstab identity, we have

$$ \begin{align*}S(\mathcal{A},\mathcal{P},x) = S(\mathcal{A},\mathcal{P},N^{\gamma})-\sum_{\substack{N^{\gamma} \leqslant p < x\\p \in \mathcal{P}}}S(\mathcal{A}_p, \mathcal{P},p),\end{align*} $$

for a parameter $\gamma \in (0,1)$ to be chosen later. We denote $S(\mathcal {A},\mathcal {P},N^{\gamma })$ by $S_1$ . If $p\mid f(a,b)$ , then $p\mid f_i(a,b)$ for some i. Therefore, we have the decomposition

$$ \begin{align*}S(\mathcal{A},\mathcal{P},x) \geqslant S_1 -\sum_{i=0}^m S_2^{(i)}- \sum_{i=m+1}^k \left(S_3^{(i)}+S_4^{(i)}\right),\end{align*} $$

where

(4.3) $$ \begin{align} \begin{aligned} S_2^{(i)} &=\sum_{\substack{N^{\gamma}\leqslant p \ll N\\p \in \mathcal{P}}}S(\mathcal{A}_p^{(i)},\mathcal{P},p),\\ S_3^{(i)}&=\sum_{\substack{N^{\gamma}\leqslant p < N^{\beta_i}\\p \in \mathcal{P}}}S(\mathcal{A}_p^{(i)}, \mathcal{P},p), \\ S_4^{(i)}&=\sum_{\substack{N^{\beta_i}\leqslant p< x\\p \in \mathcal{P}}}S(\mathcal{A}_p^{(i)}, \mathcal{P},p), \end{aligned} \end{align} $$

for parameters $\beta _i\geqslant \gamma $ to be chosen later.

4.1 The beta sieve

Like most combinatorial sieves, the beta sieve provides a mechanism to estimate sifting functions of the form $S(\mathcal {A},\mathcal {P},z)$ given arithmetic information about the related quantities $|\mathcal {A}_d| := \sum _{d\mid n}a_n$ for squarefree integers d. More specifically, we require an approximation $|\mathcal {A}_d| = |\mathcal {A}_1|g(d) + r(d)$ , where $g(d)$ is a multiplicative function supported on squarefree integers and

$$\begin{align*}R(z) := \sum_{\substack{d\leqslant z\\ d \textrm{ squarefree}}}|r(d)| \end{align*}$$

is small. Define

$$ \begin{align*}V(z) = \sum_{d \mid P(z)}\mu(d)g(d) = \prod_{p \in \mathcal{P}_{\leqslant z}}(1-g(p)).\end{align*} $$

We shall assume that for some $\kappa , L \geqslant 0$ , we have

(4.4) $$ \begin{align} V(w) \leqslant \left(\frac{\log z}{\log w}\right)^{\kappa}\left(1+\frac{L}{\log w}\right)V(z) \end{align} $$

for all $2 \leqslant w \leqslant z$ .

For some choice of sieve weights $\Lambda ^{\pm } =(\lambda _d^{\pm })_{d \geqslant 1}$ , define

$$\begin{align*}V^{\pm}(z) = \sum_{d\mid P(z)}\lambda_d^{\pm}g(d). \end{align*}$$

We recall that if $\Lambda ^{\pm }$ are upper and lower bound sieves of level z (i.e., if $\lambda ^{\pm }_d$ are supported on squarefree integers $d<z$ and

$$\begin{align*}\sum_{d\mid m}\lambda^-_d \leqslant \sum_{d\mid m}\mu(d) \leqslant \sum_{d\mid m}\lambda^+_d \end{align*}$$

for all integers m), then we have

(4.5) $$ \begin{align} \begin{aligned} S(\mathcal{A}, \mathcal{P}, z) &\leqslant |\mathcal{A}_1|V^+(z) + R(z),\\ S(\mathcal{A}, \mathcal{P}, z) &\geqslant |\mathcal{A}_1|V^-(z) - R(z). \end{aligned} \end{align} $$

The main theorem of the beta sieve we apply is given in [18, Theorem 11.12]. We record this theorem here for convenience, in the special case $s=1$ .

Theorem 4.1. Suppose $\kappa , L$ are such that the assumption (4.4) holds. Then there is a choice of upper and lower bound sieve weights $\Lambda ^{\pm }$ (taking values in $\{-1,0,1\}$ ), and explicit constants $A(\kappa ), B(\kappa ) \geqslant 0$ such that, as $z\rightarrow \infty $ , we have

(4.6) $$ \begin{align} \begin{aligned} V^+(z) &\leqslant \left(A(\kappa) + o(1)\right)V(z),\\ V^-(z) &\geqslant \left(B(\kappa) + o(1)\right)V(z). \end{aligned} \end{align} $$

Notation 4.2. Throughout the remainder of this chapter, for a sequence $\mathcal {A}$ , a set of primes $\mathcal {P}$ , a multiplicative function g, and a sifting level $z\geqslant 1$ , we define $\Lambda ^{\pm }(\mathcal {A},\mathcal {P},g,z)$ to be the corresponding upper and lower bound beta sieves with these parameters. We sometimes apply (4.6) directly without reference to a sequence, in which case the parameter $\mathcal {A}$ is omitted from the notation.

In our applications of the beta sieve, the required bounds on $R(z)$ are provided by Corollary 3.2 and Lemma 3.6. For $i\in \{0,\ldots , k\}$ , we define multiplicative functions

$$ \begin{align*} \varrho _i(d_1,d_2) &=\#\{a,b \ (\mathrm{mod}\ d_1d_2): d_1\mid f_i(a,b), d_2\mid f(a,b)\},\\ \varrho _i(d) &=\#\{a,b \ (\mathrm{mod}\ d): f_i(a,b) \equiv 0 \ (\mathrm{mod}\ d)\},\\ \varrho (d) &= \#\{a,b \ (\mathrm{mod}\ d): f(a,b) \equiv 0 \ (\mathrm{mod}\ d)\}. \end{align*} $$

We note that the function $\varrho _i(d_1,d_2)$ is the same as the function $\varrho (d_1,d_2)$ from (3.2) with $g_1(x,y) = f_i(x,y)$ and $g_2(x,y) = f(x,y)$ , but in this section, we add a subscript to keep track of the dependence on i. When $\gcd (d_1,d_2) = 1$ , we have $\varrho _i(d_1,d_2) = \varrho _i(d_1)\varrho (d_2)$ . Moreover, for any $i\in \{1,\ldots , k\}$ and any prime $p\notin S$ , we have

(4.7) $$ \begin{align} \varrho _i(p) &= \nu_i(p)(p-1)+1, \end{align} $$

(4.8) $$ \begin{align} \varrho (p) &= \nu(p)(p-1)+1, \end{align} $$

where $\nu _i(p)$ and $\nu (p)$ are as defined in (2.2) and (2.3). We define multiplicative functions

$$\begin{align*}g(p) := \frac{\varrho (p)}{p^2}, \qquad g_i(p) := \frac{\varrho _i(p)}{p^2} \end{align*}$$

and define

(4.9) $$ \begin{align} V(x)= \prod_{\substack{p\in \mathcal{P}_{\leqslant z}}}\left(1-g(p)\right), \qquad V_i(x)= \prod_{\substack{p\in \mathcal{P}^{\prime}_{\leqslant z}}}\left(1-g_i(p)\right) \end{align} $$

for $i\in \{1,\ldots , k\}$ .

4.2 Sieve dimensions

We prove in Lemma 4.3 that the functions $V,V_i$ defined above satisfy the hypothesis (4.4) with the sieve dimensions

where $\alpha , \theta _i$ and $\theta $ are as defined in (2.4), (2.5) and (2.6).

In the notation of Theorem 4.1, we write $A=A(\kappa ), B=B(\kappa )$ and $A_i = A(\kappa _i)$ . We assume throughout this section that $\kappa <1/2$ , and so $\kappa _i>1/2$ . Then A and B are defined in [18, Equations (11.62, 11.63)] (see also Section 4.9), and $A_i$ is defined in [18, Equations (11.42), (11.57)]. A table of numerical values of these constants can be found in [18, Section 11.19].

Lemma 4.3. Let $x\geqslant 1$ . For $i\in \{1,\ldots ,k\}$ , and $V,V_i$ as in (4.9), there exist constants $c,c_i>0$ such that

(4.10) $$ \begin{align} V(x)&= \frac{c}{(\log x)^{\kappa}}\left(1+O((\log x)^{-1})\right), \end{align} $$

(4.11) $$ \begin{align} V_i(x)&= \frac{c_i}{(\log x)^{\kappa_i}}\left(1+O((\log x)^{-1})\right). \end{align} $$

The asymptotic in (4.11) also holds for $i=0$ when $f_0 \not \equiv 1$ .

Proof. We follow a similar approach to [27, Lemma 4.2]. Below, we denote by C a constant which is allowed to vary from line to line. We have

(4.12) $$ \begin{align} \log V(x)&=-\sum_{p \in \mathcal{P}_{\leqslant x}}\left(\sum_{m=1}^{\infty}\frac{\varrho (p)^m}{mp^ {2m}}\right) \end{align} $$

(4.13) $$ \begin{align} &=-\sum_{p\in \mathcal{P}_{\leqslant x}}\frac{\nu(p)(p-1)+1}{p^2} + C + O((\log x)^{-1}). \end{align} $$

(4.14) $$ \begin{align} &=-\sum_{p\in \mathcal{P}_{\leqslant x}}\frac{\nu(p)}{p} + C + O((\log x)^{-1}), \end{align} $$

where in (4.14) we have used that $\nu (p) \leqslant \deg f$ for all but finitely many primes p. To estimate the sum in (4.14), we apply partial summation, together with our assumption (2.6). For $t\geqslant 2$ , we define

(4.15) $$ \begin{align} A_t = \sum_{p \in \mathcal{P}_{\leqslant t}}\nu(p). \end{align} $$

Then

(4.16) $$ \begin{align} \sum_{p \in \mathcal{P}_{\leqslant x}}\frac{\nu(p)}{p}&=\frac{A_x}{x} + \int_{2}^x \frac{A_t}{t^2}\textrm{d}t\nonumber\\ &=\kappa\int_{2}^x \frac{\pi(t)\left(1+O\left((\log t)^{-1}\right)\right)}{t^2} \textrm{d}t + O((\log x)^{-1})\nonumber\\ &=\kappa\int_{2}^x \frac{\textrm{d}t}{t\log t} + C+O((\log x)^{-1})\nonumber\\ &=\kappa\log\log x + C + O((\log x)^{-1}). \end{align} $$

We deduce (4.10) by taking the exponential of (4.16).

We can prove (4.11) in a similar way. When $i=0$ and $f_0 \not \equiv 1$ , we have $\varrho _0(p) = p$ , and so the result is a consequence of Mertens’ theorem [29, Equation (2.16)]. For any $i\in \{1,\ldots , k\}$ , we have

(4.17) $$ \begin{align} \log V_i(x)&=\sum_{p\in \mathcal{P'}_{\leqslant x}}\frac{\nu_i(p)}{p} + C + O((\log x)^{-1})\nonumber\\ &=\sum_{p\leqslant x \textrm{ prime}}\frac{\nu_i(p)}{p} - \sum_{p\in \mathcal{P}_{\leqslant x}}\frac{\nu_i(p)}{p} + C + O((\log x)^{-1}). \end{align} $$

Similarly to above, using partial summation and (2.5), we have

(4.18) $$ \begin{align}\sum_{p\in \mathcal{P}_{\leqslant x}}\frac{\nu_i(p)}{p} = \alpha\theta_i \log\log x + C+ O((\log x)^{-1}).\end{align} $$

To treat the first sum in (4.17), we define $L_i$ to be the number field generated by $f_i$ . For all but finitely many primes p, the quantity $\nu _i(p)$ is equal to the number of degree one prime ideals $\mathfrak {p}$ in $L_i$ above p. Let $\pi _{L_i}(x)$ denote the number of prime ideals $\mathfrak {p}$ in L of norm at most x. This count is dominated by degree one ideals. In fact, the number of prime ideals of degree at least $2$ enumerated by $\pi _{L_i}(x)$ is $O(x^{1/2})$ because such an ideal must lie over a rational prime $p \leqslant x^{1/2}$ and each rational prime p has at most $[L_i:\mathbb {Q}]$ prime ideals in $L_i$ lying above it. Therefore,

$$ \begin{align*}\sum_{p\leqslant x \textrm{ prime}}\nu_i(p) = \pi_{L_i}(x) + O(x^{1/2}).\end{align*} $$

Using partial summation, as above, together with the Prime ideal theorem [38], we deduce that

(4.19) $$ \begin{align} \sum_{p\leqslant x \textrm{ prime}}\frac{\nu_i(p)}{p} = \log\log x + C + O((\log x)^{-1}). \end{align} $$

Combining this with (4.18) and taking exponentials, we deduce the asymptotic in (4.11).

In the following lemma, we record three more useful estimates following similar arguments to Lemma 4.3.

Lemma 4.4. There exists constants $C,C_i>0$ such that

(4.20) $$ \begin{align} \sum_{p \in \mathcal{P}_{\leqslant x}} \frac{\varrho _i(p)}{p^2} &=(1-\kappa_i)\log\log x + C + O((\log x)^{-1}), \end{align} $$

(4.21) $$ \begin{align} \sum_{p \in \mathcal{P}^{\prime}_{\leqslant x}} \frac{\varrho _i(p)}{p^2} &=\kappa_i\log\log x + C_i + O((\log x)^{-1}), \qquad\end{align} $$

(4.22) $$ \begin{align} \sum_{p \in \mathcal{P}^{\prime}_{\leqslant x}} \frac{\varrho _i(p)}{p^2}\log p &=\kappa_i\log x +O(1). \qquad\qquad\qquad \qquad\qquad\end{align} $$

Proof. The estimates (4.20) and (4.21) are immediate consequences of (4.16), (4.18) and (4.19), together with fact that

$$ \begin{align*}\frac{\varrho _i(p)}{p^2} =\frac{(p-1)\nu_i(p)+1}{p^2}= \frac{\nu_i(p)}{p} + O(p^{-2}).\end{align*} $$

To prove (4.22), we proceed via partial summation in a very similar manner to (4.16). We recall from the Prime number theorem that

(4.23) $$ \begin{align} \pi(t) = \frac{t}{\log t}+\frac{t}{(\log t)^2} + O\left(\frac{t}{(\log t)^3}\right). \end{align} $$

For $A_t$ as defined in (4.15), we have

$$ \begin{align*} \sum_{p \in \mathcal{P}^{\prime}_{\leqslant x}}\frac{\varrho _i(p)}{p^2}\log p &= \sum_{p \in \mathcal{P}^{\prime}_{\leqslant x}}\frac{\nu_i(p)}{p}\log p + O(1)\\ &=\frac{A_x\log x}{x} - \int_{2}^x A_t\left(\frac{\log t}{t}\right)' \textrm{d}t + O(1)\\ &=\kappa_i\int_2^x \frac{(\log t -1)\pi(t)(1+O((\log t)^{-A})}{t^2} \textrm{d}t + O(1)\\ &=\kappa_i\int_{2}^x \frac{1}{t} + O\left(\frac{1}{t(\log t)^{2}}\right) \textrm{d}t + O(1) \quad (\textrm{from } (4.3))\\ &=\kappa_i\log x + O(1), \end{align*} $$

as required.

4.3 The sum $S_1$

We apply the lower bound sieve $\Lambda ^{-}(\mathcal {A}, \mathcal {P}, g, N^{\gamma })$ and the level of distribution result from Corollary 3.2 with $g_1(x,y)=1, g_2(x,y)=f(x,y), D_1=1$ and $D_2= N^{\gamma }$ . The hypotheses of Corollary 3.2 require that $\gamma <1$ . We obtain

(4.24) $$ \begin{align} S_1 \geqslant (B+o(1))XV(N^{\gamma}). \end{align} $$

By Lemma 4.3, we have

$$ \begin{align*}V(N^{\gamma})\sim \frac{c}{(\log N^{\gamma})^{\kappa}}.\end{align*} $$

For any $\epsilon>0$ , taking $\gamma $ sufficiently close to $1$ , we obtain

(4.25) $$ \begin{align} S_1 \geqslant \frac{(cB-\epsilon+o(1))X}{(\log N)^{\kappa}}. \end{align} $$

4.4 The sums $S_2^{(i)}$

We write $f_i(a,b) = pr$ , for $p \in \mathcal {P}$ and $N^{\gamma }< p \ll N$ . We apply the switching principle, which transforms the sum over p defining $S_2^{(i)}$ into a much shorter sum over the variable r.

Let $R=N^{1-\gamma }$ . The sums $S_2^{(i)}$ only involve linear factors $f_i(x,y)$ since we assume $i\in \{0,\ldots , m\}$ . Therefore, for $(a,b) \in \mathcal {R}\cap \mathbb {Z}^2$ , we have $f_i(a,b) \ll N$ , and so $|r|\ll R$ . Let $z = N^{1/3}$ . We shall take $\gamma $ arbitrarily close to 1; for now, we assume that $\gamma>2/3$ . Then by definition of $S(\mathcal {A}_p^{(i)},\mathcal {P},p)$ , we know that $\gcd (r, P(R))=1$ and $\gcd (f(a,b),P(z))=1$ .

Let $r' = |r|/\gcd (r,\Delta )$ . We now explain why $r'$ only has prime factors in $\mathcal {P}'$ . Fix $l \in S$ . Recall the assumption that $\nu _l(f(a_0,b_0)) < \nu _l(\Delta )$ . Since $f_i(a_0,b_0) \mid f(a_0,b_0)$ and $f_i(a,b) \equiv f_i(a_0,b_0) \ (\mathrm {mod}\ \Delta )$ (by the condition $C(a,b)$ ), we have

(4.26) $$ \begin{align} \nu_l(r) = \nu_l(f_i(a, b)) \leqslant \nu_l(f(a,b)) = \nu_l(f(a_0,b_0)) < \nu_l(\Delta). \end{align} $$

Therefore, $\gcd (r',\Delta ) =1$ , and by the assumption that every prime in S divides $\Delta $ , this implies $r'$ has no prime factors in S. Moreover, p is the smallest prime in $\mathcal {P}$ dividing $f_i(a,b)$ . Since $r<p$ , this means that $r'$ has no prime factors in $\mathcal {P}$ , and hence only has prime factors in $\mathcal {P}'$ .

However, $f_i(a,b)/r'$ has no prime factors in $\mathcal {P}'$ . Therefore,

(4.27) $$ \begin{align} S_2^{(i)} \leqslant \sum_{\substack{r'\ll R\\ \gcd(r',P(R)\Delta)=1}}S_2^{(i)}(r'), \end{align} $$

where

(4.28) $$ \begin{align} S_2^{(i)}(r') = \#\left\{\!\!\!(a,b) \in \mathcal{R}\cap \mathbb{Z}^2: \begin{array}{l l l} &\displaystyle C(a,b), r'\mid f_i(a,b), \\ &\displaystyle \gcd(f_i(a,b)/r', P'(z))=1\\ &\displaystyle \gcd(f(a,b),P(z))=1 \end{array} \!\!\!\!\!\right\}. \end{align} $$

Below, for convenience, we change notation from $r'$ back to r.

Let $\Lambda _1^+, \Lambda _2^+$ be upper bound sieves of level z. Defining $A(dr,e), r(dr,e)$ as in Section 3 with $g_1(x,y) = f_i(x,y)$ and $g_2(x,y) = f(x,y)$ , and writing

$$\begin{align*}m_1 = \gcd\left(\frac{f_i(a,b)}{r}, P'(z)\right), \qquad m_2= \gcd(f(a,b), P(z)), \end{align*}$$

we obtain

(4.29) $$ \begin{align} S_2^{(i)}(r) &\leqslant \sum_{\substack{(a,b) \in \mathcal{R}\cap \mathbb{Z}^2\\ C(a,b)}}\sum_{d \mid m_1}\mu(d)\sum_{e\mid m_2}\mu(e)\nonumber\\ &\leqslant \sum_{\substack{(a,b) \in \mathcal{R}\cap \mathbb{Z}^2\\ C(a,b)}}\sum_{d \mid m_1}\lambda_1^+(d)\sum_{e\mid m_2}\lambda_2^+(e)\nonumber\\ &\leqslant \sum_{\substack{d \mid P'(z)\\ \gcd(d,r)=1}}\lambda_1^+(d)\sum_{e\mid P(z)}A(dr,e)\lambda_2^+(e)\nonumber\\ &=\sum_{\substack{d \mid P'(z)\\ \gcd(d,r)=1}}\lambda_1^+(d)\sum_{e\mid P(z)}\lambda_2^+(e)\left(\frac{X\varrho (dr,e)}{(dre)^2} + r(dr,e)\right)\nonumber\\ &\leqslant \left(\frac{X\varrho _i(r)}{r^2}\sum_{\substack{d \mid P'(z)\\ \gcd(d,r) = 1}}\lambda_1^+(d)g_i(d)\sum_{\substack{e \mid P(z)}}\lambda_2^+(e)g(d)\right) + \sum_{\substack{d,e \leqslant z\\ \gcd(dr,e) = 1\\ \gcd(dre, \Delta) = 1}}|r(dr,e)|. \end{align} $$

Choose $\lambda _1^+,\lambda _2^+$ to be the beta sieves $\Lambda ^+(\mathcal {P}'\backslash \{p\mid r\}, g_i, N^{1/3}), \Lambda ^+(\mathcal {P}, g, N^{1/3})$ , respectively. Then the remainder term in (4.29) can be bounded by

$$ \begin{align*}\sum_{\substack{d,e \leqslant N^{1/3}\\ \gcd(dr,e) = 1\\ \gcd(dre, \Delta) = 1}}|r(dr,e)| \leqslant \sum_{\substack{d \ll N^{1/3}R\\e \ll N^{1/3}\\ \gcd(dr,e) = 1\\ \gcd(dre, \Delta) = 1}}|r(d,e)|,\end{align*} $$

which is negligible by Lemma 3.6 because $N^{2/3}R = N^{5/3-\gamma } \ll N^{1-\epsilon }$ .

Define a multiplicative function $h_i(r)$ supported on squarefree integers r, which is zero unless all prime factors of r are in $\mathcal {P}'$ and

(4.30) $$ \begin{align} h_i(r)=\frac{\varrho _i(r)}{r^2}\prod_{p\mid r}\left(1-\frac{\varrho _i(p)}{p^2}\right)^{-1} \end{align} $$

otherwise. Applying Theorem 4.1, we conclude that

$$ \begin{align*}S_2^{(i)}(r) \leqslant Xh_i(r)V(N^{1/3})V_i(N^{1/3})(AA_i + o(1)).\end{align*} $$

From Lemma 4.3, we obtain

$$\begin{align*}S_2^{(i)}\leqslant \frac{(cc_iAA_i+o(1))X}{(\log N^{1/3})^{\kappa + \kappa_i}}\sum_{\substack{r \ll R}}h_i(r). \end{align*}$$

To deal with the sum over $h_i(r)$ , we note that

(4.31) $$ \begin{align} \sum_{\substack{r \ll R}}h_i(r) \leqslant \prod_{\substack{p \in \mathcal{P}'\\p\ll R}}\left(1+\sum_{m=1}^{\infty} h_i(p^m)\right)= \prod_{\substack{p \in \mathcal{P}'\\p \ll R}}\left(1-g_i(p)\right)^{-1}(1+O(p^{-2})). \end{align} $$

Applying Lemma 4.3 to the product, we obtain

$$ \begin{align*}\sum_{\substack{r \ll R}}h_i(r) \ll (\log R)^{\kappa_i}.\end{align*} $$

Since $R=N^{1-\gamma }$ , we deduce that

$$ \begin{align*}S_2^{(i)} \ll \frac{cc_iAA_iX}{(\log N)^{\kappa}}\left(\frac{(1-\gamma)^{\kappa_i}}{(1/3)^{\kappa_i+\kappa}}\right).\end{align*} $$

Therefore, $S_2^{(i)}$ can be made negligible compared to $S_1$ by taking $\gamma $ arbitrarily close to 1.

4.5 The sums $S_3^{(i)}$

For a fixed prime p and an upper bound sieve $\lambda ^+$ of level z, we have

$$ \begin{align*}S(\mathcal{A}_p^{(i)},\mathcal{P}, z) \leqslant \sum_{d\mid P(z)}\lambda^+(d)A(p,d),\end{align*} $$

where $A(p,d)$ is as in Section 3 with $g_1(x,y) = f_i(x,y)$ and $g_2(x,y) = f(x,y)$ .

We first deal with the primes p in the interval $I:=(N^{\gamma }, N^{2-\gamma -\epsilon }]$ . For any $p\in I$ , let $\lambda ^+$ be the beta sieve $\Lambda ^+(\mathcal {A}_p^{(i)}, \mathcal {P}, g_i(p)g, N^{\gamma })$ . We obtain

(4.32) $$ \begin{align} S(\mathcal{A}_p^{(i)}, \mathcal{P},p) &\leqslant S(\mathcal{A}_p^{(i)}, \mathcal{P},N^{\gamma})\nonumber\\ &\leqslant (A+o(1))XV(N^{\gamma})g_i(p) + \sum_{\substack{d \leqslant N^{\gamma}\\ \gcd(d,p\Delta)=1}}|r(p,N^{\gamma})|. \end{align} $$

We apply Corollary 3.2 with $D_1 = N^{2-\gamma - \epsilon }$ and $D_2 = N^{\gamma }$ . These choices of $D_1,D_2$ satisfy the hypotheses of Corollary 3.2. Taking a sum over $p\in I$ , the contribution from the remainder term in (4.32) is negligible. We obtain

(4.33) $$ \begin{align} \sum_{\substack{p \in I\cap \mathcal{P}}}S(\mathcal{A}_p^{(i)}, \mathcal{P},p)\leqslant (A+o(1)) XV(N^{\gamma})\sum_{p \in I\cap \mathcal{P}}g_i(p). \end{align} $$

It follows from Lemma 4.4 that

$$ \begin{align*} \sum_{\substack{N^{\gamma}<p\leqslant N^{2-\gamma-\epsilon}}}g_i(p) &= \log\log N^{2-\gamma-\epsilon} - \log\log N^{\gamma} + o(1)\\ &= \log(2-\gamma - \epsilon) - \log \gamma +o(1). \end{align*} $$

Therefore, the contribution to $S_3^{(i)}$ from this range is negligible if we take $\gamma $ arbitrarily close to 1.

In the remaining range $N^{2-\gamma -\epsilon }< p <N^{\beta _i}$ , we split into dyadic intervals $(R,2R]$ . Note that for $p\in (R,2R]$ , the assumption $\gamma <1$ implies that $N^{2-\epsilon }/R <p$ , so $S(\mathcal {A}_p^{(i)}, \mathcal {P}, p) \leqslant S(\mathcal {A}_p^{(i)}, \mathcal {P}, N^{2-\epsilon }/R)$ . For each dyadic interval, we apply the beta sieve $\Lambda ^+(\mathcal {A}_p^{(i)}, \mathcal {P}, g_i(p)g, N^{2-\epsilon }/R)$ and the level of distribution result from Corollary 3.2 with $D_1=2R$ and $D_2 = N^{2-\epsilon }/R$ . At this point, we need to assume that $\beta _i<2$ for all i, so that $D_2\geqslant 1$ . We obtain

(4.34) $$ \begin{align} &\sum_{\substack{N^{2-\gamma-\epsilon}<p< N^{\beta_i}\\ p \in \mathcal{P}}}S(\mathcal{A}_p^{(i)}, \mathcal{P},p) \end{align} $$

(4.35) $$ \begin{align} &\leqslant \sum_{\substack{R \textrm{ dyadic}\\N^{2-\gamma -\epsilon}<R < N^{\beta_i}}}\sum_{\substack{p \in (R,2R]\\ p \in \mathcal{P}}}S(\mathcal{A}_p^{(i)}, \mathcal{P},N^{2-\epsilon}/R)\nonumber\\ &\leqslant (A+o(1))X\sum_{\substack{R \textrm{ dyadic}\\ N^{2-\gamma -\epsilon}<R < N^{\beta_i}}}V(N^{2-\epsilon}/R)\sum_{\substack{p \in (R,2R]\\p \in \mathcal{P}}}g_i(p)\nonumber\\ &\leqslant \frac{(cA+o(1))X}{(\log N)^{\kappa}}\sum_{\substack{ N^{2-\gamma-\epsilon} \leqslant p< N^{\beta_i}\\p \in \mathcal{P}}}\frac{g_i(p)}{(2-\epsilon - \frac{\log p}{\log N})^{\kappa}}, \end{align} $$

where the last line follows from Lemma 4.3 and the fact that $V(N^{2-\epsilon }/R)<V(N^{2-\epsilon -\log p/\log N})$ for all $p\in (R,2R]$ .

We denote the sum in (4.35) by $T(\beta _i, \kappa )$ . Since $\gamma <1$ , we have $2-\gamma -\epsilon>1$ for sufficiently small $\epsilon $ , and so we may upper bound $T(\beta _i,\kappa )$ by enlarging its range of summation to $N<p< N^{\beta _i}$ . Define

$$ \begin{align*}A(t) = \sum_{p \in \mathcal{P}_{\leqslant t}}g_i(p), \qquad h(t) = \left(2-\epsilon - \frac{\log t}{\log N}\right)^{-\kappa}.\end{align*} $$

From Lemma 4.4, we have $A(t)= (1-\kappa _i)\log \log t + C+o(1)$ for some constant C. In particular, for any $r>0$ , we have $A(N^r)-A(N) = (1-\kappa _i)\log r + o(1)$ . Applying summation by parts, followed by the substitution $t=N^s$ , we obtain

(4.36) $$ \begin{align} T(\beta_i,\kappa) &\leqslant (A(N^{\beta_i})-A(N))h(N^{\beta_i}) - \int_{N}^{N^{\beta_i}}(A(t)-A(N))h'(t) \textrm{d}t \nonumber \\ & =(A(N^{\beta_i})-A(N))h(N^{\beta_i}) - \int_{1}^{\beta_i}(A(N^s)-A(N))\frac{\partial h(N^s)}{\partial s}\textrm{d}s \nonumber \\ &=(1-\kappa_i)\log \beta_i h(N^{\beta_i}) -(1-\kappa_i)\int_{1}^{\beta_i}(\log s) \frac{\partial h(N^s)}{\partial s}\textrm{d}s + o(1) \nonumber \\ &=(1-\kappa_i)\int_{1}^{\beta_i}\frac{h(N^s)}{s}\textrm{d}s + o(1). \end{align} $$

Taking $\epsilon $ sufficiently small and redefining it appropriately, we may replace $h(N^s)$ by $(2-s)^{-\kappa }$ at the cost of adding $\epsilon $ in (4.36). Combining with (4.35), we conclude that for any $\epsilon>0$ ,

(4.37) $$ \begin{align} S_3^{(i)}\leqslant \frac{(cA+\epsilon+o(1))X}{(\log N)^{\kappa}}\cdot (1-\kappa_i) \int_{1}^{\beta_i}(2-s)^{-\kappa}\frac{\textrm{d}s}{s}. \end{align} $$

Due to the factor $1-\kappa _i = \alpha \theta _i$ appearing in the above estimate, $S_3^{(i)}$ becomes negligible compared to $S_1$ as $\alpha \rightarrow 0$ . We perform a more precise quantitative comparison in Section 4.7.

4.6 The sums $S_4^{(i)}$

We begin in a similar manner to the sums $S_2^{(i)}$ , by writing $f_i(a,b) = pr$ , for $p \in \mathcal {P}$ , where now $N^{\beta _i}\leqslant p < x$ and $R=x/N^{\beta _i}$ . Let $D_1 = N^{\eta _1}$ and $D_2 = N^{\eta _2}$ for parameters $\eta _1, \eta _2>0$ (which may depend on r and i) to be chosen later. We assume $\eta _2 < \beta _i$ so that the condition $\gcd (f(a,b), P(p)) =1$ can be replaced by the weaker condition $\gcd (f(a,b), P(N^{\eta _2}))=1$ . Proceeding as in Section 4.4, we have

(4.38) $$ \begin{align} S_4^{(i)} \leqslant \sum_{\substack{r\ll R\\ \gcd(r,P(N^{\beta_i})\Delta)=1}}S_4^{(i)}(r), \end{align} $$

where

(4.39) $$ \begin{align} S_4^{(i)}(r) = \#\left\{\!(a,b) \in \mathcal{R}\cap \mathbb{Z}^2: \begin{array}{l l l} &\displaystyle C(a,b), r\mid f_i(a,b), \\ &\displaystyle \gcd(f_i(a,b)/r, P'(N^{\eta_1}))=1\\ &\displaystyle \gcd(f(a,b),P(N^{\eta_2}))=1 \end{array} \!\!\!\!\right\}. \end{align} $$

Similarly to (4.29), for any upper bound sieves $\lambda _1^+, \lambda _2^+$ of levels $N^{\eta _1}, N^{\eta _2}$ , the quantity $S_4^{(i)}(r)$ is bounded above by

$$\begin{align*}\left(\frac{X\varrho _i(r)}{r^2}\sum_{\substack{d \mid P'(N^{\eta_1})\\ \gcd(d,r) = 1}}\lambda_1^+(d)g_i(d)\sum_{\substack{e \mid P(N^{\eta_2})}}\lambda_2^+(e)g(d)\right) + \sum_{\substack{d\leqslant N^{\eta_1}, e \leqslant N^{\eta_2}\\ \gcd(dr,e) = 1\\ \gcd(dre, \Delta) = 1}}|r(dr,e)|. \end{align*}$$

In order to ensure the error terms from applying the level of distribution result from Corollary 3.2 are negligible after summing over r, we need $\eta _1,\eta _2$ to satisfy

(4.40) $$ \begin{align} \eta_1, \eta_2>0, \quad \eta_2 \leqslant 1-\delta \qquad (\textrm{for all }r\leqslant R), \end{align} $$

(4.41) $$ \begin{align} \sum_{r\leqslant R}N^{\eta_1+\eta_2}\leqslant N^{2-\delta}. \end{align} $$

Remark 4.5. There are no $\eta _1,\eta _2$ satisfying (4.40) and (4.41) unless $R\leqslant N^{2-\delta }$ , i.e., $x \leqslant N^{2-\delta + \beta _i}$ . Since we had to assume $\beta _i<2$ in the treatment of the sums $S_3^{(i)}$ in Section 4.5, this means our approach cannot handle the case $d \geqslant 4$ , in which $f(x,y)$ has an irreducible factor of degree $\geqslant 4$ . Therefore, we proceed with the additional assumption that $d\leqslant 3$ .

Choose $\lambda ^+_1, \lambda ^+_2$ to be the beta sieves $\Lambda ^+(\mathcal {P}'\backslash \{p\mid r\}, g_i, N^{\eta _1}), \Lambda ^+(\mathcal {P}, g, N^{\eta _2})$ , respectively. Recalling the definition of $h_i(r)$ from (4.30), we obtain

(4.42) $$ \begin{align} S_4^{(i)}(r) \leqslant (AA_i+o(1))Xh_i(r)V_i(N^{\eta_1})V(N^{\eta_2}).\end{align} $$

Using Lemma 4.3 to estimate the products in (4.42), and taking a sum over r, we obtain

(4.43) $$ \begin{align} S_4^{(i)}\leqslant \frac{(cc_iAA_i+o(1))X}{(\log N)^{\kappa_i+\kappa}}\sum_{r\leqslant R}\frac{h_i(r)}{\eta_1^{\kappa_i}\eta_2^{\kappa}}. \end{align} $$

We divide the sum over r into dyadic intervals $r \in (R_1,2R_1]$ , and take $\eta _1,\eta _2$ depending only on $R_1$ and i. To obtain a good estimate for (4.43), we maximise $\eta _1^{\kappa _i}\eta _2^{\kappa }$ subject to the constraints

$$ \begin{align*}\eta_1,\eta_2>0, \qquad \eta_2\leqslant 1-\delta, \qquad \eta_1+\eta_2 \leqslant 2-\delta - \frac{\log R_1}{\log N}.\end{align*} $$

By a similar computation to [27, Section 6.5], the optimal solution is

$$ \begin{align*} \eta_1 &= \frac{\kappa_i}{\kappa+\kappa_i}\left(2-\delta-\frac{\log R_1}{\log N}\right),\\ \eta_2 &= \frac{\kappa}{\kappa+\kappa_i}\left(2-\delta-\frac{\log R_1}{\log N}\right). \end{align*} $$

We note that for $\delta>0$ sufficiently small, this solution satisfies $\eta _2 \leqslant 1-\delta $ due to the assumption $\kappa <1/2$ . Substituting this choice of $\eta _1,\eta _2$ into (4.43), we obtain

(4.44) $$ \begin{align} \sum_{r\leqslant R}\frac{h_i(r)}{\eta_1^{\kappa_i}\eta_2^{\kappa}}\leqslant \sum_{r\leqslant R}w(r,\delta)h_i(r), \end{align} $$

where

(4.45) $$ \begin{align} w(r,\delta) = \left(\frac{\kappa}{\kappa+\kappa_i}\right)^{-\kappa}\left(\frac{\kappa_i}{\kappa+\kappa_i}\right)^{-\kappa_i}\left(2-\delta -\frac{\log r}{\log N}\right)^{-(\kappa+\kappa_i)}. \end{align} $$

We treat the sum in (4.44) using partial summation, for which we require estimates for $\sum _{r\leqslant t}h_i(r)$ . For the case $d=2$ , the estimate already found in (4.31) is sufficient. Below, we find a more refined estimate which we use in the case $d=3$ .

We first consider the contribution to (4.44) from squarefree values of r. We would like to apply [18, Theorem A.5], which states that under certain hypothesis on the function $h_i(r)$ , we have

(4.46) $$ \begin{align} \sum_{\substack{m \leqslant x\\ \mu^2(m)=1}}h_i(m)= c_{h_i}(\log x)^{\kappa_i} + O((\log x)^{\kappa_i -1}), \end{align} $$

where

$$ \begin{align*}c_{h_i} = \frac{1}{\Gamma(\kappa_i+1)}\prod_p \left(1-\frac{1}{p}\right)^{\kappa_i}(1+h_i(p)).\end{align*} $$

In the following lemma, we verify that the function $h_i(r)$ satisfies the required hypotheses for [18, Theorem A.5].

Lemma 4.6. For any $x\geqslant 1$ and any $2\leqslant w< z$ , the function $h_i(r)$ satisfies the following estimates:

(4.47) $$ \begin{align} \prod_{w\leqslant p <z}(1+h_i(p)) &\ll \left(\frac{\log z}{\log w}\right)^{\kappa_i}, \end{align} $$

(4.48) $$ \begin{align} \sum_p h_i(p)^2\log p &< \infty, \end{align} $$

(4.49) $$ \begin{align} \sum_{p \leqslant x}h_i(p)\log p &= \kappa_i\log x + O(1). \end{align} $$

Proof. To prove (4.47), we note that $1+h_i(p) = \left (1-\frac {\varrho _i(p)}{p^2}\right )^{-1}$ for all $p\in \mathcal {P}'$ . The result is then immediate from Lemma 4.3. To prove (4.45), we recall that $\varrho _i(p) \ll p$ , and so $h_i(p) \ll p^{-1}$ . Therefore,

$$ \begin{align*}\sum_{p}h_i(p)^2\log p \ll \sum_{p}\frac{\log p}{p^2}< \infty.\end{align*} $$

Finally, we note that

$$ \begin{align*} \sum_{p\leqslant x}h_i(p)\log p =\sum_{p \in \mathcal{P}^{\prime}_{\leqslant x}}\frac{\varrho _i(p)}{p^2}\log p + O(1), \end{align*} $$

so that (4.49) follows by applying Lemma 4.4.

We can now evaluate the sum in (4.44) using partial summation. We obtain

$$ \begin{align*} \sum_{\substack{r\leqslant R\\ \mu^2(r) = 1}}w(r,\delta)h_i(r)&= w(R,\delta)\sum_{\substack{r\leqslant R\\ \mu^2(r)=1}}h_i(r)-\int_{1}^R \left(\sum_{\substack{r\leqslant t\\ \mu^2(r)=1}}h_i(r)\right)w'(t,\delta) \textrm{d}t\\ &=c_{h_i}\left[(\log R)^{\kappa_i}w(R,\delta)-\int_{1}^R w'(t,\delta)(\log t)^{\kappa_i}\textrm{d}t\right]\\ &\quad +o(1)\left[(\log R)^{\kappa_i}w(R,\delta)+\int_{1}^R w'(t,\delta)(\log t)^{\kappa_i}\textrm{d}t\right]\\ &= c_{h_i}\kappa_i\int_{1}^R w(t,\delta)(\log t)^{\kappa_i-1}t^{-1}\textrm{d}t+o\left((\log R)^{\kappa_i}\right)\\ &=c_{h_i}\kappa_i(\log N)^{\kappa_i}\int_{0}^{\log R/\log N}w(N^s, \delta)s^{\kappa_i -1}\textrm{d}s + o\left((\log R)^{\kappa_i}\right)\\ &\leqslant (c_{h_i}\kappa_i +o(1))(\log N)^{\kappa_i}\int_{0}^{d-\beta_i}W(s)\textrm{d}s, \end{align*} $$

where $W(s) = w(N^s, 0)s^{\kappa _i-1}.$

We now consider the contribution to (4.30) from those r which are not squarefree. Let $d_i$ denote the degree of $f_i$ . From [15, Lemma 3.1], we have $\varrho _i(p^{\alpha }) \ll p^{2\alpha (1-1/d_i)}$ for all primes $p \notin S$ and any positive integer $\alpha $ . By the multiplicativity of $h_i(r)$ , it follows that $h_i(r) \ll r^{-2/d_i +\epsilon }$ .

We recall that a positive integer n is squareful if for any prime $p\mid n$ , we also have $p^2\mid n$ . Since $d_i\leqslant 3$ , we have

$$ \begin{align*}\sum_{r \textrm{ squareful}}h_i(r) \ll \sum_{r \textrm{ squareful}}r^{-2/d_i + \epsilon} \leqslant \sum_{r \textrm{ squareful}}r^{-2/3 + \epsilon}< \infty,\end{align*} $$

where the last inequality follows from partial summation together with the fact that there are $O(M^{1/2})$ squareful positive integers less than M. Since $h_i(r)$ is supported on integers with no prime factors in S, for any $\epsilon>0$ , there exists a set of primes $S_0$ , depending only on $\epsilon $ and f, such that for any $S\supseteq S_0$ , we have

$$ \begin{align*}\sum_{\substack{r \textrm{ squareful}\\ r>1}}h_i(r) <\epsilon.\end{align*} $$

For the remainder of this section, we assume that $S\supseteq S_0$ . Proceeding as in [27, Lemma 6.2], we use that $w(r,\delta ) \ll 1$ , and decompose each non-squarefree r into $r=r_1r_2$ , where $r_1$ is squarefree and $r_2>1$ is squareful. We have

$$ \begin{align*}\sum_{\substack{r \leqslant R\\ \mu^2(r)=0}}w(r,\delta)h_i(r) \ll \sum_{\substack{r_1 \leqslant R\\ \mu^2(r_1)=1}}h_i(r_1) \sum_{\substack{r_2 \textrm{ squareful}\\right_2>1}}h_i(r_2).\end{align*} $$

Combining with (4.46), we deduce that for any $\epsilon>0$ , we have

$$ \begin{align*}\sum_{\substack{r \leqslant R\\ \mu^2(r)=0}}w(r,\delta)h_i(r) \leqslant (\epsilon + o(1))(\log N)^{\kappa_i}.\end{align*} $$

In conclusion, we have the upper bound

(4.50) $$ \begin{align} S_4^{(i)} \leqslant \frac{(cc_iAA_ic_{h_i}\kappa_i + \epsilon + o(1))X}{(\log N)^{\kappa}}\int_{0}^{d-\beta_i}W(s)\textrm{d}s, \end{align} $$

where

(4.51) $$ \begin{align} W(s)= \left(\frac{\kappa}{\kappa+\kappa_i}\right)^{-\kappa}\left(\frac{\kappa_i}{\kappa+\kappa_i}\right)^{-\kappa_i}(2-s)^{-(\kappa+\kappa_i)}s^{\kappa_i-1}. \end{align} $$

Remark 4.7. Whilst $c,c_i, c_{h_i}$ all depend on S, the ratio $\kappa _ic_ic_{h_i}AA_i/B$ of the constants in the bounds for $S_4^{(i)}$ and $S_1$ is independent of S, as seen below in (4.52). Therefore, we may indeed take $\epsilon $ to be arbitrarily small in (4.50) without sacrificing anything in our comparison of $S_1$ and $S_4^{(i)}$ . The $o(1)$ terms in (4.50) and in the bounds for the other sums $S_1$ , $S_2^{(i)}, S_3^{(i)}$ also depend on S (via the quantity L in (4.4)), but this does not matter because we may take N to be sufficiently large in terms of S.

4.7 Proof of Theorem 2.1

We now suppose that $d=2$ (i.e., $f(x,y)$ consists of irreducible factors of degree at most $2$ ). We first obtain a qualitative result by considering the limit as $\alpha \rightarrow 0$ . As $\alpha \rightarrow 0$ , we have $\kappa \rightarrow 0$ and $\kappa _i \rightarrow 1$ . Therefore, we have $A_i \rightarrow A(1)$ , which is equal to $2e^{\gamma }$ , where $\gamma = 0.57721\ldots $ is the Euler–Mascheroni constant. Moreover, by [18, Equation (11.62)], we have $A(\kappa ),B(\kappa ) \rightarrow 1$ as $\kappa \rightarrow 0$ . By a very similar computation to [27, p. 248], we have

(4.52) $$ \begin{align} c_ic_{h_i} = \frac{e^{-\gamma \kappa_i}}{\Gamma(1+\kappa_i)}. \end{align} $$

Therefore, the ratio of the constants in the bounds for $S_1$ and $S_4^{(i)}$ is $\kappa _ic_icc_{h_i}AA_i/cB$ , which is independent of S and tends to zero as $\alpha \rightarrow 0$ . Also,

$$ \begin{align*}\lim_{\alpha \rightarrow 0}W(s) = (2-s)^{-1}.\end{align*} $$

Therefore,

(4.53) $$ \begin{align} \lim_{\alpha \rightarrow 0}S_4^{(i)} \ll \frac{(B+o(1))X}{(\log N)^{\kappa}}\left(\log 2-\log \beta_i\right). \end{align} $$

For all $\epsilon>0$ , by choosing $\beta _i$ sufficiently close to $2$ , we have the bound

(4.54) $$ \begin{align} \lim_{\alpha \rightarrow 0}S_4^{(i)} \leqslant \frac{(\epsilon B+o(1)) X}{(\log N)^{\kappa}}\ll \epsilon S_1. \end{align} $$

We recall from Sections 4.4 and 4.5 that $S_2^{(i)}$ and $S_3^{(i)}$ are also negligible compared to $S_1$ as $\alpha \rightarrow 0$ . Therefore, we see that $S(\mathcal {A},\mathcal {P},x)>0$ for sufficiently small $\alpha $ and sufficiently large N.

To obtain the best quantitative bounds, the choices of $\beta _i \in (1,2)$ should be optimised so as to minimise $S_3^{(i)}+S_4^{(i)}$ . However, in practice, numerical computations suggest that the optimal choices for $\beta _i$ are extremely close to $2$ , and little is lost in taking them arbitrarily close to 2, as above. In this case, the contributions from the sums $S_4^{(i)}$ are negligible. Taking a sum over i of the estimates from (4.37) and combining with (4.25), for any $\epsilon>0$ , we have

$$ \begin{align*}S(\mathcal{A},\mathcal{P},x) \geqslant \left(\frac{(c-\epsilon+o(1))X}{(\log N)^{\kappa}}\right)\left(B-A\alpha \sum_{i=m+1}^k \theta_i \int_{1}^{2}(2-s)^{-\kappa}\frac{\textrm{d}s}{s}\right).\end{align*} $$

Let $r(\kappa ) = B/A$ . Then we have established that $S(\mathcal {A}, \mathcal {P}, x)>0$ provided that

(4.55) $$ \begin{align} r(\kappa) - \alpha\sum_{i=m+1}^k \theta_i\int_{1}^{2}(2-s)^{-\kappa}\frac{\textrm{d}s}{s}>0. \end{align} $$

We recall that $\theta = \theta _0 + \cdots +\theta _k$ and $\kappa = \alpha \theta $ . Therefore, we may replace $\alpha \sum _{i=m+1}^k\theta _i$ with the trivial upper bound $\kappa $ , after which we find by numerical computations (see Section 4.9 for details) that the largest value of $\kappa $ we can take in (4.55) is $\kappa = 0.39000\ldots $ . Thus the condition $\alpha \theta \leqslant 0.39$ is enough to ensure that $S(\mathcal {A}, \mathcal {P},x)>0$ for sufficiently large N. This completes the proof of Theorem 2.1.

4.8 Proof of Theorem 2.3

We now discuss the case $d=3$ , where $f(x,y)$ may contain irreducible factors of degree up to $3$ . We recall in the case $d=2$ , the sums $S_4^{(i)}$ could be made negligible compared to $S_1$ by choosing $\beta _i$ arbitrarily close to $2$ , due to the factor $\log 2- \log \beta _i$ appearing in (4.53). When $d=3$ , we obtain the same bound as in (4.53), but with $\log 2-\log \beta _i$ replaced with $\log 2 - \log (\beta _i-1)$ . Consequently, $S_4^{(i)}$ is no longer negligible, even when $\alpha \rightarrow 0$ and $\beta _i$ is arbitrarily close to $2$ . In fact, it can be checked that its limit as $\alpha \rightarrow 0$ and $\beta _i \rightarrow 2$ is larger than $S_1$ , and so the above methods break down in this case.

However, we recall the additional hypothesis in Theorem 2.3 that Assumption 2.2 holds. By enlarging $\mathcal {P}$ if necessary, we may assume that equality holds in (2.9) – namely,

$$\begin{align*}\mathcal{P}\backslash S = \bigcup_{j=1}^n \{p \equiv t_i \ (\mathrm{mod}\ q_i)\}. \end{align*}$$

By Dirichlet’s theorem on primes in arithmetic progressions, (2.4) holds with $\alpha \leqslant \sum _{j=1}^n \frac {1}{q_j-1} \leqslant 0.32380$ . Moreover, it follows from Lemma 5.8 (a version of the Chebotarev density theorem) that (2.5) holds for some $\theta _i \leqslant 3$ . We now explain why this choice of $\mathcal {P}$ is easier to handle than arbitrary choices of $\mathcal {P}$ of the same density $\alpha $ .

When applying the switching principle for $S_4^{(i)}$ , we wrote $f_i(a,b) = pr$ for $p \in \mathcal {P}$ . Since we may assume $q_1, \ldots , q_n \in S_0$ , the congruence condition $C(a,b)$ forces $f_i(a,b)$ to lie in a particular congruence class modulo $q_1, \ldots , q_n$ . Combined with the fact that $p \equiv t_j \ (\mathrm {mod}\ q_j)$ for some j, we see that r lies in a particular congruence class modulo $q_j$ for some $j\in \{1, \ldots , n\}$ , and this congruence class depends only on $t_j, C(a,b)$ and f. Moreover, by (4.26), we have that $\gcd (r, \Delta ) = \gcd (f_i(a_0,b_0),\Delta )$ depends only on $C(a,b)$ and f. We deduce that there exist $r_1,\ldots , r_n$ , depending only on $t_1, \ldots , t_n, C(a,b)$ and f, such that $r':= |r|/\gcd (r,\Delta )$ lies in the set

$$\begin{align*}\mathcal{T} := \{r' \in \mathbb{N}: r'\equiv r_j \ (\mathrm{mod}\ q_j) \textrm{ for some }j\in\{1, \ldots, n\}\}. \end{align*}$$

Adding this condition into (4.43) and (4.44) and changing notation from $r'$ back to r, we obtain

(4.56) $$ \begin{align} S_4^{(i)}\leqslant \frac{(cc_iAA_i+o(1))X}{(\log N)^{\kappa_i+\kappa}}\sum_{r \in \mathcal{T}_{\leqslant x}}w(r,\delta)h_i(r). \end{align} $$

As demonstrated by Irving [27, Lemma 6.1], the argument based on [18, Theorem A.5] we used to obtain (4.46) can be generalised to give

(4.57) $$ \begin{align} \sum_{\substack{r \in \mathcal{T}_{\leqslant x}\\ \mu^2(r)=1}}h_i(r) \leqslant \sum_{j=1}^n \sum_{\substack{r \leqslant x\\ r\equiv r_j \ (\mathrm{mod}\ q_j)}}h_i(r) = c_{h_i}(\log x)^{\kappa_i}\sum_{j=1}^n \frac{1}{q_j-1} + O((\log x)^{\kappa_i-1}). \end{align} $$

Proceeding as before, we deduce the same estimate for $S_4^{(i)}$ as in (4.50), but with an additional factor $\alpha _0 := \sum _{j=1}^n \frac {1}{q_j-1}$ . It is now clear qualitatively that for sufficiently large $q_1, \ldots , q_n$ (i.e., as $\alpha _0 \rightarrow 0$ ), the sums $S_4^{(i)}$ are once again negligible compared to $S_1$ .

We now make the above discussion more quantitative in order to complete the proof of Theorem 2.3. Combining everything, we see that $S(\mathcal {A},\mathcal {P},x)>0$ for sufficiently large N provided that

(4.58) $$ \begin{align}r(\kappa)-\sum_{i\,:\, \deg f_i =2}\alpha \theta_i \int_{1}^2(2-s)^{-\kappa}\frac{\textrm{d}s}{s}- \sum_{\substack{i \,:\, \deg f_i =3}}\alpha_0 A_i\kappa_ic_ic_{h_i}\int_{0}^1 W(s) \textrm{d}s>0. \end{align} $$

We denote the left-hand side of (4.58) by $F(\boldsymbol {\theta })$ . Recalling (4.51) and (4.52), we have

(4.59) $$ \begin{align} A_i\kappa_ic_ic_{h_i}\int_{0}^1 W(s) \textrm{d}s = \frac{A_i\kappa_ie^{-\gamma \kappa_i}\kappa^{-\kappa}\kappa_i^{-\kappa_i}(\kappa+\kappa_i)^{\kappa+\kappa_i}}{\Gamma(1+\kappa_i)}\int_0^1 (2-s)^{-(\kappa+\kappa_i)}s^{\kappa_i-1}\textrm{d}s. \end{align} $$

For a fixed choice of $\kappa <1/2$ , the integrand is a decreasing function of $\kappa _i$ because $0\leqslant \frac {s}{(2-s)}\leqslant 1$ for any $s\in [0,1]$ . The functions $\Gamma (1+\kappa _i)^{-1}$ , $\kappa _i^{-\kappa _i}$ and $e^{-\gamma \kappa _i}$ are also decreasing in $\kappa _i$ in the range $\kappa _i\in (1/2,1)$ . Let $t = \alpha _0 \deg f$ . Since $\kappa _i \geqslant 1-\kappa \geqslant 1-t$ , we therefore obtain an upper bound by replacing $\kappa _i$ with $1-t$ in all these terms. The remaining terms in (4.59) are all increasing in $\kappa _i$ , and we apply the trivial bound $\kappa _i \leqslant 1$ . Finally, we note that $\kappa ^{-\kappa }(\kappa +\kappa _i)^{\kappa +\kappa _i}$ is an increasing function in $\kappa $ for $\kappa <1/2$ , and so we may replace $\kappa $ by t in this expression. Therefore, (4.59) can be bounded by

(4.60) $$ \begin{align} \frac{A(1)e^{\gamma(t-1)}t^{-t}(t+1)^{t+1}}{\Gamma(2-t)}\int_{0}^1 (2-s)^{-1}s^{-\kappa}\textrm{d}s. \end{align} $$

We denote the factor outside the integral in (4.60) by $H(t)$ . The integral in (4.60) is equal to the first integral in (4.58), as can be seen by making the substitution $u=2-s$ . Therefore,

$$ \begin{align*} F(\boldsymbol{\theta})&\geqslant r(\kappa)-\left(\sum_{i\,:\,\deg f_i =2}\alpha\theta_i + \sum_{i\,:\, \deg f_i = 3} \alpha_0 H(t)\right) \int_0^1 (2-s)^{-1}s^{-\kappa}\textrm{d}s. \end{align*} $$

Recalling that $A(1)=2e^{\gamma }$ , it can be checked that $H(t)>4$ for all $t>0.2$ , and so in particular, $H(t)>2\theta _i$ whenever $\deg f_i =2$ . Moreover, r is a decreasing function of $\kappa $ , so $r(\kappa )\geqslant r(t)$ . We obtain

(4.61) $$ \begin{align} F(\boldsymbol{\theta})&\geqslant r(t)-H(t)\int_{0}^1 (2-s)^{-1}s^{-t}\textrm{d}s\left(\sum_{i\,:\,\deg f_i=2}\alpha/2 +\sum_{\substack{i:\deg f_i =3}}\alpha_0\right)\nonumber\\ &\geqslant r(t)-\frac{tH(t)}{3}\int_{0}^1 (2-s)^{-1}s^{-t}\textrm{d}s \end{align} $$

for any $t>0.2$ . We find by numerical computations that the above expression is positive provided that $t\leqslant 0.32380.$ . Since $t= \alpha _0\deg f$ , this is implied by the condition (2.9) assumed in Theorem 2.3.

4.9 Details of the numerical computations

We now explain how we obtained numerical values with guaranteed error bounds in the proofs of Theorem 2.1 and Theorem 2.3. From [18, Equations (11.62), (11.63)], we have for $0< \kappa < 1/2$ that

$$\begin{align*}A(\kappa) = \frac{2e^{\gamma\kappa}}{\Gamma(1-\kappa)}\frac{q(\kappa)}{p(\kappa)+ q(\kappa)}, \qquad B(\kappa) = \frac{2e^{\gamma \kappa}}{p(\kappa) + q(\kappa)}, \end{align*}$$

where

$$ \begin{align*} p(\kappa) &= \int_{0}^{\infty}z^{-\kappa}\exp(-z+\kappa\operatorname{Ei}(-z))\operatorname{d}z \\ q(\kappa) &=\int_{0}^{\infty}z^{-\kappa}\exp(-z-\kappa\operatorname{Ei}(-z))\operatorname{d}z \end{align*} $$

and where $\operatorname {Ei}(-z) = -\int _{z}^{\infty }\frac {e^{-u}}{u}\operatorname {d}u$ is the standard exponential integral. For any $0<\kappa < 1/2$ , the integrand defining $p(\kappa )$ is monotonically decreasing, decays exponentially as $z \rightarrow \infty $ , and is bounded above by $2$ . Similarly, the integrand defining $q(\kappa )$ is monotonically decreasing and decays exponentially as $z \rightarrow \infty $ , but diverges as $z \rightarrow 0$ . To analyse its behaviour near $0$ , we note that for $\delta \in (0,1)$ and $\kappa < 0.4$ , we have

$$ \begin{align*} &\int_{0}^{\delta}z^{-\kappa}e^{-z}\exp(-\kappa\operatorname{Ei}(-z))\operatorname{d}z\\ &\leqslant \int_{0}^{\delta}z^{-0.4}\operatorname{exp}\left(0.4\int_{z}^{\infty}\frac{e^{-u}}{u}\operatorname{d}u\right)\operatorname{d}z\\ &\leqslant \int_{0}^{\delta}z^{-0.4} \operatorname{exp}\left(0.4\int_{z}^{1}\frac{\operatorname{d}u}{u} + 0.4\int_{1}^{\infty}e^{-u}\operatorname{d}u\right)\operatorname{d}z\\ &\leqslant \int_{0}^{\delta}z^{-0.4}\exp(-0.4\log(z) + 0.4e^{-1})\operatorname{d}z\\ &\leqslant 2 \int_{0}^{\delta}z^{-0.8}\operatorname{d}z\\ &\leqslant 10\delta^{0.2}. \end{align*} $$

Therefore, we can obtain estimates for $p(\kappa ), q(\kappa )$ (and hence also $A(\kappa ), B(\kappa )$ with provable error bounds using standard numerical integration methods such as the trapezoid rule. The functions from (4.55) and (4.61) are bounded, continuous, and monotonically decreasing in $\kappa $ , and so it is straightforward to find their roots to the desired precision. This was implemented in Sage, which produced the values $0.39000, 0.32380$ correct to five decimal places in $12.3$ seconds on a standard laptop with $2$ cores.

5 Application to the Hasse principle

In this section, we apply the main sieve results (Theorem 2.1 and Theorem 2.3) obtained in Section 4 in order to prove Theorem 1.1 and Theorem 1.3.

5.1 Algebraic reduction of the problem

Let K be a number field of degree n satisfying the Hasse norm principle. In [27, Lemma 2.6], Irving turns the problem of establishing the Hasse principle for

(5.1) $$ \begin{align} f(t) = \mathbf{N}(x_1, \ldots, x_n) \neq 0 \end{align} $$

into a sieve problem. Irving assumes that $f(t) \in \mathbb {Z}[t]$ is an irreducible cubic polynomial. However, in the following result, we demonstrate that Irving’s strategy can be applied to establish a similar result for an arbitrary polynomial $f\in \mathbb {Z}[t]$ . We recall that $f(x,y)$ denotes the homogenisation of f. Throughout this section, we make the choice

(5.2) $$ \begin{align} \mathcal{P} = \{p \notin S: \textrm{ the inertia degrees of }p\textrm{ in }K/\mathbb{Q}\textrm{ are not coprime}\} \end{align} $$

for a finite set of primes S containing all ramified primes in $K/\mathbb {Q}$ .

Proposition 5.1. Suppose that (5.1) has solutions over $\mathbb {Q}_p$ for every p and over $\mathbb {R}$ . Let $\mathcal {P}$ and S be as in (5.2). Then there exists $\Delta \in \mathbb {N}$ , divisible only by primes in S, integers $a_0,b_0$ , and real numbers $r,\xi>0$ such that the following implication holds:

Suppose that $a,b$ are integers for which

1. (1) $a \equiv a_0 \ (\mathrm {mod}\ \Delta )$ and $b \equiv b_0 \ (\mathrm {mod}\ \Delta )$ ,

2. (2) $|a/b - r|<\xi $ ,

3. (3) $bf(a,b)$ has no prime factors in $\mathcal {P}$ .

Then (5.1) has a solution over $\mathbb {Q}$ .

By multiplicativity of norms, it suffices to find integers $a,b$ such that b and $f(a,b)$ are in $N_{K/\mathbb {Q}}(K^*)$ . Since K satisfies the Hasse norm principle, we have $\mathbb {Q}^*\cap N_{K/\mathbb {Q}}(I_K) = N_{K/\mathbb {Q}}(K^*)$ , where $I_K$ denotes the ideles of K. Consequently, to show that $c\in \mathbb {Q}^*$ is a norm from K, it suffices to find elements $x_v \in K_v^{*}$ for each place v of K, such that

1. (1) For all but finitely many places v of K, we have $x_v\in \mathcal {O}_v^*$ (this ensures that $(x_v) \in I_K$ ).

2. (2) For all places w of $\mathbb {Q}$ , we have

   (5.3) $$ \begin{align} \prod_{v\mid w}N_{K_v/\mathbb{Q}_w}(x_v)=c. \end{align} $$

The arguments in [27, Lemma 2.2, Lemma 2.3, Lemma 2.4] go through without changes. We summarise them below.

Lemma 5.2. Suppose $c\neq 0$ is an integer. Then

1. (1) If $p\nmid c$ and $K/\mathbb {Q}$ is unramified at p, then there exist $x_v \in \mathcal {O}_v^*$ for each place v of K above p, such that $\prod _{v\mid p}N_{K_v/\mathbb {Q}_p}(x_v)=c$ .

2. (2) Suppose that $K/\mathbb {Q}$ is unramified at p, and that the inertia degrees above p in K are coprime. Then there exist $x_v \in K_v^*$ for each place v of K above p, such that $\prod _{v\mid p}N_{K_v/\mathbb {Q}_p}(x_v)=c$ .

3. (3) Let p be a place of $\mathbb {Q}$ . Suppose that there exists $x_1, \ldots , x_n \in \mathbb {Q}_p$ such that $c =\mathbf {N}(x_1, \ldots ,x_n)$ . Then there exists $x_v \in K_v^*$ for each place v of K above p, such that $\prod _{v\mid p}N_{K_v/\mathbb {Q}_p}(x_v)=c$ .

We now give a slight generalisation of [27, Lemma 2.5] to the case of an arbitrary polynomial f.

Lemma 5.3. Let p be a prime for which $f(t) = \mathbf {N}(x_1, \ldots , x_n) \neq 0$ has a solution over $\mathbb {Q}_p$ . Then there exists $a_0,b_0 \in \mathbb {Z}$ and $l \in \mathbb {N}$ (all depending on p) such that for any $a,b \in \mathbb {Z}$ satisfying $a\equiv a_0 \ (\mathrm {mod}\ p^l), b\equiv b_0 \ (\mathrm {mod}\ p^l)$ , we have $b,f(a,b) \in \mathbf {N}(\mathbb {Q}_p^n)\backslash \{0\}$ .

Proof. We define $N = \mathbf {N}(\mathbb {Q}_p^n)\backslash \{0\}$ . Let $t_1 \in \mathbb {Q}_p$ be such that $f(t_1) = \mathbf {N}(x_1, \ldots , x_n) \neq 0$ has a solution over $\mathbb {Q}_p$ . Choose $a_1, b_1 \in \mathbb {Z}_p$ such that $\nu _p(b_1)$ is a multiple of n, and $a_1/b_1 = t_1$ . Then $b_1 \in N$ and $f(a_1,b_1) \in N$ .

The set $N \subseteq \mathbb {Q}_p$ is open, and so $N\times N \subseteq \mathbb {Q}_p^2$ is open. Moreover, the map $\varphi : \mathbb {Q}_p^2 \rightarrow \mathbb {Q}_p^2$ sending $(a,b)$ to $(f(a,b),b)$ is continuous in the p-adic topology. Therefore, the set $\varphi ^{-1}(N\times N)$ is open, and contains the element $(a_1,b_1)$ . Hence, there is a small p-adic ball with centre $(a_1,b_1)$ , all of whose elements $(a,b)$ satisfy $b, f(a,b) \in N$ . This ball can be described by congruence conditions $a\equiv a_0 \ (\mathrm {mod}\ p^l), b\equiv b_0 \ (\mathrm {mod}\ p^l)$ for a sufficiently large integer l, as claimed in the lemma.

Proof of Proposition 5.1

The proof closely follows the argument in [27, Lemma 2.6]. By the Hasse norm principle, to find solutions to (5.1), it suffices to find integers $a,b$ such that properties (1) and (2) stated before Lemma 5.2 hold with $c=b$ and $c=f(a,b)$ . We divide the places of $\mathbb {Q}$ into four sets:

1. (1) $p\in S$ . Here, Lemma 5.3 gives congruence conditions $a \equiv a_{0,p} \ (\mathrm {mod}\ p^l)$ , $b \equiv b_{0,p} \ (\mathrm {mod}\ p^l)$ which ensure that $b, f(a,b) \in N(\mathbb {Q}_p^n)\backslash \{0\}$ . By part (3) of Lemma 5.2, we deduce that property (2) stated before Lemma 5.2 holds with $c=b$ and $c=f(a,b)$ . The congruence conditions at each prime $p \in S$ can be merged into one congruence condition $a \equiv a_0 \ (\mathrm {mod}\ \Delta ), b \equiv b_0 \ (\mathrm {mod}\ \Delta )$ using the Chinese remainder theorem.

2. (2) $p\notin S$ and $p\notin \mathcal {P}$ . If $p \nmid b$ , we apply part (1) of Lemma 5.2 with $c=b$ . If $p\mid b$ , we apply part (2) of Lemma 5.2 with $c=b$ . The same argument works for $f(a,b)$ by choosing $c=f(a,b)$ .

3. (3) $p \in \mathcal {P}$ . Since we are assuming that $bf(a,b)$ has no prime factors in $\mathcal {P}$ , for these primes, part (1) of Lemma 5.2 applies with $c=b$ and $c=f(a,b)$ .

4. (4) $p=\infty $ . We follow a similar argument to Lemma 5.3. We may assume that (5.1) is everywhere locally soluble, so in particular, there exists $r \in \mathbb {R}$ such that $f(r) \in \mathbf {N}(\mathbb {R}^n)\backslash \{0\}$ . Since f is continuous and $\mathbf {N}(\mathbb {R}^n)\backslash \{0\}$ is open in the Euclidean topology, we can find $\xi>0$ such that $f(t) \in \mathbf {N}(\mathbb {R}^n)\backslash \{0\}$ whenever $|t-r|<\xi $ . Clearly, solubility of $f(t) = \mathbf {N}(\mathbf {x}) \neq 0$ and $f(-t) = \mathbf {N}(\mathbf {x}) \neq 0$ over $\mathbb {Q}$ are equivalent; consequently, we may assume $r>0$ . Suppose in addition that $t\in \mathbb {Q}$ , and write $t=a/b$ for $a,b \in \mathbb {N}$ . Since b is positive, it is automatically in $\mathbf {N}(\mathbb {R}^n)\backslash \{0\}$ . By multiplicativity of norms, we conclude that $b,f(a,b) \in \mathbf {N}(\mathbb {R}^n)\backslash \{0\}$ . The condition (5.3) now follows from part (3) of Lemma 5.2.

Example 5.4. Let $f(t) = (t^2-2)(-t^2+3)$ and $K = \mathbb {Q}(i)$ , so that

$$ \begin{align*} f(x,y) &= (x^2-2y^2)(-x^2+3y^2)\\ N_{K/\mathbb{Q}}(u,v) &= u^2+v^2. \end{align*} $$

It is known that there is a Brauer–Manin obstruction to the Hasse principle for the equation $(t^2-2)(-t^2+3) = u^2+v^2 \neq 0$ by the work of Iskovskikh [28]. However, Proposition 5.1 still applies.

When $p\equiv 1 \ (\mathrm {mod}\ 4)$ , the prime p splits in $K/\mathbb {Q}$ , and so the inertia degrees of p in $K/\mathbb {Q}$ are coprime. However, when $p \equiv 3 \ (\mathrm {mod}\ 4)$ , the prime p is inert in $K/\mathbb {Q}$ and has degree 2, so the inertia degrees are not coprime. Therefore, we have $\mathcal {P} = \{p: p\equiv 3 \ (\mathrm {mod}\ 4)\}$ . We choose $S=\{2\}$ . In this example, it can be checked that the congruence conditions $a \equiv 8 \ (\mathrm {mod}\ 16)$ and $b \equiv 1 \ (\mathrm {mod}\ 16)$ are sufficient. Finally, for the infinite place, we just have the condition $f(a,b)>0$ . The sieve problem we obtain is to find integers $a,b$ such that

1. (1) $a \equiv 8 \ (\mathrm {mod}\ 16), b \equiv 1 \ (\mathrm {mod}\ 16)$ ,

2. (2) $f(a,b)>0$ ,

3. (3) $f(a,b)$ has no prime factors $p \equiv 3 \ (\mathrm {mod}\ 4)$ .

We remark that $f(a/b) = b^{-4}f(a,b)$ , and since $2=[K:\mathbb {Q}]$ divides $4$ , $b^{-4}$ is automatically a norm from K. This explains why in (3) above, we can consider prime factors of $f(a,b)$ rather than $bf(a,b)$ .

An integer is the sum of two squares if and only if it is non-negative and all prime factors $p \equiv 3 \ (\mathrm {mod}\ 4)$ occur with an even exponent. The above conditions are stronger than this, so the algebraic reduction performed in Proposition 5.1 is consistent with what we already knew for this example.

Since the Hasse principle fails for this example, we know that the above sieve problem cannot have a solution. This is indeed the case since condition (1) implies that $-a^2+3b^2 \equiv 3 \ (\mathrm {mod}\ 4)$ , and so $f(a,b)$ must contain a prime factor $p \equiv 3 \ (\mathrm {mod}\ 4)$ .

The fact that the above sieve problem has no solutions also does not contradict Theorem 2.1. For a prime $p>3$ , we have that $\nu _1(p)$ is 2 if $p \equiv \pm 1 \ (\mathrm {mod}\ 8)$ and zero otherwise, and $\nu _2(p)$ is 2 if $p \equiv \pm 1\ (\mathrm {mod}\ 12)$ and zero otherwise. Consequently, by Dirichlet’s theorem on primes in arithmetic progressions, for $i\in \{1,2\}$ , asymptotically one half of the primes $p\in \mathcal {P}$ have $\nu _i(p) = 2$ and half have $\nu _i(p) = 0$ . We conclude that $\theta _1 = \theta _2 = 1$ , so that $\theta = 2$ . Theorem 2.1 therefore requires the density of $\mathcal {P}$ to be less than $0.39000\ldots /\theta =0.19503\ldots $ . However, the density of $\mathcal {P}$ here is $1/2$ , and so Theorem 2.1 does not apply to this example.

5.2 The Chebotarev density theorem

In order to apply the sieve results from Section 2 to prove Theorem 1.1 and Theorem 1.3, we need to compute the densities (2.4) and (2.5) for the choice of $\mathcal {P}$ given in (5.2). For more complicated number fields K, such as when $K/\mathbb {Q}$ is non-abelian, it is not possible to write down the set $\mathcal {P}$ explicitly in terms of congruence conditions. However, we can still compute the densities (2.4) and (2.5) by appealing to the Chebotarev density theorem, which can be viewed as a vast generalisation of Dirichlet’s theorem on primes in arithmetic progressions.

Let $K/\mathbb {Q}$ be a number field of degree n. Let p be a prime, unramified in $K/\mathbb {Q}$ . We can factorise the ideal $(p)$ as $(p) = \mathfrak {p}_1\cdots \mathfrak {p}_r$ , where $\mathfrak {p}_1, \ldots , \mathfrak {p}_r$ are distinct prime ideals in $\mathcal {O}_K$ . The splitting type of p in $K/\mathbb {Q}$ is the partition $(a_1, \ldots ,a_r)$ of n, where $a_i$ is the inertia degree of $\mathfrak {p}_i$ (i.e., $N(\mathfrak {p}_i) = p^{a_i}$ ). Equivalently, the splitting type is the list of degrees of the irreducible factors of the minimum polynomial of $K/\mathbb {Q}$ , when factorised modulo p.

Suppose first that $K/\mathbb {Q}$ is Galois, with Galois group G. Then G acts transitively on $\{\mathfrak {p}_1, \ldots , \mathfrak {p}_r\}$ . Fix $i \in \{1, \ldots , r\}$ . The Decomposition group $D_{\mathfrak {p}_i}$ is the stabilizer of $\mathfrak {p}_i$ under this action. Note that $D_{\mathfrak {p}_i}$ is cyclic, and there is an isomorphism

$$ \begin{align*}\psi_i : D_{\mathfrak{p}_i} \rightarrow \mathrm{Gal}((\mathcal{O}_K/\mathfrak{p}_i)/(\mathbb{Z}/(p))).\end{align*} $$

The group $\mathrm {Gal}((\mathcal {O}_K/\mathfrak {p}_i)/(\mathbb {Z}/(p)))$ is generated by the Frobenius element defined by $x\mapsto x^p$ , which has order $a_i$ . Let $\sigma _i$ denote the preimage of the Frobenius element under $\psi _i$ . We define the Artin symbol

$$ \begin{align*}\left[\frac{K/\mathbb{Q}}{p}\right] = \{\sigma_1, \ldots, \sigma_r\}.\end{align*} $$

The Artin symbol is a conjugacy class of G. Indeed, all the $\mathfrak {p}_i$ ’s lie in the same orbit of G (there is only one orbit as G acts transitively). Stabilisers of points in the same orbit of an action are conjugate, and so all the $D_{\mathfrak {p}_i}$ are conjugate.

We now come to the statement of the Chebotarev density theorem. For a conjugacy class C of G, we let $\pi _C(x)$ denote the number of primes $p\leqslant x$ whose Artin symbol is equal to C. We define the (natural) density of a set of primes $\mathcal {P}$ to be

$$ \begin{align*}\lim_{x \rightarrow \infty}\left( \frac{\#\{p \in \mathcal{P}: p \leqslant x\}}{\pi(x)} \right),\end{align*} $$

if such a limit exists.

Theorem 5.5 (Chebotarev density theorem)

Let C be a conjugacy class of G. The density of primes p for which the Artin symbol is equal to C is $\#C/\#G$ .

In order to obtain the explicit error terms from (2.4) and (2.5), we need an effective version of the Chebotarev density theorem. The following lemma is a straightforward consequence of a more refined result due to Lagarias and Odlyzko [33, Theorem 1].

Theorem 5.6 (Effective Chebotarev density theorem)

For any $A\geqslant 1$ , we have

$$ \begin{align*}\pi_C(x) =\pi(x)\left(\frac{\#C}{\#G} + O_A((\log x)^{-A})\right),\end{align*} $$

where the implied constant may depend on $K,C$ and A.

We now consider the non-Galois case. As above, let $(p) = \mathfrak {p}_1\cdots \mathfrak {p}_r$ be the factorisation of $(p)$ in $\mathcal {O}_K$ . Let $\widehat {K}$ denote the Galois closure of K, and let $G= \mathrm {Gal}(\widehat {K}/\mathbb {Q})$ . This time, there is no action of G on $\{\mathfrak {p}_1, \ldots , \mathfrak {p}_r\}$ because the $\mathfrak {p}_i$ ’s could split further in $\widehat {K}$ , and elements of G could permute prime factors which are above different $\mathfrak {p}_i$ ’s.

To get around this, we define $H=\mathrm {Gal}(\widehat {K}/K)$ , and instead consider the action of G on the set X of left cosets of H in G. For an element $\sigma \in G$ , the cyclic group $\langle \sigma \rangle $ generated by $\sigma $ acts by left multiplication on X. The sizes of the orbits of this action form a partition of $[G:H] =[K:\mathbb {Q}] = n$ . Moreover, it can be checked that conjugate elements of G give the same orbit sizes, so we can associate a single partition of n with each Artin symbol $\left ([\frac {\widehat {K}/\mathbb {Q}}{p}\right ]$ . The following fact relating this partition with the splitting type of p can be found in [30, Ch. 3, Proposition 2.8].

Lemma 5.7. Let $\sigma \in \left [\frac {\widehat {K}/\mathbb {Q}}{p}\right ]$ . Then p has splitting type $(a_1,\ldots , a_r)$ in $K/\mathbb {Q}$ if and only if the action of $\langle \sigma \rangle $ on X has orbit sizes $(a_1, \ldots , a_r)$ .

Let $K=\mathbb {Q}(\alpha )$ , and let g be the minimum polynomial of $\alpha $ . We can also view $\langle \sigma \rangle $ as acting on the set of n roots of g in $\widehat {K}$ . By definition of H, we have that $\sigma \alpha = \sigma '\alpha $ if and only if $\sigma H = \sigma ' H$ . It follows that the orbit sizes of $\langle \sigma \rangle $ acting on X are the same as the orbit sizes of $\langle \sigma \rangle $ acting on the roots of g, which in turn are the cycle lengths of $\sigma $ viewed as a permutation on the n roots of g in $\widehat {K}$ . The set of $\sigma \in G$ with cycle lengths $(a_1, \ldots , a_r)$ is a union $\bigcup _{i=1}^s C_i$ of conjugacy classes $C_i$ . We may now apply Theorem 5.6 to each of these conjugacy classes separately. Putting everything together, we have the following result on densities of splitting types in non-Galois extensions.

Lemma 5.8. Let K be a number field of degree n over $\mathbb {Q}$ , and let $\widehat {K}$ denote its Galois closure. Let $G= \mathrm {Gal}(\widehat {K}/\mathbb {Q})$ , viewed as a permutation group on the n roots of the minimum polynomial of K in $\widehat {K}$ . For a partition $\mathbf {a} = (a_1, \ldots , a_r)$ of n, let $\mathcal {P}(\mathbf {a})$ denote the set of primes with splitting type $\mathbf {a}$ in $K/\mathbb {Q}$ , and let $T(\mathbf {a})$ denote the proportion of elements of G with cycle shape $\mathbf {a}$ . Then for any $A\geqslant 1$ ,

$$ \begin{align*}\#\{p \in \mathcal{P}(\mathbf{a}): p \leqslant x\}= \pi(x)\left(T(\mathbf{a}) + O_A((\log x)^{-A})\right).\end{align*} $$

5.3 Proof of Theorem 1.1 and Theorem 1.3

Proof of Theorem 1.1

Since K satisfies the Hasse norm principle, we may apply the algebraic reduction from Proposition 5.1. Therefore, it suffices to show that conditions (1)–(3) from Proposition 5.1 hold for $\mathcal {P}$ as defined in (5.2), and for some choice of S containing all the ramified primes in $K/\mathbb {Q}$ . Let $F(x,y)$ be the binary form obtained from $yf(x,y)$ after removing any repeated factors. Clearly, to prove $bf(a,b)$ is free from prime factors in $\mathcal {P}$ , it suffices to prove $F(a,b)$ is, and so we may replace $bf(a,b)$ with $F(a,b)$ in Condition (3) of Proposition 5.1. We apply Theorem 2.1 to the binary form $F(x,y)$ , which has nonzero discriminant. We choose S to be the union of the ramified primes in $K/\mathbb {Q}$ and the set $S_0$ from Theorem 2.1, and we make the choice $\mathcal {R}=\mathcal {B}N$ , where $\mathcal {B}$ takes the form (2.7) for the parameters $r,\xi>0$ coming from the application of Proposition 5.1.

It remains only to check that (2.4) and (2.5) hold with $\alpha \theta \leqslant 0.39$ . By Lemma 5.8, (2.4) holds with $\alpha = T(G)$ . We now compute the quantity $\theta $ . We claim that $\theta \leqslant k+j+1$ . We recall that $\theta $ is a sum over the quantities $\theta _i$ associated to each irreducible factor of f, plus an additional term $\theta _0=1$ coming from the homogenising factor $f_0(x,y) = y$ . We may therefore reduce to the case where f is itself irreducible of degree at most $2$ , with the goal of proving that

$$ \begin{align*}\theta \leqslant \begin{cases} 3, & \textrm {if }f\textrm{ is quadratic and }L\subseteq \widehat{K},\\ 2, & \textrm{if }f\textrm{ otherwise}, \end{cases} \end{align*} $$

where L denotes the number field generated by f.

If f is linear, then $\nu _f(p)=1$ for all $p\notin S$ , and so $\theta = \theta _0+ 1 = 2$ , as required. We now consider the case where f is an irreducible quadratic. Let

$$ \begin{align*}\nu_f(p) = \#\{t \ (\mathrm{mod}\ p): f(t) \equiv 0 \ (\mathrm{mod}\ p)\}.\end{align*} $$

If $L\subseteq \widehat {K}$ , then Lemma 5.8 could be applied to compute $\theta $ , with the desired error terms from (2.6). However, we apply the trivial bound $\nu _f(p)\leqslant 2$ for $p \notin S$ since it is not possible to improve on this in general. We therefore obtain $\theta \leqslant \theta _0+ 2 = 1+2 = 3$ , which is satisfactory.

We now assume that $L\not \subseteq \widehat {K}$ . We want to show that $\theta =2$ , or equivalently that $\nu _f(p)=1$ on average over $p \in \mathcal {P}$ . Let $M = \widehat {K}L$ be the compositum of $\widehat {K}$ and L. Since $\widehat {K}\cap L =\mathbb {Q}$ , we have by [34, Ch. VI, Theorem 1.14] that $M/\mathbb {Q}$ is Galois, and

$$ \begin{align*}\mathrm{Gal}(M/\mathbb{Q}) \cong \mathrm{Gal}(L/\mathbb{Q}) \times \mathrm{Gal}(\widehat{K}/\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times \mathrm{Gal}(\widehat{K}/\mathbb{Q}).\end{align*} $$

We have $\nu _f(p) = 2$ if p is split in L, and $\nu _f(p) = 0$ if p is inert in L, and so

(5.4) $$ \begin{align} \theta = 1+ 2\lim_{x \rightarrow \infty}\left(\frac{\#\{p\leqslant x: p \in \mathcal{P}, \,p \textrm{ split in }L \}}{\#\{p\leqslant x: p \in \mathcal{P}\}}\right). \end{align} $$

Let $\sigma '=(\tau , \sigma )$ be an element of $\mathrm {Gal}(M/\mathbb {Q})$ , where $\tau \in \mathrm {Gal}(L/\mathbb {Q})$ and $\sigma \in \mathrm { Gal}(\widehat {K}/\mathbb {Q})$ . Applying Lemma 5.7, the primes $p\in \mathcal {P}$ correspond to the $\sigma '$ for which $\sigma $ has non-coprime cycle lengths (so these primes have density $T(G)$ as mentioned above). If in addition, the prime p is split in L, then we require that $\tau = \textrm {id}$ . Therefore, by Lemma 5.8, asymptotically as $x\rightarrow \infty $ , one half of the primes in $\mathcal {P}$ are also split in L. We conclude from (5.4) that (2.6) holds with $\theta = 1+ 2(1/2) = 2$ .

Proof of Theorem 1.3

We begin in the same manner as in the proof of Theorem 1.1, by appealing to Proposition 5.1 to reduce to a sieve problem. The binary form $F(x,y)$ has degree at most one higher than the degree of f. Therefore, we deduce from Theorem 2.3 that the Hasse principle holds for (1.1) provided that $(\deg f +1)\sum _{j=1}^n \frac {1}{q_j-1}\leqslant 0.32380$ .

Example 5.9. We consider the example $K=\mathbb {Q}(2^{1/q})$ discussed by Irving [27], where q is prime. Since K has prime degree, it satisfies the Hasse norm principle by work of Bartels [1]. We now compute $G=\mathrm { Gal}(\widehat {K}/\mathbb {Q})$ . The minimum polynomial of $K/\mathbb {Q}$ is $x^q-2$ , which has roots $\{\beta , \beta \omega , \ldots , \beta \omega ^{q-1}\}$ , where $\omega $ is a primitive qth root of unity and $\beta $ is the real root of $x^q-2$ . We identify these roots with $\{0,\ldots , q-1\}$ in the obvious way. An element $\sigma \in G$ is determined by the image of $0$ and $1$ since $\beta , \beta \omega $ multiplicatively generate all the other roots. Therefore, $\sigma $ takes the form $\sigma _{a,b}:x \mapsto ax+b$ for some $a \in \mathbb {F}_q^{\times }, b \in \mathbb {F}_q$ . Conversely, the maps $\sigma _{1,b}$ correspond to the q different embeddings K into $\widehat {K}$ , and the maps $\sigma _{a,0}$ for $a \in \mathbb {F}_q^{\times }$ are elements of $\mathrm {Gal}(\widehat {K}/K) \leqslant G$ . Combining these, we see that $\sigma _{a,b}\in G$ for any $a \in \mathbb {F}_q^{\times }, b \in \mathbb {F}_q$ . We conclude that $G \cong \operatorname {AGL}(1,q)$ , the group of affine linear transformations on $\mathbb {F}_q$ .

When $a=1$ and $b\neq 0$ , $\sigma _{a,b}$ is a q-cycle. For all other choices of $a,b$ , the equation $ax+b=x$ has a solution $x \in \mathbb {F}_q$ , and so $\sigma _{a,b}$ has a fixed point. Therefore, $T(G) = (q-1)/\#G = 1/q$ . From this, we see that Irving’s choice of $\mathcal {P} = \{p \notin S: p \equiv 1 \ (\mathrm {mod}\ q)\}$ is not quite optimal,because it has density $\alpha = 1/(q-1)$ , whereas the set of primes we actually need to sift out has density $T(G) = 1/q$ . In fact, we can see directly that even when $p\equiv 1 \ (\mathrm {mod}\ q)$ , there is sometimes a solution to $x^q -2 \equiv 0 \ (\mathrm {mod}\ p)$ (e.g., $q=3$ , $p = 31$ , $x= 4$ ). However, it can be checked that even with this smaller sieve dimension, we are still not able to handle the cases $q=5$ or $q=3$ when f is an irreducible cubic.

We remark that we can replace the number $2$ by any positive integer r such that $x^q-r$ is irreducible in the above example. (A necessary and sufficient condition for irreducibility of $x^q-r$ is given in [31, Theorem 8.16].) We can still take a sifting set $\mathcal {P}$ contained in $\{p\notin S: p\equiv 1\ (\mathrm {mod}\ q)\}$ and with density $1/q$ , and the Galois group is still $\operatorname {AGL}(1,q)$ , and so generalising to $x^q-r$ does not affect the analysis.

5.4 Proof of Corollary 1.2

We now consider the case when $[K:\mathbb {Q}]=n$ and $G = S_n$ , with a view to proving Corollary 1.2. Such number fields automatically satisfy the Hasse principle by the work of Kunyavskiı̆ and Voskresenskiı̆ [32]. To ease notation, we shall write $T(n)$ in place of $T(S_n)$ . In the following lemma, we find an estimate for $T(n)$ .

Lemma 5.10. For all $n\geqslant 1$ , we have

(5.5) $$ \begin{align} T(n) &= 1-\sum_{k\mid n}\frac{\mu(k)\Gamma((n+1)/k)}{\Gamma(1/k)\Gamma(n/k+1)}, \end{align} $$

(5.6) $$ \begin{align} T(n) &< \frac{2}{\sqrt{\pi}}n^{1/r-1}\omega(n), \end{align} $$

where r is the smallest prime factor of n and $\omega (n)$ is the number of prime factors of n.

Proof. Define

$$ \begin{align*}T_k(n) = \frac{1}{n!}\#\{\sigma \in S_n: \textrm{ the cycle lengths of }\sigma \textrm{ are all divisible by } k\}.\end{align*} $$

By Möbius inversion, we have $T(n) = 1-\sum _{k\mid n}\mu (k)T_k(n)$ . We now find an explicit formula for $T_k(n)$ . For $j\geqslant 1$ , let $a_{jk}$ denote the number of cycles of length $jk$ in $\sigma $ . The cycle lengths of $\sigma $ are all a multiple of k if and only if $\sum _{j=1}^{n/k} jka_{jk} = n$ . We apply the well-known formula for the number of permutations of $S_n$ with a given cycle shape to obtain

(5.7) $$ \begin{align}T_k(n) &= \frac{1}{n!}\sum_{\substack{a_{k}, a_{2k}, \ldots, a_{n}\\ \sum_{j=1}^{n/k}jka_{jk}=n}}\frac{n!}{\prod_{j=1}^{n/k}(jk)^{a_{jk}}a_{jk}!}\nonumber\\ &=\sum_{\substack{b_1,\ldots, b_{n/k}\\ \sum_{j=1}^{n/k}jb_j=n/k}}\frac{1}{\prod_{j=1}^{n/k}(jk)^{b_j}b_j!}\nonumber\\ &=\sum_{i=1}^{n/k}k^{-i}\sum_{\substack{b_1, \ldots, b_{n/k}\\ \sum_{j=1}^{n/k}jb_j=n/k\\ \sum_{j=1}^{n/k}b_j = i}}\frac{1}{\prod_{j=1}^{n/k}j^{b_j}b_j!}\nonumber\\ &= \frac{1}{m!}\sum_{i = 1}^{m}k^{-i}c(m,i), \end{align} $$

where $m = n/k$ and $c(m,i)$ is the number of $\sigma ' \in S_m$ with exactly i cycles. The quantity $c(m,i)$ is called the Stirling number of the first kind. In order to evaluate (5.7), we follow the argument from [17, Example II.12]. We define a bivariate generating function

(5.8)

By [17, Proposition II.4], we have

$$ \begin{align*}\sum_{m=0}^{\infty}\frac{z^m}{m!}c(m,i) = \frac{1}{i!}\left(\log\left(\frac{1}{1-z}\right)\right)^i.\end{align*} $$

Therefore,

$$ \begin{align*}P(w,z) = \sum_{i=0}^{\infty}\frac{w^i}{i!}\left(\log\left(\frac{1}{1-z}\right)\right)^i = \exp\left(w\log\left(\frac{1}{1-z}\right)\right)=(1-z)^{-w}.\end{align*} $$

Applying the Binomial theorem, we find that the $z^m$ coefficient of $P(w,z)$ is equal to $w(w+1)\cdots (w+m-1)/m!$ . However, if we substitute $w= 1/k$ , the $z^m$ coefficient of (5.8) is precisely (5.7). We conclude that

$$ \begin{align*} \frac{1}{m!}\sum_{i=1}^m k^{-i}c(m,i) &= (1/k)(1+1/k)\cdots (m-1+1/k)/m!\\ &=\frac{\Gamma(m+1/k)}{\Gamma(1/k)\Gamma(m+1)}, \end{align*} $$

which completes the proof of (5.5).

We now establish the upper bound in (5.6). A basic bound on the gamma function is that for any real $s\in (0,1)$ and any positive real number x, we have

$$ \begin{align*}x^{1-s} < \frac{\Gamma(x+1)}{\Gamma(x+s)} < (1+x)^{1-s}.\end{align*} $$

Applying this with $x=m$ and $s=1/k$ , we have

$$ \begin{align*}\frac{\Gamma(m + 1/k)}{\Gamma(m+1)}< m^{1/k - 1}.\end{align*} $$

Moreover, we have

$$ \begin{align*}\frac{1}{\Gamma(1/k)} = \frac{1}{\Gamma(1+1/k)}\frac{\Gamma(1+1/k)}{\Gamma(1/k)}\leqslant \frac{2}{\sqrt{\pi}k},\end{align*} $$

since $\Gamma (1+1/k)$ for integers $k\geqslant 2$ achieves its minimum at $k=2$ , where we have $\Gamma (1+1/k)=\Gamma (3/2) = \sqrt {\pi }/2$ . We conclude that

$$ \begin{align*}T_k(n)< \frac{2}{\sqrt{\pi}k}(n/k)^{1/k-1}= \frac{2}{\sqrt{\pi}}n^{1/k-1}k^{-1/k}<\frac{2}{\sqrt{\pi}}n^{1/k-1}.\end{align*} $$

Taking a sum over $k=p$ prime, we obtain

$$ \begin{align*}T(n) < \sum_{p\mid n}T_p(n) \leqslant \frac{2}{\sqrt{\pi}}n^{1/r-1}\omega(n),\end{align*} $$

as required.

Proof of Corollary 1.2

We recall the setting of Corollary 1.2. We assume that $G=S_n$ , and f is a product of two quadratics generating a biquadratic extension L of $\mathbb {Q}$ . We apply Theorem 2.1 to the binary form $\prod _{i=0}^2 f_i(x,y)$ , where $f_0(x,y) = y$ and $f_1(x,y), f_2(x,y)$ are the homogenisations of the two quadratic factors of f. We also assume that $L\cap \widehat {K} = \mathbb {Q}$ , and so by the proof of Theorem 1.1, we have $\theta _0= \theta _1 = \theta _2 = 1$ , and $\theta = 3$ . By maximising the value of $\kappa $ in (4.55) directly, we find by numerical computations that the largest value of $\kappa $ we can take here is $0.42214\ldots $ . (The slight improvement over $\kappa \leqslant 0.39$ for the general case comes from computing $\alpha \sum _{i=m+1}^k \theta _i = 2\alpha $ in our example, whilst in the proof of Theorem 1.1, we applied the trivial bound $\alpha \sum _{i=m+1}^k \theta _i \leqslant \kappa = 3\alpha $ .) Hence, the Hasse principle holds for $f(t) = \mathbf {N}(\mathbf {x})\neq 0$ provided that $T(n) \leqslant 0.42214\ldots /3 = 0.14071\ldots $ .

We use the upper bound (5.6) from Lemma 5.10 to reduce the n for which $T(n)\geqslant 0.14071\ldots $ to finitely many cases, and then the exact formula (5.5) to find precisely which n satisfy $T(n)\geqslant 0.14071\ldots $ . We find that $T(n)\leqslant 0.14071\ldots $ unless $n\in \{2,3,\ldots , 10, 12,14,15,16,18,20,22,24,26,28,30,36,42,48\}.$

6 Application to the Harpaz–Wittenberg conjecture

In this section, we apply the sieve result from Theorem 2.1 to prove Theorem 1.4. We recall the statement of [23, Conjecture 9.1] (we shall only work with the ground field $\mathbb {Q}$ ).

Conjecture 6.1 (Harpaz, Wittenberg)

Let $P_1, \ldots , P_n \in \mathbb {Q}[t]$ be pairwise distinct irreducible monic polynomials. Let $k_i=\mathbb {Q}[t]/(P_i(t))$ be the corresponding number fields. Let $a_i\in k_i$ denote the class of t. For each $i\in \{1,\ldots , n\}$ , let $L_i/k_i$ be a finite extension, and let $b_i \in k_i^*$ . Let $S_0$ be a finite set of places of $\mathbb {Q}$ including the archimedean place, and all finite places above which, for some i, either $b_i$ is not a unit or $L_i/k_i$ is ramified. For each $v\in S_0$ , fix an element $t_v \in \mathbb {Q}_v$ . Suppose that for every $i\in \{1,\ldots , n\}$ and every $v\in S_0$ , there exists $x_{i,v} \in (L_i \otimes _{\mathbb {Q}} \mathbb {Q}_v)^*$ such that $b_i(t_v-a_i) = N_{L_i\otimes _{\mathbb {Q}} \mathbb {Q}_v/k_i \otimes _{\mathbb {Q}} \mathbb {Q}_v}(x_{i,v})$ in $k_i \otimes _{\mathbb {Q}} \mathbb {Q}_v$ . Then there exists $t_0 \in \mathbb {Q}$ satisfying the following conditions.

1. (1) $t_0$ is arbitrarily close to $t_v$ for all $v \in S_0$ .

2. (2) For every $i\in \{1,\ldots , n\}$ and every place $\mathfrak {p}$ of $k_i$ with $\operatorname {ord}_{\mathfrak {p}}(t_0-a_i)>0$ , either $\mathfrak {p}$ lies above a place of $S_0$ or the field $L_i$ possesses a place of degree $1$ over $\mathfrak {p}$ .

We remark that below, the $b_i$ and $x_{i,v}$ appearing in Conjecture 6.1 do not play a role, and so in the cases that Theorem 1.4 applies, it establishes a stronger version of Conjecture 6.1, where the assumption on the existence of the elements $x_{i,v}$ is removed. We discuss this further in Section 6.1.

We can reduce Conjecture 6.1 to a sieve problem as follows. Let $f_i(x,y) = c_iN_{k_i/\mathbb {Q}}(x-a_iy)$ , where $c_i \in \mathbb {Q}$ is chosen such that the coefficients of $f_i(x,y)$ are coprime integers. Then $f_i(x,y)$ is an irreducible polynomial in $\mathbb {Z}[x,y]$ .

Below, we suppose that S is a finite set of primes containing all primes in $S_0$ and all primes dividing any of the denominators $c_1, \ldots , c_n$ . For $i\in \{1,\ldots , n\}$ , we define $\mathcal {P}_i$ to be the set of primes $p \notin S$ , such that for some place $\mathfrak {p}$ of $k_i$ above p, $L_i$ does not possess a place of degree $1$ above $\mathfrak {p}$ .

Lemma 6.2. Let $k_1, \ldots , k_n$ and $L_1, \ldots , L_n$ and $S_0$ be as in Conjecture 6.1, and let $\mathcal {P}_i$ and $f_i(x,y)$ be as defined above. Suppose that there exists a finite set of primes $S\supset S_0\backslash \{\infty \}$ such that for any congruence condition $C(x,y)$ on $x,y$ modulo an integer $\Delta $ with only prime factors in S, and any real numbers $r, \xi>0$ , there exists $x_0,y_0 \in \mathbb {N}$ such that

1. (i) $C(x_0,y_0)$ holds,

2. (ii) $|x_0/y_0 - r| < \xi $ ,

3. (iii) $f_i(x_0,y_0)$ has no prime factors in $\mathcal {P}_i$ for all $i\in \{1, \ldots , n\}$ .

Then Conjecture 6.1 holds for this choice of $k_1,\ldots ,k_n,L_1, \ldots , L_n$ and $S_0$ .

Proof. From [23, Remark 9.3 (iii)], we are free to adjoin to $S_0$ a finite number of places, and so we may assume that $S_0 = S\cup \{\infty \}$ . We may assume that the value $t_{\infty }$ prescribed in Conjecture 6.1 is non-negative, and let $t_0 = x_0/y_0$ . (If alternatively $t_{\infty } < 0$ , then we instead choose $t_0 = -x_0/y_0$ and proceed similarly.) Then property (1) of Conjecture 6.1 immediately follows from (i) and (ii) by appropriate choices of $C(x,y), r$ and $\xi $ . Let $\mathfrak {p}$ be a place of $k_i$ above a prime $p \notin S$ , satisfying $\operatorname {ord}_{\mathfrak {p}}(t_0-a_i)>0$ . Then

$$ \begin{align*}f_i(x_0,y_0) = y_0^{\deg f_i} f_i(x_0/y_0,1) = y_0^{\deg f_i}c_i N_{k_i/\mathbb{Q}}(t_0-a_i).\end{align*} $$

Now $\operatorname {ord}_{\mathfrak {p}}(t_0-a_i)>0$ implies that $\operatorname {ord}_p(N_{k_i/\mathbb {Q}}(t_0-a_i))>0$ . Since $p \notin S$ , we have $\operatorname {ord}_p(y_0c_i) \geqslant 0$ , and so $p\mid f_i(x_0,y_0)$ . By (iii), we have $p \notin \mathcal {P}_i$ , and so by construction of $\mathcal {P}_i$ , we deduce that property (2) of Conjecture 6.1 holds.

In view of Lemma 6.2, we let

and we aim to show that the sifting function

(6.1)

is positive for sufficiently large N. We do not attempt here to generalise Theorem 2.1 to deal with different sifting sets $\mathcal {P}_i$ for each i, but instead define below in (6.14) a set $\mathcal {P} \supseteq \bigcup _{i=1}^n \mathcal {P}_i$ and replace each of the conditions $\gcd (f_i(x_0,y_0), P_i(x))=1$ with $\gcd (f_i(x_0,y_0), P(x))=1$ .

Proof of Theorem 1.4

We recall that $L_i$ is the compositum $k_iM_i$ , for a number field $M_i$ which is linearly disjoint to $k_i$ over $\mathbb {Q}$ . Consequently, $[L_i:k_i] = [M_i:\mathbb {Q}]$ . Writing $M_i = \mathbb {Q}(\beta _i)$ using the primitive element theorem, we therefore have that the minimum polynomial of $\beta _i$ over $\mathbb {Q}$ and over $k_i$ coincide. We denote this minimum polynomial by $g_i$ .

Let $\mathfrak {p}$ denote a place of $k_i$ . For all but finitely many places, the inertia degrees of the places of $L_i$ above $\mathfrak {p}$ are the degrees of $g_i$ when factored modulo $\mathfrak {p}$ . If $g_i$ has a root modulo p, then it has a root modulo every $\mathfrak {p}\mid p$ , and so $p \notin \mathcal {P}_i$ . Therefore, for suitably chosen S, we have

(6.2)

Define $\mathcal {P} = \bigcup _{i=1}^n \widetilde {\mathcal {P}}_i$ . Clearly, to show that the sifting function from (6.1) is positive, it suffices to show the sifting function $S(\mathcal {A}, \mathcal {P},x)$ (in the notation of (4.2)) is positive.

By the Chebotarev density theorem (Lemma 5.8), the sets $\widetilde {\mathcal {P}_i}$ have density $\alpha _i = T_i$ , where $T_i$ is defined in (1.4). The sets $\widetilde {\mathcal {P}_i}$ are examples of Frobenian sets as defined by Serre [41, Section 3.3.1], which roughly means that away from S their membership is determined Artin symbols of some Galois extension (here $\widehat {M_i}$ ). It follows from [41, Proposition 3.7 b)] that the intersection of Frobenian sets is Frobenian. Since $\mathcal {P}$ is a disjoint union of such intersections, we conclude from Theorem 5.5 that the set $\mathcal {P}$ does indeed have a density. We shall bound its density trivially by $\sum _{i=1}^n \alpha _i$ .

We now bound the value of $\theta $ defined in (2.6). Here, we have already defined $f_i$ to be a binary form, and so no additional term $\theta _0$ coming from homogenisation is required. We apply the trivial estimate $\nu _i(p) \leqslant \deg f_i = [k_i:\mathbb {Q}]$ for all $p \notin S$ . We conclude that $\theta \leqslant \sum _{i=1}^n[k_i:\mathbb {Q}]=d$ . Combining Lemma 6.2 and Theorem 2.1 completes the proof of part (1) of Theorem 1.4.

We now turn to the cubic case. Assumption 2.2 holds for our choice $\mathcal {P}$ and f. As in Section 4.8, we conclude that for sufficiently large N, provided that $t\leqslant 0.32380$ , where $t = \deg f \sum _{i=1}^n 1/(q_i-1)$ . Rearranging, and recalling $\deg f=d$ , we complete the proof of part (2) of Theorem 1.4.

Remark 6.3. In contrast to Section 5, now $\mathrm {Gal}(\widehat {M_i}/\mathbb {Q})=S_n$ is not a case we can handle because there the proportion of fixed point free elements is $1-1/e$ as $n\rightarrow \infty $ (where $e=2.718\ldots $ is Euler’s constant), which is much too large.

For a permutation group G acting on $X=\{1, \ldots , k\}$ , we define $h(G)$ to be the proportion of elements of G with no fixed point. The family of of groups G for which $h(G)$ is smallest are the Frobenius groups. These are the groups where G has a nontrivial element fixing one point of X, but no nontrivial elements fixing more than one point of X. We state two known results about Frobenius groups.

Lemma 6.4 [43, Theorem 1]

Any Frobenius group can be realised as a Galois group over $\mathbb {Q}$ .

Lemma 6.5 [3, Theorem 3.1]

Let G be a transitive permutation group on k letters.

1. (1) We have $h(G) \geqslant 1/k$ , with equality if and only if G is a Frobenius group of order $k(k-1)$ and k is a prime power.

2. (2) In all other cases, $h(G) \geqslant 2/k$ .

Proof of Corollary 1.5

As computed in Example 5.9, we have that $G_i :=\mathrm {Gal}(\widehat {M_i}/\mathbb {Q})$ is isomorphic to the group $\operatorname {AGL}(1,q_i)$ of affine linear transformations on $\mathbb {F}_{q_i}$ . This is a Frobenius group of order $q_i(q_i-1)$ . By Lemma 6.5, we have $T_i = h(G_i) = 1/q_i$ . (This also agrees with our computation in Example 5.9.) If $[k_i:\mathbb {Q}]\leqslant 2$ for all i, we can therefore apply part (1) of Theorem 1.4 provided that $\sum _{i=1}^n 1/q_i \leqslant 0.39/d$ . Moreover, for all $i\in \{1,\ldots , n\}$ , the sifting sets $\mathcal {P}_i$ are contained in $\{p\notin S: p \equiv 1 \ (\mathrm {mod}\ q_i)\}$ . Indeed, when $p \not \equiv 1 \ (\mathrm {mod}\ q_i)$ , the qth power map on $\mathbb {F}_p^{\times }$ is a bijection, and so $x^{q_i}-r_i$ has a root modulo p.

The minimum polynomial $x^{q_i}-r_i$ has a root modulo p for all but finitely many $p \not \equiv 1 \ (\mathrm {mod}\ q_i)$ , and these finitely many exceptional primes can be included in $S_0$ . Therefore, we can apply part (2) of Theorem 1.4 provided that $\sum _{i=1}^n 1/(q_i-1) \leqslant 0.32380/d$ .

6.1 The hypothesis on $b_i$

Given that the quantities $b_i$ play no role in Theorem 1.4, it is natural to ask under what circumstances we should expect a stronger version of Conjecture 6.1 to hold, without the hypothesis on the $b_i$ . The following proposition demonstrates that the hypothesis remains unchanged after passing to maximal abelian subextensions of each $L_i/k_i$ .

Proposition 6.6. Let $L/k$ be an extension of number fields. Let $L'/k$ be the maximal abelian subextension of $L/k$ . Let S be a sufficiently large set of places of k, including all archimedean places. For each $v\in S$ , fix an element $b_v \in k_v^{*}$ . Then the following are equivalent:

1. (1) There exists an S-unit $b \in k$ such that for all $v\in S$ , $b/b_v$ is in the image of the norm map $(L\otimes _k k_v)^* \rightarrow k_v^*$ .

2. (2) There exists an S-unit $b \in k$ such that for all $v\in S$ , $b/b_v$ is in the image of the norm map $(L'\otimes _k k_v)^* \rightarrow k_v^*$ .

We prove Proposition 6.6 at the end of this section. We now demonstrate how the lack of the hypothesis on $b_i$ in Theorem 1.4 is explained by Proposition 6.6. We shall choose S to consist of all places that lie above a finite set of places $S_0$ of k, which corresponds to the set $S_0$ from Conjecture 6.1. We recall that due to [23, Remark 9.3 (iii)], we are free to make $S_0$ , and hence S, large enough that Proposition 6.6 applies. We apply Proposition 6.6 with $L=L_i$ and $k=k_i$ for each extension $L_i/k_i$ from Conjecture 6.1, and with $b_v = 1/(t_v-a_i)$ . Clearly, if $L/k$ contains no nontrivial abelian subextensions (so that $L'=k$ ), then Condition (2) above is trivially satisfied, and so Proposition 6.6 implies that the hypothesis on $b_i$ in Conjecture 6.1 is vacuous. To complete the argument, it suffices to show that in the setting of Theorem 1.4, we have $L_i^{\prime } = k_i$ for all i. In fact, in the following lemma, we show that the hypotheses of Theorem 1.4 force the stronger property that the $L_i/k_i$ contain no nontrivial Galois subextensions.

Lemma 6.7. Suppose that $L/k$ is one of the extensions $L_i/k_i$ from Theorem 1.4, and let $T=T_i$ be as in (1.4). Suppose that $T[k:\mathbb {Q}]< 1/2$ . Then $L/k$ has no nontrivial Galois subextensions.

Proof. We recall from the proof of Theorem 1.4 that $T\geqslant \alpha $ , where $\alpha $ is the natural density of the set $\mathcal {P}$ of primes $p\notin S$ such that there is some place $\mathfrak {p}$ of k above p for which L does not possess a place of degree 1 above $\mathfrak {p}$ . For $x\geqslant 1$ , we have the trivial bound

$$ \begin{align*}T\pi(x) &\geqslant \#(\mathcal{P}_{\leqslant x})\\ &\geqslant \frac{1}{[k:\mathbb{Q}]}\#\{\mathfrak{p} \subseteq \mathcal{O}_k: N(\mathfrak{p})\leqslant x, L \textrm{ has no degree one place above }\mathfrak{p}\}, \end{align*} $$

since there are at most $[k:\mathbb {Q}]$ prime ideals $\mathfrak {p}$ above each p, and $N(\mathfrak {p})\geqslant p$ .

Suppose that $N/k$ is a Galois subextension of $L/k$ . In order for L to possess a degree one place above $\mathfrak {p}$ , so must N. However, since $N/k$ is Galois, N possesses a place of degree one above $\mathfrak {p}$ if and only in $\mathfrak {p}$ splits completely in N, and by the Chebotarev density theorem, this occurs with density $1/[N:k]$ . We deduce that

$$ \begin{align*}T\pi(x)&\geqslant \frac{1}{[k:\mathbb{Q}]}\#\{\mathfrak{p} \subseteq \mathcal{O}_k: N(\mathfrak{p})\leqslant x, N \textrm{ has no degree one place above }\mathfrak{p}\}\\ &\geqslant \frac{\#\{\mathfrak{p} \subseteq \mathcal{O}_k: N(\mathfrak{p}) \leqslant x\}}{[k:\mathbb{Q}]}\left(1-\frac{1}{[N:k]}\right). \end{align*} $$

Taking a limit at $x\rightarrow \infty $ and applying the prime ideal theorem, we conclude that

$$ \begin{align*}T\geqslant \frac{1}{[k:\mathbb{Q}]}\left(1-\frac{1}{[N:k]}\right).\end{align*} $$

The assumption $T[k:\mathbb {Q}]<1/2$ therefore implies that $N=k$ .

Let T denote the norm one torus associated to $L/k$ , which is the algebraic group over k defined by the the equation $\mathbf {N}_{L/k}(\mathbf {x}) = 1$ . We have a short exact sequence

(6.3) $$ \begin{align} 1 \rightarrow T \rightarrow R_{L/k}\mathbb{G}_m \xrightarrow{N_{L/k}} \mathbb{G}_m \rightarrow 1, \end{align} $$

where $R_{L/k}$ denotes the Weil restriction. Let $T_{\overline {k}} = T\times _k \overline {k}$ and $\mathbb {G}_{m,\overline {k}}\cong \overline {k}^{\times }$ . We define the character group of T to be $\widehat {T} = \operatorname {Hom}(T_{\overline {k}},\mathbb {G}_{m,\overline {k}})$ , viewed as a Galois module via the natural action of $\mathrm {Gal}(\overline {k}/k)$ .

Let $\mathcal {O}_{S}$ denote the ring of S-integers of k, and $\mathcal {O}_{L, S_L}$ the ring of $S_L$ -integers of L, where $S_L$ consists of all places of L above a place in S. We include in S all places of k which ramify in L. Then $\mathcal {O}_{L,S_L}/\mathcal {O}_{S}$ is étale, and so the equation $\mathbf {N}_{\mathcal {O}_{L,S_L}/\mathcal {O}_{S}}(\mathbf {z}) = 1$ defines a model $\mathcal {T}$ of T over $\mathcal {O}_S$ . Similarly to (6.3), we have a short exact sequence

(6.4) $$ \begin{align} 1 \rightarrow \mathcal{T} \rightarrow R_{\mathcal{O}_{L,S_L}/\mathcal{O}_S}\mathbb{G}_{m,\mathcal{O}_{L,S_L}} \xrightarrow{N_{L/k}} \mathbb{G}_{m,\mathcal{O}_S} \rightarrow 1. \end{align} $$

Let $k_S$ denote the maximal subextension of $\overline {k}/k$ which is unramified at all places not contained in S. Below, we shall work with profinite group cohomology of . We note that L is a subextension of $k_S$ since S is assumed to contain all ramified places of $L/k$ . We may therefore define $G_{L,S} = \mathrm {Gal}(k_S/L)$ . Let $A_S$ denote the integral closure of $\mathcal {O}_S$ in $k_S$ . The natural action of $\mathrm {Gal}(\overline {k}/k)$ on $\widehat {T}$ factors through $G_S$ , so $\widehat {T}$ can be viewed as a $G_S$ -module. The character group $\operatorname {Hom}(\mathcal {T}_{A_S}, \mathbb {G}_{m, A_S})$ is nothing more than $\widehat {T}$ as a $G_S$ -module, so we shall henceforth denote it by $\widehat {T}$ .

Lemma 6.8. Let $\varphi :H^1(G_S, \mathbb {Q}/\mathbb {Z}) \rightarrow H^1(G_{L,S},\mathbb {Q}/\mathbb {Z})$ be the restriction map induced by the inclusion $G_{L,S} \hookrightarrow G_S$ . Then there is an exact sequence

(6.5) $$ \begin{align} 0 \rightarrow H^1(G_S,\widehat{T}) \rightarrow H^1(G_S,\mathbb{Q}/\mathbb{Z}) \xrightarrow{\varphi} H^1(G_{L,S},\mathbb{Q}/\mathbb{Z}). \end{align} $$

Proof. We begin by taking character groups of the short exact sequence (6.4), or in other words, applying the contravariant functor $\operatorname {Hom}(-, \mathbb {G}_{m,A_S})$ . we obtain a short exact sequence

(6.6) $$ \begin{align} 0 \rightarrow \mathbb{Z} \rightarrow \mathbb{Z}[L/k] \rightarrow \widehat{T} \rightarrow 0, \end{align} $$

where $\mathbb {Z}[L/k]\cong \mathbb {Z}[\mathcal {O}_{L,S_L}/\mathcal {O}_{S}]$ denotes the free abelian group generated by the k-linear embeddings $L\hookrightarrow \overline {k}$ . We now take group cohomology of (6.6) to obtain a long exact sequence

(6.7) $$ \begin{align} \cdots \rightarrow H^1(G_S,\mathbb{Z}[L/k]) \rightarrow H^1(G_S,\widehat{T}) \rightarrow H^2(G_S,\mathbb{Z}) \rightarrow H^2(G_S,\mathbb{Z}[L/k]) \rightarrow \cdots. \end{align} $$

For any groups $H\leqslant G$ , any H-module N and any integer $i\geqslant 0$ , Shapiro’s lemma [37, Proposition 1.11] states that $H^i(H,N)\cong H^i(G,\operatorname {Coind}^G_H(N))$ , where $\operatorname {Coind}^G_H(N) = \operatorname {Hom}_{\mathbb {Z}[H]}(\mathbb {Z}[G],N)$ denotes the coinduced module. We apply Shapiro’s lemma to the first and last terms in the exact sequence (6.7) by choosing $G=G_S, H=G_{L,S}$ and $N=\mathbb {Z}$ . Then

$$ \begin{align*}\operatorname{Coind}^G_H(N) = \operatorname{Hom}_{\mathbb{Z}[G_{L,S}]}(\mathbb{Z}[G_S],\mathbb{Z})\cong \mathbb{Z}[G_S/G_{L,S}] \cong \mathbb{Z}[L/k],\end{align*} $$

and so we conclude that $H^i(G_S,\mathbb {Z}[L/k]) \cong H^i(G_{L,S},\mathbb {Z})$ for all $i\geqslant 0$ . Moreover, $H^1(G_{L,S},\mathbb {Z})$ consists of all continuous group homomorphisms $G_{L,S} \rightarrow \mathbb {Z}$ . Since $G_{L,S}$ is compact, and $\mathbb {Z}$ is discrete, we have $H^1(G_{L,S},\mathbb {Z}) = 0$ . Therefore, we obtain from (6.7) the exact sequence

(6.8) $$ \begin{align} 0 \rightarrow H^1(G_S,\widehat{T}) \rightarrow H^2(G_S,\mathbb{Z}) \rightarrow H^2(G_{L,S},\mathbb{Z}). \end{align} $$

Now, for any subextension E of $k_S/k$ , we have $H^2(G_{E,S},\mathbb {Z}) \cong H^1(G_{E,S}, \mathbb {Q}/\mathbb {Z})$ . To see this, we take group cohomology of the exact sequence $0 \rightarrow \mathbb {Z} \rightarrow \mathbb {Q} \rightarrow \mathbb {Q}/\mathbb {Z} \rightarrow 0$ to obtain a long exact sequence

$$ \begin{align*} \cdots \rightarrow H^1(G_{E,S},\mathbb{Q}) \rightarrow H^1(G_{E,S}, \mathbb{Q}/\mathbb{Z}) \rightarrow H^2(G_{E,S},\mathbb{Z}) \rightarrow H^2(G_{E,S},\mathbb{Q}) \rightarrow \cdots, \end{align*} $$

and note that $H^1(G_{E,S},\mathbb {Q}) = H^2(G_{E,S},\mathbb {Q}) = 0$ since $G_{E,S}$ is a profinite group [39, Proposition 1.6.2 (c)]. Therefore, applying this fact with $E=k$ and $E=L$ , we have an exact sequence

(6.9) $$ \begin{align} 0 \rightarrow H^1(G_S,\widehat{T}) \rightarrow H^1(G_S,\mathbb{Q}/\mathbb{Z}) \xrightarrow{\psi} H^1(G_{L,S},\mathbb{Q}/\mathbb{Z}). \end{align} $$

To complete the proof, we need to show that the map $\psi $ we have obtained from the above argument is equal to the map $\varphi $ defined in the lemma. We consider the diagram

Here, $\delta ^{-1}$ denotes the inverse of the connecting homomorphism, and $\operatorname {sh}$ denotes the Shapiro map, (i.e., the isomorphism from the above application of Shapiro’s lemma). By definition, $\psi $ comes from applying these isomorphisms to the map $H^2(G_S, \mathbb {Z}) \rightarrow H^2(G_S,\mathbb {Z}[L/k])$ from (6.7), which is the map $i_*$ in the notation of [39, Proposition 1.6.5]. Therefore, by [39, Proposition 1.6.5], the middle horizontal arrow is just the restriction map. Finally, restriction maps commute with connecting homomorphisms [39, Proposition 1.5.2], and so $\psi $ is the restriction homomorphism induced by the inclusion $G_{L,S} \hookrightarrow G_S$ , as required.

In what follows, we denote by $\operatorname {Cl}_{S_L}(L)$ the $S_L$ -ideal class group (i.e., the quotient of the usual class group $\operatorname {Cl}(L)$ by the classes of all prime ideals in $S_L$ ). Since $\operatorname {Cl}(L)$ is finite, by adjoining finitely many primes to S, we may assume that $\operatorname {Cl}_{S_L}(L)=0$ .

Lemma 6.9. Let S be as above. If $a\in k^*$ lies in the image of the norm map $(k_v\otimes _k L)^* \rightarrow k_v^*$ for all $v\notin S$ , then there exists $y\in N_{L/k}(L^*)$ such that $ay \in \mathcal {O}_S^{\times }$ .

Proof. Fix a place $v\notin S$ at which a is not a unit. Let $w_1\cdots w_r$ be the factorisation of v into prime ideals in $\mathcal {O}_L$ , and let $c_i = [L_{w_i}: k_v]$ be the corresponding inertia degrees. (Since $L/k$ is unramified outside S, the ideals $w_1, \ldots , w_r$ are distinct.) We define $c=\gcd (c_1, \ldots , c_r)$ . The image of the norm map $(k_v\otimes _k L)^* \rightarrow k_v^*$ consists of the elements of $k_v^*$ whose valuation is divisible by c. In particular, we may write $v(a) = \sum _{i=1}^r n_ic_i$ for some integers $n_1, \ldots , n_r$ . Since $\operatorname {Cl}_{S_L}(L)=0$ , we can find an element $z_v\in L$ such that $w_i(z_v) = n_i$ for all $i\in \{1,\ldots , r\}$ , and such that $z_v$ is a unit at all other places not in $S_L$ . Let $y_v = N_{L/k}(z_v)$ . Then $v(y_v) = \sum _{i=1}^r n_i c_i = v(a)$ , and $y_v$ is a unit at all other places outside S. The result now follows by taking y to be the product of the elements $y_v^{-1}$ over the (finitely many) places $v\notin S$ at which a is not a unit.

Proof of Proposition 6.6

The implication $(1)\implies (2)$ is trivial, so we only need to prove $(2)\implies (1)$ . Taking Galois cohomology of (6.3), we obtain a long exact sequence

(6.10) $$ \begin{align} 1 \rightarrow T(k) \rightarrow L^{\times} \xrightarrow{N_{L/k}} k^{\times} \xrightarrow{\delta_k} H^1(k,T)\rightarrow H^1(k,R_{L/k}\mathbb{G}_m)\rightarrow \cdots. \end{align} $$

Hilbert’s Theorem 90 [11, Theorem 1.3.2] implies that $H^1(k,R_{L/k}\mathbb {G}_m)$ is trivial, and so we can naturally identify $H^1(k,T)$ with $\frac {k^{\times }}{N_{L/k}(L^{\times })}$ . In a similar way, we can identify $H^1(k_v, T)$ with the quotient of $k_v^*$ by the image of the norm map $(L\otimes _k k_v)^* \rightarrow k_v^*$ . We can also take group cohomology of the short exact sequence (6.4) to obtain a long exact sequence

(6.11) $$ \begin{align} 1 \rightarrow \mathcal{T}(\mathcal{O}_{S}) \rightarrow \mathcal{O}_{L,S_L}^{\times} \rightarrow \mathcal{O}_{S}^{\times} \xrightarrow{\delta_{\mathcal{O}_S}} H^1(G_S,\mathcal{T}_{A_S}) \rightarrow H^1(G_S, A_{S_L}^{\times}) \rightarrow \cdots, \end{align} $$

where $A_{S_L} = A_S \otimes _{\mathcal {O}_S} \mathcal {O}_{L, S_L}$ is the integral closure of $\mathcal {O}_{L,S_L}$ in $k_S$ . (We remark that the ring $A_S$ is denoted by $\mathcal {O}_S$ in [39].) By [39, Proposition 8.3.11 (ii)], we have $H^1(G_S, A_{S_L}^{\times }) \cong \operatorname {Cl}_{S_L}(L)$ , which we recall is trivial due to our choice of S. Therefore, we may identify $H^1(G_S, \mathcal {T}_{A_S})$ with $\frac {\mathcal {O}_{S}^{\times }}{\mathcal {O}_{S}^{\times } \cap N_{L/k}(L^{\times })}$ .

Let $v\in S$ , and let $\overline {k}_v$ denote an algebraic closure of $k_v$ . Fixing an embedding $k_S \hookrightarrow \overline {k_v}$ determines a surjection $\mathrm {Gal}(\overline {k_v}/k_v) \rightarrow G_S$ , which induces restriction maps $H^1(G_S, \mathcal {T}_{A_S}) \rightarrow H^1(k_v, T)$ and $H^1(k, T) \rightarrow H^1(k_v, T)$ called localisation maps. Below, we denote these maps by $\operatorname {res}_v$ . Under the above identifications, we obtain a commutative diagram

(6.12)

where $(L \otimes _k k_v)^* \rightarrow k_v^*$ denotes the norm map, and the bottom arrows are induced by the inclusions $\mathcal {O}_S^{\times } \hookrightarrow k \hookrightarrow k_v^*$ . Consequently, we can reformulate Condition (1) from Proposition 6.6 as

1. (3) The class of $(b_v)_{v \in S}$ in $\prod _{v \in S}H^1(k_v, T)$ belongs to the image of the map $\prod _{v \in S}\operatorname {res}_v$ .

We assume that S is also large enough that $\operatorname {Cl}_{S_{L'}}(L')=0$ , so that the above argument gives a similar reformulation of (2), but with the torus $T'$ associated to $L'/k$ in place of T.

Let $(-)^{\vee }=\operatorname {Hom}(-,\mathbb {Q}/\mathbb {Z})$ . Poitou–Tate duality [39, Theorem 4.20 b)] gives an exact sequence

(6.13) $$ \begin{align} H^1(k,T) \xrightarrow{\operatorname{res}} \sideset{}{'}\prod H^1(k_v, T) \xrightarrow{\xi} H^1(k, \widehat{T})^{\vee}. \end{align} $$

Here, the restricted product is over all places v of k, with the added assumption that the specified element of $H^1(k_v,T)$ comes from $H^1(\mathcal {O}_v, T)$ for all but finitely many v, and the first map is induced by the residue maps $\operatorname {res}_v$ for each place v of k.

Let $\iota : \prod _{v\in S}H^1(k_v, T) \hookrightarrow \sideset {}{'}\prod H^1(k_v, T)$ be defined by adding trivial classes at the places $v\notin S$ , and let $\xi _S = \xi \circ \iota $ . We now explain how to deduce from (6.13) an exact sequence of the form

(6.14) $$ \begin{align} H^1(G_S, \mathcal{T}_{A_S}) \xrightarrow{\prod_{v\in S}\operatorname{res}_v} \prod_{v \in S}H^1(k_v,T) \xrightarrow{\xi_S} H^1(G_S,\widehat{T})^{\vee}. \end{align} $$

Suppose that $b= (b_v)_{v\in S} \in \ker (\xi _S)$ . Then $\iota (b) \in \ker \xi $ . By the exactness of (6.13), this implies $\iota (b) =\operatorname {res}(a)$ for some $a\in H^1(k,T)$ . Moreover, we know that $\operatorname {res}_v(a)$ is trivial at all places $v\notin S$ . Recalling the identifications from (6.12), this means that a lies in the image of the norm map $(k_v \otimes _L k)^* \rightarrow k_v^*$ for all $v\notin S$ . By Lemma 6.9, there is a representative of the class $a\in H^1(k,T)$ which is contained in $\mathcal {O}_S^{\times }$ . Therefore, a in fact lies in $H^1(G_S, \mathcal {T}_{A_S})$ . It follows that $\prod _{v\in S}\operatorname {res}_v(a) = b$ , establishing the exactness of (6.14). Hence, condition (1) from the proposition is equivalent to the condition $(b_v)_{v\in S} \in \ker \xi _S$ .

The map $\xi _S$ is induced by the localisation maps $\prod _{v \in S}\operatorname {res}_v:H^1(G_S, \widehat {T}) \rightarrow \prod _{v\in S}H^1(k_v,\widehat {T})$ and the pairing

$$ \begin{align*}\prod_{v\in S}\left(H^1(k_v,T)\times H^1(k_v,\widehat{T})\right)\xrightarrow{\cup}\prod_{v\in S}H^2(k_v, \mathbb{G}_m)\xrightarrow{\sum_{v \in S}\operatorname{inv}_v} \mathbb{Q}/\mathbb{Z},\end{align*} $$

where $\cup $ denotes the cup product applied at each place $v\in S$ and $\operatorname {inv}_v$ denotes the local invariant map, as defined in [39, pp. 156].

Consider the diagram

where all the horizontal arrows are induced by the map $T\xrightarrow {N_{L/L'}} T'$ . (The norm map $N_{L/L'}: R_{L/k}\mathbb {G}_m \rightarrow R_{L'/k}\mathbb {G}_m$ restricts to a map $T \rightarrow T'$ because $N_{L/k}(z) = N_{L/L'}(N_{L'/k}(z))$ for any $z \in L$ .) This diagram commutes, thanks to the functorality properties of the localisation maps $\operatorname {res}_v$ [39, Proposition 1.5.2], and the projection formula for the cup product [39, Proposition 1.4.2]. The composition of the vertical arrows are the map $\xi _S$ and the corresponding map $\xi ^{\prime }_{S}$ for $T'$ . Therefore, we obtain a commutative diagram

Applying Lemma 6.8, we have $H^1(G_S,\widehat {T}) \cong \ker \varphi $ , where $\varphi $ is as defined in the exact sequence (6.5). As discussed in the paragraphs preceding Lemma A.5, $\varphi $ sends cyclic subextensions $M/k$ of $k_S$ (with a given choice of generator for $\mathrm {Gal}(M/k)$ ) to their compositum $LM/L$ , and so elements of $\ker \varphi $ correspond to cyclic subextensions of $L/k$ . However, since $L'/k$ is the maximal abelian subextension of $L/k$ , it contains all cyclic subextensions of $L/k$ , and so the map $H^1(G_S, \widehat {T'}) \rightarrow H^1(G_S, \widehat {T})$ is surjective. Since $\mathbb {Q}/\mathbb {Z}$ is a divisible group, it is an injective object in the category of abelian groups, and so the contravariant functor $\operatorname {Hom}(-,\mathbb {Q}/\mathbb {Z})$ is exact. We deduce that $\theta $ is an injection. Therefore, if $\xi ^{\prime }_S((b_v)_{v\in S})=0$ , then $\xi _S((b_v)_{v\in S})=0$ , so $(b_v)_{v\in S} \in \ker \xi ^{\prime }_S \implies (b_v)_{v \in S} \in \ker \xi _S$ . Recalling (6.14), this means that $(2) \implies (1)$ .