4.1 Sums of multiplicative functions

After the various reduction steps in the later sections, the count in which we are interested will be expressed as a character sum. For $\ell > 2$, the main term will occur when the combinations of characters are principal. In such cases, we will repeatedly call upon the following general theorem of Koukoulopoulos [Reference KoukoulopoulosKou19, Theorem 13.2] based on the earlier work of Granville and Koukoulopoulos [Reference Granville and KoukoulopoulosGK19].

Theorem 4.1 [Reference Granville and KoukoulopoulosGK19, Theorem 1]

Let $Q \geq 2$ be a parameter and let $f$ be a multiplicative function such that there exists $\alpha \in \mathbb {C}$ with

(4.1)\begin{equation} \sum_{p \leq x} f(p) \log p = \alpha x + O_A\biggl( \frac{x}{(\log x)^A} \biggr) \quad (x \geq Q) \end{equation}

for all $A>0$. Moreover, suppose, for all $n$, that $| f(n) | \leq \tau _k(n)$ for some positive real number $k$. Fix $\epsilon > 0$ and $J \in \mathbb {Z}_{\geq 1}$. Then we have

\[ \sum_{n \leq x} f(n) = x \sum_{j = 0}^{J - 1} c_j\frac{(\log x)^{\alpha - j - 1}}{\Gamma(\alpha - j)} + O\biggl(\frac{x (\log Q)^{2k + J - 1}}{(\log x)^{J + 1 - \text{Re}(\alpha)}}\biggr) \]

for $x \geq e^{(\log Q)^{1 + \epsilon }}$ and some explicit constants $c_j$. The implied constant depends at most on $k$, $J$, $\epsilon$ and the implied constant in (4.1) for $A$ large enough in terms of $k$, $J$ and $\epsilon$ only. Furthermore, we have

\[ c_0 = \prod_p \biggl( 1 + \frac{f(p)}{p} + \frac{f(p^2)}{p^2} + \cdots\biggr)\biggl(1 - \frac{1}{p} \biggr)^{\alpha} \]

and $c_j \ll _{j, k} (\log Q)^{j + 2k}$.

4.2 An abstract large sieve

To handle some of the combinations of characters with large modulus which are non-principal, we will use a number field large sieve. Let $K$ be a number field and let $\ell$ be a prime number. If $\mathfrak {f}$ is an ideal, we write $S_{\mathfrak {f}}$ for the subset of $\alpha \in O_K$ coprime with $\mathfrak {f}$. We also write $N(w)$ for the absolute norm of an element. Let $M \geq 1$ be an integer. Suppose that we are given a map

\[ \gamma : S_{M O_K\!} \times S_{M O_K} \rightarrow \{0\} \cup \{\zeta_{\ell}^i : i = 0, \ldots, \ell - 1\} \]

and a subset $A_{\text {bad}}$ of $\mathbb {Z}_{\geq 0}$ satisfying the following properties.

1. (P1) Multiplicativity: we have

   \[ \gamma (w, z_1z_2) = \gamma (w, z_1) \gamma (w, z_2)\quad {\rm for\ all}\ w,\ z_1\ {\rm and}\ z_2 \]
   and
   \[ \gamma (w_1w_2, z) = \gamma (w_1, z) \gamma (w_2, z)\quad {\rm for\ all}\ w_1,\ w_2\ {\rm and}\ z. \]

2. (P2) Periodicity: if $z_1, z_2, w \in S_{M O_K}$ satisfy $z_1 \equiv z_2 \bmod N(w)$ and $z_1 \equiv z_2 \bmod M$, then we have $\gamma (w, z_1) = \gamma (w, z_2)$. Furthermore, if $N(w) \not \in A_{\text {bad}}$, then we have

   \[ \sum_{\substack{\xi \bmod MN(w) \\ \gcd(\xi, M) = (1)}} \gamma(w, \xi) = 0. \]

3. (P3) Bad count: we have

   \[ \sum_{\substack{n \in A_{\text{bad}} \\ n \leq X}} 1 \leq C_1 X^{1 - C_2} \]
   for some absolute constants $C_1 > 0$ and $0 < C_2 < 1$.

Decompose

\[ O_K^\ast = T \oplus V, \]

where $T$ is torsion and $V$ is free. Such a decomposition is not unique, but we will fix one such decomposition. Fix a fundamental domain $\mathcal {D} \subseteq O_K$ as in [Reference Koymans and MilovicKM19, § 3.3] for the action of $V$ on $O_K$. We will recite the properties of the fundamental domain that we need.

We will consider bilinear sums of the type

\[ B(X, Y, \delta, \epsilon, t_1, t_2) = \sum_{\substack{w \in t_1 \mathcal{D}(X) \\ w \equiv \delta \bmod M}} \sum_{\substack{z \in t_2 \mathcal{D}(Y) \\ z \equiv \epsilon \bmod M}} \alpha_w \beta_z \gamma(w, z), \]

where $(\alpha _w)_w$ and $(\beta _z)_z$ are sequences of complex numbers bounded in absolute value by $1$, $\delta$ and $\epsilon$ are invertible congruence classes modulo $M$, $t_1$ and $t_2$ are fixed elements of $T$ (so $t_i \mathcal {D}$ is a translate of the fundamental domain) and $X, Y \geq 2$ are real numbers. Here we use the notation $(t_i \mathcal {D})(X)$ for the subset of $\alpha \in t_i \mathcal {D}$ with $N(\alpha ) \leq X$.

Proposition 4.3 Assume that $X \leq Y$. Then we have

\[ |B(X, Y, \delta, \epsilon, t_1, t_2)| \ll \bigl(X^{({-C_2})/{3n}} + Y^{({-1})/{6n}}\bigr) XY (\log XY)^{C_K}, \]

where $n = [K : \mathbb {Q}]$ and $C_K$ is a constant depending only on $K$. The implied constant depends only on $K$, $M$ and the constants $C_1, C_2$.

Since $0 < C_2 < 1$, we achieve a power saving in both $X$ and $Y$. Careful scrutiny of the proof shows that the constant $C_K$ may be taken to depend at most on $n$. However, the same cannot be said of the implicit constant, which will likely depend on the regulator of $K$ with the current argument. This will not be a cause for concern; however, as in our application this field will be a fixed cyclotomic field of the form $\mathbb {Q}(\zeta _\ell )$.

Proof of Proposition 4.3 The argument is a minor generalisation of [Reference Koymans and MilovicKM19, Proposition 3.6]. Pick an integer $k \geq 1$ that we will optimise later. We start the proof by applying Hölder's inequality to

\[ 1 = \frac{k - 1}{k} + \frac{1}{k}, \]

which gives

\begin{align*} |B(X, Y, \delta, \epsilon, t_1, t_2)|^k &\leq \Bigg(\sum_{\substack{w \in t_1 \mathcal{D}(X) \\ w \equiv \delta \bmod M}} |\alpha_w| \cdot \Bigg|\sum_{\substack{z \in t_2 \mathcal{D}(Y) \\ z \equiv \epsilon \bmod M}} \beta_z \gamma(w, z)\Bigg|\Bigg)^k\\ &\leq \Bigg(\sum_{\substack{w \in t_1 \mathcal{D}(X) \\ w \equiv \delta \bmod M}} |\alpha_w|^{{k}/({k - 1})}\Bigg)^{k - 1} \cdot \Bigg(\sum_{\substack{w \in t_1 \mathcal{D}(X)\\ w \equiv \delta \bmod M}} \Bigg|\sum_{\substack{z \in t_2 \mathcal{D}(Y) \\ z \equiv \epsilon \bmod M}} \beta_z \gamma(w, z)\Bigg|^k\Bigg)\\ &\ll X^{k - 1} \cdot \sum_{\substack{w \in t_1 \mathcal{D}(X) \\ w \equiv \delta \bmod M}} \Bigg|\sum_{\substack{N(z) \leq Y^k}} \beta'_z \gamma(w, z)\Bigg|, \end{align*}

where

\[ \beta'_z := \sum_{\substack{z_1 \cdot \ldots \cdot z_k = z \\ z_i \in t_2 \mathcal{D}(Y) \\ z_i \equiv \epsilon \bmod M}} \beta_{z_1} \cdot \ldots \cdot \beta_{z_k}. \]

Here we used property (P1) to expand the $k$-fold product.

Fix an integral basis $\eta _1, \ldots, \eta _n$ of $O_K$, where $n = [K : \mathbb {Q}]$. We call an element $z$ $(C, Y)$-well-balanced if

(4.2)\begin{align} z = a_1\eta_1 + \ldots + a_n\eta_n, \quad a_i \in \mathbb{Z}, \end{align}

implies that $|a_i| \leq C Y^{1/n}$. From the construction of the fundamental domain $\mathcal {D}(Y)$ (see the second property of Lemma 4.2), it follows that there exists a constant $C > 0$ depending only on $k$, $K$ and the choice of integral basis such that $\beta '_z = 0$ if $z$ is not $(C, Y^k)$-well-balanced. Fix such a choice of absolute constant $C$. Then we may assume from now on that we are summing over all $z$ such that $N(z) \leq Y^k$ and $z$ is $(C, Y^k)$-well-balanced. Write $\mathcal {B}(Y, C)$ for the set of $z \in O_K$ such that $|a_i| \leq CY^{1/n}$ upon expanding $z$ as in (4.2) and such that $z$ is coprime with $M$. For the remainder of this proof, all our implied constants may depend on $K$, $M$, $k$, $C_1$, $C_2$, our choice of $C$ and our choice of integral basis. We rewrite

\begin{align*} \sum_{\substack{w \in t_1 \mathcal{D}(X) \\ w \equiv \delta \bmod M}} \Bigg|\sum_{\substack{N(z) \leq Y^k \\ z \in \mathcal{B}(Y^k, C)}} \beta'_z \gamma(w, z)\Bigg| &= \sum_{\substack{w \in t_1 \mathcal{D}(X) \\ w \equiv \delta \bmod M}} \epsilon(w) \sum_{\substack{N(z) \leq Y^k \\ z \in \mathcal{B}(Y^k, C)}} \beta'_z \gamma(w, z)\\ &= \sum_{\substack{N(z) \leq Y^k \\ z \in \mathcal{B}(Y^k, C)}} \beta'_z \sum_{\substack{w \in t_1 \mathcal{D}(X) \\ w \equiv \delta \bmod M}} \epsilon(w) \gamma(w, z), \end{align*}

where $\epsilon (w)$ are complex numbers of absolute value $1$. We now drop the condition $N(z) \leq Y^k$. This does not change the sum, since $\beta '_z = 0$ if $N(z) > Y^k$. Then the Cauchy–Schwarz inequality yields

\begin{align*} |B(X, Y, \delta, \epsilon, t_1, t_2)|^{2k} &\ll X^{2k - 2} \cdot \Biggl(\sum_{\substack{z \in \mathcal{B}(Y^k, C)}} |\beta'_z|^2\Biggr)\\ &\quad \cdot \Biggl(\sum_{\substack{z \in \mathcal{B}(Y^k, C)}} \sum_{\substack{w_1 \in t_1 \mathcal{D}(X) \\ w_1 \equiv \delta \bmod M}} \sum_{\substack{w_2 \in t_1 \mathcal{D}(X) \\ w_2 \equiv \delta \bmod M}} \epsilon(w_1) \epsilon(w_2) \gamma(w_1w_2, z)\Biggr) \end{align*}

thanks to property (P1). We bound the former sum by

\[ \sum_{\substack{z \in \mathcal{B}(Y^k, C)}} |\beta'_z|^2 \ll Y^k (\log Y)^{C_{K, k}}, \]

where $C_{K, k}$ is an effectively computable constant depending only on $K$ and $k$. For the latter sum we invert the order of summation to get

\[ \sum_{\substack{w_1 \in t_1 \mathcal{D}(X) \\ w_1 \equiv \delta \bmod M}} \sum_{\substack{w_2 \in t_1 \mathcal{D}(X) \\ w_2 \equiv \delta \bmod M}} \epsilon(w_1) \epsilon(w_2) \sum_{z \in \mathcal{B}(Y^k, C)} \gamma(w_1w_2, z). \]

We have the estimates

\[ \sum_{z \in \mathcal{B}(Y^k, C)} \gamma(w_1w_2, z) \ll \begin{cases} Y^k, & \mbox{if } N(w_1w_2) \in A_{\text{bad}} ,\\ \sum_{i = 1}^n X^{2i} Y^{k(1 - {i}/{n})}, & \mbox{if } N(w_1w_2) \not \in A_{\text{bad}}. \end{cases} \]

Indeed, the first inequality is just the trivial bound. For the second inequality, we use property (P2) and split $\mathcal {B}(Y^k, C)$ into boxes of side length $MN(w_1w_2) \leq MX^2$.

From now on we shall take $k \geq 2n$. Hence, we have $Y^{k/n} \geq X^2$, since we assumed that $Y \geq X$. Therefore, the last bound can be simplified to $X^2 Y^{k(1 - 1/n)}$. Thanks to property (P3), we get the bound

\[ \sum_{\substack{w_1 \in t_1 \mathcal{D}(X) \\ w_1 \equiv \delta \bmod M}} \sum_{\substack{w_2 \in t_1 \mathcal{D}(X) \\ w_2 \equiv \delta \bmod M}} \mathbf{1}_{N(w_1w_2) \in A_{\text{bad}}} \ll_{K, k, C_1} X^{2 - 2C_2} (\log X)^{C_{K, k}} \]

for a potentially different effectively computable constant $C_{K, k}$ depending only on $K$ and $k$. This shows that

\begin{align*} &\sum_{\substack{w_1 \in t_1 \mathcal{D}(X) \\ w_1 \equiv \delta \bmod M}} \sum_{\substack{w_2 \in t_1 \mathcal{D}(X) \\ w_2 \equiv \delta \bmod M}} \epsilon(w_1) \epsilon(w_2) \sum_{z \in \mathcal{B}(Y^k, C)} \gamma(w_1w_2, z) \\ &\qquad \ll\big(X^{2 - 2C_2} Y^k + X^4 Y^{k(1 - 1/n)}\big) (\log XY)^{C_{K, k}}. \end{align*}

We conclude that

\[ |B(X, Y, \delta, \epsilon, t_1, t_2)|^{2k} \ll \bigl(X^{2k - 2C_2} Y^{2k} + X^{2k + 2} Y^{2k - {k}/{n}}\bigr) (\log XY)^{C_{K, k}}. \]

We take $k = 3n$, which depends only on $K$, to finish the proof of the proposition.

5.1 The main term: the reduction step

To prepare for our application of Theorem 5.2, we start by fixing all the variables $v_a$ with $a \in A - A[\ell ]$. The induced homomorphism $\varphi : G_\mathbb {Q} \rightarrow A \rightarrow A/A[\ell ]$ depends only on the $v_a$ with $a \in A - A[\ell ]$. Define for $a \in A$ the set

\[ \text{Adm}(a) := \{b \in A : \text{Hom}(\wedge^2(A), \mathbb{Q}/\mathbb{Z}) \rightarrow \text{Hom}(\wedge^2(\langle a, b \rangle), \mathbb{Q}/\mathbb{Z}) \text{ is zero}\}. \]

Write $\widetilde {\pi }: A \rightarrow A/A[\ell ]$ for the natural quotient map. We make the following assumptions.

1. (P1) The homomorphism $\varphi$ is surjective. Write $K$ for the fixed field of $\varphi$, which is an $A/A[\ell ]$-extension of $\mathbb {Q}$.

2. (P2) For all $p \mid v_a$, we have $\varphi (\text {Frob}_p) \in \widetilde {\pi }(\text {Adm}(a))$. Here and henceforth we make a fixed choice of a prime ideal $\mathfrak {p}$ of $\overline {\mathbb {Z}}$ above $p$ and we also make a fixed choice of an element $\text {Frob}_p \in \{\sigma : \sigma (\mathfrak {p}) = \mathfrak {p}\} \subseteq \mathrm {Gal}(\overline {\mathbb {Q}}/\mathbb {Q})$ lifting the Frobenius automorphism of $\overline {\mathbb {Z}}/\mathfrak {p} \cong \overline {\mathbb {F}_p}$.

3. (P3) There exists an $A$-extension of $\mathbb {Q}$ containing $K$ that satisfies weak approximation. We remark that this condition in fact implies (P1) and (P2).

If these conditions are not met, we may freely ignore the variables $v_a$, since they do not contribute to the counting function. There exists at least one tuple $v_a$ satisfying (P1), (P2) and (P3) by an application of [Reference Frei, Loughran and NewtonFLN18, Proposition 5.5] with $k = \mathbb {Q}$ and $G = A$. Indeed, weak approximation certainly holds if all decomposition groups are cyclic. We will use this later on to guarantee that our leading constant $C_{\text {weak}}$ is strictly greater than zero.

We will now work towards applying Theorem 5.2. We take $M = \mathfrak {f}(K)^2$, where $\mathfrak {f}(K)$ is the conductor of $K$. There exists a subgroup $G$ of $(\mathbb {Z}/M\mathbb {Z})^\ast$ such that $p$ splits completely in $K$ if and only if $p \bmod M \in G$. Then for $a \in A[\ell ] - \{0\}$ we take

\[ H_a := \bigcup_{b \in \widetilde{\pi}(\text{Adm}(a))} \text{FrobInv}(b) \cdot G, \]

where $\text {FrobInv}(b)$ is any prime $p$, not dividing $M$, with Artin symbol in $K$ equal to $b$. By choosing $C$ sufficiently large in terms of the abelian group $A$, we see that the condition $[(\mathbb {Z}/M\mathbb {Z})^\ast : G] \leq C$ in Theorem 5.2 holds.

We take $B = A[\ell ] \cap \ell A$. We have to check that $|H_a| = |H_{a'}|$ for all $a, a' \in A[\ell ] - B$. This is equivalent to showing that

(5.4)\begin{equation} |\widetilde{\pi}(\text{Adm}(a))| = |\widetilde{\pi}(\text{Adm}(a'))| \end{equation}

for all $a, a' \in A[\ell ] - B$. If $\langle a \rangle = \langle a' \rangle$, then the result is correct. So now suppose that $\langle a \rangle \neq \langle a' \rangle$. We claim that there exists an automorphism $\psi$ of $A$ such that $\psi (a) = a'$.

Let us first show that the claim implies (5.4). To this end, take an automorphism $\psi$ that sends $a$ to $a'$. Consider the induced map $\psi ^\ast : \text {Adm}(a) \rightarrow \text {Adm}(a')$, which sends $b$ to $\psi (b)$. To check that this is well-defined, we further claim that

\[ \text{Hom}(\wedge^2(A), \mathbb{Q}/\mathbb{Z}) \rightarrow \text{Hom}(\wedge^2(\langle a, b \rangle), \mathbb{Q}/\mathbb{Z}) \text{ is zero} \]

if and only if

\[ \text{Hom}(\wedge^2(A), \mathbb{Q}/\mathbb{Z}) \rightarrow \text{Hom}(\wedge^2(\langle \psi(a), \psi(b) \rangle), \mathbb{Q}/\mathbb{Z}) \text{ is zero}. \]

The former statement is equivalent to all alternating maps $\lambda : A \times A \rightarrow \mathbb {Q}/\mathbb {Z}$ satisfying $\lambda (a, b) = 0$, while the latter statement is equivalent to all alternating maps $\lambda : A \times A \rightarrow \mathbb {Q}/\mathbb {Z}$ satisfying $\lambda (\psi (a), \psi (b)) = 0$. As $\lambda$ runs through all alternating maps, so does $\lambda (\psi (-), \psi (-))$ and therefore the above statements are all equivalent. This shows that $\psi ^\ast$ is well-defined. It is now readily verified that $\psi ^\ast$ is a bijection. We next observe that $\text {Adm}(a)$ is a subgroup of $A$. Therefore, to establish (5.4), it suffices to show that $|\text {Adm}(a) \cap A[\ell ]| = |\text {Adm}(a') \cap A[\ell ]|$. But this is true because $\psi$ is an automorphism that restricts to an isomorphism between $\text {Adm}(a)$ and $\text {Adm}(a')$. We conclude that the claim indeed implies (5.4).

Let us now prove the claim. If the injection

\[ 0 \rightarrow \langle a, a' \rangle \rightarrow A \]

is split, then it is straightforward to construct the desired bijection $\psi$. So now suppose that the above injection is not split. Because $a, a' \not \in B$, this means that there exist $n, m \in \mathbb {Z}$ both not divisible by $\ell$ such that $na + ma' \in \ell A$. Using that $a \not \in B$, we may decompose $A$ as an internal direct sum $\langle a \rangle \oplus C$ for some subgroup $C$ of $A$. Now consider the homomorphism $\psi$ that is the identity on $C$ and sends $a$ to $a'$.

It remains to prove that $\psi$ is surjective. By construction we have that $C \subseteq \text {im}(\psi )$ and that $a' \in \text {im}(\psi )$. Therefore, it suffices to show that $a \in \text {im}(\psi )$. By the relation $na + ma' \in \ell A$ and the inclusion $\ell A \subseteq C$, we deduce that $na \in \text {im}(\psi )$. Because $n$ is coprime to $\ell$, we conclude that $a \in \text {im}(\psi )$. We have proven the claim and thus (5.4).

We will now construct a congruence function $(f, g)$ for $M$. Suppose that $p \neq 2$ is unramified in $K$. Then, if $p$ ramifies in $\text {Par}(\mathbf {v})$, we have $p \mid v_a$ for some $a \in A[\ell ] - \{0\}$. Take ${S = \langle a \rangle \in \mathcal {S}}$, which has dimension one. For weak approximation to hold, we certainly must have that $\text {Frob}_{K/\mathbb {Q}}(p) \in \widetilde {\pi }(\text {Adm}(a))$ (or equivalently $p \bmod M \in H_a$), which we will assume from now on. We will now distinguish two cases.

Case I: $a \in B$. In this case, Theorem 3.4 yields that $\text {Frob}_p$ automatically lands in $\text {Adm}(a)$. Correspondingly, we take $f(p, S) = A/S$.

Case II: $a \not \in B$. In this case Theorem 3.4 tells us that $\text {Adm}(a) = \ell A + \langle a \rangle$. Define $\mathbf {x}_Q$ to be the unique tuple that equals $v_a$ for $a \in A - A[\ell ]$ and equals $1$ for $a \in A[\ell ] - \{0\}$. We claim that there exists an element $h(p, S) \in A[\ell ]$ such that

(5.5)\begin{equation} \text{Par}(\mathbf{x}_Q)(\text{Frob}_p) + h(p, S) \in \text{Adm}(a). \end{equation}

By construction of $\mathbf {x}_Q$ we have that $\widetilde {\pi } \circ \text {Par}(\mathbf {x}_Q) = \varphi$. Therefore, it follows from $\text {Frob}_{K/\mathbb {Q}}(p) \in \widetilde {\pi }(\text {Adm}(a))$ that (5.5) is true after applying $\widetilde {\pi }$, establishing the claim. Then we take $f(p, S)$ to be the coset $h(p, S) + B$. One may now directly verify that condition (5.1) is satisfied.

5.2 The bad primes

Next consider a prime $p \neq 2$ that divides the conductor of $K$, which implies that $p \mid v_a$ for some $a \not \in A[\ell ]$. Let $g(p)$ be the subset of $a \in A[\ell ]$ satisfying

\[ \text{Par}(\mathbf{x}_Q)(\text{Frob}_p) + a \in \text{Adm}(a). \]

We deduce from property (P2) of $K$ that $g(p)$ is non-empty.

5.3 Completing the reduction step

We now rewrite

\[ \sum_{\substack{\mathbf{v} = (v_a)_{a \in A - \{0\}} \in \mathcal{B} \\ \text{Disc}(\text{Par}(\mathbf{v})) \leq X \\ \prod_{a \not \in A[\ell]} v_a \leq (\log X)^{C_3}}} \mathbf{1}_{\text{WA holds}} \]

as

\[ \sum_{\substack{(v_a)_{a \in A - A[\ell]} \\ \\ \prod_{a \not \in A[\ell]} v_a \leq (\log X)^{C_3}}} \sum_{(c_a)_{a \in A[\ell] - \{0\}} \in (\mathbb{Z}/d(\ell)\mathbb{Z})^{A[\ell] - \{0\}}} \mathbf{1}_{(c_a) \text{ nice}} \sum_{\substack{\mathbf{w} = (w_a)_{a \in A[\ell] - \{0\}} \\ \mathbf{v} \in \mathcal{B} \\ \text{Disc}(\text{Par}(\mathbf{v})) \leq X \\ w_a \equiv c_a \bmod d(\ell) \\ p \mid v_a \Rightarrow p \bmod M \in H_a}} \mathbf{1}_{\text{WA holds}}, \]

where $\mathbf {v}$ is the tuple obtained by concatenating $(v_a)$ and $(w_a)$ and where we will soon define when $(c_a)$ is nice.

Fix the tuple $\mathbf {v}_Q = (v_a)_{a \in A - A[\ell ]}$ satisfying assumptions (P1), (P2) and (P3) and fix a tuple $(c_a)_{a \in A[\ell ] - \{0\}}$. We first claim that the condition $\mathbf {1}_{\text {WA holds}}$ may be replaced by $\mathbf {1}_{\text {Par}(\mathbf {w}) f\text {-correct}}$. First of all, we remark that $D_p$ is certainly cyclic if $p$ is unramified in $\text {Par}(\mathbf {v})$. At the odd ramified places this is true by construction of $f$. Finally, whether the natural restriction map $\text {Hom}(\wedge ^2(A), \mathbb {Q}/\mathbb {Z}) \rightarrow \text {Hom}(\wedge ^2(\text {im}(G_{\mathbb {Q}_2}), \mathbb {Q}/\mathbb {Z})$ is zero is entirely determined by $\mathbf {v}_Q = (v_a)_{a \in A - A[\ell ]}$ and $(c_a)_{a \in A[\ell ] - \{0\}}$. Indeed, if $\ell > 2$, then this map is always zero as $2$ is necessarily unramified in $A$. If $\ell = 2$, then this follows from the fact that the restrictions of $\psi _{v_a, 2, 1}$ and $\psi _{v_b, 2, 1}$ to $G_{\mathbb {Q}_2}$ are equal if $v_a \equiv v_b \bmod 16$ (recall that $d(2) = 16$) combined with (2.1). Now we simply define $(c_a)$ to be nice if the above map is zero.

Let us now explicate the condition $\mathbf {v} = (v_a)_{a \in A - \{0\}} \in \mathcal {B}$ in terms of the variables in $A[\ell ] - \{0\}$. One directly checks that

\[ \mathbf{v} \in \mathcal{B} \Longleftrightarrow \langle a \in A - \{0\} : v_a \neq 1 \rangle = A \]

for all $\mathbf {v} \in \mathcal {A}$. Therefore, since $\varphi$ is already assumed to be surjective by (P1), we have that there exists a subspace $T$, containing $B$ and depending on $\mathbf {v}_Q = (v_a)_{a \in A - A[\ell ]}$, such that

(5.6)\begin{equation} \mathbf{v} \in \mathcal{B} \Longleftrightarrow T + \langle a \in A[\ell] - \{0\} : v_a \neq 1 \rangle = A[\ell]. \end{equation}

Next we compare $\text {Disc}(\text {Par}(\mathbf {v}))$ with $\text {Disc}(\text {Par}(\mathbf {w}))$. To do so, we compute the $p$-adic valuation of the discriminant using Theorem 2.3 for primes $p \nmid 2|A|$. Locally at $2$, we have just proven that the restriction of $\text {Par}(\mathbf {v})$ is unramified if $\ell > 2$ and completely determined by $c_a$ and $\mathbf {v}_Q = (v_a)_{a \in A - A[\ell ]}$ if $\ell = 2$. For the odd places $p \mid |A|$, we compute the $p$-adic valuation of the discriminant using Theorem 2.4. More precisely, if $\text {Par}(\sigma _p)$ is the identity, then $v_p(\text {Disc}(\text {Par}(\mathbf {v}))) = 0$. If instead $\text {Par}(\sigma _p) \in A - A[\ell ]$, then $v_p(\text {Disc}(\text {Par}(\mathbf {v})))$ is determined by $\mathbf {v}_Q$ as a consequence of Theorem 2.4. Finally, if $\text {Par}(\sigma _p) \in A[\ell ] - \{0\}$, then the proof of Theorem 2.4 shows that $v_p(\text {Disc}(\text {Par}(\mathbf {v})))$ equals ${|A|}/{\ell }$ multiplied by the $p$-adic valuation of the discriminant of the unique degree $\ell$ subfield of $\mathbb {Q}_p(\zeta _{p^\infty })$. Therefore, the conductor–discriminant formula demonstrates the validity of

\[ v_p(\text{Disc}(\text{Par}(\mathbf{v}))) = \begin{cases} |A| \cdot \biggl(1 - \dfrac{1}{\ell}\biggr), & \text{if } \ell \neq p, \\ 2|A| \cdot \biggl(1 - \dfrac{1}{\ell}\biggr), & \text{if } \ell = p. \end{cases} \]

Therefore, there exists $C_4 > 0$, depending only on $c_a$ and $\mathbf {v}_Q = (v_a)_{a \in A - A[\ell ]}$, such that

\[ \text{Disc}(\text{Par}(\mathbf{v})) = C_4 \cdot \Bigg(\prod_{a \in A[\ell] - \{0\}} \Delta(|w_a|)\Bigg)^{({|A| \cdot (\ell - 1)})/{\ell}}. \]

We are now ready to apply Theorem 5.2 to the innermost sum. We shall do so with every choice of $S \subseteq A[\ell ] - \{0\}$ satisfying $T + S = A[\ell ]$, which allows us to detect the condition (5.6) using inclusion–exclusion. Here we make essential use of the fact that the leading constant $C_{\text {lead}}$ from Theorem 5.2 is independent of $S$.

5.4 The exponent of the logarithm in the main term

Let us compute $\alpha (A)$, the exponent of the logarithm appearing in Theorem 1.2. When we apply Theorem 5.2, the exponent of the logarithm, denoted by $\alpha$ in the theorem statement, is equal to

\[ \sum_{a \in B - \{0\}} \frac{|\text{Lift}(H_a)|}{\varphi(\text{lcm}(M, \ell))} + \sum_{a \in A - B} \frac{|B|}{\ell^{n - 1}} \cdot \frac{|\text{Lift}(H_a)|}{\varphi(\text{lcm}(M, \ell))} \]

with $n$ the dimension of $A[\ell ]$ as an $\mathbb {F}_{\ell }$-vector space. Since $\ell$ is the smallest prime divisor of $A$, it follows that $\mathbb {Q}(\zeta _{\ell })$ and $K$ are disjoint extensions of $\mathbb {Q}$. Therefore, we have

\[ \frac{|\text{Lift}(H_a)|}{\varphi(\text{lcm}(M, \ell))} = \frac{|H_a|}{(\ell - 1) \varphi(M)}. \]

By construction, we have that

\[ \frac{|H_a|}{\varphi(M)} = \frac{|\widetilde{\pi}(\text{Adm}(a))| \cdot |G|}{\varphi(M)} = \frac{|\widetilde{\pi}(\text{Adm}(a))|}{[K : \mathbb{Q}]} = \frac{|\widetilde{\pi}(\text{Adm}(a))|}{|A/A[\ell]|}. \]

Finally, it follows from Theorem 3.4 that

\[ \frac{|\widetilde{\pi}(\text{Adm}(a))|}{|A/A[\ell]|} = \begin{cases} \dfrac{|\text{Adm}(a)|}{|A|}, & \text{if } a \in B - \{0\} ,\\ \dfrac{|\text{Adm}(a)|}{|A|} \cdot \dfrac{\ell^{n - 1}}{|B|}, & \text{if } a \in A - B. \end{cases} \]

This shows that $\alpha (A)$ is the correct exponent.

5.5 The leading constant

To finish the proof, we now apply Theorem 5.2 for each tuple $\mathbf {v}_Q = (v_a)_{a \in A - A[\ell ]}$ such that the associated map $\varphi : G_\mathbb {Q} \rightarrow A/A[\ell ]$ satisfies (P1), (P2) and (P3). Every such tuple $\mathbf {v}_Q$ together with a choice of $\mathbf {c} = (c_a)_{a \in A[\ell ] - \{0\}}$ gives a leading constant that we denote by $C_{\text {lead}}(\mathbf {v}_Q, \mathbf {c})$ and also a constant $C_4(\mathbf {v}_Q, \mathbf {c}) > 0$ satisfying

\[ \text{Disc}(\text{Par}(\mathbf{v})) = C_4(\mathbf{v}_Q, \mathbf{c}) \cdot \Bigg(\prod_{a \in A[\ell] - \{0\}} \Delta(|w_a|)\Bigg)^{({|A| \cdot (\ell - 1)})/{\ell}}. \]

Then we take

\[ C_{\text{weak}} = \sum_{\mathbf{v}_Q} \sum_\mathbf{c} \frac{C_{\text{lead}}(\mathbf{v}_Q, \mathbf{c})}{C_4(\mathbf{v}_Q, \mathbf{c})}. \]

Because the exponent in the discriminant for all variables outside $A[\ell ]$ is bigger than the exponent for the variables in $A[\ell ]$, one sees that the sum

\[ \sum_{\mathbf{v}_Q} \sum_\mathbf{c} \frac{1}{C_4(\mathbf{v}_Q, \mathbf{c})} \]

converges and also may be truncated (to those $\mathbf {v}_Q$ satisfying that $M \leq (\log X)^{C_5}$) with an acceptable error term. Since the leading constant $C_{\text {lead}}(\mathbf {v}_Q, \mathbf {c})$ is uniformly bounded, we conclude that the sum defining $C_{\text {weak}}$ may also be truncated. Hence, the contribution from the error term in Theorem 5.2 is negligible.

In order to show that $C_{\text {weak}} > 0$, it suffices to show that $C_{\text {lead}}(\mathbf {v}_Q, \mathbf {c}) > 0$ for some choice of $\mathbf {v}_Q$ and $\mathbf {c}$. We choose a splitting of $A[\ell ]$ as $A[\ell ] = B \oplus B_{\text {comp}}$ and a splitting $A = \widetilde {B} \oplus B_{\text {comp}}$ with $B \subseteq \widetilde {B}$. By [Reference Frei, Loughran and NewtonFLN18, Proposition 5.5], we may find a surjective homomorphism $\varphi ': G_\mathbb {Q} \rightarrow \widetilde {B}$ such that all decomposition groups are cyclic. In particular, $\varphi '$ satisfies weak approximation. We extend $\varphi '$ to a homomorphism $\varphi '': G_\mathbb {Q} \rightarrow A$ by sending $g$ to $(\varphi '(g), 0)$, where we have implicitly used our splitting $A = \widetilde {B} \oplus B_{\text {comp}}$.

Now we apply Theorem 5.2 with $M$ equal to the absolute discriminant of $\varphi ''$, $B = A[\ell ] \cap \ell A$ as above, $f(p, S) = B$ for $S$ intersecting $B$ trivially, $g(p) = \{0\}$ and $c_a = 1$. Now twisting $\varphi ''$ with such multicyclic extensions gives a new $A$-extension satisfying weak approximation. This forces $C_{\text {lead}}(\mathbf {v}_Q, \mathbf {c}) > 0$ and therefore $C_{\text {weak}} > 0$.

7.1 Multiquadratic case

Define

\[ \mathcal{I} := \{(S, T) : S, T \in \mathbb{F}_2^n, S \not \in B, \langle \gamma, T \rangle = 0\ \forall \gamma \in B_S\}. \]

For a subspace $V$ of $\mathbb {F}_2^n$ we write

\[ V^\top = \{S \in \mathbb{F}_2^n : \langle S, v \rangle = 0 \ \forall v \in V\} \]

for the complement under the pairing $\langle \cdot, \cdot \rangle$. We have the following crucial lemma.

Lemma 7.1 Let $X \subseteq \mathcal {I}$ with $|X| \geq 2^n - |B|$. Suppose that

(7.1)\begin{equation} \langle S_1, T_2 \rangle + \langle S_2, T_1 \rangle = 0 \end{equation}

for all $(S_1, T_1), (S_2, T_2) \in X$. Then we have

\[ X = \{(S, f(S)) : S \in \mathbb{F}_2^n - B\} \]

for some function $f: \mathbb {F}_2^n - B \rightarrow B^\top$ which is alternating with respect to the bilinear pairing $\langle \cdot, \cdot \rangle$, in the sense that

\[ \langle S, f(T) \rangle = \langle T, f(S) \rangle \quad\text{and}\quad \langle S, f(S) \rangle = 0. \]

Proof. Denote by $\pi _1$ and $\pi _2$ the natural projection maps from $\mathcal {I}$ to $\mathbb {F}_2^n$. Write $V_1$ for the subspace generated by $\pi _1(X)$ and write $V_2$ for the subspace generated by $\pi _2(X)$. By construction of $\mathcal {I}$ we have that

(7.2)\begin{equation} V_2 \subseteq B^\top. \end{equation}

By the pigeonhole principle there exists some $T_0 \in \pi _2(X)$ such that

(7.3)\begin{equation} |\pi_2^{-1}(T_0) \cap X| \geq \frac{2^n - |B|}{|V_2|}. \end{equation}

List the elements of $\pi _2^{-1}(T_0) \cap X$ as

\[ (S_1, T_0), \ldots, (S_\alpha, T_0). \]

Suppose that there exists $T_1 \in \pi _2(X)$ such that

\[ \langle S_i, T_1 \rangle + \langle S_j, T_1 \rangle = 1 \]

for some $1 \leq i, j \leq \alpha$. We claim that this contradicts (7.1). Indeed, take such a $T_1$ and such $i, j$. Let $U$ be such that $(U, T_1) \in X$. Then either $(U, T_1)$ and $(S_i, T_0)$ contradict (7.1) or $(U, T_1)$ and $(S_j, T_0)$ do.

Therefore, we may assume from now on that for all $T_1 \in \pi _2(X)$ and all $1 \leq i, j \leq \alpha$,

\[ \langle S_i, T_1 \rangle + \langle S_j, T_1 \rangle = 0. \]

We conclude that $S_i - S_j \in V_2^\top$. We now claim that $V_1$ contains $V_2^\top$, so it contains in particular $B$ by (7.2). Since we have $S_i - S_j \in V_2^\top$, we now consider the elements

\[ \{S_j - S_1 : 2 \leq j \leq \alpha\}. \]

This gives $\alpha - 1 = |\pi _2^{-1}(T_0) \cap X| - 1$ non-zero elements of $V_2^\top \cap V_1$. Therefore, we get from (7.3) that

\[ |V_2^\top \cap V_1| \geq \alpha \geq \frac{2^n - |B|}{|V_2|} = |V_2^\top| - \frac{|B|}{|V_2|} \]

after also taking into account the zero element of $V_2^\top \cap V_1$. If $B$ has codimension zero or one, then the lemma is trivial. Otherwise we have

\[ |V_2^\top| - \frac{|B|}{|V_2|} \geq |V_2^\top| - \frac{|V_2^\top|}{4}. \]

We conclude that

\[ |V_2^\top \cap V_1| \geq \frac{3|V_2^\top|}{4}, \]

which readily implies the claim. Thanks to the claim we see that $V_1$ contains $B$.

We next claim that $V_1$ equals $\mathbb {F}_2^n$. We now fix some $S_0 \in \pi _1(X)$. Arguing as before, we see that

\[ \langle S_1, T_i \rangle + \langle S_1, T_j \rangle = 0 \]

for all $S_1 \in \pi _1(X)$ and all $i$ and $j$ such that $(S_0, T_i), (S_0, T_j) \in X$. In particular, we deduce that $|\pi _1^{-1}(S_0) \cap X| \leq |V_1^\top |$. We now sum to obtain

\[ 2^n - |B| \leq |X| = \sum_{S_0 \in \pi_1(X)} |\pi_1^{-1}(S_0) \cap X| \leq \sum_{S_0 \in \pi_1(X)} |V_1^\top| \leq (|V_1| - |B|) \cdot |V_1^\top| = 2^n - |B| \cdot |V_1^\top|, \]

because $V_1$ contains $B$. But this is only possible if $|V_1^\top | = 1$ or equivalently $V_1 = \mathbb {F}_2^n$. Therefore, there exists a function $f: \mathbb {F}_2^n - B \rightarrow B^\top$ such that

\[ X = \{(S, f(S)) : S \in \mathbb{F}_2^n - B\}. \]

It is readily verified that $f$ must be alternating, completing the proof.

7.2 Multicyclic case

Define

\[ \mathcal{I} := \{(a, b) : a, b \in \mathbb{F}_\ell^n, a \not \in B, \langle \gamma, b \rangle = 0\ \forall \gamma \in B_a\}, \]

where we recall that $\langle \cdot, \cdot \rangle$ is the standard bilinear form. We have the following crucial lemma.

Lemma 7.2 Let $X \subseteq \mathcal {I}$ with $|X| \geq \ell ^n - |B|$. Suppose that

(7.4)\begin{equation} \langle a_2, b_1 \rangle = 0 \end{equation}

for all $(a_1, b_1), (a_2, b_2) \in X$. Then we have

\[ X = \{(a, 0) : a \in \mathbb{F}_\ell^n - B\}. \]

Proof. The proof is similar to the proof of Lemma 7.1. We write $\pi _1$ and $\pi _2$ for the natural projection maps from $\mathcal {I}$ to $\mathbb {F}_\ell ^n$. We denote by $N$ the number of elements in the subspace generated by $\pi _2(X)$. If $N = 1$, then we have

\[ X = \{(a, 0) : a \in \mathbb{F}_\ell^n - B\}. \]

From now on we may and will assume that $N > 1$ and seek a contradiction.

Take some $b_0 \in \pi _2(X)$ and suppose that $|\pi _2^{-1}(b_0) \cap X| \geq (\ell ^n - |B|)/N$. The existence of such a $b_0$ is guaranteed by the pigeonhole principle and our assumption $|X| \geq \ell ^n - |B|$. We enumerate the elements of $\pi _2^{-1}(b_0) \cap X$ as

\[ (a_1, b_0), \ldots, (a_k, b_0) \]

with $k \geq (\ell ^n - |B|)/N$. Then we have, thanks to (7.4), the equality

\[ \langle a_i, b \rangle = 0 \]

for all $b \in \pi _2(X)$ and all $1 \leq i \leq k$. Since $\langle \cdot, \cdot \rangle$ is non-degenerate and $N$ is the cardinality of the subspace generated by $\pi _2(X)$, it follows that there exists a subspace $V$ of dimension $n - \log _\ell N$ containing $B$ such that $a_i \in V$ for all $1 \leq i \leq k$. Furthermore, we know that the $a_i$ are not in $B$. This gives the inclusion

\[ \{a_i : 1 \leq i \leq k\} \subseteq V \setminus B \]

and therefore the bound $k \leq {\ell ^n}/{N} - |B|$. Therefore, we conclude that

\[ \frac{\ell^n}{N} - |B| \geq k = |\pi_2^{-1}(b_0) \cap X| \geq \frac{\ell^n - |B|}{N}, \]

which is a contradiction for $N > 1$.

8.1 Large variables

We split the character sum $N(X)$ in two subsums

\[ N(X) = N_{\text{small}}(X) + N_{\text{large}}(X), \]

where $N_{\text {small}}(X)$ is by definition the contribution to $N(X)$, where at most $\ell ^n - |B| - 1$ of the variables $w_{a, b}$ are large, and $N_{\text {large}}(X)$ is by definition the remaining contribution. We will make use of the following well-known lemma.

Lemma 8.1 Let $\kappa, C > 0$ be fixed real numbers. Then we have the bounds

\[ \sum_{\substack{1 \leq n \leq x \\ p \mid n \Rightarrow p \bmod M \in H}} \mu^2(n) \kappa^{\omega(n)} \ll_{\kappa, C} x (\log x)^{({|H| \cdot \kappa})/{\varphi(M)} - 1} \]

and

\[ \sum_{\substack{1 \leq n \leq x \\ p \mid n \Rightarrow p \bmod M \in H}} \frac{\mu^2(n) \kappa^{\omega(n)}}{n} \ll_{\kappa, C} (\log x)^{({|H| \cdot \kappa})/{\varphi(M)}} \]

for all $M \leq (\log x)^C$ and all subsets $H$ of $(\mathbb {Z}/M\mathbb {Z})^\ast$.

Proof. The first bound follows from Theorem 4.1 with

\[ f(n) := \mu^2(n) \kappa^{\omega(n)} \mathbf{1}_{p \mid n \Rightarrow p \bmod M \in H}, \quad \alpha := \frac{|H| \cdot \kappa}{\varphi(M)}, \quad k := \kappa, \quad Q := \exp\bigl((\log x)^{\min({1}/{4\kappa}, {1}/{2})}\bigr) \]

and $J := 1$, $\epsilon := 1/10$. Assumption (4.1) is guaranteed by the Siegel–Walfisz theorem. The second bound follows from partial summation and the first bound.

After choosing the constant $A_1 > 0$ to be sufficiently small, it follows from Lemma 8.1 that we have the bound

\[ N_{\text{small}}(X) = O\bigl(X (\log X)^{\alpha - 1 - \delta}\bigr) \]

for some $\delta > 0$. It is precisely at this step that we make fundamental use of the assumptions $|H_a| = |H_{a'}|$ for all $a, a' \in A - B$ and $[(\mathbb {Z}/M\mathbb {Z})^\ast : H_a] \leq C$.

8.2 Linked variables

We will now turn our attention to $N_{\text {large}}(X)$. We say that an integer $x$ is medium if

\[ |x| > (\log X)^{A_2}, \]

where $A_2 > 0$ is a large constant to be chosen later. We also split the sum $N_{\text {large}}(X)$ into two subsums, namely

\[ N_{\text{large}}(X) = N_{\text{linked}}(X) + N_{\text{main}}(X). \]

Here $N_{\text {linked}}(X)$ is the contribution to $N_{\text {large}}(X)$ for which the following assertions hold.

• – If $\ell > 2$, then there exists $(a_1, b_1), (a_2, b_2) \in \mathcal {I}$ such that the following conditions hold:

  1. (1) we have $\langle a_2, b_1 \rangle \neq 0$ or $\langle a_1, b_2 \rangle \neq 0$;

  2. (2) we have that $w_{a_1, b_1}$ and $w_{a_2, b_2}$ are both medium or we have that $|w_{a_1, b_1}|, |w_{a_2, b_2}| > 1$ and one of the $w_{a_i, b_i}$ is large.

• – If $\ell = 2$, then there exists $(a_1, b_1), (a_2, b_2) \in \mathcal {I}$ such that the following conditions hold:

  1. (1) we have $\langle a_2, b_1 \rangle + \langle a_1, b_2 \rangle = 1$;

  2. (2) we have that $w_{a_1, b_1}$ and $w_{a_2, b_2}$ are both medium or we have that $|w_{a_1, b_1}|, |w_{a_2, b_2}| > 2$ and one of the $w_{a_i, b_i}$ is large.

Furthermore, $N_{\text {main}}(X)$ is by definition the remaining contribution.

Our next goal is to bound $N_{\text {linked}}(X)$. Our two principal tools are the large sieve, as presented in Proposition 4.3, and the Siegel–Walfisz theorem over number fields as presented in the main theorem of [Reference GoldsteinGol70].

8.2.1 Equidistribution with the large sieve

We will now bound $N_{\text {linked}}(X)$. We will first suppose that there exist $(a_1, b_1), (a_2, b_2) \in \mathcal {I}$ with $w_{a_1, b_1}$ and $w_{a_2, b_2}$ both medium and satisfying the aforementioned conditions. So fix such a choice of $(a_1, b_1)$ and $(a_2, b_2)$. Define $\mathcal {I}' := \mathcal {I} - \{(a_1, b_1), (a_2, b_2)\}$. The corresponding contribution to $N_{\text {linked}}(X)$ is bounded by

(8.1)\begin{align} &\sum_{(v_a)_{a \in B - \{0\}}} \sum_{(w_{a, \bullet})} \sum_{(w_{a, b})_{(a, b) \in \mathcal{I'}}} \ell^{-m \sum_{(a, b) \in \mathcal{I}'} \tilde{\omega}(w_{a, b})}\nonumber\\ &\quad \times\Bigg|\sum_{\substack{w_{a_1, b_1}, w_{a_2, b_2} \nonumber\\ \Delta(|w_{a_1, b_1} w_{a_2, b_2}|) \leq {X}/({\prod_a \Delta(|v_a|) \prod_{(a, b) \in \mathcal{I}'} \Delta(|w_{a, b}|)})}} \hspace{-2cm} \alpha_{w_{a_1, b_1}} \beta_{w_{a_2, b_2}} \psi_{w_{a_2, b_2}, \ell, 1}^{\langle b_1, a_2 \rangle}(\text{Frob}_{w_{a_1, b_1}}\!) \psi_{w_{a_1, b_1}, \ell, 1}^{\langle b_2, a_1 \rangle}(\text{Frob}_{w_{a_2, b_2}}\!)\Bigg|, \end{align}

where $\alpha _{w_{a_1, b_1}}$ and $\beta _{w_{a_2, b_2}}$ are complex numbers of absolute value bounded by $1$ depending only on respectively $w_{a_1, b_1}$ and $w_{a_2, b_2}$ (and $w_{a, b}$ for $(a, b) \in \mathcal {I}'$). By changing the coefficients if necessary, we may assume from now on that $w_{a_1, b_1}$ and $w_{a_2, b_2}$ are coprime to $\ell$.

We now work towards our goal of applying Proposition 4.3. We take $K = \mathbb {Q}(\zeta _{\ell })$ and take the $M$ of Proposition 4.3 to be a sufficiently large power of $\ell$. Critically, the field $K$ depends only on the abelian group $A$. Write $(\cdot /\cdot )_{\mathbb {Q}(\zeta _{\ell }), \ell }$ for the $\ell$th power residue symbol in $\mathbb {Q}(\zeta _{\ell })$. We define

(8.2)\begin{equation} \gamma(w, z) := \biggl(\frac{N_{\mathbb{Q}(\zeta_{\ell})/\mathbb{Q}}(w)}{z}\biggr)_{\mathbb{Q}(\zeta_{\ell}), \ell}^{\langle b_1, a_2 \rangle} \biggl(\frac{N_{\mathbb{Q}(\zeta_{\ell})/\mathbb{Q}}(z)}{w}\biggr)_{\mathbb{Q}(\zeta_{\ell}), \ell}^{\langle b_2, a_1 \rangle}. \end{equation}

Note that if we define $\widetilde {\gamma }(w, z) := \gamma (z, w)$, then $\widetilde {\gamma }(w, z)$ is still of the form (8.2). Therefore, if we check properties (P1)–(P3) for all $\gamma (w, z)$, then these properties will also hold for $\widetilde {\gamma }(w, z)$. This allows us to circumvent the condition $X \leq Y$ in Proposition 4.3 by applying Proposition 4.3 to $\gamma (w, z)$ or $\widetilde {\gamma }(w, z)$ depending on whether $X$ or $Y$ is larger. Let us now verify (P1)–(P3).

Property (P1) is clear. We take $A_{\text {bad}}$ to be the set of squarefull integers. In particular, property (P3) is immediate for $C_2 = 1/2$ if we take $C_1$ sufficiently large. The first part of property (P2) follows from reciprocity and the periodicity of power residue symbols provided that we take the $M$ from Proposition 4.3 to be a sufficiently large power of $\ell$. It remains to prove the final part of property (P2).

To this end, fix some $w$. Then the application $z \mapsto \gamma (w, z)$ is a multiplicative character with period $M N_{\mathbb {Q}(\zeta _{\ell })/\mathbb {Q}}(w)$ by assumption. Therefore, by orthogonality of characters, it suffices to show that the character $z \mapsto \gamma (w, z)$ is not the principal character. By assumption, we have that $N_{\mathbb {Q}(\zeta _{\ell })/\mathbb {Q}}(w)$ is not squarefull. Therefore, we may take a prime ideal $\mathfrak {p}$ of $\mathbb {Q}(\zeta _{\ell })$ of degree one that divides $w$ such that none of the conjugates of $\mathfrak {p}$ divides $w$. By the Chinese remainder theorem, we may find an element $z \in \mathbb {Z}[\zeta _{\ell }]$ such that

\[ z \equiv 1 \bmod M, \quad z \equiv 1 \bmod \mathfrak{q} \text{ for all } \mathfrak{q} \mid N_{\mathbb{Q}(\zeta_{\ell})/\mathbb{Q}}(w) \text{ with } \mathfrak{q} \neq \mathfrak{p}, \quad z \equiv \alpha \bmod \mathfrak{p}, \]

where $\alpha$ is any generator of the cyclic group $(\mathbb {Z}[\zeta _{\ell }]/\mathfrak {p})^\ast \cong \mathbb {F}_p^\ast$. Using our assumptions on $\langle b_1, a_2 \rangle$ and $\langle b_2, a_1 \rangle$, it is not hard to show now that $z \mapsto \gamma (w, z)$ is not the principal character. We have finished checking that $\gamma (\cdot, \cdot )$ satisfies all the conditions of Proposition 4.3.

However, in (8.1), we are at the moment summing over rational integers and not over elements of $\mathbb {Z}[\zeta _{\ell }]$. Therefore, we aim to replace the sum over the integers by a sum taking place in $\mathbb {Q}(\zeta _{\ell })$. The following lemma is critical.

Lemma 8.2 Fix a root of unity $\zeta _\ell \in \mathbb {C}$ inducing an identification between $\mathbb {F}_\ell$ and $\langle \zeta _\ell \rangle$ by sending $1$ to $\zeta _\ell$. Let $p$ be a prime number such that $p \equiv 1 \bmod \ell$. Then there are canonical bijections

\begin{align*} \{\phi \in \text{Epi}(G_\mathbb{Q}, \mathbb{F}_\ell) : \phi \text{ ramified only at } p\} &\leftrightarrow \{\text{Dirichlet characters modulo } p \text{ of order } \ell\}\\ &\leftrightarrow \{\text{prime ideals of } \mathbb{Q}(\zeta_\ell) \text{ above } p\}. \end{align*}

Proof. Given a map $\phi \in \text {Epi}(G_\mathbb {Q}, \mathbb {F}_\ell )$ that is only ramified at $p$, class field theory implies that $\phi$ factors through $\mathrm {Gal}(\mathbb {Q}(\zeta _p)/\mathbb {Q})$, which is canonically isomorphic to $(\mathbb {Z}/p\mathbb {Z})^\ast$. Therefore, we may associate to $\phi$ an epimorphism from $(\mathbb {Z}/p\mathbb {Z})^\ast$ to $\mathbb {F}_\ell$. Using our identification between $\mathbb {F}_\ell$ and $\langle \zeta _\ell \rangle$, $\phi$ induces a map $(\mathbb {Z}/p\mathbb {Z})^\ast \rightarrow \mathbb {C}^\ast$. Extending $\phi$ in the usual way to $\mathbb {Z}$ gives a Dirichlet character modulo $p$ of order equal to $\ell$. This defines the first bijection.

For the second bijection, suppose that we are given an ideal $\mathfrak {p}$ of $\mathbb {Q}(\zeta _\ell )$ above $p$. Since $\mathfrak {p}$ has degree one, one readily verifies that the map

\[ n \mapsto \biggl(\frac{n}{\mathfrak{p}}\biggr)_{\mathbb{Q}(\zeta_\ell), \ell} \]

is a Dirichlet character. One directly shows that this is a bijection as well, completing the proof of the lemma.

By Lemma 8.2, the choice of $\psi _{p, \ell, 1}$ from § 2 is equivalent to choosing a prime ideal $\mathfrak {p}$ of $\mathbb {Q}(\zeta _{\ell })$ above $p$. Observe that $p$ indeed splits in $\mathbb {Q}(\zeta _{\ell })$, because we have the congruence $p \equiv 1 \bmod \ell$ if $\psi _{p, \ell, 1}$ is a homomorphism. We write this unique ideal $\mathfrak {p}$ as $\text {Pref}_\ell (p)$. We extend the definition of $\text {Pref}_\ell (p)$ multiplicatively to a function $\text {Pref}_\ell (x)$ for all squarefree integers $x$ supported in primes congruent to $1$ modulo $\ell$.

We fix a set of integral ideals $I_1, \ldots, I_t$ representing every ideal class of $\text {Cl}(\mathbb {Q}(\zeta _{\ell }))$. We now split (8.1) into $t^2$ sums, where we insert the additional condition that

\[ \text{Pref}_\ell(w_{a_1, b_1}\!) \sim I_{s_1}, \quad \text{Pref}_\ell(w_{a_2, b_2}\!) \sim I_{s_2}, \]

where we fixed some integers $1 \leq s_1, s_2 \leq t$. We introduce new variables $w$ and $z$ in the fundamental domain of $K$ such that

\[ (w) I_{s_1} = \text{Pref}_\ell(w_{a_1, b_1}\!), \quad (z) I_{s_2} = \text{Pref}_\ell(w_{a_2, b_2}\!). \]

We also define new coefficients

\[ \alpha_w = \begin{cases} \alpha_{N_{\mathbb{Q}(\zeta_{\ell})/\mathbb{Q}}(w I_{s_1}\!)} \cdot \biggl(\dfrac{N_{\mathbb{Q}(\zeta_{\ell})/\mathbb{Q}}(w)}{I_{s_2}}\biggr)_{\mathbb{Q}(\zeta_{\ell}), \ell}^{\langle b_2, a_1 \rangle} \cdot \biggl(\dfrac{N_{\mathbb{Q}(\zeta_{\ell})/\mathbb{Q}}(I_{s_2}\!)}{w}\biggr)_{\mathbb{Q}(\zeta_{\ell}), \ell}^{\langle b_1, a_2 \rangle}, & \text{if } (w) I_{s_1} \in \text{Im}(\text{Pref}_\ell),\\ 0, & \text{otherwise} \end{cases} \]

and

\[ \beta_z = \begin{cases} \beta_{N_{\mathbb{Q}(\zeta_{\ell})/\mathbb{Q}}(z I_{s_2})} \cdot \biggl(\dfrac{N_{\mathbb{Q}(\zeta_{\ell})/\mathbb{Q}}(z)}{I_{s_1}}\biggr)_{\mathbb{Q}(\zeta_{\ell}), \ell}^{\langle b_1, a_2 \rangle} \cdot \biggl(\frac{N_{\mathbb{Q}(\zeta_{\ell})/\mathbb{Q}}(I_{s_1}\!)}{z}\biggr)_{\mathbb{Q}(\zeta_{\ell}), \ell}^{\langle b_2, a_1 \rangle}, & \text{if } (z) I_{s_2} \in \text{Im}(\text{Pref}_\ell),\\ 0, & \text{otherwise}. \end{cases} \]

Then the inner sum of (8.1) becomes $t^2$ sums of the form

(8.3)\begin{equation} \frac{1}{(\ell - 1)^2} \bigg|\sum_w \sum_z \alpha_w \beta_z \gamma(w, z) \bigg|, \end{equation}

where we divide by ${1}/{(\ell - 1)^2}$, because the fundamental domain of $\mathbb {Q}(\zeta _{\ell })$ contains $\ell - 1 = \varphi (\ell )$ generators for each principal ideal.

Finally, there is one more barrier to overcome before we are ready to apply Proposition 4.3. The implicit condition in (8.3) is that

\[ N_{\mathbb{Q}(\zeta_{\ell})/\mathbb{Q}}(wz) \leq B \]

for some bound $B > 0$, while Proposition 4.3 applies only to box shapes given by $N_{\mathbb {Q}(\zeta _{\ell })/\mathbb {Q}}(w) \leq W$ and $N_{\mathbb {Q}(\zeta _{\ell })/\mathbb {Q}}(z) \leq Z$. To this end, we split $w$ and $z$ into intervals of the form

\[ W \leq N_{\mathbb{Q}(\zeta_{\ell})/\mathbb{Q}}(w) \leq W \biggl(1 + \frac{1}{(\log X)^{A_3}}\biggr), \quad Z \leq N_{\mathbb{Q}(\zeta_{\ell})/\mathbb{Q}}(z) \leq Z \biggl(1 + \frac{1}{(\log X)^{A_3}}\biggr). \]

This does not cover the entire region, but for sufficiently large $A_3$ one may bound the resulting leftover trivially. Inserting the bound of Proposition 4.3 for each such sum into (8.1) and summing trivially shows that $N_{\text {linked}}(X)$ ends up in the error term, in the case where $w_{a_1, b_1}$ and $w_{a_2, b_2}$ are both medium, upon choosing $A_2$ sufficiently large in terms of $A_3$.