3.1. Properties of ideals

For any fractional ideal $\mathfrak {a}$ in K, one defines the dual ideal

$$ \begin{align*}\hat{\mathfrak{a}} = \{\alpha\in K: \operatorname{\mathrm{Tr}}_{K/\mathbb{Q}}(\alpha x)\in \mathbb{Z} \mbox{ for all } x\in \mathfrak{a}\}. \end{align*} $$

In particular, $\hat {\mathfrak {a}}= \mathfrak {a}^{-1}\mathfrak {d}^{-1}$ , where $ \mathfrak {d} = \{\alpha \in K: \alpha \hat {\mathfrak {o}}\subseteq \mathfrak {o}\} $ denotes the different ideal of K and is itself an integral ideal. One notes that $\hat {\mathfrak {o}}=\mathfrak {d}^{-1}$ . Furthermore, we have $\hat {\mathfrak {a}}\subseteq \hat {\mathfrak {b}}$ if and only if $\mathfrak {b}\subseteq \mathfrak {a}$ . An additional integral ideal featuring in our work is the denominator ideal

$$ \begin{align*}\mathfrak{a}_\gamma=\{ \alpha\in \mathfrak{o}: \alpha \gamma\in \mathfrak{o}\}, \end{align*} $$

associated to any $\gamma \in K$ . Recall that $\operatorname {\mathrm {N}} \mathfrak {a}=|\mathfrak {o}/\mathfrak {a}|$ is the ideal norm of any integral ideal $\mathfrak {a}$ . One important property of the ideal norm is that $\operatorname {\mathrm {N}}\mathfrak {a}^{\tau }=\operatorname {\mathrm {N}} \mathfrak {a}$ for any $\tau \in \operatorname {\mathrm {Gal}}(K/\mathbb {Q}).$ (This follows from the isomorphism $\mathfrak {o}/\mathfrak {a}\to \mathfrak {o}/\mathfrak {a}^{\tau }$ given by $\alpha \mapsto \alpha ^{\tau }$ .) Furthermore, we have $\operatorname {\mathrm {N}}\mathfrak {a}\in \mathfrak {a}$ for any integral ideal $\mathfrak {a}$ .

We will write $(\mathfrak {a},\mathfrak {b})=\mathfrak {a}+\mathfrak {b}$ for the greatest common divisor of two integral ideals $\mathfrak {a},\mathfrak {b}\subset \mathfrak {o}$ . When these ideals are coprime, meaning that $\mathfrak {a}+\mathfrak {b}=\mathfrak {o}$ , we shall adopt the abuse of notation $(\mathfrak {a},\mathfrak {b})=1$ . We close this section by recording the following basic result.

We shall also require a version of the Chinese remainder theorem, as in [Reference Skinner12, Lemma 3].

Lemma 3.2. Suppose that $\mathfrak {a},\mathfrak {a}_1,\mathfrak {a}_2$ are integral ideals such that $\mathfrak {a}=\mathfrak {a}_1\mathfrak {a}_2$ , with $\mathfrak {a}_1$ and $\mathfrak {a}_2$ coprime. Let $\alpha _1, \alpha _2\in \mathfrak {o}$ satisfy $\mathrm {ord}_{\mathfrak {p}}(\alpha _1)=\mathrm {ord}_{\mathfrak {p}}(\mathfrak {a}_1)$ and $\mathrm {ord}_{\mathfrak {p}}(\alpha _2)=\mathrm {ord}_{\mathfrak {p}}(\mathfrak {a}_2)$ , for all $\mathfrak {p}\mid \mathfrak {a}$ . Then

$$ \begin{align*}\mathfrak{o}/\mathfrak{a}=\{\alpha_1 \mu +\alpha_2 \beta: \beta\in \mathfrak{o}/\mathfrak{a}_1, ~\mu\in \mathfrak{o}/\mathfrak{a}_2\}. \end{align*} $$

3.2. Construction of primitive characters

Let $\psi (\cdot )=\exp (2\pi i \operatorname {\mathrm {Tr}}_{K/\mathbb {Q}}(\cdot ))$ be a character on K. The following result gives a way to construct primitive characters $\mathfrak {o}/\mathfrak {b}\to \mathbb {C}$ .

We proceed to define a particularly convenient additive character modulo $\mathfrak {b}$ . Associated to any nonzero integral ideal $\mathfrak {b}$ is the subset $\mathfrak {F}(\mathfrak {b})\subset K$ given by

$$ \begin{align*}\mathfrak{F}(\mathfrak{b})= \hspace{-0.05cm} \left\{\frac{g}{\alpha}\in K: \begin{array}{l} \exists ~\text{prime ideal } \mathfrak{p}_1 \text{ with } \operatorname{\mathrm{N}} \mathfrak{p}_1 \ll \operatorname{\mathrm{N}} \mathfrak{b} \text{ s.t.}\\ \qquad\text{(i)}. \quad (\alpha)=\mathfrak{b}\mathfrak{d}\mathfrak{p}_1\\ \qquad\text{(ii)}. \quad g\in \mathfrak{p}_1\cap\mathbb{Z} \text{ with } ((g),\ \mathfrak{b}^{\tau}\mathfrak{d})=1 \ \forall~ \tau \in \operatorname{\mathrm{Gal}}(K/\mathbb{Q})\\ \qquad\text{(iii)}. \quad \exists \ \mathfrak{e}\mid \mathfrak{d} \text{ s.t. } (\mathfrak{d}/\mathfrak{e},\mathfrak{b})=1 \text{ and } \mathfrak{a}_{g/\alpha}=\mathfrak{b}\mathfrak{e} \end{array} \hspace{-0.2cm} \right\}. \end{align*} $$

Note that condition (i) implies that $\alpha \in \mathfrak {b}\mathfrak {d}$ and condition (ii) implies that $\mathfrak {p}_1\nmid \mathfrak {b}^{\tau }\mathfrak {d}$ for any $\tau \in \operatorname {\mathrm {Gal}}(K/\mathbb {Q})$ . We may now record a variant of [Reference Browning and Vishe2, Lemma 2.3], which shows that $\mathfrak {F}(\mathfrak {b})\neq \emptyset $ for any choice of $\mathfrak {b}$ .

Proof. We consider the integral ideal $\mathfrak {c}=\mathfrak {b}\mathfrak {d}$ . Observe that $\mathfrak {d}^{\tau }=\mathfrak {d}$ for all $\tau \in \operatorname {\mathrm {Gal}}(K/\mathbb {Q})$ since $\mathfrak {d}=\hat {\mathfrak {o}}^{-1}$ and the trace is invariant under the action of the Galois group. Taking $\varepsilon =1$ in Lemma 3.1(ii), we can find $\alpha \in \mathfrak {c}$ and a prime ideal $\mathfrak {p}_1$ coprime to $\mathfrak {c}^{\tau }=\mathfrak {b}^{\tau }\mathfrak {d}$ , for every $\tau \in \operatorname {\mathrm {Gal}}(K/\mathbb {Q})$ , with $\operatorname {\mathrm {N}}\mathfrak {p}_1 \ll \operatorname {\mathrm {N}} \mathfrak {b}$ and such that $(\alpha )=\mathfrak {c}\mathfrak {p}_1$ . It follows from Lemma 3.1(i) that there exists $\nu \in \mathfrak {p}_1$ such that $((\nu ), \mathfrak {c}^{\tau })=1$ for any $\tau \in \operatorname {\mathrm {Gal}}(K/\mathbb {Q})$ . But this implies that $g=N_{K/\mathbb {Q}}(\nu )$ is coprime to $\mathfrak {c}^{\tau }$ , for any $\tau \in \operatorname {\mathrm {Gal}}(K/\mathbb {Q})$ , with $g\in \mathfrak {p}_1$ . We will show that $\mathfrak {c}=\mathfrak {a}_\gamma $ with $\gamma =g/\alpha $ , after which an application of Lemma 3.3 with $\mathfrak {e}=\mathfrak {d}$ will complete the proof. To check the claim we note that

$$ \begin{align*}\beta\in \mathfrak{a}_\gamma \Leftrightarrow \gamma\beta\in \mathfrak{o} \Leftrightarrow (g\beta)\subset (\alpha)\Leftrightarrow (\alpha)=\mathfrak{b}\mathfrak{d}\mathfrak{p}_1\mid (g\beta) \Leftrightarrow \beta\in \mathfrak{b}\mathfrak{d} \end{align*} $$

since $\mathfrak {p}_1\mid (g)$ and $\mathfrak {b}\mathfrak {d}$ is coprime with $(g)$ .

3.3. The G-invariant ideal and an important $\mathbb {Z}$ -module

Let F be a generalised quadratic form, as in Definition 1.1. Let $G=G_F\subset \operatorname {\mathrm {Gal}}(K/\mathbb {Q})$ be the subset of automorphisms $\tau \in \operatorname {\mathrm {Gal}}(K/\mathbb {Q})$ that actually appear in F. Note that $G=\{\mathrm {id}\}$ if and only if F is a standard quadratic form. For any integral ideal $\mathfrak {b}$ , we define the G-invariant ideal to be

(3.1) $$ \begin{align} {{}^{{G}}\mathfrak{b}}=\bigcap_{\tau\in G} \mathfrak{b}^{\tau^{-1}}. \end{align} $$

This is the least common multiple of the ideals $\mathfrak {b}^{\tau ^{-1}}$ for $\tau \in G$ .

Next, associated to our generalised quadratic form F is a generalised bilinear form

(3.2) $$ \begin{align} B(X_1,\dots,X_n;Y_1,\dots,Y_n)= \sum_{1\leqslant i,j\leqslant n} \sum_{\tau,\tau'\in \operatorname{\mathrm{Gal}}(K/\mathbb{Q})}c_{i,j,\tau,\tau'} X_i^{\tau} Y_j^{\tau'}. \end{align} $$

This defines a map $K^n\times K^n\to K$ , with

$$ \begin{align*}B(\mathbf{x};\mathbf{u}+\mathbf{v})=B(\mathbf{x};\mathbf{u})+B(\mathbf{x};\mathbf{v}) \quad \text{and}\quad B(\mathbf{u}+\mathbf{v};\mathbf{y})=B(\mathbf{u};\mathbf{y})+B(\mathbf{v};\mathbf{y}), \end{align*} $$

for any vectors $\mathbf {x},\mathbf {y},\mathbf {u},\mathbf {v}\in K^n$ . (However, this fails to be a bilinear form on $K^n$ since $B(\lambda \mathbf {x};\mathbf {y})$ , $B(\mathbf {x};\lambda \mathbf {y})$ and $\lambda B( \mathbf {x};\mathbf {y})$ needn’t be equal for $\lambda \in K$ .)

For any ideal $\mathfrak {b}\subset \mathfrak {o}$ , let

$$ \begin{align*}{\mathcal H}_{\mathfrak{b}}=\left\{ {\mathbf h} \in {\mathfrak o}^n: F({\mathbf a} + {\mathbf h}) \equiv F({\mathbf a}) \mbox{ mod } {\mathfrak{b}} \text{ for all } {\mathbf a} \in {\mathfrak o}^n\right\}. \end{align*} $$

This is an additive group, and it is clear that ${{}^{{G}}\mathfrak {b}}^n\subset {\mathcal H}_{\mathfrak {b}} \subset {\mathfrak o}^n$ , where ${{}^{{G}}\mathfrak {b}}$ is the G-invariant ideal defined in Equation (3.1). By testing the hypothesis with ${\mathbf a} \equiv \mathbf {0} \mbox { mod } {{}^{{G}}\mathfrak {b}}$ , we have $F({\mathbf h}) \equiv 0 \mbox { mod } {\mathfrak {b}}$ for any ${\mathbf h}\in {\mathcal H}_{\mathfrak {b}}$ . Hence,

(3.3) $$ \begin{align} {\mathcal H}_{\mathfrak{b}}=\left\{ {\mathbf h} \in {\mathfrak o}^n: 2B({\mathbf a};\mathbf{h})\in \mathfrak{b} \text{ for all } {\mathbf a} \in {\mathfrak o}^n\right\}. \end{align} $$

We claim that ${\mathcal H}_{\mathfrak {b}}$ has the structure of a finitely generated ${\mathbb Z}$ -module. To see this, let ${\mathbf e}_i$ be the ith unit vector, for $1\leqslant i\leqslant n$ . Observe that $(\operatorname {\mathrm {N}} {\mathfrak b}) {\omega }_j{\mathbf e}_i\in {\mathcal H}_{\mathfrak b}$ for all $1\leqslant i\leqslant n$ and $1\leqslant j\leqslant d$ . Hence, the image of ${\mathcal H}_{\mathfrak b}$ under the isomorphism ${\mathfrak o}^n \cong {\mathbb Z}^{nd}$ is a lattice of full rank and, thus, finitely generated as a ${\mathbb Z}$ -module.

The set ${\mathcal H}_{\mathfrak {b}}$ will emerge naturally in our analysis of certain key exponential sums, and it will be important to have an estimate for its index in $\mathfrak {o}^n$ . In the special case (1.3), it will be easier to calculate ${\mathcal H}_{\mathfrak b}$ directly, but for now we content ourselves with proving a general bound. In the following lemma, we consider the coefficient matrix $(c_{i,j,\tau ,\tau '})_{(i,\tau )\times (j,\tau ')}$ of a generalised quadratic form as a $nd\times nd$ matrix.

Lemma 3.5. There is a constant $C_1>0$ , depending only on F, such that for all $\mathfrak {b}$ we have

$$ \begin{align*}|{\mathfrak o}^n/{\mathcal H}_{\mathfrak b}|\leqslant C_1 (\operatorname{\mathrm{N}} {\mathfrak b})^{\operatorname{\mathrm{rank}}(c_{i,j,\tau,\tau'})}.\end{align*} $$

Moreover, there is an integral ideal $\mathfrak {d}_1$ such that one can take $C_1=1$ for all ideals $\mathfrak {b}$ with $({\mathfrak b},\mathfrak {d}_1)=1$ .

Proof. Let ${\Delta }= \operatorname {\mathrm {rank}} (c_{i,j,\tau ,\tau '})_{(i,\tau )\times (j,\tau ')}$ . Let ${\mathcal S}\subset \{1,\ldots , n\}\times \operatorname {\mathrm {Gal}}(K/\mathbb {Q})$ be a subset of indices such that the vectors $(c_{i,j,\tau ,\tau '})_{(j,\tau ')}$ , $(i,\tau )\in {\mathcal S}$ , are linearly independent and $|{\mathcal S}|$ is maximal. Then, for any $(k,\sigma )\in \{1,\ldots , n\}\times \operatorname {\mathrm {Gal}}(K/\mathbb {Q})$ there are numbers $a_{i,\tau }^{(k,\sigma )}$ (for $(i,\tau )\in {\mathcal S}$ ) such that

$$ \begin{align*}c_{k,j,\sigma,\tau'}= \sum_{(i,\tau)\in {\mathcal S}} a_{i,\tau}^{(k,\sigma)}c_{i,j,\tau,\tau'},\end{align*} $$

for all $1\leqslant j\leqslant n$ and $\tau '\in \operatorname {\mathrm {Gal}}(K/\mathbb {Q})$ . Let ${\alpha }\in {\mathfrak o}$ such that ${\alpha } a_{i,\tau }^{(k,\sigma )}\in {\mathfrak o}$ for all $(i,\tau )\in {\mathcal S}$ and $(k,\sigma )\in \{1,\ldots , n\}\times \operatorname {\mathrm {Gal}}(K/\mathbb {Q})$ . Now, set

$$ \begin{align*}{\mathcal H}_{\mathfrak b}' = \left\{{\mathbf h}\in {\mathfrak o}^n: \sum_{1\leqslant j\leqslant n} \sum_{\tau'\in \operatorname{\mathrm{Gal}}(K/\mathbb{Q})} c_{i,j,\tau,\tau'}h_j^{\tau'}\in ({\alpha}){\mathfrak b},\, \forall (i,\tau)\in {\mathcal S}\right\}.\end{align*} $$

Observe that ${\mathcal H}_{\mathfrak b}^{\prime }\subset {\mathcal H}_{\mathfrak b}$ . Moreover, if ${\mathfrak b}$ and $({\alpha })$ are coprime, then the ideal $({\alpha })$ may be omitted in the definition of ${\mathcal H}_{\mathfrak b}^{\prime }$ .

Finally, we observe that there is an injection

$$ \begin{align*} \begin{aligned} \psi: {\mathfrak o}^n/{\mathcal H}_{\mathfrak b}^{\prime} &\rightarrow ({\mathfrak o}/{\alpha}{\mathfrak b})^{{\Delta}},\quad [{\mathbf h}]\mapsto \left( \sum_{1\leqslant j\leqslant n}\sum_{\tau'\in \operatorname{\mathrm{Gal}}(K/\mathbb{Q})}c_{i,j,\tau,\tau'}h_j^{\tau'}\right)_{(i,\tau)\in {\mathcal S}}. \end{aligned} \end{align*} $$

Hence, $|{\mathfrak o}^n/{\mathcal H}_{\mathfrak b}^{\prime }|\leqslant (\operatorname {\mathrm {N}} ({\alpha }{\mathfrak b}))^{{\Delta }},$ which suffices since ${\mathcal H}_{\mathfrak b}^{\prime }\subset {\mathcal H}_{\mathfrak b}$ .

We now wish to provide an alternative upper bound involving ${\mathcal H}_{\mathfrak b}$ under a suitable assumption on the generalised quadratic form.

Definition 3.6. We say that $F(X_1,\dots ,X_n)$ is admissible if there exist vectors

$$ \begin{align*}{\mathbf v}_1,\ldots, {\mathbf v}_n\in K^n \end{align*} $$

such that $B({\mathbf v}_i;{\mathbf h})=0$ for all $1\leqslant i\leqslant n$ if and only if ${\mathbf h}=\mathbf {0}$ .

In this language, a standard quadratic form is admissible if and only if it is nonsingular. We may now prove the following result.

Lemma 3.7. Assume that F is admissible. Then there exists a constant $C_2>0$ , depending only on F, such that

$$ \begin{align*}|{\mathcal H}_{\mathfrak b}/{{}^{{G}}\mathfrak{b}}^n|\leqslant C_2\frac{(\operatorname{\mathrm{N}}{{}^{{G}}\mathfrak{b}})^n}{(\operatorname{\mathrm{N}}{\mathfrak b})^n} .\end{align*} $$

Moreover, there exists an integral ideal $\mathfrak {d}_2$ such that one can take $C_2=1$ for all ideals ${\mathfrak b}$ with $({\mathfrak b},\mathfrak {d}_2)=1$ .

We can use this result to get information about the index of ${\mathcal H}_{\mathfrak b}$ in $\mathfrak {o}^n$ via the identity

(3.4) $$ \begin{align} |\mathfrak{o}^n/{\mathcal H}_{\mathfrak b}||{\mathcal H}_{\mathfrak b}/{{}^{{G}}\mathfrak{b}}^n| =|\mathfrak{o}^n/{{}^{{G}}\mathfrak{b}}^n|=(\operatorname{\mathrm{N}} {{}^{{G}}\mathfrak{b}})^n. \end{align} $$

Lemma 3.7 is of the expected magnitude, which we can see by considering the case of the standard diagonal quadratic form $F(\boldsymbol {X})=\sum _{i=1}^n c_iX_i^2$ , for example, with nonzero $c_1,\dots ,c_n\in \mathfrak {o}$ . In this case, $G=\{\mathrm {id}\}$ and ${{}^{{G}}\mathfrak {b}}=\mathfrak {b}$ . It therefore follows that $|{\mathcal H}_{\mathfrak {b}}/{{}^{{G}}\mathfrak {b}}^n|\ll 1$ since ${\mathcal H}_{\mathfrak {b}}=(2c_1)^{-1}\mathfrak {b}\times \cdots \times (2c_n)^{-1}\mathfrak {b}$ .

Proof of Lemma 3.7.

Let ${\mathbf v}_1,\dots , {\mathbf v}_n$ be a set of vectors as in Definition 3.6. By scaling these vectors with a rational integer, we may assume that ${\mathbf v}_i\in {\mathfrak o}^n$ for all $1\leqslant i\leqslant n$ . We define the auxiliary set

$$ \begin{align*}\widetilde{{\mathcal H}}_{\mathfrak b}=\{{\mathbf h}\in {\mathfrak o}^n: 2B({\mathbf v}_i;\ {\mathbf h})\in {\mathfrak b},\ \forall 1\leqslant i\leqslant n\},\end{align*} $$

and observe that ${\mathcal H}_{\mathfrak b}\subset \widetilde {{\mathcal H}}_{\mathfrak b}$ . Next, consider the map

$$ \begin{align*} \begin{aligned} \varphi: {\mathfrak o}^n &\rightarrow {\mathfrak o}^n , \quad {\mathbf h} \mapsto (2B({\mathbf v}_i;{\mathbf h}))_{1\leqslant i\leqslant n}, \end{aligned} \end{align*} $$

which is injective by the definition of admissibility in Definition 3.6. Let ${\Gamma }$ be the image of ${\mathfrak o}^n$ under the map $\varphi $ . Then $\varphi $ induces an isomorphism

$$ \begin{align*}{\mathfrak o}^n/\widetilde{{\mathcal H}}_{\mathfrak b} \cong {\Gamma}/({\mathfrak b}^n\cap {\Gamma}).\end{align*} $$

Note that ${\Gamma }$ only depends on $B(\boldsymbol {X};\boldsymbol {Y})$ and vectors ${\mathbf v}_1,\dots ,{\mathbf v}_n$ and hence can be taken to be independent of the ideal ${\mathfrak b}$ . As in Equation (3.4), we therefore obtain

$$ \begin{align*}|{\mathcal H}_{\mathfrak b}/{{}^{{G}}\mathfrak{b}}^n|\leqslant |\widetilde{{\mathcal H}_{\mathfrak b}}/{{}^{{G}}\mathfrak{b}}^n| = \frac{|{\mathfrak o}^n/{{}^{{G}}\mathfrak{b}}^n|}{|{\mathfrak o}^n/\widetilde{{\mathcal H}}_{\mathfrak b}|} = \frac{(\operatorname{\mathrm{N}}{{}^{{G}}\mathfrak{b}})^n}{|{\Gamma} /({\mathfrak b}^n\cap {\Gamma})|} \leqslant C_{\Gamma} \frac{(\operatorname{\mathrm{N}}{{}^{{G}}\mathfrak{b}})^n}{(\operatorname{\mathrm{N}}{\mathfrak b})^n},\end{align*} $$

where $C_{\Gamma }$ is a constant only depending on ${\Gamma }$ . Moreover, there is an ideal $\mathfrak {d}_2$ such that $|{\Gamma }/({\mathfrak b}^n\cap {\Gamma })|=(\operatorname {\mathrm {N}}{\mathfrak b})^n$ whenever $(\mathfrak {d}_2,{\mathfrak b})=1$ . This completes the proof of the lemma.

4.1. The embedded system

Let $F(X_1,\dots ,X_n)$ be a generalised quadratic form, and let $\{\omega _1,\dots ,\omega _d\}$ be a $\mathbb {Z}$ -basis for $\mathfrak {o}$ . We have seen in Equation (1.2) how there is a descended system $\{Q_1,\dots ,Q_d\}$ of quadratic forms, that is associated to F via

$$ \begin{align*}F(X_1,\dots,X_n)=\sum_{1\leqslant i\leqslant d}{\omega_i}Q_i(\mathbf{U}_1,\dots,\mathbf{U}_d), \end{align*} $$

with variables $\mathbf {U}_l=(U_{l1},\dots ,U_{ln})$ for $1\leqslant l\leqslant d$ .

We will need to be able to relate the descended system to the embedded system, which amounts to how $F(\mathbf {x})$ embeds in V for given $\mathbf {x}\in K^n$ . We extend $F:K^n\to K$ to get a map $V^n\to V$ , through the identification of K with V. Associated to $\mathbf {x}$ is the vector $ (\mathbf {x}^{(1)}, \dots , \mathbf {x}^{(d)}), $ with $\mathbf {x}^{(l)}\in \mathbb {R}^n$ for $1\leqslant l\leqslant d$ . Let $l\in \{1,\dots ,d\}$ . To any $\tau \in \operatorname {\mathrm {Gal}}(K/\mathbb {Q})$ may be associated a unique integer $l_\tau \in \{1,\dots ,d\}$ such that Equation (1.4) holds. Then we define

(4.2) $$ \begin{align} F^{(l)}(\mathbf{x}^{(1)},\dots,\mathbf{x}^{(d)})= \sum_{1\leqslant i,j\leqslant n} \sum_{\tau,\tau'\in \operatorname{\mathrm{Gal}}(K/\mathbb{Q})}c_{i,j,\tau,\tau'}^{(l)}x_i^{(l_{\tau^{-1}})} x_j^{(l_{\tau^{\prime -1}})}, \end{align} $$

where $c_{i,j,\tau ,\tau }^{\prime (l)}=\rho _l(c_{i,j,\tau ,\tau '})\in \mathbb {R}$ . With this notation, we have

$$ \begin{align*}\rho_l(F(\mathbf{x}))=F^{(l)}(\mathbf{x}^{(1)},\dots,\mathbf{x}^{(d)}). \end{align*} $$

Thus, $\rho _l(F(\mathbf {x}))$ is a real quadratic form in the $dn$ variables $\mathbf {x}^{(1)},\dots ,\mathbf {x}^{(d)}$ . We call $\{F^{(1)},\dots ,F^{(d)}\}$ the embedded system. In particular, it is clear that $N_{K/\mathbb {Q}}(F(\mathbf {x}))=\operatorname {\mathrm {Nm}}(F(\mathbf {x}))$ and

(4.3) $$ \begin{align} \operatorname{\mathrm{Tr}}(vF(\mathbf{x}))= \sum_{1\leqslant l \leqslant d} v_l \rho_l\left(F(\mathbf{x})\right), \end{align} $$

for any $v=(v_1,\dots ,v_d)\in V$ and $\mathbf {x}\in V^n$ , identities that we shall often make use of in our analysis of the exponential integrals in §4.5.

Note that if F is a standard quadratic form, then $\rho _l(F(\mathbf {x}))=F^{(l)}(\mathbf {x}^{(l)})$ for $1\leqslant l\leqslant d$ . One positive effect of this is that the relevant oscillatory integrals factorise into a product of d integrals, one for each embedding. The situation is much more complicated for generalised quadratic forms since there is usually no such factorisation.

Let $\mathbf {A}=(\omega _j^{(i)})_{1\leqslant i,j\leqslant d}$ , where $\omega _j^{(i)}=\rho _i(\omega _j)$ . Then $(\det \mathbf {A})^2=D_K$ . Moreover, on recalling that $\mathbf {x}=\omega _1\mathbf {u}_1+\cdots +\omega _d\mathbf {u}_d$ , we have

(4.4) $$ \begin{align} \begin{pmatrix} \mathbf{x}^{(1)}\\ \vdots \\ \mathbf{x}^{(d)} \end{pmatrix} = \mathbf{W} \begin{pmatrix} \mathbf{u}_1\\ \vdots \\ \mathbf{u}_d \end{pmatrix}, \end{align} $$

where $\mathbf {W}$ is the $dn\times dn$ block matrix

(4.5)

Switching appropriate rows and columns takes $\mathbf {W}$ to $\mathrm {Diag}(\mathbf {A},\dots ,\mathbf {A})$ , whence $\det \mathbf {W}=(\det \mathbf {A})^{n}=D_K^{n/2}$ . In particular, it follows that

$$ \begin{align*}F^{(l)}(\mathbf{x}^{(1)},\dots,\mathbf{x}^{(d)}) =\sum_{1\leqslant i\leqslant d} \omega_i^{(l)} Q_i(\mathbf{u}_1\dots,\mathbf{u}_d), \end{align*} $$

for any $1\leqslant l\leqslant d$ , under the transformation (4.4).

4.2. Construction of the weight W

We assume that the descended system is of codimension d and has a nonsingular real point. This means that there exists $\underline {\boldsymbol {\xi }}=(\boldsymbol {\xi }_1,\dots ,\boldsymbol {\xi }_d)\in \mathbb {R}^{dn}$ such that $J_{Q_1,\dots , Q_d}(\underline {\boldsymbol {\xi }})$ has rank d, where

$$ \begin{align*}J_{Q_1,\dots,Q_d}= \left(\frac{\partial }{\partial X_j^{(k)}}Q_l\right)_{\substack{1\leqslant l\leqslant d\\ 1\leqslant k\leqslant d, 1\leqslant j\leqslant n} } \end{align*} $$

is the associated $d\times dn$ Jacobian matrix. Define the smooth weight function

$$ \begin{align*}w(x)=\begin{cases} e^{-1/(1-x^2)} &\text{ if } |x|<1,\\ 0&\text{ if } |x|\geqslant 1, \end{cases} \end{align*} $$

and let $\delta>0$ be a small parameter. In this paper, we shall work with the weight function $W:V^n\to \mathbb {R}_{\geqslant 0}$ , which is given by

$$ \begin{align*}W(\mathbf{x})=w(\delta^{-1}|\mathbf{W}^{-1}\mathbf{x}-\underline{\boldsymbol{\xi}}|), \end{align*} $$

where $\mathbf {x}$ is identified with $(\mathbf {x}^{(1)},\dots ,\mathbf {x}^{(d)})$ , and where $\mathbf {W}$ is the matrix in Equation (4.5). It is clear that W is infinitely differentiable and that it is supported on the region $|\mathbf {W}^{-1}\mathbf {x}-\underline {\boldsymbol {\xi }}| \leqslant \delta $ . Ultimately, we will want to work with a value of $\delta $ that is sufficiently small but which still satisfies $1\ll \delta \leqslant 1$ for an absolute implied constant.

4.3. Poisson summation

It follows from Equation (4.1) that

(4.6) $$ \begin{align} \begin{aligned} &N_{W}(F,N;P) \\ &\quad =\frac{c_{Q}}{Q^{2d}}\sum_{\mathfrak{b}}\ \ {\sideset{}{{}^{*}}\sum_{\sigma{\,(\operatorname{\mathrm{mod}}{{\mathfrak{b}}})}}} \sum_{\mathbf{x}\in \mathfrak{o}^{n}} \sigma(F(\mathbf{x})-N)W(\mathbf{x}/P)h\left(\frac{\operatorname{\mathrm{N}}\mathfrak{b}}{Q^d} ,\frac{|N_{K/\mathbb{Q}}(F(\mathbf{x})-N)|}{Q^{2d}}\right), \end{aligned} \end{align} $$

for any $Q\geqslant 1$ . Here, the constant $c_Q$ satisfies $ c_Q=1+O_A(Q^{-A}), $ for any $A>0$ . Furthermore, we have $h(x,y)\ll x^{-1}$ for all y and $h(x,y)\neq 0$ only if $x\leqslant \max \{1,2|y|\} $ .

In our work, we will take $Q=P$ and we henceforth follow the convention that the implied constant in any estimate involving W is allowed to depend implicitly on the parameters that enter into its definition of $\mathcal {W}_n(V)$ in [Reference Browning and Vishe2, §2.2]. Likewise, the integer N and the number field K are considered fixed once and for all so that all implied constants are allowed to depend implicitly on N and K. In view of the fact that $h(x,y)\neq 0$ only if $x\leqslant \max (1,2|y|) $ , it is clear that the sum over $\mathfrak {b} $ is restricted to $\operatorname {\mathrm {N}} \mathfrak {b}\ll Q^{d}=P^d $ .

If F were a standard quadratic form over $\mathfrak {o}$ , we would proceed by breaking the sum over $\mathbf {x}$ into residue classes modulo $\mathfrak {b}$ before executing an application of Poisson summation. This would ultimately lead to an expression of the form [Reference Browning and Vishe2, Thm. 5.1]. For generalised quadratic forms F, this route is not directly accessible, since for given $\mathbf {a},\mathbf {h}\in \mathfrak {o}^n$ and any primitive character $\sigma $ modulo $\mathfrak {b}$ , one may have $\sigma (F(\mathbf {a}+\mathbf {h}))\neq \sigma (F(\mathbf {a}))$ even when $\mathbf {h}\in \mathfrak {b}^n$ . In this way, we see that a special role will be played by the set ${\mathcal H}_{\mathfrak {b}}$ that was introduced in §3.3.

Lemma 4.1. We have

$$ \begin{align*}N_{W}(F,N;P)= \frac{c_{P}P^{(n-2)d}}{D_{K}^{n/2}} \sum_{\operatorname{\mathrm{N}}\mathfrak{b}\ll P^d} \sum_{\substack{\mathbf{m}\in \widehat{{{}^{{G}}\mathfrak{b}} }^n}} (\operatorname{\mathrm{N}} {{}^{{G}}\mathfrak{b}} )^{-n} S_{\mathfrak{b}} (N;\mathbf{m}) I_{\mathfrak{b}} (N/P^2;P\mathbf{m}), \end{align*} $$

where the sum over $\mathfrak {b}$ is over nonzero integral ideals and

$$ \begin{align*} S_{\mathfrak{b}} (N;\mathbf{m}) &= {\sideset{}{{}^{*}}\sum_{\sigma{\,(\operatorname{\mathrm{mod}}{\mathfrak{b}})}}} \sum_{\mathbf{a}{\,(\operatorname{\mathrm{mod}}{{}^{G}\mathfrak{b}})}} \sigma(F(\mathbf{a})-N)\psi(\mathbf{m}. \mathbf{a}),\\ I_{\mathfrak{b}} (t;\mathbf{k}) &= \int_{ V^{n}}W(\mathbf{x})h\left(\frac{\operatorname{\mathrm{N}} \mathfrak{b}}{P^d} ,|\operatorname{\mathrm{Nm}}(F(\mathbf{x})-t)|\right)\psi\left(-\mathbf{k}.\mathbf{x} \right)\mathrm{d}\mathbf{x}. \end{align*} $$

Proof. Our approach is based on breaking the $\mathbf {x}$ -sum in Equation (4.6) into residue classes modulo ${{}^{{G}}\mathfrak {b}}$ . Since $Q=P$ and ${{}^{{G}}\mathfrak {b}}^n\subset {\mathcal H}_{\mathfrak {b}}$ , it follows that this sum equals

$$ \begin{align*}\sum_{\mathbf{a}\in (\mathfrak{o}/{{}^{{G}}\mathfrak{b}} )^n}\sigma(F(\mathbf{a})-N) \sum_{\mathbf{x}\in {{{}^{{G}}\mathfrak{b}}}^n}W\left((\mathbf{x}+\mathbf{a})/P\right) h\left(\frac{\operatorname{\mathrm{N}} \mathfrak{b}}{P^d} ,\frac{|N_{K/\mathbb{Q}}(F(\mathbf{x}+\mathbf{a})-N)|}{P^{2d}}\right), \end{align*} $$

for any primitive character $\sigma $ modulo $\mathfrak {b}$ . We apply the multidimensional Poisson summation formula (cf. [Reference Browning and Vishe2, §5]). Recalling that K is totally real, we find that the inner $\mathbf {x}$ -sum is equal to

$$ \begin{align*} \frac{1}{D_K^{n/2}(\operatorname{\mathrm{N}} {{}^{{G}}\mathfrak{b}} )^{n}} \sum_{\mathbf{m}\in \widehat{{{}^{{G}}\mathfrak{b}} }^n} \psi(\mathbf{m}.\mathbf{a}) \int_{V^{n}} W(\mathbf{x}/P) h\left(\frac{\operatorname{\mathrm{N}} \mathfrak{b}}{P^d}, \frac{|\operatorname{\mathrm{Nm}}(F(\mathbf{x})-N)|}{P^{2d}}\right) \psi(-\mathbf{m}.\mathbf{x})\mathrm{d} \mathbf{x}, \end{align*} $$

where we recall that $\widehat {{{}^{{G}}\mathfrak {b}} }={{}^{{G}}\mathfrak {b}}^{-1}\mathfrak {d}^{-1}$ is the dual of ${{}^{{G}}\mathfrak {b}}$ . Putting everything together in Equation (4.6), we have therefore established that

$$ \begin{align*}N_{W}(F,N;P)= \frac{c_{P}}{D_{K}^{n/2}P^{2d}} \sum_{\operatorname{\mathrm{N}}\mathfrak{b}\ll P^d} \sum_{\mathbf{m}\in \widehat{{{}^{{G}}\mathfrak{b}} }^n}(\operatorname{\mathrm{N}} {{}^{{G}}\mathfrak{b}} )^{-n}S_{\mathfrak{b}} (N;\mathbf{m}) \tilde I_{\mathfrak{b}} (\mathbf{m}), \end{align*} $$

with $S_{\mathfrak {b}} (N;\mathbf {m})$ as in the statement of the lemma and

$$ \begin{align*}\tilde I_{\mathfrak{b}} (\mathbf{m})= \int_{ V^{n}}W(\mathbf{x}/P)h\left(\frac{\operatorname{\mathrm{N}} \mathfrak{b}}{P^d} ,\frac{|\operatorname{\mathrm{Nm}}(F(\mathbf{x})-N)|}{P^{2d}}\right)\psi\left(-\mathbf{m}.\mathbf{x} \right)\mathrm{d}\mathbf{x}. \end{align*} $$

A simple change of variables yields $\tilde I_{\mathfrak {b}} (\mathbf {m})=P^{dn} I_{\mathfrak {b}} (N/P^2;P\mathbf {m})$ , as required.

4.4. The exponential sum

We proceed by discussing $S_{\mathfrak {b}} (N;\mathbf {m})$ in Lemma 4.1, for $\mathbf {m}\in \widehat {{}^{{G}}\mathfrak {b}}^n$ . Let $\gamma =g/\alpha \in \mathfrak {F}(\mathfrak {b})$ be as in Lemma 3.4. Then we have

$$ \begin{align*}\sideset{}{{}^{*}}\sum_{\sigma{\,(\operatorname{\mathrm{mod}}{{\mathfrak{b}}})}}\sigma(x) = \sum_{a\in (\mathfrak{o}/\mathfrak{b})^{*}} \psi( \gamma a x), \end{align*} $$

for any $x\in \mathfrak {o}$ . It follows that

(4.7) $$ \begin{align} S_{\mathfrak{b}} (N;\mathbf{m}) =\sum_{a\in (\mathfrak{o}/\mathfrak{b})^{*}} \psi(-\gamma a N) \sum_{\mathbf{x}{\,(\operatorname{\mathrm{mod}}{{{{}^{{G}}\mathfrak{b}}}})}}\psi\left( \gamma a F(\mathbf{x})+ \mathbf{m}. \mathbf{x}\right). \end{align} $$

Our work hinges upon the following upper bound for this sum.

Lemma 4.2. We have

$$ \begin{align*}|S_{\mathfrak{b}} (N;\mathbf{m})| \leqslant |({\mathfrak o}/{\mathfrak{b}})^{*}| |{\mathcal H}_{\mathfrak{b}}/{{}^G\mathfrak{b} }^n|^{1/2}|{\mathfrak o}/{{}^G\mathfrak{b} }|^{n/2}, \end{align*} $$

where ${\mathcal H}_{\mathfrak {b}}$ is given by Equation (3.3).

Proof. For fixed $a\in ({\mathfrak o}/{\mathfrak {b}})^{*}$ , we have

$$ \begin{align*} \begin{aligned} \Bigg| \sum_{\mathbf{x} {\,(\operatorname{\mathrm{mod}}{{{}^G\mathfrak{b} }})}}\psi({\gamma} a&F(\mathbf{x})+{\mathbf m}.\mathbf{x})\Bigg|^2 \\ &= \sum_{{\mathbf h} {\,(\operatorname{\mathrm{mod}}{{{}^G\mathfrak{b} }})} }\sum_{{\mathbf u} {\,(\operatorname{\mathrm{mod}}{{{}^G\mathfrak{b} }})}} \psi ({\gamma} a(F({\mathbf u}+{\mathbf h})-F({\mathbf u}))+{\mathbf m}.{\mathbf h})\\ &\leqslant \sum_{{\mathbf h}{\,(\operatorname{\mathrm{mod}}{{{}^G\mathfrak{b} }})}} \Bigg| \sum_{{\mathbf u} {\,(\operatorname{\mathrm{mod}}{{{}^G\mathfrak{b} }})}} \psi (2{\gamma} a B({\mathbf u};{\mathbf h}))\Bigg| \end{aligned} \end{align*} $$

in the notation of Equation (3.2). We observe that the function ${\mathbf u} \mapsto \psi (2{\gamma } a B({\mathbf u};{\mathbf h}))$ is a character modulo ${{}^G\mathfrak {b} }^n$ , and it is the trivial character precisely when

$$\begin{align*}2{\gamma} a B({\mathbf u};{\mathbf h}) \in {\mathfrak{d}}^{-1}, \qquad \forall {\mathbf u} \in (\mathfrak{o}/{{}^G\mathfrak{b} })^n.\end{align*}$$

We rewrite ${\gamma }$ in the form ${\gamma }=g/{\alpha }$ with $({\alpha })={\mathfrak {b}}{\mathfrak {d}}{\mathfrak p}_1$ for some prime ideal ${\mathfrak p}_1$ and $g\in {\mathfrak p}_1\cap {\mathbb Z}$ with the property that $((g),{\mathfrak {d}}{{}^G\mathfrak {b} })=1$ . Thus, the above condition is equivalent to the condition

$$\begin{align*}2 g a B({\mathbf u};{\mathbf h}) \in (\alpha) {\mathfrak{d}}^{-1} = {\mathfrak{b}} {\mathfrak p}_1, \qquad \forall {\mathbf u} \in ({\mathfrak o}/{{}^G\mathfrak{b} })^n.\end{align*}$$

Since $a \in ( {\mathfrak o}/{\mathfrak {b}})^{*}$ , $g\in {\mathfrak p}_1$ and $((g),{\mathfrak {d}}{{}^G\mathfrak {b} })=1$ , this is equivalent to saying that

$$\begin{align*}2 B({\mathbf u};{\mathbf h}) \in {\mathfrak{b}}, \qquad \forall {\mathbf u} \in ({\mathfrak o}/ {{}^G\mathfrak{b} })^n.\end{align*}$$

Finally, since this condition on $\mathbf {u}$ is invariant modulo ${{}^G\mathfrak {b} }^n$ , this is equivalent to the condition $2 B({\mathbf u};{\mathbf h}) \in {\mathfrak {b}}$ , for all ${\mathbf u} \in {\mathfrak o}^n$ , which is equivalent to specifying that ${\mathbf h} \in {\mathcal H}_{\mathfrak {b}}$ , by Equation (3.3). The statement of the lemma now follows.

It is straightforward to show that $S_{\mathfrak {b}}(N;\mathbf {m})$ vanishes unless $\mathbf {m}$ satisfies additional constraints, as demonstrated in the following result.

Proof. Returning to the definition of $S_{\mathfrak {b}}(N;\mathbf {m})$ in Lemma 4.1 and noting that ${{}^{{G}}\mathfrak {b}}^n\subset {\mathcal H}_{\mathfrak {b}}\subset \mathfrak {o}^n$ , we may write

$$ \begin{align*}S_{\mathfrak{b}} (N;\mathbf{m}) ={\sideset{}{{}^{*}}\sum_{\sigma{\,(\operatorname{\mathrm{mod}}{{\mathfrak{b}}})}}}\sum_{\mathbf{a}\in \mathfrak{o}^n/{{\mathcal H}_{\mathfrak{b}}}} \sigma(-N ) \sum_{\mathbf{h}\in {\mathcal H}_{\mathfrak{b}}/{{}^{{G}}\mathfrak{b}}^n} \sigma(F(\mathbf{a}))\psi(\mathbf{m}. \mathbf{a})\psi(\mathbf{m}. \mathbf{h}). \end{align*} $$

However, orthogonality of characters gives

$$\begin{align*}\sum_{{\mathbf h} \in {\mathcal H}_{\mathfrak{b}}/{{}^G\mathfrak{b} }^n} \psi( \mathbf{m}.{\mathbf h}) = \begin{cases} |{\mathcal H}_{\mathfrak{b}}/{{}^G\mathfrak{b} }^n| & \text{if } {\mathbf m}.{\mathbf h} \in {\mathfrak{d}}^{-1}~\forall {\mathbf h} \in {\mathcal H}_{\mathfrak{b}}/{{}^{{G}}\mathfrak{b}}^n, \\ 0 & \text{otherwise}. \end{cases} \end{align*}$$

Since we automatically have $\mathbf {m}.\mathbf {h}\in \mathfrak {d}^{-1}$ for any $\mathbf {m} \in \widehat {{{}^{{G}}\mathfrak {b}} }^n$ and $\mathbf {h}\in {{}^{{G}}\mathfrak {b}}^n$ , the statement of the lemma follows.

We shall also need to establish a multiplicativity property for the exponential sums. This is achieved in the following result.

Lemma 4.5. Let ${\mathfrak {b}}$ be a nonzero integral ideal, and suppose that $\mathfrak {b}=\mathfrak {b}_1\mathfrak {b}_2$ for integral ideals $\mathfrak {b}_1,\mathfrak {b}_2$ such that $ \gcd (\operatorname {\mathrm {N}}\mathfrak {b}_1,\operatorname {\mathrm {N}}\mathfrak {b}_2)=1. $ Then, for any $N\in \mathfrak {o}$ and any $\mathbf {m}\in \widehat {{{}^{{G}}\mathfrak {b}}}^n$ , we have

$$ \begin{align*}S_{\mathfrak{b}}(N;{\mathbf m})= S_{\mathfrak{b}_1} (\overline{\operatorname{\mathrm{N}}\mathfrak{b}_2}^2N;(\operatorname{\mathrm{N}}\mathfrak{b}_2)\mathbf{m}) S_{\mathfrak{b}_2} (\overline{\operatorname{\mathrm{N}}\mathfrak{b}_1}^2N;(\operatorname{\mathrm{N}}\mathfrak{b}_1)\mathbf{m}). \end{align*} $$

Proof. According to Lemma 3.4, there exists $\gamma =g/\alpha \in \mathfrak {F}(\mathfrak {b})$ such that $\psi (\gamma \cdot )$ is a primitive character modulo $\mathfrak {b}$ . Then, Equation (4.7) implies that

$$ \begin{align*}S_{\mathfrak{b}} (N;\mathbf{m}) = \sum_{a\in (\mathfrak{o}/\mathfrak{b})^{*}} \psi(-\gamma a N) \sum_{\mathbf{x}{\,(\operatorname{\mathrm{mod}}{{{{}^{{G}}\mathfrak{b}}}})}}\psi\left( \gamma a F(\mathbf{x})+ \mathbf{m}. \mathbf{x}\right). \end{align*} $$

Let us write $\operatorname {\mathrm {N}}\mathfrak {b}_i\hspace{-0.5pt}=\hspace{-0.5pt}b_i$ for $i=1,2$ . The assumption $\gcd (b_1,b_2)\hspace{-0.5pt}=\hspace{-0.5pt}1$ implies that $({{}^{{G}}\mathfrak {b}}_1,{{}^{{G}}\mathfrak {b}}_2)\hspace{-0.5pt}=\hspace{-0.5pt}1$ . Moreover, we have $b_1\in \mathfrak {b}_1$ , $b_2\in \mathfrak {b}_2$ and

(4.8) $$ \begin{align} ((b_1),\mathfrak{b}_2)=((b_2),\mathfrak{b}_1)=1. \end{align} $$

According to Lemma 3.1(i), we find elements $\lambda , \mu \in \mathfrak {o}$ such that $\mathrm {ord}_{\mathfrak {p}}(\lambda )=\mathrm {ord}_{\mathfrak {p}}(\mathfrak {b}_1)$ and $\mathrm {ord}_{\mathfrak {p}}(\mu )=\mathrm {ord}_{\mathfrak {p}}(\mathfrak {b}_2)$ for all $\mathfrak {p} \mid {{}^{{G}}\mathfrak {b}}_1{{}^{{G}}\mathfrak {b}}_2$ . It follows from the Chinese remainder theorem, in the form Lemma 3.2, that we can write $a=\mu b+\lambda c$ for $b {\,(\operatorname {\mathrm {mod}}{{\mathfrak {b}_1}})}$ and $c {\,(\operatorname {\mathrm {mod}}{{\mathfrak {b}_2}})}$ . Likewise, we claim that we can write $\mathbf {x}=b_2\mathbf {b}+b_1\mathbf {c}$ , for $\mathbf {b} {\,(\operatorname {\mathrm {mod}}{{{{}^{{G}}\mathfrak {b}}_1}})}$ and $\mathbf {c} {\,(\operatorname {\mathrm {mod}}{{{{}^{{G}}\mathfrak {b}}_2}})}$ . To prove the claim it suffices to show that there is an isomorphism $\mathfrak {o}/{{}^{{G}}\mathfrak {b}}_1\times \mathfrak {o}/{{}^{{G}}\mathfrak {b}}_2\to \mathfrak {o}/{{}^{{G}}\mathfrak {b}}$ , given by $(u,v)\mapsto b_2u+b_1v$ . This map is clearly well defined since $b_1\in {{}^{{G}}\mathfrak {b}}_1$ and $b_2\in {{}^{{G}}\mathfrak {b}}_2$ . Moreover, injectivity follows from the coprimality conditions $((b_2),{{}^{{G}}\mathfrak {b}}_1)=((b_1),{{}^{{G}}\mathfrak {b}}_2)=1$ , which are a direct consequence of Equation (4.8). The claim follows, since the cardinalities are the same, by the Chinese remainder theorem.

In summary, on observing that $b_1\in {{}^{{G}}\mathfrak {b}}_1$ and $b_2\in {{}^{{G}}\mathfrak {b}}_2$ , it follows that

$$ \begin{align*} S_{\mathfrak{b}}(N;\mathbf{m}) =~&\sum_{\substack{b\in (\mathfrak{o}/\mathfrak{b}_1)^{*}\\ c\in (\mathfrak{o}/\mathfrak{b}_2)^{*}}} \psi(-\gamma (\mu b+\lambda c) N)\\ &\quad \times \sum_{\substack{\mathbf{b}{\,(\operatorname{\mathrm{mod}}{{{{}^{{G}}\mathfrak{b}}_1}})}\\ \mathbf{c}{\,(\operatorname{\mathrm{mod}}{{{{}^{{G}}\mathfrak{b}}_2}})}}} \psi\left( \gamma (\mu b+\lambda c)F(b_2\mathbf{b}+b_1\mathbf{c})+ \mathbf{m}.(b_2\mathbf{b}+b_1\mathbf{c})\right)\\ =~&\sum_{b\in (\mathfrak{o}/\mathfrak{b}_1)^{*}} \psi(-\gamma \mu b N)\sum_{\mathbf{b}{\,(\operatorname{\mathrm{mod}}{{{{}^{{G}}\mathfrak{b}}_1}})}}\psi\left( \gamma \mu b_2^2b F(\mathbf{b})+b_2\mathbf{m}. \mathbf{b}\right) \\ &\quad\times \sum_{c\in (\mathfrak{o}/\mathfrak{b}_2)^{*}}\psi(-\gamma \lambda c N)\sum_{\mathbf{c}{\,(\operatorname{\mathrm{mod}}{{{{}^{{G}}\mathfrak{b}}_2}})}} \psi\left( \gamma \lambda b_1^2 c F(\mathbf{c})+ b_1\mathbf{m} .\mathbf{c}\right). \end{align*} $$

We claim that $\psi (\gamma \mu b_2^2\cdot )$ defines a primitive character modulo $\mathfrak {b}_1$ . For this, we note that

$$ \begin{align*}\beta\in \mathfrak{a}_{\gamma\mu b_2^2} \Leftrightarrow \gamma \mu b_2^2 \beta\in \mathfrak{o} \Leftrightarrow (g \mu b_2^2\beta)\subset (\alpha)\Leftrightarrow \mathfrak{b}_1\mathfrak{d}\mid (\mu b_2^2\mathfrak{b}_2^{-1})(g\mathfrak{p}_1^{-1})(\beta) \end{align*} $$

since $b_2\in \mathfrak {b}_2$ and $g\in \mathfrak {p}_1$ . Now, $(g)$ is coprime to $\mathfrak {b}_1\mathfrak {d}$ and $(\mu b_2)$ is coprime to $\mathfrak {b}_1$ . Thus, it follows that

$$ \begin{align*}\beta\in \mathfrak{a}_{\gamma\mu b_2^2} \Leftrightarrow \beta\in \mathfrak{b}_1\mathfrak{e}, \end{align*} $$

where $\mathfrak {e}=\mathfrak {d}/(\mathfrak {d},\mu b_2^2\mathfrak {b}_2^{-1})$ . Clearly, $\mathfrak {e}\mid \mathfrak {d}$ . We claim that $(\mathfrak {d}/\mathfrak {e},\mathfrak {b}_1)=1$ . To see this, note that $\mathfrak {d}/\mathfrak {e}$ is equal to the common divisor $(\mathfrak {d}, \mu b_2^2 \mathfrak {b}_2^{-1})$ . Now, $\mu $ is coprime to $\mathfrak {b}_1$ and so is $\mathfrak {b}_2$ . Hence, the common divisor of these ideals most be coprime to the ideal $\mathfrak {b}_1$ , as claimed. Thus, Lemma 3.3 establishes the claim that $\psi (\gamma \mu b_2^2\cdot )$ is a primitive character modulo $\mathfrak {b}_1$ . It follows that

$$ \begin{align*}\sum_{b\in (\mathfrak{o}/\mathfrak{b}_1)^{*}} \psi(-\gamma \mu b N) \sum_{\mathbf{b}{\,(\operatorname{\mathrm{mod}}{{{{}^{{G}}\mathfrak{b}}_1}})}}\psi\left( \gamma \mu b_2^2b F(\mathbf{b})+b_2\mathbf{m}. \mathbf{b}\right) = S_{\mathfrak{b}_1} (\overline{\operatorname{\mathrm{N}}\mathfrak{b}_2}^2N;(\operatorname{\mathrm{N}}\mathfrak{b}_2)\mathbf{m}), \end{align*} $$

where $\overline {\operatorname {\mathrm {N}}\mathfrak {b}_2}$ is the multiplicative inverse of $\operatorname {\mathrm {N}}\mathfrak {b}_2$ modulo $\mathfrak {b}_1$ . Similarly,

$$ \begin{align*}\sum_{c\in (\mathfrak{o}/\mathfrak{b}_2)^{*}} \psi(-\gamma \lambda c N) \sum_{\mathbf{c}{\,(\operatorname{\mathrm{mod}}{{{{}^{{G}}\mathfrak{b}}_2}})}} \psi\left( \gamma \lambda b_1^2 c F(\mathbf{c})+ b_1\mathbf{m}.\mathbf{c}\right)= S_{\mathfrak{b}_2} (\overline{\operatorname{\mathrm{N}}\mathfrak{b}_1}^2N;(\operatorname{\mathrm{N}}\mathfrak{b}_1)\mathbf{m}), \end{align*} $$

from which the lemma follows.

4.5. The exponential integral

In this section, we discuss the exponential integral $I_{\mathfrak {b}}(t;\mathbf {k})$ that appears in Lemma 4.1 for given $t\in V$ and $\mathbf {k}\in V^n$ . It will be convenient to set

$$ \begin{align*}0<\rho=\frac{\operatorname{\mathrm{N}} \mathfrak{b}}{P^d}\ll 1, \end{align*} $$

with which notation we have

$$ \begin{align*}I_{\mathfrak{b}}(t;\mathbf{k})=\int_{ V^{n}}W(\mathbf{x})h\left(\rho ,|\operatorname{\mathrm{Nm}}(F(\mathbf{x})-t)|\right)\psi\left(-\mathbf{k}.\mathbf{x} \right)\mathrm{d}\mathbf{x}. \end{align*} $$

We now bring into play the work in [Reference Browning and Vishe2, §6]. It follows from an application of Fourier inversion, as in [Reference Browning and Vishe2, Eq. (6.3)], that there exists a function $p_\rho (v):V\to \mathbb {C}$ such that

(4.9) $$ \begin{align} I_{\mathfrak{b}}(t;\mathbf{k})=\int_{ V} p_\rho(v) \psi(-vt) K(v,\mathbf{k}) \mathrm{d} v, \end{align} $$

where

(4.10) $$ \begin{align} K(v,\mathbf{k})=\int_{ V^{n}}W(\mathbf{x})\psi\left(vF(\mathbf{x})-\mathbf{k}.\mathbf{x} \right)\mathrm{d}\mathbf{x}. \end{align} $$

In our analysis, it will be useful to have the notion of a height function on V. Accordingly, we define $\mathfrak {H}:V\to \mathbb {R}_{\geqslant 1}$ via

$$ \begin{align*}\mathfrak{H}(v)=\prod_{l=1}^d \max\{1,|v_l|\}, \end{align*} $$

for $v=(v_1,\dots ,v_d)\in V$ . In the closing stages of our argument, we will need to estimate integrals involving powers of $\mathfrak {H}(v)$ over various regions in V. First, it follows from [Reference Browning and Vishe2, Lemma 5.3] that

(4.11) $$ \begin{align} \int_V \mathfrak{H}(v)^{\alpha}\mathrm{d} v \ll 1 \quad \text{ if } \alpha<-1. \end{align} $$

We can use this to deduce two further bounds that will play important roles.

For any $A\geqslant 1$ and $\varepsilon>0$ , we claim that

(4.12) $$ \begin{align} \int_{\{v\in V: \mathfrak{H}(v)\geqslant A\}} \mathfrak{H}(v)^{\alpha}\mathrm{d} v \ll A^{\alpha+1+\varepsilon} \quad \text{ if } \alpha<-1. \end{align} $$

If $\alpha <-1$ , then we can clearly assume that $\varepsilon <-\alpha -1$ . But then the conditions of integration imply that $(\mathfrak {H}(v)/A)^{-\alpha -1-\varepsilon }\geqslant 1$ , whence

$$ \begin{align*}\int_{\{v\in V: \mathfrak{H}(v)\geqslant A\}} \mathfrak{H}(v)^{\alpha}\mathrm{d} v \leqslant A^{\alpha+1+\varepsilon} \int_{V} \mathfrak{H}(v)^{-1-\varepsilon}\mathrm{d} v \ll A^{\alpha+1+\varepsilon} \end{align*} $$

by Equation (4.11).

Next, for any $B\geqslant 1$ and $\varepsilon>0$ , we claim that

(4.13) $$ \begin{align} \int_{\{v\in V: \mathfrak{H}(v)\leqslant B\}} \mathfrak{H}(v)^{\alpha} \mathrm{d} v \ll B^{\alpha+1+\varepsilon} \quad \text{ if } \alpha\geqslant -1. \end{align} $$

To see this, we note that $(B/\mathfrak {H}(v))^{\alpha +1+\varepsilon }\geqslant 1$ , under the conditions of the integral, if $\alpha \geqslant -1$ . But then

$$ \begin{align*}\int_{\{v\in V: \mathfrak{H}(v)\leqslant B\}} \mathfrak{H}(v)^{\alpha} \mathrm{d} v \leqslant B^{\alpha+1+\varepsilon} \int_{V} \mathfrak{H}(v)^{-1-\varepsilon}\mathrm{d} v \ll B^{\alpha+1+\varepsilon} \end{align*} $$

by Equation (4.11).

Returning to the function $p_\rho (v)$ in Equation (4.9), the following result summarises its key properties and is extracted from [Reference Browning and Vishe2, Lemmas 6.3 and 6.4].

Lemma 4.7. For any $\varepsilon>0$ , we have $ p_\rho (v)\ll P^{\varepsilon }, $ for any $v\in V$ . Moreover, for any $\varepsilon>0$ and $A\geqslant 1$ , we have

$$ \begin{align*}p_\rho(v)\ll_A \rho^{-1} \left(\rho^{-1}P^{\varepsilon} \mathfrak{H}(v)^{-1}\right)^A. \end{align*} $$

Recall here that $\rho>0$ . The next result is a straightforward consequence of the previous result, once combined with Equation (4.9) and the bound

$$ \begin{align*}|K(v,\mathbf{k})|\leqslant \int_{ V^{n}}W(\mathbf{x})\mathrm{d}\mathbf{x}\ll 1, \end{align*} $$

which follows from the fact that W is compactly supported.

Corollary 4.8. Let $\varepsilon>0$ . Let $t\in V$ and $\mathbf {k}\in V^n$ . Then

$$ \begin{align*}I_{\mathfrak{b}}(t;\mathbf{k})\ll_A P^{\varepsilon} \int_{ \mathcal{U}} |K(v,\mathbf{k})| \mathrm{d} v +P^{-A}, \end{align*} $$

for any $A\geqslant 1$ , where

$$ \begin{align*}\mathcal{U}=\mathcal{U}_\varepsilon=\left\{v\in V: \mathfrak{H}(v)\leqslant \frac{P^{d+\varepsilon}}{\operatorname{\mathrm{N}}\mathfrak{b}}\right\}. \end{align*} $$

It is interesting to pause and reflect on the corresponding situation for cubic forms G over a number field K that was considered in [Reference Browning and Vishe2], recalling that we are assuming K to be totally real in our setting. In [Reference Browning and Vishe2], crucial use was made of the fact that the integral over $\mathbf {x}$ factors as

$$ \begin{align*}\prod_{1\leqslant l\leqslant d} \int_{\mathbb{R}^{n}}W^{(l)}(\mathbf{x}^{(l)})e\left(v^{(l)}G^{(l)} (\mathbf{x}^{(l)})-\mathbf{k}^{(l)}.\mathbf{x}^{(l)} \right)\mathrm{d}\mathbf{x}^{(l)} \end{align*} $$

since $\operatorname {\mathrm {Tr}} (vG(\mathbf {x}) )=\sum _{l=1}^{d}v^{(l)}G^{(l)} (\mathbf {x}^{(l)})$ , where $G^{(l)}=\rho _l(G)$ is a cubic form over $\mathbb {R}$ . We have chosen our main example Equation (1.3) in order that a similar property holds. Such a factorisation is not necessarily enjoyed for arbitrary generalised quadratic forms F, however, and it seems very difficult to analyse the integrals $K(v,\mathbf {k})$ in generic situations.

Define

(4.14) $$ \begin{align} \mathcal{Q}(\mathbf{x}^{(1)},\dots,\mathbf{x}^{(d)})= \sum_{1\leqslant l \leqslant d} v_l F^{(l)}(\mathbf{x}^{(1)},\dots,\mathbf{x}^{(d)}), \end{align} $$

for fixed $v\in V$ , where $F^{(l)}$ is the quadratic form (4.2). Thus, $\mathcal {Q}$ is a quadratic form over $\mathbb {R}$ in $dn$ variables. Let us write, temporarily, $\underline {\mathbf {x}}=(\mathbf {x}^{(1)},\dots ,\mathbf {x}^{(d)})$ and $ \underline {\mathbf {k}}=(\mathbf {k}^{(1)},\dots ,\mathbf {k}^{(d)}). $ Then, in the light of Equation (4.3), we may write

(4.15) $$ \begin{align} K(v,\mathbf{k}) =\int_{\mathbb{R}^{dn}}W(\underline{\mathbf{x}}) e\left(\mathcal{Q}(\underline{\mathbf{x}})-\underline{\mathbf{k}}. \underline{\mathbf{x}}\right)\mathrm{d}\underline{\mathbf{x}}. \end{align} $$

A general study of these exponential integrals has been carried out by Heath-Brown and Pierce [Reference Heath-Brown and Pierce6, Lemma 3.1]. Assuming that the support of W is contained in $[-1,1]^{dn}$ , we may appeal to their work, which we record here for the convenience of the reader.

Lemma 4.9. Let $\mathcal {Q}\in \mathbb {R}[X_1,\dots ,X_m]$ be a quadratic form with coefficients of maximum modulus $\|\mathcal {Q}\|$ and eigenvalues $\rho _1,\dots ,\rho _m$ . Let $\boldsymbol {\lambda }\in \mathbb {R}^m$ , and suppose that $w:\mathbb {R}^m\to \mathbb {R}$ is any smooth weight function supported on $[-1,1]^m$ . Then

$$ \begin{align*}\int_{\mathbb{R}^m} w(\mathbf{u}) e\left(\mathcal{Q}(\mathbf{u})-\boldsymbol{\lambda}.\mathbf{u}\right) \mathrm{d} \mathbf{u} \ll _w \prod_{i=1}^m \min\left\{1,|\rho_i|^{-1/2}\right\}. \end{align*} $$

Furthermore, if $|\boldsymbol {\lambda }|\geqslant 4\|\mathcal {Q}\|$ , then the integral is $O_{w,A} (|\boldsymbol {\lambda }|^{-A})$ for any $A\geqslant 1$ .

We will apply this result with $\boldsymbol {\lambda }=\underline {\mathbf {k}}$ and with the real quadratic form in Equation (4.14). Note that $\|\mathcal {Q}\|\ll | v|$ . Next, define

$$ \begin{align*}{\mathcal{F}}(v)=\det \left( \sum_{1\leqslant l \leqslant d} v_l \mathbf{M}^{(l)}\right), \end{align*} $$

where $\mathbf {M}^{(l)}$ is the $dn\times dn$ matrix associated to $F^{(l)}$ . The function ${\mathcal {F}}(v)$ is a real form of degree $dn$ in the variables $v_1,\dots ,v_d$ . The following estimate is a direct consequence of Lemma 4.9.

Unfortunately, it appears difficult to extract anything useful from the second bound, unless the generalised quadratic form is assumed to have extra structure.

4.6. Contribution from the trivial character

In this section, we study the overall contribution from the vector $\mathbf {m}=\mathbf {0}$ in the expression for $N_{W}(F,N;P)$ in Lemma 4.1. This contribution is

$$ \begin{align*}M(P)= \frac{P^{(n-2)d}}{D_{K}^{n/2}} \sum_{ \substack{0\neq \mathfrak{b}\subset \mathfrak{o}\\ \operatorname{\mathrm{N}}\mathfrak{b}\ll P^d} } (\operatorname{\mathrm{N}} {{}^{{G}}\mathfrak{b}} )^{-n} S_{\mathfrak{b}} (N;\mathbf{0}) I_{\mathfrak{b}} (N/P^2;\mathbf{0}) +O_A(P^{-A}) \end{align*} $$

in the notation of that result.

It will ease notation if we put $t=N/P^2\in \mathbb {R}$ . Assuming that the descended system has codimension d, we begin by analysing the exponential integral $I_{\mathfrak {b}}(t;\mathbf {0})$ , writing

(4.16) $$ \begin{align} I_{\mathfrak{b}} (t;\mathbf{0}) = \int_{ V^{n}}W(\mathbf{x})h\left(\rho ,|\operatorname{\mathrm{Nm}}(F(\mathbf{x})-t)|\right)\mathrm{d}\mathbf{x} =\int_{ V} f(v)h\left(\rho ,|\operatorname{\mathrm{Nm}}(v)|\right)\mathrm{d}\mathbf{v}, \end{align} $$

where $\rho =\operatorname {\mathrm {N}}\mathfrak {b}/P^d$ and

$$ \begin{align*}f(v)=\int_{\substack{\mathbf{x}\in V^n\\ F(\mathbf{x})-t=v}} W(\mathbf{x})\mathrm{d} \mathbf{x}, \end{align*} $$

where, by an abuse of notation, $\mathrm {d} \mathbf {x}$ is the surface measure obtained by eliminating d variables from the equation $F(\mathbf {x})-t=v$ . We shall think of $f(y)$ as a function of $\mathbf {y}=(y_1,\dots ,y_d)$ on $\mathbb {R}^d$ , in which t is fixed and bounded absolutely. The following result summarises its main properties.

Lemma 4.11. Assume that the descended system has codimension d. There exist positive constants $C,C_0,C_1,\dots $ such that the function $f: \mathbb {R}^d\to \mathbb {R}$ is a smooth weight function that is supported on $[-C,C]^d$ and satisfies

$$ \begin{align*}\left|\frac{\partial^{i_1+\cdots+i_d}}{\partial y_1^{i_1}\cdots \partial y_d^{i_d} }f (\mathbf{y})\right| \leqslant C_{i_1+\cdots+i_d} \end{align*} $$

for any $\mathbf {y}\in [-C,C]^d$ and any $i_1,\dots ,i_d\geqslant 0$ . The constants $C,C_0,C_1,\dots $ depend only on the coefficients of F and the parameter $\delta $ in the definition of W.

Proof. In the course of the proof, it will be convenient to write $\underline {\mathbf {s}}=(\mathbf {s}_1,\dots ,\mathbf {s}_d)$ , $\underline {\mathbf {u}}=(\mathbf {u}_1,\dots ,\mathbf {u}_d)$ , $\underline {\boldsymbol {\xi }}=(\boldsymbol {\xi }_1,\dots ,\boldsymbol {\xi }_d)$ and $\mathbf {t}=(t_1,\dots ,t_d)$ . Recall the definition of the weight function W in §4.2 for a suitable fixed $\underline {\boldsymbol {\xi }}\in \mathbb {R}^{dn}$ . Making the change of variables in Equation (4.4), we see that

(4.17) $$ \begin{align} f(\mathbf{y}) =D_K^{-n/2} \int_{\substack{\underline{\mathbf{u}}\in \mathbb{R}^{dn}\\ Q_l(\underline{\mathbf{u}})-\tau_l=w_l}} w(\delta^{-1}|\underline{\mathbf{u}}-\underline{\boldsymbol{\xi}}|) \mathrm{d} \underline{\mathbf{u}}, \end{align} $$

where ${\boldsymbol \tau }=\mathbf {A}^{-1} \mathbf {t}$ and $\mathbf {w}=\mathbf {A}^{-1}\mathbf {y}$ . It will clearly suffice to prove the properties recorded in the lemma for the integral on the right-hand side, $\widetilde {f}({\mathbf {w}})$ say, regarded as a function of $\mathbf {w}$ . Making the change of variables $\underline {\mathbf {s}}=\underline {\mathbf {u}}-\underline {\boldsymbol {\xi }}$ , we have

$$ \begin{align*}\widetilde f(\mathbf{w}) = \int_{\substack{\underline{\mathbf{s}}\in \mathbb{R}^{dn}\\ Q_l(\underline{\mathbf{s}}+\underline{\boldsymbol{\xi}})-\tau_l=w_l}} w(\delta^{-1}|\underline{\mathbf{s}}|) \mathrm{d} \underline{\mathbf{s}}. \end{align*} $$

It is now clear that $\widetilde {f}(\mathbf {w})=0$ unless $\mathbf {w}\in [-C,C]^d$ for suitable $C>0$ .

Next, we recall that $J_{Q_1,\dots ,Q_d}(\underline {\boldsymbol {\xi }})$ has rank d. We may assume without loss of generality that

$$ \begin{align*}\det \left(\frac{\partial }{\partial U_{j1}}Q_i(\underline{\boldsymbol{\xi}})\right)_{\substack{1\leqslant i,j\leqslant d}}\neq 0. \end{align*} $$

Let $\varphi :\mathbb {R}^{dn}\to \mathbb {R}^{dn}$ be given by

$$ \begin{align*}\underline{\mathbf{s}} \mapsto \left(Q_1(\underline{\mathbf{s}}+\underline{\boldsymbol{\xi}})-\tau^{(1)},s_{1,2},\dots,s_{1,n},\dots , Q_d(\underline{\mathbf{s}}+\underline{\boldsymbol{\xi}})-\tau^{(d)},s_{d,2},\dots,s_{d,n} \right). \end{align*} $$

The implicit function theorem implies that there exist open subsets $W',W\subset \mathbb {R}^{dn}$ with $\underline {\mathbf {0}}\in W'$ and $\varphi (\underline {\mathbf {0}})\in W$ such that $\varphi :W'\to W$ is a bijection and has differentiable inverse $\varphi ^{-1}$ on W. It is now clear that we wish to choose $\delta>0$ small enough to ensure that $\underline {\mathbf {s}}\in W'$ whenever $|\underline {\mathbf {s}}|\leqslant \delta $ .

We may now conclude that

$$ \begin{align*}\widetilde{f}(\mathbf{w}) = \int_{\substack{\underline{\mathbf{s}}'\in \mathbb{R}^{d(n-1)}}} \partial_1 \varphi^{-1} w(\delta^{-1} |\left(s_{1,1},\dots,s_{d,n} \right)|) \mathrm{d} \underline{\mathbf{s}}', \end{align*} $$

where $\underline {\mathbf {s}}'= (s_{1,2},\dots ,s_{1,n},\dots ,s_{d,2},\dots ,s_{d,n}) $ , and $s_{1,1},s_{2,1},\dots , s_{d,1}$ are implicitly given by $\underline {\mathbf {s}}'$ and $\mathbf {w}$ , and

$$ \begin{align*}{\partial_1 \varphi^{-1}=\left. \det \left(\frac{\partial (\varphi^{-1})_{in+1-n}}{\partial w_j} \right)_{1\leqslant i,j\leqslant d}\right|}_{(s_{1,1},\dots , s_{d,n})} \end{align*} $$

is the associated Jacobian. Since $\varphi ^{-1}$ is smooth, this implies that $\widetilde {f}(\mathbf {w})$ is infinitely differentiable and that its partial derivatives satisfy the bound claimed in the lemma.

Now, it follows from Corollary 4.8 that for $t=N/P^2\in \mathbb {R}$ , and $\varepsilon $ fixed as in the corollary, we have

$$ \begin{align*}I_{\mathfrak{b}}(t;\mathbf{0})\ll \frac{P^{d+2\varepsilon}}{\operatorname{\mathrm{N}}\mathfrak{b}}. \end{align*} $$

Furthermore, in view of Lemma 4.11, it follows from Equation (4.16) and [Reference Browning and Vishe2, Lemma 4.1] that

$$ \begin{align*}I_{\mathfrak{b}} (t;\mathbf{0}) = \sqrt{D_K} f(0) +O_A\left(\left(\frac{\operatorname{\mathrm{N}}\mathfrak{b}}{P^d}\right)^A\right), \end{align*} $$

for any $A\geqslant 0$ , where

$$ \begin{align*}f(0) = \int_{\substack{\underline{\mathbf{x}}\in \mathbb{R}^{dn}\\ F^{(l)}(\underline{\mathbf{x}})=t_l}} W(\underline{\mathbf{x}})\mathrm{d} \underline{\mathbf{x}}, \end{align*} $$

if $\underline {\mathbf {x}}=(\mathbf {x}_1,\dots ,\mathbf {x}_d)$ . According to Equation (4.17), we have $f(0)= D_K^{-n/2} \sigma _\infty (t),$ where

(4.18) $$ \begin{align} \sigma_\infty(t)= \int_{\substack{\underline{\mathbf{u}}\in \mathbb{R}^{dn}\\ Q_l(\underline{\mathbf{u}})=\tau_l}} w(\delta^{-1}|\underline{\mathbf{u}}-\underline{\boldsymbol{\xi}}|) \mathrm{d} \underline{\mathbf{u}} \end{align} $$

is the usual singular integral for the descended system. In particular, arguing as in Davenport and Lewis [Reference Davenport and Lewis3, §6], a standard argument ensures that $\sigma _\infty (t)>0$ since $\underline {\boldsymbol {\xi }}$ is a nonsingular real point on the descended system.

We summarise our preliminary treatment of the main term in the following result.

Lemma 4.12. Let $\varepsilon>0$ . Then, for any $A\geqslant 1$ , we have

$$ \begin{align*} M(P)=~& \frac{P^{(n-2)d} }{D_{K}^{n-1/2}} \sum_{\substack{0\neq \mathfrak{b}\subset \mathfrak{o}\\ \operatorname{\mathrm{N}}\mathfrak{b}\ll P^d}} (\operatorname{\mathrm{N}} {{}^{{G}}\mathfrak{b}} )^{-n} S_{\mathfrak{b}} (N;\mathbf{0}) \left(\sigma_\infty(N/P^2) +O_A\left(\left(\frac{\operatorname{\mathrm{N}}\mathfrak{b}}{P^d}\right)^A\right)\right) \\ &\qquad +O_A(P^{-A}), \end{align*} $$

where $\sigma _\infty (N/P^2)>0$ is given by Equation (4.18).

In order to proceed further, it is clear that one requires a good enough upper bound for $S_{\mathfrak {b}}(N;\mathbf {0})$ in order to show that the error term is satisfactory and the sum over $\mathfrak {b}$ can be extended to infinity. Such a bound is available for admissible F, thanks to Corollary 4.3. Although we omit details, one can use it to prove that

$$ \begin{align*}M(P)= \frac{\sigma_\infty(N/P^2) }{D_{K}^{n-1/2}} P^{(n-2)d} \sum_{\substack{0\neq \mathfrak{b}\subset \mathfrak{o}}} (\operatorname{\mathrm{N}} {{}^{{G}}\mathfrak{b}} )^{-n} S_{\mathfrak{b}} (N;\mathbf{0}) +O(P^{dn/2+\varepsilon}), \end{align*} $$

for any admissible F such that $n\geqslant 5$ . In the setting of Theorems 1.4 and 1.5, we shall produce better bounds for $S_{\mathfrak {b}} (N;\mathbf {0})$ which allow such a deduction under milder hypotheses.

We close this section with a formal analysis of the singular series

$$ \begin{align*}\mathfrak{S}(N)=\sum_{(0)\neq \mathfrak{b}\subset \mathfrak{o}} (\operatorname{\mathrm{N}}{{}^{{G}}\mathfrak{b}})^{-n}S_{\mathfrak{b}} (N;\mathbf{0}), \end{align*} $$

ignoring issues of convergence. This is summarised in the following result.

Lemma 4.13. Assume that $\mathfrak {S}(N)$ is absolutely convergent. Then

$$ \begin{align*}\mathfrak{S}(N)=\prod_p \lim_{\ell\to \infty} p^{-d\ell(n-1)}\#\left\{\mathbf{x}\in (\mathfrak{o}/p^\ell\mathfrak{o})^n: F(\mathbf{x})\equiv N{\,(\operatorname{\mathrm{mod}}{{p^\ell}})}\right\}. \end{align*} $$

We have $\mathfrak {S}(N)>0$ if the shifted descended system has a nonsingular p-adic solution for every prime p.

Proof. We may write

$$ \begin{align*}\mathfrak{S}(N) =\sum_{k=1}^\infty \sum_{\substack{ \mathfrak{b}\subset \mathfrak{o}\\ \operatorname{\mathrm{N}}\mathfrak{b}=k }} (\operatorname{\mathrm{N}}{{}^{{G}}\mathfrak{b}})^{-n}S_{\mathfrak{b}} (N;\mathbf{0}) =\sum_{k=1}^\infty S(k), \end{align*} $$

say. It follows from Corollary 4.6 that $S(k_1k_2)=S(k_1)S(k_2)$ if $k_1,k_2$ are coprime integers. Hence,

$$ \begin{align*}\mathfrak{S}(N)=\prod_p \sum_{j\geqslant 0}S(p^j). \end{align*} $$

Since K is Galois, we may assume that p admits a factorisation $(p)=(\mathfrak {p}_1\cdots \mathfrak {p}_r)^e$ , with $\operatorname {\mathrm {N}} \mathfrak {p}_1=\cdots =\operatorname {\mathrm {N}}\mathfrak {p}_r=p^f$ . Let $\ell \geqslant 0$ , and let $I_\ell $ denote the set of integral ideals $\mathfrak {b}=\mathfrak {p}_1^{k_1}\cdots \mathfrak {p}_r^{k_r}$ , with $0\leqslant k_i\leqslant \ell e$ for $1\leqslant i\leqslant r$ . Then the union of $I_\ell $ over $\ell \geqslant 0$ exactly matches the set of integral ideals whose norm is a power of p. Hence,

$$ \begin{align*}\mathfrak{S}(N)=\prod_p \lim_{\ell\to \infty} \sum_{\substack{ \mathfrak{b}\in I_\ell}} (\operatorname{\mathrm{N}}{{}^{{G}}\mathfrak{b}})^{-n}S_{\mathfrak{b}} (N;\mathbf{0}). \end{align*} $$

It follows from Equation (4.7) that

$$ \begin{align*} S_{\mathfrak{b}} (N;\mathbf{0}) &=\sum_{a\in (\mathfrak{o}/\mathfrak{b})^{*}} \psi(-\gamma a N) \sum_{\mathbf{x}{\,(\operatorname{\mathrm{mod}}{{{{}^{{G}}\mathfrak{b}}}})}}\psi\left( \gamma a F(\mathbf{x})\right)\\ &=\frac{(\operatorname{\mathrm{N}}{{}^{{G}}\mathfrak{b}})^{n}}{p^{\ell dn}} \sum_{a\in (\mathfrak{o}/\mathfrak{b})^{*}} \psi(-\gamma a N) \sum_{\mathbf{x}{\,(\operatorname{\mathrm{mod}}{{p^\ell\mathfrak{o}}})}}\psi\left( \gamma a F(\mathbf{x})\right), \end{align*} $$

on extending the inner sum to a sum over elements of $(\mathfrak {o}/p^\ell \mathfrak {o})^n$ . Hence, on rearranging, we obtain

$$ \begin{align*}\sum_{\substack{ \mathfrak{b}\in I_\ell}} (\operatorname{\mathrm{N}}{{}^{{G}}\mathfrak{b}})^{-n}S_{\mathfrak{b}} (N;\mathbf{0})= \frac{1}{p^{\ell dn}} \sum_{\mathbf{x}{\,(\operatorname{\mathrm{mod}}{{p^\ell\mathfrak{o}}})}} E(p), \end{align*} $$

where

$$ \begin{align*}E(p)= \sum_{\substack{ \mathfrak{b}\in I_\ell}} \sum_{a\in (\mathfrak{o}/\mathfrak{b})^{*}} \psi\left(\gamma a (F(\mathbf{x})-N)\right). \end{align*} $$

We claim that $\psi (\gamma a\cdot )$ runs over all characters modulo $p^\ell $ , as a runs over $(\mathfrak {o}/\mathfrak {b})^{*}$ and $\mathfrak {b}$ runs over $I_\ell $ . On one hand, since $a\in (\mathfrak {o}/\mathfrak {b})^{*}$ , each character $\psi (\gamma a\cdot )$ is a primitive character modulo $\mathfrak {b}$ . Since $\mathfrak {b}=\mathfrak {p}^{k_1}\cdots \mathfrak {p}_r^{k_r}$ , for $k_1,\dots ,k_r\leqslant \ell e$ and $p^\ell =(\mathfrak {p}_1\cdots \mathfrak {p}_r)^{\ell e}$ , we conclude that each such character induces a character modulo $p^\ell $ . In order to complete the proof of the claim, it remains to show that we get all $p^{\ell d}$ characters modulo $p^\ell $ this way. But the number of characters is precisely

$$ \begin{align*} \sum_{\mathfrak{b}\in I_\ell} \operatorname{\mathrm{N}}\mathfrak{b} \prod_{\mathfrak{p}\mid \mathfrak{b}} \left(1-\frac{1}{\operatorname{\mathrm{N}}\mathfrak{p}}\right) &= \sum_{\mathfrak{b}\in I_\ell} p^{f(k_1+\cdots+k_r)}\prod_{\mathfrak{p}\mid \mathfrak{b}} \left(1-\frac{1}{p^f}\right) \\ &=\prod_{1\leqslant i\leqslant r} \left(1+\sum_{1\leqslant k\leqslant \ell e} p^{fk} \left(1-\frac{1}{p^f}\right)\right) \\ &=p^{\ell d}, \end{align*} $$

as required.

We may now conclude from orthogonality of characters that

$$ \begin{align*}E(p)=\begin{cases} p^{\ell d} & \text{ if } F(\mathbf{x})\equiv N{\,(\operatorname{\mathrm{mod}}{{p^\ell}})},\\ 0 & \text{ otherwise,} \end{cases} \end{align*} $$

from which the first part of the lemma follows. The second part is standard. Using Equation (1.2), the solubility of $F(\mathbf {x})- N$ in $\mathfrak {o}/p^\ell \mathfrak {o}$ can be reduced to the solubility of a shifted descended system $Q_i(\mathbf {u}_1,\dots ,\mathbf {u}_d) -N_i$ modulo primes powers, for $1\leqslant i\leqslant d$ , where we have written $N=\omega _1N_1+\cdots +\omega _d N_d$ . Arguing as in work of Birch [Reference Birch1, Lemma 7.1], for example, the existence of nonsingular p-adic zeros of this system is enough to deduce that $\mathfrak {S}(N)>0$ . The details of this will not be repeated here.

4.7. Contribution from the nontrivial characters

In this section, we make some initial steps in the treatment of the contribution from the nonzero vectors $\mathbf {m}$ in the asymptotic formula for $N_W(F,N;P)$ in Lemma 4.1. This contribution is

$$ \begin{align*}\ll P^{(n-2)d} E(N;P), \end{align*} $$

where

(4.19) $$ \begin{align} E(N;P)= \sum_{\substack{0\neq \mathfrak{b}\subset \mathfrak{o}\\ \operatorname{\mathrm{N}}\mathfrak{b}\ll P^d}} \sum_{\substack{\mathbf{0}\neq \mathbf{m}\in \widehat{{{}^{{G}}\mathfrak{b}} }^n}} (\operatorname{\mathrm{N}} {{}^{{G}}\mathfrak{b}} )^{-n} |S_{\mathfrak{b}} (N;\mathbf{m})| |I_{\mathfrak{b}} (N/P^2;P\mathbf{m})|. \end{align} $$

The primary task is to establish conditions under which there is an absolute constant $\Delta>0$ such that $E(N;P)=O(P^{-\Delta })$ .

We now place ourselves in the context of the generalised quadratic forms (1.3) and make some initial steps that will be common to Theorems 1.3–1.5. It will be convenient to consider the overall contribution from $\mathfrak {b}$ such that $\operatorname {\mathrm {N}}\mathfrak {b}$ and $\operatorname {\mathrm {N}}{{}^{{G}}\mathfrak {b}}$ are constrained to lie in dyadic intervals. Note that $\operatorname {\mathrm {N}}\mathfrak {b}\ll P^d$ and $\operatorname {\mathrm {N}}{{}^{{G}}\mathfrak {b}}\leqslant (\operatorname {\mathrm {N}}\mathfrak {b})^2$ since $\#G=2$ . Accordingly, we let $X,Y$ be parameters such that

(4.20) $$ \begin{align} 1\leqslant X\leqslant Y\leqslant X^2, \quad X\ll P^d. \end{align} $$

We then write $E(N;P;X,Y)$ for the overall contribution to $E(N;P)$ from nonzero ideals $\mathfrak {b}\subset \mathfrak {o}$ for which

$$ \begin{align*}X\leqslant \operatorname{\mathrm{N}}\mathfrak{b} <2X \quad \text{ and } \quad Y\leqslant \operatorname{\mathrm{N}}{{}^{{G}}\mathfrak{b}} <2Y. \end{align*} $$

We denote by $\mathcal {B}(X,Y)$ the set of all such ideals. On summing over dyadic intervals for $X,Y$ satisfying Equation (4.20), it will suffice to establish the existence of $\Delta>0$ such that

(4.21) $$ \begin{align} E(N;P;X,Y)=O(P^{-\Delta}), \end{align} $$

for any $X,Y$ satisfying Equation (4.20).

It follows from Corollary 4.8 that

$$ \begin{align*}I_{\mathfrak{b}} (N/P^2;P\mathbf{m}) \ll_A P^{\varepsilon} \int_{\mathcal{U}} |K(u,P\mathbf{m})|\mathrm{d} u +P^{-A}, \end{align*} $$

for any $A\geqslant 1$ , where $K(u,P\mathbf {m})$ is given by Equation (4.10) and

(4.22) $$ \begin{align} \mathcal{U}= \left\{u\in V: \mathfrak{H}(u)\leqslant \frac{P^{d+\varepsilon}}{\operatorname{\mathrm{N}}\mathfrak{b}}\right\}. \end{align} $$

Hence, Equation (4.19) yields

(4.23) $$ \begin{align} \begin{aligned} E(N;P;X,Y)\ll_A P^{-A}+ P^{\varepsilon} Y^{-n} \sum_{\substack{\mathfrak{b}\in \mathcal{B}(X,Y)}} \sum_{\substack{0\neq \mathbf{m}\in \widehat{{{}^{{G}}\mathfrak{b}} }^n}} |S_{\mathfrak{b}} (N;\mathbf{m})| \int_{\mathcal{U}} |K(u,P\mathbf{m})|\mathrm{d} u , \end{aligned} \end{align} $$

for any $A\geqslant 1$ , where $\mathcal {B}(X,Y)$ is the set of nonzero ideals $\mathfrak {b}\subset \mathfrak {o}$ for which $ X\leqslant \operatorname {\mathrm {N}}\mathfrak {b} <2X$ and $Y\leqslant \operatorname {\mathrm {N}}{{}^{{G}}\mathfrak {b}} <2Y$ .