# Generalised quadratic forms over totally real number fields

Tim Browning  IST Austria
Am Campus 1
3400 Klosterneuburg
Austria<EMAIL_ADDRESS>,  Lillian B. Pierce  Department of Mathematics
Duke University
Durham NC 27708
USA<EMAIL_ADDRESS>and  Damaris Schindler  Göttingen University
Bunsenstraße 3–5
37073 Göttingen
Germany<EMAIL_ADDRESS>
###### Abstract.

We introduce a new class of generalised quadratic forms over totally real
number fields, which is rich enough to capture the arithmetic of arbitrary
systems of quadrics over the rational numbers. We explore this connection
through a version of the Hardy–Littlewood circle method over number fields.

###### 2010 Mathematics Subject Classification:

11P55 (11D09, 14G05)

###### Contents

  1. 1 Introduction
  2. 2 Generalised quadratic forms and the descended system
  3. 3 Recap from algebraic number theory
  4. 4 Enter the circle method
  5. 5 Homogeneous case: proof of Theorems 1.3 and 1.4
  6. 6 Inhomogeneous case: proof of Theorem 1.5

##  1\. Introduction

The study of quadratic forms over number fields is a rich and highly developed
area of mathematics. Let $K$ be a number field of degree $d\geqslant 2$ over
$\mathbb{Q}$ and let

$Q(X_{1},\dots,X_{n})=\sum_{1\leqslant i,j\leqslant n}c_{i,j}X_{i}X_{j}$

be a non-singular quadratic form, with symmetric coefficients
$c_{i,j}\in\mathfrak{o}_{K}$. For given $N\in\mathfrak{o}_{K}$, it is very
natural to ask about the solubility of

$Q(x_{1},\dots,x_{n})=N,$

with $x_{1},\dots,x_{n}\in\mathfrak{o}_{K}$. If $n\geqslant 4$, a number field
version of the Hardy–Littlewood circle method is capable of establishing the
Hasse principle for these equations. When $n\geqslant 5$ this follows from
work of Skinner [13], and for $n=4$ it is carried out by Helfrich in a 2015
PhD thesis [8].

In this paper we shall introduce the notion of a generalised quadratic form
over $K$ and ask about the Hasse principle in this new setting. We shall
always assume that $K/\mathbb{Q}$ is a Galois extension of degree $d$ that is
totally real. (Our methods can handle arbitrary number fields, but doing so
causes extra notational complexity and gives no new insight into the
arithmetic of generalised quadratic forms.) We may now make the following
definition.

######  Definition 1.1.

Let $n\geqslant 2$. A generalised quadratic form is given by

$F(X_{1},\dots,X_{n})=\sum_{1\leqslant i,j\leqslant
n}\sum_{\tau,\tau^{\prime}\in\operatorname{Gal}(K/\mathbb{Q})}c_{i,j,\tau,\tau^{\prime}}X_{i}^{\tau}X_{j}^{\tau^{\prime}},$

for symmetric coefficients
$c_{i,j,\tau,\tau^{\prime}}=c_{j,i,\tau^{\prime},\tau}\in\mathfrak{o}_{K}$.

We will be interested in the set of
$(x_{1},\dots,x_{n})\in\mathfrak{o}_{K}^{n}$ for which

$F(x_{1},\dots,x_{n})=N,$

for given $N\in\mathfrak{o}_{K}$, in which case $x_{i}^{\tau}$ should be
interpreted as the conjugate of $x_{i}$ under
$\tau\in\operatorname{Gal}(K/\mathbb{Q})$. Definition 1.1 encompasses standard
integral quadratic forms over $\mathfrak{o}_{K}$ and forms defined using norms
and traces. For example, let $\operatorname{Tr}_{K/\mathbb{Q},H}:K\to K$ be
the partial trace, defined via
$\operatorname{Tr}_{K/\mathbb{Q},H}(u)=\sum_{\tau\in H}u^{\tau}$, for any
subset $H\subset\operatorname{Gal}(K/\mathbb{Q})$. Then, a natural
generalisation of the question about representing elements of
$\mathfrak{o}_{K}$ as a sum of squares is to ask about the existence of
$\mathbf{x}\in\mathfrak{o}_{K}^{n}$ such that

$\operatorname{Tr}_{K/\mathbb{Q},H}(x_{1}^{2})+\dots+\operatorname{Tr}_{K/\mathbb{Q},H}(x_{n}^{2})=N,$
(1.1)

for given $N\in\mathfrak{o}_{K}$ and a given subset
$H\subset\operatorname{Gal}(K/\mathbb{Q})$.

The coefficients of a generalised quadratic form $F(X_{1},\dots,X_{n})$ form a
$dn\times dn$ matrix
$\mathbf{M}=(c_{i,j,\tau,\tau^{\prime}})_{(i,\tau)\times(j,\tau^{\prime})}$.
In the generic setting we might expect this matrix to have full rank, but
there are many cases of interest where the rank is much smaller. For example,
standard quadratic forms produce a coefficient matrix $\mathbf{M}$, which
after reordering rows and columns, contains a $n\times n$ block matrix in the
upper left corner and has zeros everywhere else. Our methods break down in the
completely generic situation, and so our interest in this paper lies at the
opposite end of the spectrum, in which the rank of $\mathbf{M}$ is not much
bigger than $n$.

Let $W:(K\otimes_{\mathbb{Q}}\mathbb{R})^{n}\to\mathbb{R}_{\geqslant 0}$ be a
smooth weight function, whose precise construction is deferred until §4. Our
main results will comprise of asymptotic formulae for sums of the shape

$N_{W}(F,N;P)=\sum_{\begin{subarray}{c}\mathbf{x}\in\mathfrak{o}_{K}^{n}\\\
F(\mathbf{x})=N\end{subarray}}W(\mathbf{x}/P),$

as $P\rightarrow\infty$, for given $N\in\mathfrak{o}_{K}$ and suitable
generalised quadratic forms $F$. When $N=0$, we shall simply write
$N_{W}(F;P)=N_{W}(F,0;P)$.

###  1.1. Homogeneous setting

Of particular interest is the case $N=0$, which we now assume. For standard
quadratic forms $Q\in\mathfrak{o}_{K}[X_{1},\dots,X_{n}]$, studying non-
trivial zeros of $Q$ over $\mathfrak{o}_{K}$ is equivalent to studying
$K$-rational points on the smooth quadric $X\subset\mathbb{P}_{K}^{n-1}$ cut
out by $Q=0$. This, in turn, can be accessed via the Weil restriction (or
restriction of scalars). The Weil restriction $R_{K/\mathbb{Q}}X$ is an
algebraic variety whose set of $\mathbb{Q}$-points is canonically in bijection
with the $K$-rational points of $X$. In the setting where
$Q\in\mathfrak{o}_{K}[X_{1},\dots,X_{n}]$ is a non-singular quadratic form,
the Weil restriction $R_{K/\mathbb{Q}}X$ is a smooth complete intersection of
$d$ quadrics in $\mathbb{P}_{\mathbb{Q}}^{dn-1}$, all of which are defined
over $\mathbb{Q}$. However, the set of complete intersections that arise in
this way is a very limited subset of the family of all smooth codimension $d$
complete intersections of quadrics over $\mathbb{Q}$ in
$\mathbb{P}_{\mathbb{Q}}^{dn-1}$. Our first result shows that, after Weil
restriction, the space of generalised quadratic forms is rich enough to
capture the arithmetic over $\mathbb{Q}$ of arbitrary codimension $d$ complete
intersections of quadrics in $\mathbb{P}_{\mathbb{Q}}^{M-1}$, provided that
$d\mid M$.

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 $\mathfrak{o}_{K}$.
Any element $\mathbf{x}\in\mathfrak{o}_{K}^{n}$ can be written
$\mathbf{x}=\omega_{1}\mathbf{u}_{1}+\cdots+\omega_{d}\mathbf{u}_{d}$ for
$(\mathbf{u}_{1},\dots,\mathbf{u}_{d})\in\mathbb{Z}^{dn}$. Taking the Weil
restriction corresponds to writing down a set of quadratic forms
$Q_{1},\dots,Q_{d}\in\mathbb{Z}[\mathbf{U}_{1},\dots,\mathbf{U}_{d}]$, in $dn$
variables, such that

$F(X_{1},\dots,X_{n})=\sum_{1\leqslant i\leqslant
d}{\omega_{i}}Q_{i}(\mathbf{U}_{1},\dots,\mathbf{U}_{d}).$ (1.2)

We henceforth call $\\{Q_{1},\dots,Q_{d}\\}$ the descended system. We shall
prove the following result in §2.

######  Theorem 1.2.

Let $K/\mathbb{Q}$ be a Galois extension of degree $d$. Then there is a
bijection between the space of generalised quadratic forms in $n$ variables
over $K$ and systems of $d$ rational quadratic forms in $dn$ variables.

The question of $\mathfrak{o}_{K}$-solubility for a generalised quadratic form
therefore becomes a question of $\mathbb{Z}$-solubility for the descended
system and vice versa. It presents an intriguing challenge to gain insight
into smooth codimension $d$ complete intersections of quadrics in
$\mathbb{P}_{\mathbb{Q}}^{M-1}$ over $\mathbb{Q}$ by working with generalised
quadratic forms. It follows from work of Birch [1] that the usual
Hardy–Littlewood asymptotic formula holds for systems of quadrics over
$\mathbb{Q}$, provided that $M>B+2d(d+1)$, where $B$ is the affine dimension
of the “Birch singular locus” of the descended system. (Note that one can take
$B\leqslant d-1$ when the descended system is a smooth complete intersection.)
Breakthrough work of Rydin Myerson [11] handles smooth codimension $d$
complete intersections of quadrics in $\mathbb{P}_{\mathbb{Q}}^{M-1}$ when
$M\geqslant 9d$. The latter result is particularly significant, since it
allows one to handle arbitrary generalised quadratic forms over $K$ in
$n\geqslant 9$ variables, provided that the descended system defines a smooth
complete intersection of codimension $d$.

Our main results will concern a special class of generalised quadratic forms,
in which only one non-trivial automorphism appears, and in which the
conjugated variables separate completely from the unconjugated variables.
These examples are chosen to represent a first step on the way to a fuller
understanding of generalised quadratic forms, and yet exhibit enough features
that make them untreatable by other methods. In the light of Theorem 1.2, a
complete understanding of generalised quadratic forms must lie rather deep.

Let $Q\in\mathfrak{o}_{K}[X_{1},\dots,X_{n}]$ and
$R\in\mathfrak{o}_{K}[X_{1},\dots,X_{m}]$ be quadratic forms in $n$ and $m$
variables, respectively, for $1\leqslant m\leqslant n$. The generalised
quadratic forms we shall treat take the shape

$F(X_{1},\dots,X_{n})=Q(X_{1},\dots,X_{n})+R(X_{1}^{\tau},\dots,X_{m}^{\tau}),$
(1.3)

for a fixed non-trivial automorphism
$\tau\in\operatorname{Gal}(K/\mathbb{Q})$. Let $\rho_{1},\dots,\rho_{d}$ be
the $d$ distinct embeddings of $K$ into $\mathbb{R}$, where we recall that $K$
is totally real. For each $1\leqslant l\leqslant d$, we define $l_{\tau}$
through the relation

$\rho_{l_{\tau}}\tau=\rho_{l}.$ (1.4)

Suppose that $\mathbf{A}$ is the $n\times n$ symmetric matrix defining $Q$ and
that $\mathbf{B}$ is the $n\times n$ symmetric matrix given by the condition
that its upper left $m\times m$ submatrix defines $R$, with all other entries
equal to $0$. For any $1\leqslant l\leqslant d$, we shall write
$\mathbf{A}^{(l)}$ and $\mathbf{B}^{(l)}$ for the $l$-th embeddings of
$\mathbf{A}$ and $\mathbf{B}$, respectively. We make the following key
hypotheses about $\mathbf{A}$ and $\mathbf{B}$.

######  Assumption 1.

Assume that the descended system

$Q_{1}(\mathbf{U}_{1},\dots,\mathbf{U}_{d})=\dots=Q_{1}(\mathbf{U}_{1},\dots,\mathbf{U}_{d})=0$

has codimension $d$ in $\mathbb{P}^{dn-1}$. Furthermore, assume that
$\det\mathbf{A}\neq 0$ and that the upper left $m\times m$ submatrix of
$\mathbf{B}$ is non-singular.

Our first result deals with the special case $m=1$.

######  Theorem 1.3.

Let $K/\mathbb{Q}$ be a totally real Galois extension of degree $d\geqslant
2$. Suppose that $m=1$ and that Assumption 1 holds. Assume that
$\det(\mathbf{A}^{(l)}+t\mathbf{B}^{(l_{\tau})})$ is a constant polynomial in
$t$, for each $1\leqslant l\leqslant d$, where $l_{\tau}$ is defined via
(1.4). Let $n\geqslant 6$ and assume that the descended system has non-
singular points everywhere locally. Then there exist constants $c>0$ and
$\Delta>0$ such that

$N_{W}(F;P)=cP^{(n-2)d}+O(P^{(n-2)d-\Delta}).$

The implied constants in our work are always allowed to depend on $K$ and $F$.
The generalised quadratic form
$2X_{1}X_{2}+a(X_{1}^{\tau})^{2}+\tilde{Q}(X_{3},\ldots,X_{n})$ meets the
hypotheses of the theorem, for example, where
$\tilde{Q}\in\mathfrak{o}_{K}[X_{3},\ldots,X_{n}]$ is a non-singular quadratic
form and $a\in\mathfrak{o}_{K}$ is non-zero.

We are also able to prove an asymptotic formula for $N_{W}(F;P)$ for arbitrary
$m\geqslant 1$, provided we make additional assumptions about the matrices
$\mathbf{A}$ and $\mathbf{B}$.

######  Assumption 2.

For all $1\leqslant l\leqslant d$, assume that
$\operatorname{rank}(\mathbf{A}^{(l)}+t\mathbf{B}^{(l_{\tau})})\geqslant n-1$,
for all $t\in\mathbb{R}$, where $l_{\tau}$ is defined via (1.4).

######  Assumption 3.

For all $1\leqslant l\leqslant d$, assume that
$\det(\mathbf{A}^{(l)}+t\mathbf{B}^{(l_{\tau})})$ has degree at least $m-1$,
viewed as a polynomial in $t$.

When $m=1$ and $\det(\mathbf{A}^{(l)}+t\mathbf{B}^{(l_{\tau})})$ has degree
exactly $0$ in Assumption 3, we see that Assumption 2 is implied by Assumption
1, since then
$\operatorname{rank}(\mathbf{A}^{(l)}+t\mathbf{B}^{(l_{\tau})})=\operatorname{rank}(\mathbf{A}^{(l)})=n$.
For general $m\geqslant 1$, Assumption 2 is similar to one that is commonly
made in the study of pairs of quadratic forms. Indeed, suppose one is given
two matrices $A,B\in M_{n\times n}(L)$ over an algebraically closed field $L$
of characteristic not equal to $2$, with associated quadratic forms $Q_{A}$
and $Q_{B}$. It follows from Reid’s thesis [10, Prop. 2.1] that the rank of
any element in the pencil $\lambda A+\mu B$, with $(\lambda,\mu)\neq(0,0)$, is
never smaller than $n-1$, provided the intersection $Q_{A}=Q_{B}=0$ is non-
singular as a projective variety, and of the expected dimension. In our
situation, by contrast, we only look at the pencil
$\mathbf{A}^{(l)}+t\mathbf{B}^{(l_{\tau})}$, since the matrix
$\mathbf{B}^{(l_{\tau})}$ has rank $m$ by construction. (We shall relate this
situation to the properties of an appropriate singular locus in Lemma 5.1
below.)

We are now ready to reveal our main result in the homogeneous setting.

######  Theorem 1.4.

Let $K/\mathbb{Q}$ be a totally real Galois extension of degree $d\geqslant
2$. Suppose that Assumptions 1–3 hold, and that $n>3m+4-4m/d.$ Assume that the
descended system has non-singular points everywhere locally. Then there exist
constants $c>0$ and $\Delta>0$ such that

$N_{W}(F;P)=cP^{(n-2)d}+O(P^{(n-2)d-\Delta}).$

On taking $m=1$, we note that this result subsumes Theorem 1.3 when
$n\geqslant 7$. If one makes further assumptions on $Q$ one can do even
better. Suppose, for example, that the last $n-m$ variables split off from
$Q$, so that

$Q(X_{1},\dots,X_{n})=Q_{1}(X_{1},\dots,X_{m})+Q_{2}(X_{m+1},\dots,X_{n}),$

for quadratic forms $Q_{1}$ and $Q_{2}$ over $\mathfrak{o}_{K}$. Then it seems
likely that a classical version of the circle method can be employed. On
summing trivially over the first $m$-variables of the associated exponential
sums, one would be left with handling an exponential sum in $n-m$ variables
involving $Q_{2}.$ If $Q_{2}$ has rank at least $5$, then Skinner’s treatment
over number fields [13] would yield the necessary saving. This ought to allow
$n\geqslant m+5$ in the statement of Theorem 1.4 if
$Q(0,\dots,0,X_{m+1},\dots,X_{n})$ has rank at least $5$.

###  1.2. Inhomogeneous setting

We now assume that $N\in\mathfrak{o}_{K}$ is non-zero. Then we may write
$N=\omega_{1}N_{1}+\cdots+\omega_{d}N_{d}$, where
$N_{1},\dots,N_{d}\in\mathbb{Z}$ are not all zero. We shall henceforth call
$\\{Q_{1}-N_{1},\dots,Q_{d}-N_{d}\\}$ the shifted descended system, where
$Q_{1},\dots,Q_{d}$ are obtained from $F$ via (1.2), continuing to call
$\\{Q_{1},\dots,Q_{d}\\}$ the associated descended system.

Our next result demonstrates that sharper results are available if $N\neq 0$
and $Q,R$ are both diagonal. Suppose that

$F(X_{1},\dots,X_{n})=a_{1}X_{1}^{2}+\cdots+a_{n}X_{n}^{2}+\sum_{i=1}^{m}b_{i}(X_{i}^{\tau})^{2},$
(1.5)

for $1\leqslant m\leqslant n$ and non-zero
$a_{1},\dots,a_{n},b_{1},\dots,b_{m}\in\mathfrak{o}_{K}$, and where
$\tau\in\operatorname{Gal}(K/\mathbb{Q})$ is a fixed non-trivial automorphism.
Taking $m=n$ and $a_{i}=b_{i}=1$ for $1\leqslant i\leqslant n$, we are led to
an instance of the partial trace problem in (1.1) with
$H=\\{\mathrm{id},\tau\\}$. We will prove the following result.

######  Theorem 1.5.

Let $K/\mathbb{Q}$ be a totally real Galois extension of degree $d\geqslant
2$. Assume that $N\in\mathfrak{o}_{K}$ is non-zero and that $n\geqslant m+4.$
Suppose that the descended system has codimension $d$ and a non-singular real
point and that the shifted descended system has non-singular points over
$\mathbb{Z}_{p}$ for every prime $p$. Then there exist constants $c>0$ and
$\Delta>0$ such that

$N_{W}(F,N;P)=cP^{(n-2)d}+O(P^{(n-2)d-\Delta}).$

The implied constant in this result is allowed to depend on $N$, in addition
to $K$ and $F$. In order to illustrate our result, take the quadratic number
field $K=\mathbb{Q}(\sqrt{2})$ in (1.5) and assume that
$a_{1},\dots,a_{n},b_{1},\dots,b_{m}\in\mathbb{Z}$ are all non-zero. Then it
follows from Theorem 1.5 that our work treats the shifted descended system

$\displaystyle\sum_{i=1}^{m}(a_{i}+b_{i})u_{i}^{2}+2\sum_{i=1}^{m}(a_{i}+b_{i})v_{i}^{2}+\sum_{i=m+1}^{n}a_{i}(u_{i}^{2}+2v_{i}^{2})$
$\displaystyle=N_{1},$ $\displaystyle
2\sum_{i=1}^{m}(a_{i}-b_{i})u_{i}v_{i}+2\sum_{i=m+1}^{n}a_{i}u_{i}v_{i}$
$\displaystyle=N_{2},$

when $n\geqslant m+4$ and $N_{1},N_{2}\in\mathbb{Z}$ are not both zero.

###  1.3. Some words on the proof

Let $F(X_{1},\dots,X_{n})$ be a generalised quadratic form defined over
$\mathfrak{o}_{K}$ and let $N\in\mathfrak{o}_{K}$. Our analysis of
$N_{W}(F,N;P)$ relies on a Fourier-analytic interpretation of the indicator
function

$\delta_{K}(\alpha)=\begin{cases}1,&\mbox{if $\alpha=0$},\\\ 0,&\mbox{if
$\alpha\in\mathfrak{o}_{K}\setminus\\{0\\}$}.\end{cases}$ (1.6)

Browning and Vishe [2, Thm 1.2] have extended to arbitrary number fields the
smooth $\delta$-function technology of Duke–Friedlander–Iwaniec [4], as later
refined by Heath-Brown [5]. This will underpin the work in this paper,
affording us the opportunity to extract non-trivial savings, in the spirit of
Kloosterman’s method, in the proof of Theorem 1.5. We will be led to an
expression for $N_{W}(F,N;P)$ in (4.6), involving an infinite sum over non-
zero integral ideals $\mathfrak{b}$. The next stage is to apply Poisson
summation, but an obstacle arises from the fact that it is no longer possible
to break into residue classes modulo $\mathfrak{b}$ for generalised quadratic
forms $F$. Instead we shall break into residue classes modulo a larger ideal
${{}^{{G}}\mathfrak{b}}$, which is the least common multiple of the ideals
$\mathfrak{b}^{\tau^{-1}}$, as $\tau$ ranges over the automorphisms that
actually occur in $F$. Poisson summation then leads to the analysis of certain
exponential sums $S_{\mathfrak{b}}(N;\mathbf{m})$ and oscillatory integrals
$I_{\mathfrak{b}}(N;\mathbf{m})$, which are indexed by
$\mathfrak{b}\subset\mathfrak{o}_{K}$ and suitable vectors $\mathbf{m}\in
K^{n}$. While the treatment of $S_{\mathfrak{b}}(N;\mathbf{m})$ is relatively
standard, the main challenge is to understand
$I_{\mathfrak{b}}(N;\mathbf{m})$. When $F$ is a standard quadratic form, these
integrals factorise into a product of $d$ oscillatory integrals, one for each
of the $d$ real embeddings of $K$. This reduces the problem to looking at
oscillatory integrals over $\mathbb{R}^{n}$. For generic generalised quadratic
forms, it seems very difficult to obtain the kind of cancellation one needs
for the method to go through, for the relevant oscillatory integrals over
$\mathbb{R}^{dn}$.

We now summarise the contents of the paper. In §2 we shall prove Theorem 1.2
by spelling out the connection between generalised quadratic forms over $K$
and descended systems over $\mathbb{Q}$. In §3 we collect together some useful
facts from algebraic number theory. The rest of the paper will be concerned
with estimating the size of the counting function $N_{W}(F,N;P)$, as
$P\to\infty$. In order to facilitate future investigation we shall present
most of the arguments for arbitrary generalised quadratic forms in §4. Next,
in §5 we shall specialise to the case (1.3) and $N=0$, in order to deduce
Theorems 1.3 and 1.4. Finally, §6 will deal with Theorem 1.5, which pertains
to the diagonal generalised quadratic form (1.5) and $N\neq 0$.

######  Acknowledgements.

The authors are grateful to Jayce Getz for asking questions that set this
project in motion. T.B. was supported by FWF grant P 32428-N35 and by a grant
from the Institute for Advanced Study School of Mathematics. L.B.P. was
partially supported by NSF DMS-2200470 and DMS-1652173, and thanks the
Hausdorff Centre for Mathematics for hosting research visits.

##  2\. Generalised quadratic forms and the descended system

In this section we shall prove Theorem 1.2, by making explicit the
correspondence between generalised quadratic forms $F$ and the descended
system of $d$ quadratic forms over ${\mathbb{Q}}$ in $dn$ variables. Let
$K/\mathbb{Q}$ be a degree $d$ Galois number field, which (in this section
only) need not be totally real. Assume that we are given a set of coefficients
$(c_{i,j,\tau,\tau^{\prime}})$ of a generalised quadratic form, with
$c_{i,j,\tau,\tau^{\prime}}=c_{j,i,\tau^{\prime},\tau}$ for all $1\leqslant
i,j\leqslant n$ and $\tau,\tau^{\prime}\in\operatorname{Gal}(K/\mathbb{Q})$.
We can write each coefficient $c_{i,j,\tau,\tau^{\prime}}\in K$ with respect
to the basis $\\{{\omega}_{1},\ldots,{\omega}_{d}\\}$ as
$c_{i,j,\tau,\tau^{\prime}}=\sum_{k=1}^{d}c_{i,j,\tau,\tau^{\prime}}^{(k)}{\omega}_{k}.$
We proceed to compute the descended system explicitly, by writing
$X_{i}=\sum_{k=1}^{d}U_{k,i}{\omega}_{k},$ for $1\leqslant i\leqslant n$. Then

$F(X_{1},\dots,X_{n})=\sum_{1\leqslant i,j\leqslant
n}\sum_{\tau,\tau^{\prime}\in\operatorname{Gal}(K/\mathbb{Q})}\sum_{1\leqslant
l,m,k\leqslant
d}c_{i,j,\tau,\tau^{\prime}}^{(k)}{\omega}_{k}U_{l,i}{\omega}_{l}^{\tau}U_{m,j}{\omega}_{m}^{\tau^{\prime}}.$

Let $\\{\rho_{1},\ldots,\rho_{d}\\}$ be a dual basis of
$\\{{\omega}_{1},\ldots,{\omega}_{d}\\}$ with respect to the trace, so that
$(\operatorname{Tr}_{K/\mathbb{Q}}(\rho_{i}\omega_{j}))_{i,j}$ is the identity
matrix and any $\alpha\in K$ can be written in the form
$\alpha=\sum_{p=1}^{d}\operatorname{Tr}_{K/\mathbb{Q}}(\alpha\rho_{p})\omega_{p}$.
Thus $F(X_{1},\ldots,X_{n})$ is equal to

$\sum_{p=1}^{d}{\omega}_{p}\sum_{1\leqslant i,j\leqslant n}\sum_{1\leqslant
l,m\leqslant
d}U_{l,i}U_{m,j}\operatorname{Tr}_{K/\mathbb{Q}}\left(\rho_{p}\sum_{1\leqslant
k\leqslant
d}\sum_{\tau,\tau^{\prime}\in\operatorname{Gal}(K/\mathbb{Q})}c_{i,j,\tau,\tau^{\prime}}^{(k)}{\omega}_{k}{\omega}_{l}^{\tau}{\omega}_{m}^{\tau^{\prime}}\right)$

and we arrive at our descended system (1.2), with

$Q_{p}(\underline{\mathbf{U}})=\sum_{1\leqslant i,j\leqslant
n}\sum_{1\leqslant l,m\leqslant d}\beta_{p,l,i,m,j}U_{l,i}U_{m,j},$

for rational coefficients

$\displaystyle\beta_{p,l,i,m,j}$
$\displaystyle=\operatorname{Tr}_{K/\mathbb{Q}}\left(\rho_{p}\sum_{1\leqslant
k\leqslant
d}\sum_{\tau,\tau^{\prime}\in\operatorname{Gal}(K/\mathbb{Q})}c_{i,j,\tau,\tau^{\prime}}^{(k)}{\omega}_{k}{\omega}_{l}^{\tau}{\omega}_{m}^{\tau^{\prime}}\right)$
$\displaystyle=\sum_{1\leqslant k\leqslant
d}\sum_{\tau,\tau^{\prime}\in\operatorname{Gal}(K/\mathbb{Q})}c_{i,j,\tau,\tau^{\prime}}^{(k)}\operatorname{Tr}_{K/\mathbb{Q}}(\rho_{p}{\omega}_{k}{\omega}_{l}^{\tau}{\omega}_{m}^{\tau^{\prime}}).$

By construction, the coefficients $\beta_{p,l,i,m,j}$ satisfy
$\beta_{p,l,i,m,j}=\beta_{p,m,j,l,i}$, for all $1\leqslant p,l,m\leqslant d$
and $1\leqslant i,j\leqslant n$. Moreover, they depend linearly on the given
set of coefficients $(c_{i,j,\tau,\tau^{\prime}}^{(k)})$. Now the space of all
tuples $(c_{i,j,\tau,\tau^{\prime}}^{(k)})$ of rational numbers satisfying the
symmetry relation
$c_{i,j,\tau,\tau^{\prime}}^{(k)}=c_{j,i,\tau^{\prime},\tau}^{(k)}$ can be
parametrised by ${\mathbb{Q}}^{\frac{1}{2}dn(dn+1)d}$. Similarly, the space of
all symmetric rational tuples $(\beta_{p,l,i,m,j})$ is naturally parametrised
by ${\mathbb{Q}}^{\frac{1}{2}dn(dn+1)d}$. We define the map

$\Phi:{\mathbb{Q}}^{\frac{1}{2}dn(dn+1)d}\rightarrow{\mathbb{Q}}^{\frac{1}{2}dn(dn+1)d},\quad(c_{i,j,\tau,\tau^{\prime}}^{(k)})\mapsto(\beta_{p,l,i,m,j}).$

We claim that this map is an injective linear map. This implies that there is
a bijection between generalised quadratic forms in $n$ variables and systems
of $d$ rational quadratic forms in $nd$ variables, as claimed in Theorem 1.2.

To check the claim we assume that $\beta_{p,l,i,m,j}=0$ for all $1\leqslant
p,l,m\leqslant d$ and $1\leqslant i,j\leqslant n$. By the non-degeneracy of
the trace as a bilinear form, we deduce that

$\sum_{\tau,\tau^{\prime}\in\operatorname{Gal}(K/\mathbb{Q})}c_{i,j,\tau,\tau^{\prime}}{\omega}_{l}^{\tau}{\omega}_{m}^{\tau^{\prime}}=0,\quad
1\leqslant i,j\leqslant n,\ 1\leqslant l,m\leqslant d.$

Note that the matrix $({\omega}_{l}^{\tau})_{\begin{subarray}{c}1\leqslant
l\leqslant d\\\ \tau\in\operatorname{Gal}(K/\mathbb{Q})\end{subarray}}$ is of
maximal rank, and hence we obtain

$\sum_{\tau^{\prime}\in\operatorname{Gal}(K/\mathbb{Q})}c_{i,j,\tau,\tau^{\prime}}{\omega}_{m}^{\tau^{\prime}}=0,\quad
1\leqslant i,j\leqslant n,\ \tau\in\operatorname{Gal}(K/\mathbb{Q}),\
1\leqslant m\leqslant d.$

Applying the same argument again, we finally obtain

$c_{i,j,\tau,\tau^{\prime}}=0,\quad 1\leqslant i,j\leqslant n,\
\tau,\tau^{\prime}\in\operatorname{Gal}(K/\mathbb{Q}),$

and hence $c_{i,j,\tau,\tau^{\prime}}^{(k)}=0$ for all $1\leqslant k\leqslant
d$.

##  3\. Recap from algebraic number theory

In this section we collect together some of the facts about algebraic number
fields that are important in our work. As usual $K/\mathbb{Q}$ is a totally
real Galois extension of degree $d$. We shall henceforth write
$\mathfrak{o}=\mathfrak{o}_{K}$ for its ring of integers. In §3.1 and §3.2 we
recall some facts about ideals and discuss the construction of primitive
characters modulo ideals, respectively. The need to deal with generalised
quadratic forms naturally leads to two basic objects that can be associated to
a given integral ideal $\mathfrak{b}$ in $K$, both of which depend on the
particular generalised quadratic form we are working with and will be
introduced in §3.3.

###  3.1. Properties of ideals

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

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

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

$\mathfrak{a}_{\gamma}=\\{\alpha\in\mathfrak{o}:\alpha\gamma\in\mathfrak{o}\\},$

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

######  Lemma 3.1.

Let $\varepsilon>0$ and let $\mathfrak{b},\mathfrak{c}$ be integral ideals.
Then

  1. (i)

there exists $\alpha\in\mathfrak{b}$ such that
$\mathrm{ord}_{\mathfrak{p}}(\alpha)=\mathrm{ord}_{\mathfrak{p}}(\mathfrak{b})$
for every prime ideal $\mathfrak{p}\mid\mathfrak{c}$;

  2. (ii)

there exists $\alpha\in\mathfrak{b}$ and an unramified prime ideal
$\mathfrak{p}$ coprime to $\mathfrak{b}^{\tau}$ for all
$\tau\in\operatorname{Gal}(K/\mathbb{Q})$, with
$\operatorname{N}\mathfrak{p}\ll(\operatorname{N}\mathfrak{b})^{\varepsilon}$,
such that $(\alpha)=\mathfrak{b}\mathfrak{p}$.

###### Proof.

Part (i) is [2, Lemma 2.2(i)] and part (ii) follows from an obvious
modification to the proof of [2, Lemma 2.2(ii)]. ∎

We shall also require a version of the Chinese remainder theorem, as in [12,
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

$\mathfrak{o}/\mathfrak{a}=\\{\alpha_{1}\mu+\alpha_{2}\beta:\beta\in\mathfrak{o}/\mathfrak{a}_{1},~{}\mu\in\mathfrak{o}/\mathfrak{a}_{2}\\}.$

###  3.2. Construction of primitive characters

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

######  Lemma 3.3.

Let $\sigma_{0}(\cdot)=\psi(\gamma\cdot):K\to\mathbb{C}$, for any $\gamma\in
K$, and let $\mathfrak{b}\subsetneq\mathfrak{o}$ be an integral ideal. Then
$\sigma_{0}$ is a non-trivial primitive additive character modulo
$\mathfrak{b}$ if and only if $\mathfrak{a}_{\gamma}=\mathfrak{b}\mathfrak{e}$
for some $\mathfrak{e}\mid\mathfrak{d}$ such that
$(\mathfrak{d}/\mathfrak{e},\mathfrak{b})=1$.

###### Proof.

We begin by showing that $\sigma_{0}$ is an additive character modulo
$\mathfrak{b}$ if and only if
$\mathfrak{b}\mathfrak{d}\subset\mathfrak{a}_{\gamma}$. But $\sigma_{0}$ is an
additive character modulo $\mathfrak{b}$ if and only if
$\sigma_{0}(x+z)=\sigma_{0}(x)$ for all $x\in\mathfrak{o}$ and
$z\in\mathfrak{b}$. But this happens if and only if $\gamma
z\in\hat{\mathfrak{o}}$ for all $z\in\mathfrak{b}$, which is if and only if
$\mathfrak{b}\mathfrak{d}\subset\mathfrak{a}_{\gamma}$. This establishes the
claim.

Now suppose that $\mathfrak{b}\mathfrak{d}\subset\mathfrak{a}_{\gamma}$, which
means that $\mathfrak{a}_{\gamma}\mid\mathfrak{b}\mathfrak{d}$. Thus there is
an integral ideal $\mathfrak{h}$ such that
$\mathfrak{b}\mathfrak{d}=\mathfrak{a}_{\gamma}\mathfrak{h}$. We wish to show
that $\sigma_{0}$ is primitive if and only if $\mathfrak{h}\mid\mathfrak{d}$
with $(\mathfrak{h},\mathfrak{b})=1$. To do so we note that $\sigma_{0}$ is
primitive if and only if
$\mathfrak{a}_{\gamma}\nmid\mathfrak{b}_{1}\mathfrak{d}$ for all
$\mathfrak{b}_{1}\mid\mathfrak{b}$ with $\mathfrak{b}_{1}\neq\mathfrak{b}$.
Indeed, if $\mathfrak{a}_{\gamma}\mid\mathfrak{b}_{1}\mathfrak{d}$ for some
proper divisor $\mathfrak{b}_{1}\mid\mathfrak{b}$, then $\gamma
z\in\hat{\mathfrak{o}}$ for every $z\in\mathfrak{b}_{1}$, which would mean
that $\sigma_{0}$ is a character modulo $\mathfrak{b}_{1}$. Suppose that
$\sigma_{0}$ is primitive and suppose that there is a prime ideal
$\mathfrak{p}\mid\mathfrak{h}$ such that $\mathfrak{p}\mid\mathfrak{b}$.
Writing $\mathfrak{h}^{\prime}=\mathfrak{h}\mathfrak{p}^{-1}$ and
$\mathfrak{b}^{\prime}=\mathfrak{b}\mathfrak{p}^{-1}$ it follows that
$\mathfrak{b}^{\prime}\mathfrak{d}=\mathfrak{a}_{\gamma}\mathfrak{h}^{\prime}$,
whence $\mathfrak{a}_{\gamma}\mid\mathfrak{b}^{\prime}\mathfrak{d}$, which is
a contradiction. Thus $\mathfrak{h}$ is coprime to $\mathfrak{b}$ and we must
have $\mathfrak{h}\mid\mathfrak{d}$. Suppose now that
$\mathfrak{b}\mathfrak{d}=\mathfrak{a}_{\gamma}\mathfrak{h}$ for some
$\mathfrak{h}\mid\mathfrak{d}$ such that $(\mathfrak{h},\mathfrak{b})=1$. We
wish to deduce that $\sigma_{0}$ is primitive, for which we suppose for a
contradiction that there exists a proper divisor
$\mathfrak{b}_{1}\mid\mathfrak{b}$ such that
$\mathfrak{a}_{\gamma}\mid\mathfrak{b}_{1}\mathfrak{d}$. Writing
$\mathfrak{b}=\mathfrak{b}_{1}\mathfrak{b}_{2}$ and recalling that
$\mathfrak{b}\mathfrak{d}=\mathfrak{a}_{\gamma}\mathfrak{h}$, we deduce that
$\mathfrak{h}=(\mathfrak{a}_{\gamma}^{-1}\mathfrak{b}_{1}\mathfrak{d})\mathfrak{b}_{2}$,
whence $\mathfrak{b}_{2}\mid\mathfrak{h}$, which is impossible since
$(\mathfrak{h},\mathfrak{b})=1$.

Finally we note that $\sigma_{0}$ is a trivial character if and only if
$\gamma\in\hat{\mathfrak{o}}$, which is equivalent to
$\mathfrak{a}_{\gamma}\supseteq\mathfrak{d}$. This is clearly impossible for
any primitive character $\sigma_{0}$ modulo a proper ideal
$\mathfrak{b}\subsetneq\mathfrak{o}$, since
$(\mathfrak{d}/\mathfrak{e},\mathfrak{b})=1$ if
$\mathfrak{a}_{\gamma}=\mathfrak{b}\mathfrak{e}$. ∎

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

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

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
$\tau\in\operatorname{Gal}(K/\mathbb{Q})$. We may now record a variant of [2,
Lemma 2.3], which shows that $\mathfrak{F}(\mathfrak{b})\neq\emptyset$ for any
choice of $\mathfrak{b}$.

######  Lemma 3.4.

Let $\mathfrak{b}\subsetneq\mathfrak{o}$ be a non-zero ideal. Then there
exists $\gamma\in\mathfrak{F}(\mathfrak{b})$ such that $\psi(\gamma\cdot)$
defines a non-trivial primitive additive character modulo $\mathfrak{b}$.

###### Proof.

We consider the integral ideal $\mathfrak{c}=\mathfrak{b}\mathfrak{d}$.
Observe that $\mathfrak{d}^{\tau}=\mathfrak{d}$ for all
$\tau\in\operatorname{Gal}(K/\mathbb{Q})$, 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
$\tau\in\operatorname{Gal}(K/\mathbb{Q})$, with
$\operatorname{N}\mathfrak{p}_{1}\ll\operatorname{N}\mathfrak{b}$, 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
$\tau\in\operatorname{Gal}(K/\mathbb{Q})$. But this implies that
$g=N_{K/\mathbb{Q}}(\nu)$ is coprime to $\mathfrak{c}^{\tau}$, for any
$\tau\in\operatorname{Gal}(K/\mathbb{Q})$, 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

$\beta\in\mathfrak{a}_{\gamma}\Leftrightarrow\gamma\beta\in\mathfrak{o}\Leftrightarrow(g\beta)\subset(\alpha)\Leftrightarrow(\alpha)=\mathfrak{b}\mathfrak{d}\mathfrak{p}_{1}\mid(g\beta)\Leftrightarrow\beta\in\mathfrak{b}\mathfrak{d},$

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
$G=G_{F}\subset\operatorname{Gal}(K/\mathbb{Q})$ be the subset of
automorphisms $\tau\in\operatorname{Gal}(K/\mathbb{Q})$ 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

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

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

$B(X_{1},\dots,X_{n};Y_{1},\dots,Y_{n})=\sum_{1\leqslant i,j\leqslant
n}\sum_{\tau,\tau^{\prime}\in\operatorname{Gal}(K/\mathbb{Q})}c_{i,j,\tau,\tau^{\prime}}X_{i}^{\tau}Y_{j}^{\tau^{\prime}}.$
(3.2)

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

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

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

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

This is an additive group and it is clear that
${{}^{{G}}\mathfrak{b}}^{n}\subset{\mathscr{H}}_{\mathfrak{b}}\subset{\mathfrak{o}}^{n}$,
where ${{}^{{G}}\mathfrak{b}}$ is the $G$-invariant ideal defined in (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 ${\mathbf{h}}\in{\mathscr{H}}_{\mathfrak{b}}$. Hence

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

We claim that ${\mathscr{H}}_{\mathfrak{b}}$ has the structure of a finitely
generated ${\mathbb{Z}}$-module. To see this, let ${\mathbf{e}}_{i}$ be the
$i$th unit vector, for $1\leqslant i\leqslant n$. Observe that
$(\operatorname{N}{\mathfrak{b}}){\omega}_{j}{\mathbf{e}}_{i}\in{\mathscr{H}}_{\mathfrak{b}}$
for all $1\leqslant i\leqslant n$ and $1\leqslant j\leqslant d$. Hence the
image of ${\mathscr{H}}_{\mathfrak{b}}$ 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 ${\mathscr{H}}_{\mathfrak{b}}$ 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 ${\mathscr{H}}_{\mathfrak{b}}$ directly, but for now we
content ourselves with proving a general bound. In the following lemma we
consider the coefficient matrix
$(c_{i,j,\tau,\tau^{\prime}})_{(i,\tau)\times(j,\tau^{\prime})}$ 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

$|{\mathfrak{o}}^{n}/{\mathscr{H}}_{\mathfrak{b}}|\leqslant
C_{1}(\operatorname{N}{\mathfrak{b}})^{\operatorname{rank}(c_{i,j,\tau,\tau^{\prime}})}.$

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

$c_{k,j,\sigma,\tau^{\prime}}=\sum_{(i,\tau)\in{\mathscr{S}}}a_{i,\tau}^{(k,\sigma)}c_{i,j,\tau,\tau^{\prime}},$

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

${\mathscr{H}}_{\mathfrak{b}}^{\prime}=\left\\{{\mathbf{h}}\in{\mathfrak{o}}^{n}:\sum_{1\leqslant
j\leqslant
n}\sum_{\tau^{\prime}\in\operatorname{Gal}(K/\mathbb{Q})}c_{i,j,\tau,\tau^{\prime}}h_{j}^{\tau^{\prime}}\in({\alpha}){\mathfrak{b}},\,\forall(i,\tau)\in{\mathscr{S}}\right\\}.$

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

Finally, we observe that there is an injection

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

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

We now wish to provide an alternative upper bound involving
${\mathscr{H}}_{\mathfrak{b}}$, 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

${\mathbf{v}}_{1},\ldots,{\mathbf{v}}_{n}\in K^{n}$

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
non-singular. 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

$|{\mathscr{H}}_{\mathfrak{b}}/{{}^{{G}}\mathfrak{b}}^{n}|\leqslant
C_{2}\frac{(\operatorname{N}{{}^{{G}}\mathfrak{b}})^{n}}{(\operatorname{N}{\mathfrak{b}})^{n}}.$

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
${\mathscr{H}}_{\mathfrak{b}}$ in $\mathfrak{o}^{n}$ via the identity

$|\mathfrak{o}^{n}/{\mathscr{H}}_{\mathfrak{b}}||{\mathscr{H}}_{\mathfrak{b}}/{{}^{{G}}\mathfrak{b}}^{n}|=|\mathfrak{o}^{n}/{{}^{{G}}\mathfrak{b}}^{n}|=(\operatorname{N}{{}^{{G}}\mathfrak{b}})^{n}.$
(3.4)

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

###### 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

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

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

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

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

${\mathfrak{o}}^{n}/\widetilde{{\mathscr{H}}}_{\mathfrak{b}}\cong{\Gamma}/({\mathfrak{b}}^{n}\cap{\Gamma}).$

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 (3.4), we therefore obtain

$|{\mathscr{H}}_{\mathfrak{b}}/{{}^{{G}}\mathfrak{b}}^{n}|\leqslant|\widetilde{{\mathscr{H}}_{\mathfrak{b}}}/{{}^{{G}}\mathfrak{b}}^{n}|=\frac{|{\mathfrak{o}}^{n}/{{}^{{G}}\mathfrak{b}}^{n}|}{|{\mathfrak{o}}^{n}/\widetilde{{\mathscr{H}}}_{\mathfrak{b}}|}=\frac{(\operatorname{N}{{}^{{G}}\mathfrak{b}})^{n}}{|{\Gamma}/({\mathfrak{b}}^{n}\cap{\Gamma})|}\leqslant
C_{\Gamma}\frac{(\operatorname{N}{{}^{{G}}\mathfrak{b}})^{n}}{(\operatorname{N}{\mathfrak{b}})^{n}},$

where $C_{\Gamma}$ is a constant only depending on ${\Gamma}$. Moreover, there
is an ideal $\mathfrak{d}_{2}$ such that
$|{\Gamma}/({\mathfrak{b}}^{n}\cap{\Gamma})|=(\operatorname{N}{\mathfrak{b}})^{n}$
whenever $(\mathfrak{d}_{2},{\mathfrak{b}})=1$. This completes the proof of
the lemma. ∎

##  4\. Enter the circle method

Our primary tool in this paper is a number field version of the
Hardy–Littlewood circle method to interpret the function $\delta_{K}$ in
(1.6). Let $K$ be a totally real Galois extension of $\mathbb{Q}$ of degree
$d$. Let $Q\geqslant 1$ and let $\alpha\in\mathfrak{o}$. Then we shall use the
version worked out by Browning and Vishe [2, Thm. 1.2]. This states that there
exists a positive constant $c_{Q}=1+O_{A}(Q^{-A})$, for any $A>0$, and an
infinitely differentiable function
$h:(0,\infty)\times\mathbb{R}\rightarrow\mathbb{R}$ such that

$\delta_{K}(\alpha)=\frac{c_{Q}}{Q^{2d}}\sum_{(0)\neq\mathfrak{b}\subseteq\mathfrak{o}}~{}\sideset{}{{}^{*}}{\sum}_{\sigma\,(\operatorname{mod}{\mathfrak{b}})}\sigma(\alpha)h\left(\frac{\operatorname{N}\mathfrak{b}}{Q^{d}},\frac{|\operatorname{N}_{K/\mathbb{Q}}(\alpha)|}{Q^{2d}}\right),$
(4.1)

where $\operatorname{N}\mathfrak{b}=|\mathfrak{o}/\mathfrak{b}|$ denotes the
norm of the ideal $\mathfrak{b}$ and the notation
$\sum^{*}_{\sigma\,(\operatorname{mod}{\mathfrak{b}})}$ means that the sum is
taken over primitive additive characters modulo $\mathfrak{b}$. Furthermore,
we have $h(x,y)\ll x^{-1}$ and $h(x,y)\neq 0$ only if
$x\leqslant\max\\{1,2|y|\\}$.

We fix some notation, before proceeding further. Let $D_{K}$ be the
discriminant of $K$ and note that $D_{K}>0$, since $K$ is totally real. Let
$\rho_{1},\dots,\rho_{d}:K\hookrightarrow\mathbb{R}$ be the distinct real
embeddings of $K$, and let
$V=K\otimes_{\mathbb{Q}}\mathbb{R}\cong\mathbb{R}^{d}$. There is a canonical
embedding $K\hookrightarrow V$ given by
$\alpha\mapsto(\rho_{1}(\alpha),\dots,\rho_{d}(\alpha))$. We identify $K$ with
its image in $V$. If $v=(v_{1},\dots,v_{d})\in V$ then we extend the norm and
trace on $K$ to get functions $\operatorname{Nm}(v):V\to\mathbb{R}$ and
$\operatorname{Tr}(v):V\to\mathbb{R}$, with

$\operatorname{Nm}(v)=\prod_{l=1}^{d}v_{l},\quad\operatorname{Tr}(v)=\sum_{l=1}^{d}v_{l}.$

We extend the absolute value on $\mathbb{R}$ to give a norm on $V$ via
$|v|=\max_{1\leqslant l\leqslant d}|v_{l}|$, which we extend to $V^{n}$ in the
obvious way.

Let $N\in\mathfrak{o}$ and let $F(X_{1},\dots,X_{n})$ be a generalised
quadratic form defined over $\mathfrak{o}$. Our central concern is with the
asymptotic behaviour of the sum

$N_{W}(F,N;P)=\sum_{\begin{subarray}{c}\mathbf{x}\in\mathfrak{o}^{n}\\\
F(\mathbf{x})=N\end{subarray}}W(\mathbf{x}/P),$

as $P\rightarrow\infty$, for $W\in\mathscr{W}_{n}^{+}(V)$, where
$\mathscr{W}_{n}^{+}(V)$ is the class of smooth weight functions described in
[2, §2.2]. Our goal in this section is to lay some groundwork that will be
useful for Theorems 1.3–1.5, but which applies to arbitrary generalised
quadratic forms.

First, in §4.1 we shall discuss the link between the descended system
associated to $F$ and the “embedded system” that arises from looking at all of
the different real embeddings of $F$. In §4.2 we shall construct the weight
function $W$ that features in our counting function $N_{W}(F,N;P)$. In §4.3 we
shall combine (4.1) with Poisson summation, in order to arrive at a
preliminary expression for $N_{W}(F,N;P)$ in Lemma 4.1. In §4.4 we make some
preliminary investigations into exponential sums, and similarly for
exponential integrals in §4.5. In §4.6 we shall discuss the main term that
comes from the trivial character after Poisson summation is applied. Finally,
in §4.7 we shall make some initial observations concerning the contribution
from the non-trivial characters.

###  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 (1.2) how there is a descended system
$\\{Q_{1},\dots,Q_{d}\\}$ of quadratic forms, that is associated to $F$ via

$F(X_{1},\dots,X_{n})=\sum_{1\leqslant i\leqslant
d}{\omega_{i}}Q_{i}(\mathbf{U}_{1},\dots,\mathbf{U}_{d}),$

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 $\tau\in\operatorname{Gal}(K/\mathbb{Q})$ may be
associated a unique integer $l_{\tau}\in\\{1,\dots,d\\}$ such that (1.4)
holds. Then we define

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

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

$\rho_{l}(F(\mathbf{x}))=F^{(l)}(\mathbf{x}^{(1)},\dots,\mathbf{x}^{(d)}).$

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 $N_{K/\mathbb{Q}}(F(\mathbf{x}))=\operatorname{Nm}(F(\mathbf{x}))$ and

$\operatorname{Tr}(vF(\mathbf{x}))=\sum_{1\leqslant l\leqslant
d}v_{l}\rho_{l}\left(F(\mathbf{x})\right),$ (4.3)

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

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

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

$\mathbf{W}=\begin{pmatrix}\omega_{1}^{(1)}\mathbf{I}_{n}&\omega_{2}^{(1)}\mathbf{I}_{n}&\dots&\omega_{d}^{(1)}\mathbf{I}_{n}\\\
\omega_{1}^{(2)}\mathbf{I}_{n}&\omega_{2}^{(2)}\mathbf{I}_{n}&\dots&\omega_{d}^{(2)}\mathbf{I}_{n}\\\
{4}\\\
\omega_{1}^{(d)}\mathbf{I}_{n}&\omega_{2}^{(d)}\mathbf{I}_{n}&\dots&\omega_{d}^{(d)}\mathbf{I}_{n}\end{pmatrix}.$
(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

$F^{(l)}(\mathbf{x}^{(1)},\dots,\mathbf{x}^{(d)})=\sum_{1\leqslant i\leqslant
d}\omega_{i}^{(l)}Q_{i}(\mathbf{u}_{1}\dots,\mathbf{u}_{d}),$

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 non-
singular 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

$J_{Q_{1},\dots,Q_{d}}=\left(\frac{\partial}{\partial
X_{j}^{(k)}}Q_{l}\right)_{\begin{subarray}{c}1\leqslant l\leqslant d\\\
1\leqslant k\leqslant d,1\leqslant j\leqslant n\end{subarray}}$

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

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

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

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

where $\mathbf{x}$ is identified with
$(\mathbf{x}^{(1)},\dots,\mathbf{x}^{(d)})$, and where $\mathbf{W}$ is the
matrix in (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 (4.1) that

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

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 the definition of
$\mathscr{W}_{n}(V)$ in [2, §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 $\operatorname{N}\mathfrak{b}\ll Q^{d}=P^{d}$.

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 [2, 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 ${\mathscr{H}}_{\mathfrak{b}}$, that was introduced in
§3.3.

######  Lemma 4.1.

We have

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

where the sum over $\mathfrak{b}$ is over non-zero integral ideals and

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

###### Proof.

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

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

for any primitive character $\sigma$ modulo $\mathfrak{b}$. We apply the
multi-dimensional Poisson summation formula (cf. [2, §5]). Recalling that $K$
is totally real, we find that the inner $\mathbf{x}$-sum is equal to

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

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 (4.6),
we have therefore established that

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

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

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

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

$\sideset{}{{}^{*}}{\sum}_{\sigma\,(\operatorname{mod}{\mathfrak{b}})}\sigma(x)=\sum_{a\in(\mathfrak{o}/\mathfrak{b})^{*}}\psi(\gamma
ax),$

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

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

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

######  Lemma 4.2.

We have

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

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

###### Proof.

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

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

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

$2{\gamma}aB({\mathbf{u}};{\mathbf{h}})\in{\mathfrak{d}}^{-1},\qquad\forall{\mathbf{u}}\in(\mathfrak{o}/{{}^{G}\mathfrak{b}})^{n}.$

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

$2gaB({\mathbf{u}};{\mathbf{h}})\in(\alpha){\mathfrak{d}}^{-1}={\mathfrak{b}}{\mathfrak{p}}_{1},\qquad\forall{\mathbf{u}}\in({\mathfrak{o}}/{{}^{G}\mathfrak{b}})^{n}.$

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

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

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

######  Corollary 4.3.

Assume that $F$ is admissible, in the sense of Definition 3.6. Let
${\mathfrak{b}}$ be an integral ideal and let ${\mathbf{m}}\in K^{n}$. Then
$S_{\mathfrak{b}}(N;{\mathbf{m}})\ll(\operatorname{N}\mathfrak{b})^{1-n/2}(\operatorname{N}{{}^{{G}}\mathfrak{b}})^{n}.$

###### Proof.

This follows from combining Lemmas 3.7 and 4.2. ∎

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.

######  Lemma 4.4.

We have $S_{\mathfrak{b}}(N;\mathbf{m})=0$ unless
$\mathbf{m}.\mathbf{h}\in\mathfrak{d}^{-1}$ for all
$\mathbf{h}\in\mathscr{H}_{\mathfrak{b}}$.

###### Proof.

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

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

However, orthogonality of characters gives

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

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 non-zero integral ideal and suppose that
$\mathfrak{b}=\mathfrak{b}_{1}\mathfrak{b}_{2}$ for integral ideals
$\mathfrak{b}_{1},\mathfrak{b}_{2}$, such that
$\gcd(\operatorname{N}\mathfrak{b}_{1},\operatorname{N}\mathfrak{b}_{2})=1.$
Then, for any $N\in\mathfrak{o}$ and any
$\mathbf{m}\in\widehat{{{}^{{G}}\mathfrak{b}}}^{n}$, we have

$S_{\mathfrak{b}}(N;{\mathbf{m}})=S_{\mathfrak{b}_{1}}(\overline{\operatorname{N}\mathfrak{b}_{2}}^{2}N;(\operatorname{N}\mathfrak{b}_{2})\mathbf{m})S_{\mathfrak{b}_{2}}(\overline{\operatorname{N}\mathfrak{b}_{1}}^{2}N;(\operatorname{N}\mathfrak{b}_{1})\mathbf{m}).$

###### 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, (4.7) implies that

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

Let us write $\operatorname{N}\mathfrak{b}_{i}=b_{i}$ for $i=1,2$. The
assumption $\gcd(b_{1},b_{2})=1$ implies that
$({{}^{{G}}\mathfrak{b}}_{1},{{}^{{G}}\mathfrak{b}}_{2})=1$. Moreover, we have
$b_{1}\in\mathfrak{b}_{1}$, $b_{2}\in\mathfrak{b}_{2}$ and

$((b_{1}),\mathfrak{b}_{2})=((b_{2}),\mathfrak{b}_{1})=1.$ (4.8)

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 $b\,(\operatorname{mod}{\mathfrak{b}_{1}})$ and
$c\,(\operatorname{mod}{\mathfrak{b}_{2}})$. Likewise, we claim that we can
write $\mathbf{x}=b_{2}\mathbf{b}+b_{1}\mathbf{c}$, for
$\mathbf{b}\,(\operatorname{mod}{{{}^{{G}}\mathfrak{b}}_{1}})$ and
$\mathbf{c}\,(\operatorname{mod}{{{}^{{G}}\mathfrak{b}}_{2}})$. 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 $(u,v)\mapsto b_{2}u+b_{1}v$. 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 (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

$\displaystyle S_{\mathfrak{b}}(N;\mathbf{m})=~{}$
$\displaystyle\sum_{\begin{subarray}{c}b\in(\mathfrak{o}/\mathfrak{b}_{1})^{*}\\\
c\in(\mathfrak{o}/\mathfrak{b}_{2})^{*}\end{subarray}}\psi(-\gamma(\mu
b+\lambda c)N)$
$\displaystyle\quad\times\sum_{\begin{subarray}{c}\mathbf{b}\,(\operatorname{mod}{{{}^{{G}}\mathfrak{b}}_{1}})\\\
\mathbf{c}\,(\operatorname{mod}{{{}^{{G}}\mathfrak{b}}_{2}})\end{subarray}}\psi\left(\gamma(\mu
b+\lambda
c)F(b_{2}\mathbf{b}+b_{1}\mathbf{c})+\mathbf{m}.(b_{2}\mathbf{b}+b_{1}\mathbf{c})\right)$
$\displaystyle=~{}$
$\displaystyle\sum_{b\in(\mathfrak{o}/\mathfrak{b}_{1})^{*}}\psi(-\gamma\mu
bN)\sum_{\mathbf{b}\,(\operatorname{mod}{{{}^{{G}}\mathfrak{b}}_{1}})}\psi\left(\gamma\mu
b_{2}^{2}bF(\mathbf{b})+b_{2}\mathbf{m}.\mathbf{b}\right)$
$\displaystyle\quad\times\sum_{c\in(\mathfrak{o}/\mathfrak{b}_{2})^{*}}\psi(-\gamma\lambda
cN)\sum_{\mathbf{c}\,(\operatorname{mod}{{{}^{{G}}\mathfrak{b}}_{2}})}\psi\left(\gamma\lambda
b_{1}^{2}cF(\mathbf{c})+b_{1}\mathbf{m}.\mathbf{c}\right).$

We claim that $\psi(\gamma\mu b_{2}^{2}\cdot)$ defines a primitive character
modulo $\mathfrak{b}_{1}$. For this we note that

$\beta\in\mathfrak{a}_{\gamma\mu b_{2}^{2}}\Leftrightarrow\gamma\mu
b_{2}^{2}\beta\in\mathfrak{o}\Leftrightarrow(g\mu
b_{2}^{2}\beta)\subset(\alpha)\Leftrightarrow\mathfrak{b}_{1}\mathfrak{d}\mid(\mu
b_{2}^{2}\mathfrak{b}_{2}^{-1})(g\mathfrak{p}_{1}^{-1})(\beta),$

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

$\beta\in\mathfrak{a}_{\gamma\mu
b_{2}^{2}}\Leftrightarrow\beta\in\mathfrak{b}_{1}\mathfrak{e},$

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

$\sum_{b\in(\mathfrak{o}/\mathfrak{b}_{1})^{*}}\psi(-\gamma\mu
bN)\sum_{\mathbf{b}\,(\operatorname{mod}{{{}^{{G}}\mathfrak{b}}_{1}})}\psi\left(\gamma\mu
b_{2}^{2}bF(\mathbf{b})+b_{2}\mathbf{m}.\mathbf{b}\right)=S_{\mathfrak{b}_{1}}(\overline{\operatorname{N}\mathfrak{b}_{2}}^{2}N;(\operatorname{N}\mathfrak{b}_{2})\mathbf{m}),$

where $\overline{\operatorname{N}\mathfrak{b}_{2}}$ is the multiplicative
inverse of $\operatorname{N}\mathfrak{b}_{2}$ modulo $\mathfrak{b}_{1}$.
Similarly,

$\sum_{c\in(\mathfrak{o}/\mathfrak{b}_{2})^{*}}\psi(-\gamma\lambda
cN)\sum_{\mathbf{c}\,(\operatorname{mod}{{{}^{{G}}\mathfrak{b}}_{2}})}\psi\left(\gamma\lambda
b_{1}^{2}cF(\mathbf{c})+b_{1}\mathbf{m}.\mathbf{c}\right)=S_{\mathfrak{b}_{2}}(\overline{\operatorname{N}\mathfrak{b}_{1}}^{2}N;(\operatorname{N}\mathfrak{b}_{1})\mathbf{m}),$

from which the lemma follows. ∎

######  Corollary 4.6.

Let ${\mathfrak{b}}$ be a non-zero integral ideal and suppose that
$\mathfrak{b}=\mathfrak{b}_{1}\mathfrak{b}_{2}$ for integral ideals
$\mathfrak{b}_{1},\mathfrak{b}_{2}$, such that
$\gcd(\operatorname{N}\mathfrak{b}_{1},\operatorname{N}\mathfrak{b}_{2})=1.$
Then
$S_{\mathfrak{b}}(N;\mathbf{0})=S_{\mathfrak{b}_{1}}(N;\mathbf{0})S_{\mathfrak{b}_{2}}(N;\mathbf{0}).$

###### Proof.

On making an obvious change of variables to the $a$-sum and the
$\mathbf{x}$-sum in (4.7), we note that
$S_{\mathfrak{b}}(c^{2}N;\mathbf{0})=S_{\mathfrak{b}}(N;\mathbf{0})$ for any
$c\in\mathbb{Z}$ which is coprime to $\mathfrak{b}$. The statement now follows
from an application of Lemma 4.5. ∎

###  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

$0<\rho=\frac{\operatorname{N}\mathfrak{b}}{P^{d}}\ll 1,$

with which notation we have

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

We now bring into play the work in [2, §6]. It follows from an application of
Fourier inversion, as in [2, Eq. (6.3)], that there exists a function
$p_{\rho}(v):V\to\mathbb{C}$ such that

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

where

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

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

$\mathfrak{H}(v)=\prod_{l=1}^{d}\max\\{1,|v_{l}|\\},$

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 [2, Lemma 5.3] that

$\int_{V}\mathfrak{H}(v)^{\alpha}\mathrm{d}v\ll 1\quad\text{ if $\alpha<-1$}.$
(4.11)

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

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

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

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

by (4.11).

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

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

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

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

by (4.11).

Returning to the function $p_{\rho}(v)$ in (4.9), the following result
summarises its key properties and is extracted from [2, 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

$p_{\rho}(v)\ll_{A}\rho^{-1}\left(\rho^{-1}P^{\varepsilon}\mathfrak{H}(v)^{-1}\right)^{A}.$

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

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

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

$I_{\mathfrak{b}}(t;\mathbf{k})\ll_{A}P^{\varepsilon}\int_{\mathscr{U}}|K(v,\mathbf{k})|\mathrm{d}v+P^{-A},$

for any $A\geqslant 1$, where

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

It is interesting to pause and reflect on the corresponding situation for
cubic forms $G$ over a number field $K$ that was considered in [2], recalling
that we are assuming $K$ to be totally real in our setting. In [2], crucial
use was made of the fact that the integral over $\mathbf{x}$ factors as

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

since
$\operatorname{Tr}(vG(\mathbf{x}))=\sum_{l=1}^{d}v^{(l)}G^{(l)}(\mathbf{x}^{(l)})$,
where $G^{(l)}=\rho_{l}(G)$ is a cubic form over $\mathbb{R}$. We have chosen
our main example (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

$\mathscr{Q}(\mathbf{x}^{(1)},\dots,\mathbf{x}^{(d)})=\sum_{1\leqslant
l\leqslant d}v_{l}F^{(l)}(\mathbf{x}^{(1)},\dots,\mathbf{x}^{(d)}),$ (4.14)

for fixed $v\in V$, where $F^{(l)}$ is the quadratic form (4.2). Thus
$\mathscr{Q}$ 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 (4.3), we may write

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

A general study of these exponential integrals has been carried out by Heath-
Brown and Pierce [6, 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 $\mathscr{Q}\in\mathbb{R}[X_{1},\dots,X_{m}]$ be a quadratic form with
coefficients of maximum modulus $\|\mathscr{Q}\|$ 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

$\int_{\mathbb{R}^{m}}w(\mathbf{u})e\left(\mathscr{Q}(\mathbf{u})-\boldsymbol{\lambda}.\mathbf{u}\right)\mathrm{d}\mathbf{u}\ll_{w}\prod_{i=1}^{m}\min\left\\{1,|\rho_{i}|^{-1/2}\right\\}.$

Furthermore, if $|\boldsymbol{\lambda}|\geqslant 4\|\mathscr{Q}\|$ 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 (4.14). Note that $\|\mathscr{Q}\|\ll|v|$.
Next, define

${\mathscr{F}}(v)=\det\left(\sum_{1\leqslant l\leqslant
d}v_{l}\mathbf{M}^{(l)}\right),$

where $\mathbf{M}^{(l)}$ is the $dn\times dn$ matrix associated to $F^{(l)}$.
The function ${\mathscr{F}}(v)$ 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.

######  Corollary 4.10.

Assume $|\mathbf{k}|\gg|v|$. Then $K(v,\mathbf{k})\ll_{A}|\mathbf{k}|^{-A},$
for any $A\geqslant 1$. Moreover,
$K(v,\mathbf{k})\ll\min\\{1,|{\mathscr{F}}(v)|^{-1/2}\\}$, for any
$\mathbf{k}\in V^{n}$.

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

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

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

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

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

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

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

$\left|\frac{\partial^{i_{1}+\cdots+i_{d}}}{\partial
y_{1}^{i_{1}}\cdots\partial y_{d}^{i_{d}}}f(\mathbf{y})\right|\leqslant
C_{i_{1}+\cdots+i_{d}}$

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 (4.4), we see that

$f(\mathbf{y})=D_{K}^{-n/2}\int_{\begin{subarray}{c}\underline{\mathbf{u}}\in\mathbb{R}^{dn}\\\
Q_{l}(\underline{\mathbf{u}})-\tau_{l}=w_{l}\end{subarray}}w(\delta^{-1}|\underline{\mathbf{u}}-\underline{\boldsymbol{\xi}}|)\mathrm{d}\underline{\mathbf{u}},$
(4.17)

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

$\widetilde{f}(\mathbf{w})=\int_{\begin{subarray}{c}\underline{\mathbf{s}}\in\mathbb{R}^{dn}\\\
Q_{l}(\underline{\mathbf{s}}+\underline{\boldsymbol{\xi}})-\tau_{l}=w_{l}\end{subarray}}w(\delta^{-1}|\underline{\mathbf{s}}|)\mathrm{d}\underline{\mathbf{s}}.$

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

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

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

$\underline{\mathbf{s}}\mapsto\left(Q_{1}(\underline{\mathbf{s}}+\underline{\boldsymbol{\xi}})-\tau^{(1)},s_{1,2},\dots,s_{1,n},\dots,Q_{d}(\underline{\mathbf{s}}+\underline{\boldsymbol{\xi}})-\tau^{(d)},s_{d,2},\dots,s_{d,n}\right).$

The implicit function theorem implies that there exist open subsets
$W^{\prime},W\subset\mathbb{R}^{dn}$ with $\underline{\mathbf{0}}\in
W^{\prime}$ and $\varphi(\underline{\mathbf{0}})\in W$, such that
$\varphi:W^{\prime}\to W$ 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 $\underline{\mathbf{s}}\in W^{\prime}$ whenever
$|\underline{\mathbf{s}}|\leqslant\delta$.

We may now conclude that

$\widetilde{f}(\mathbf{w})=\int_{\begin{subarray}{c}\underline{\mathbf{s}}^{\prime}\in\mathbb{R}^{d(n-1)}\end{subarray}}\partial_{1}\varphi^{-1}w(\delta^{-1}|\left(s_{1,1},\dots,s_{d,n}\right)|)\mathrm{d}\underline{\mathbf{s}}^{\prime},$

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

$\partial_{1}\varphi^{-1}=\left.\det\left(\frac{\partial(\varphi^{-1})_{in+1-n}}{\partial
w_{j}}\right)_{1\leqslant i,j\leqslant d}\right|_{(s_{1,1},\dots,s_{d,n})}$

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

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

Furthermore, in view of Lemma 4.11, it follows from (4.16) and [2, Lemma 4.1]
that

$I_{\mathfrak{b}}(t;\mathbf{0})=\sqrt{D_{K}}f(0)+O_{A}\left(\left(\frac{\operatorname{N}\mathfrak{b}}{P^{d}}\right)^{A}\right),$

for any $A\geqslant 0$, where

$f(0)=\int_{\begin{subarray}{c}\underline{\mathbf{x}}\in\mathbb{R}^{dn}\\\
F^{(l)}(\underline{\mathbf{x}})=t_{l}\end{subarray}}W(\underline{\mathbf{x}})\mathrm{d}\underline{\mathbf{x}},$

if $\underline{\mathbf{x}}=(\mathbf{x}_{1},\dots,\mathbf{x}_{d})$. According
to (4.17), we have $f(0)=D_{K}^{-n/2}\sigma_{\infty}(t),$ where

$\sigma_{\infty}(t)=\int_{\begin{subarray}{c}\underline{\mathbf{u}}\in\mathbb{R}^{dn}\\\
Q_{l}(\underline{\mathbf{u}})=\tau_{l}\end{subarray}}w(\delta^{-1}|\underline{\mathbf{u}}-\underline{\boldsymbol{\xi}}|)\mathrm{d}\underline{\mathbf{u}}$
(4.18)

is the usual singular integral for the descended system. In particular,
arguing as in Davenport and Lewis [3, §6], a standard argument ensures that
$\sigma_{\infty}(t)>0$, since $\underline{\boldsymbol{\xi}}$ is a non-singular
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

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

where $\sigma_{\infty}(N/P^{2})>0$ is given by (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

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

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

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

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

######  Lemma 4.13.

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

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

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

###### Proof.

We may write

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

say. It follows from Corollary 4.6 that $S(k_{1}k_{2})=S(k_{1})S(k_{2})$ if
$k_{1},k_{2}$ are coprime integers. Hence

$\mathfrak{S}(N)=\prod_{p}\sum_{j\geqslant 0}S(p^{j}).$

Since $K$ is Galois we may assume that $p$ admits a factorisation
$(p)=(\mathfrak{p}_{1}\cdots\mathfrak{p}_{r})^{e}$, with
$\operatorname{N}\mathfrak{p}_{1}=\cdots=\operatorname{N}\mathfrak{p}_{r}=p^{f}$.
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

$\mathfrak{S}(N)=\prod_{p}\lim_{\ell\to\infty}\sum_{\begin{subarray}{c}\mathfrak{b}\in
I_{\ell}\end{subarray}}(\operatorname{N}{{}^{{G}}\mathfrak{b}})^{-n}S_{\mathfrak{b}}(N;\mathbf{0}).$

It follows from (4.7) that

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

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

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

where

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

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

$\displaystyle\sum_{\mathfrak{b}\in
I_{\ell}}\operatorname{N}\mathfrak{b}\prod_{\mathfrak{p}\mid\mathfrak{b}}\left(1-\frac{1}{\operatorname{N}\mathfrak{p}}\right)$
$\displaystyle=\sum_{\mathfrak{b}\in
I_{\ell}}p^{f(k_{1}+\cdots+k_{r})}\prod_{\mathfrak{p}\mid\mathfrak{b}}\left(1-\frac{1}{p^{f}}\right)$
$\displaystyle=\prod_{1\leqslant i\leqslant r}\left(1+\sum_{1\leqslant
k\leqslant\ell e}p^{fk}\left(1-\frac{1}{p^{f}}\right)\right)$
$\displaystyle=p^{\ell d},$

as required.

We may now conclude from orthogonality of characters that

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

from which the first part of the lemma follows. The second part is standard.
Using (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
$N=\omega_{1}N_{1}+\cdots+\omega_{d}N_{d}$. Arguing as in work of Birch [1,
Lemma 7.1], for example, the existence of non-singular $p$-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 non-trivial characters

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

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

where

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

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 $\operatorname{N}\mathfrak{b}$ and
$\operatorname{N}{{}^{{G}}\mathfrak{b}}$ are constrained to lie in dyadic
intervals. Note that $\operatorname{N}\mathfrak{b}\ll P^{d}$ and
$\operatorname{N}{{}^{{G}}\mathfrak{b}}\leqslant(\operatorname{N}\mathfrak{b})^{2}$,
since $\\#G=2$. Accordingly, we let $X,Y$ be parameters such that

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

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

$X\leqslant\operatorname{N}\mathfrak{b}<2X\quad\text{ and }\quad
Y\leqslant\operatorname{N}{{}^{{G}}\mathfrak{b}}<2Y.$

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

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

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

It follows from Corollary 4.8 that

$I_{\mathfrak{b}}(N/P^{2};P\mathbf{m})\ll_{A}P^{\varepsilon}\int_{\mathscr{U}}|K(u,P\mathbf{m})|\mathrm{d}u+P^{-A},$

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

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

Hence (4.19) yields

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

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

##  5\. Homogeneous case: proof of Theorems 1.3 and 1.4

We begin by proving a general result about rank drop in pencils of quadratic
forms in situations where one of the matrices has much smaller rank. It
parallels the basic fact in Reid’s thesis [10, Prop. 2.1] about rank drop in
pencils $\nu_{1}A+\nu_{2}B$, for suitable $n\times n$ matrices $A,B$, and
shows how Assumption 2 can be deduced from an appropriate hypothesis about the
shape of the associated singular locus.

######  Lemma 5.1.

Let $L$ be an algebraically closed field of characteristic not equal to $2$,
and let $m<n$. Consider two matrices $A,B\in M_{n\times n}(L)$ such that $B$
has only non-zero entries in the upper left $m\times m$ submatrix, which we
also assume to be non-singular. Let $\det(A)\neq 0$. Assume that all singular
points of the intersection of the two quadratic forms associated to $A$ and
$B$ have the shape $(0,\mathbf{x}^{\prime\prime})$ with
$\mathbf{x}^{\prime\prime}=(x_{m+1},\ldots,x_{n})$, and that the intersection
has codimension $2$. Then we have

$\operatorname{rank}(A+\lambda B)\geqslant n-1,\quad\forall\lambda\in L.$

###### Proof.

Assume that there is some $\lambda\in L$ with

$\operatorname{rank}(A+\lambda B)\leqslant n-2.$

Let $V_{0}\subset\mathbb{A}^{n}$ be the affine subspace given by the kernel of
$A+\lambda B$. Then $\dim V_{0}\geqslant 2$. Let
$\mathbb{P}(V_{0})=V\subset\mathbb{P}^{n-1}$ and let
$Q_{B}\subset\mathbb{P}^{n-1}$ be the quadric given by the matrix $B$. Then
$\dim V\geqslant 1$ and $\dim Q_{B}=n-2$ as projective varieties. We deduce
that the intersection $V\cap Q_{B}$ is non-empty. Consider a point
$\mathbf{x}=(\mathbf{x}^{\prime},\mathbf{x}^{\prime\prime})\in
L^{n}\setminus\\{0\\}$ in the affine cone of $V\cap Q_{B}$, where
$\mathbf{x}^{\prime}\in L^{m}$ and ${\mathbf{x}}^{\prime\prime}\in L^{n-m}$.
Then we deduce that

$0={\mathbf{x}}^{t}(A+\lambda
B){\mathbf{x}}={\mathbf{x}}^{t}A{\mathbf{x}}+\lambda{\mathbf{x}}^{t}B{\mathbf{x}}={\mathbf{x}}^{t}A{\mathbf{x}}.$

We deduce that ${\mathbf{x}}$ lies on the quadric given by $A$ and as it is in
the kernel of $A+\lambda B$, it is a singular point of the intersection
$Q_{A}\cap Q_{B}$. We claim that ${\mathbf{x}}^{\prime}\neq 0$, i.e.
${\mathbf{x}}$ is not of the shape $(0,{\mathbf{x}}^{\prime\prime})$. Assume
for a moment that ${\mathbf{x}}=(0,{\mathbf{x}}^{\prime\prime})$. Note that

$0=(A+\lambda
B)(0,{\mathbf{x}}^{\prime\prime})=A(0,{\mathbf{x}}^{\prime\prime}).$

This is a contradiction to $A$ being non-singular. Hence we found a singular
point of the intersection $Q_{A}\cap Q_{B}$ which is not of the form
$(0,{\mathbf{x}}^{\prime\prime})$. ∎

The main aim of this section is to carry out the proof of Theorems 1.3 and
1.4, which corresponds to taking $N=0$ and

$F(X_{1},\dots,X_{n})=Q(X_{1},\dots,X_{n})+R(X_{1}^{\tau},\dots,X_{m}^{\tau}),$

as in (1.3). Suppose that $\mathbf{A}$ is the $n\times n$ symmetric matrix
defining $Q$ and that $\mathbf{B}$ is the $n\times n$ symmetric matrix given
by the condition that its upper left $m\times m$ submatrix defines $R$, with
all other entries are equal to $0$. We may proceed under the assumption that
Assumptions 1–3 hold.

We have two tasks remaining. The first is to show that the sum over
$\mathfrak{b}$ in Lemma 4.12 can be extended to infinity, with acceptable
error, and the second is to prove that (4.21) holds. We’ll need some more
preparations for estimating the relevant exponential sum in Lemma 4.12 and
(4.23). Recalling the definition (3.3) of ${\mathscr{H}}_{\mathfrak{b}}$, we
lower bound its index in $\mathfrak{o}^{n}$.

######  Lemma 5.2.

There exist non-zero constants
$\kappa_{1},\dots,\kappa_{n},\tilde{\kappa}_{1},\dots,\tilde{\kappa}_{m}\in
K$, depending only on $F$ and $K$, such that

${\mathscr{H}}_{\mathfrak{b}}\subseteq(\kappa_{1}\mathfrak{b}\cap\tilde{\kappa}_{1}\mathfrak{b}^{\tau^{-1}})\times\cdots\times(\kappa_{m}\mathfrak{b}\cap\tilde{\kappa}_{m}\mathfrak{b}^{\tau^{-1}})\times\kappa_{m+1}\mathfrak{b}\times\cdots\times\kappa_{n}\mathfrak{b}.$

Moreover, we have
$\kappa_{1}^{-1},\dots,\kappa_{n}^{-1},\tilde{\kappa}_{1}^{-1},\dots,\tilde{\kappa}_{m}^{-1}\in\mathfrak{o}$.

###### Proof.

Assume that $\mathbf{A}$ has symmetric entries $a_{i,j}\in\mathfrak{o}$, for
$1\leqslant i,j\leqslant n$, and that $\mathbf{B}$ has symmetric entries
$b_{i,j}\in\mathfrak{o}$, for $1\leqslant i,j\leqslant m$. Then the associated
bilinear form takes the shape

$B(X_{1},\dots,X_{n};Y_{1},\dots,Y_{n})=\sum_{i,j\leqslant
n}a_{i,j}X_{i}Y_{j}+\sum_{i,j\leqslant m}b_{i,j}X_{i}^{\tau}Y_{j}^{\tau}.$

Now $\mathbf{h}\in\mathscr{H}_{\mathfrak{b}}$ if and only if
$2B(\mathbf{h},\mathbf{k})\in\mathfrak{b}$ for all
$\mathbf{k}\in\mathfrak{o}^{n}$. Let $\omega_{1},\ldots,\omega_{d}$ be an
integral basis of $\mathfrak{o}$ with ${\omega}_{1}=1$. Let
$l\in\\{1,\ldots,d\\}$ and $j\in\\{1,\dots,n\\}$ and consider a vector
$\mathbf{k}$ such that the $j$-th entry is equal to ${\omega}_{l}$ and all
other entries are equal to zero. Then the condition
$B(\mathbf{h},\mathbf{k})\in\mathfrak{b}$ implies that

$2\omega_{l}\sum_{i=1}^{n}a_{i,j}h_{i}+2{\omega}_{l}^{\tau}\sum_{i=1}^{m}b_{i,j}h_{i}^{\tau}\in\mathfrak{b},\quad
1\leqslant l\leqslant d,\ 1\leqslant j\leqslant n.$

As the matrix $({\omega}_{l}^{\tau})_{1\leqslant l\leqslant
d,\tau\in\operatorname{Gal}(K/\mathbb{Q})}$ is invertible, this implies that
there exists $\beta\in K$ with $\beta^{-1}\in\mathfrak{o}$ such that

$\sum_{i=1}^{n}a_{i,j}h_{i}\in\beta\mathfrak{b},\quad\sum_{i=1}^{m}b_{i,j}h_{i}^{\tau}\in\beta\mathfrak{b},\quad
1\leqslant j\leqslant n.$

Thus we find that $\mathbf{A}{\mathbf{h}}\in(\beta\mathfrak{b})^{n}$ and
$\mathbf{B}(\mathbf{h}^{\prime})^{\tau}\in(\beta\mathfrak{b})^{m}$, where
$\mathbf{h}^{\prime}=(h_{1},\dots,h_{m}).$ As both matrices $\mathbf{A}$ and
$\mathbf{B}$ are non-singular, this implies that

$\mathbf{h}\in\frac{1}{(\det\mathbf{A})}(\beta\mathfrak{b})^{n},\quad\mathbf{h}^{\prime}\in\frac{1}{(\det\mathbf{B})^{\tau^{-1}}}(\beta^{\tau^{-1}}\mathfrak{b}^{\tau^{-1}})^{m}.$

Putting these together, the statement of the lemma easily follows. ∎

######  Corollary 5.3.

Let $N\in\mathfrak{o}$ and let $F$ be given by (1.3). Suppose that Assumption
1 holds. Then

$S_{\mathfrak{b}}(N;\mathbf{m})\ll(\operatorname{N}\mathfrak{b})^{1-(n-m)/2}(\operatorname{N}{{}^{{G}}\mathfrak{b}})^{n-m/2}.$

Moreover, $S_{\mathfrak{b}}(N;\mathbf{m})=0$ unless
$m_{i}\in\mathfrak{d}^{-1}\mathfrak{b}^{-1}$, for $m<i\leqslant n$.

###### Proof.

It follows from Lemma 5.2 that
$|\mathfrak{o}^{n}/{\mathscr{H}}_{\mathfrak{b}}|\gg(\operatorname{N}{{}^{{G}}\mathfrak{b}})^{m}(\operatorname{N}\mathfrak{b})^{n-m}$.
Thus (3.4) implies that

$|{\mathscr{H}}_{\mathfrak{b}}/{{}^{{G}}\mathfrak{b}}^{n}|=\frac{|{\mathfrak{o}}^{n}/{{}^{{G}}\mathfrak{b}}^{n}|}{|{\mathfrak{o}}^{n}/{{\mathscr{H}}}_{\mathfrak{b}}|}\ll\frac{(\operatorname{N}{{}^{{G}}\mathfrak{b}})^{n}}{(\operatorname{N}{{}^{{G}}\mathfrak{b}})^{m}(\operatorname{N}\mathfrak{b})^{n-m}}=\left(\frac{\operatorname{N}{{}^{{G}}\mathfrak{b}}}{\operatorname{N}{\mathfrak{b}}}\right)^{n-m}.$

Inserting this into Lemma 4.2 yields the desired upper bound. We have already
observed in Lemma 4.4 that $S_{\mathfrak{b}}(N;\mathbf{m})=0$ unless
$\mathbf{m}.\mathbf{h}\in\mathfrak{d}^{-1}$ for all
$\mathbf{h}\in\mathscr{H}_{\mathfrak{b}}$. Noting that
${{}^{{G}}\mathfrak{b}}^{m}\times\mathfrak{b}^{n-m}\subset\mathscr{H}_{\mathfrak{b}}$,
the second part easily follows. ∎

Returning to Lemma 4.12, it immediately follows from this that the overall
contribution from the tail $\operatorname{N}\mathfrak{b}\gg P^{d}$ is

$\displaystyle\ll
P^{(n-2)d}\sum_{\begin{subarray}{c}\mathfrak{b}\subset\mathfrak{o}\\\
\operatorname{N}\mathfrak{b}\gg
P^{d}\end{subarray}}(\operatorname{N}{{}^{{G}}\mathfrak{b}})^{-n}|S_{\mathfrak{b}}(N;\mathbf{0})|$
$\displaystyle\ll
P^{(n-2)d}\sum_{\begin{subarray}{c}\mathfrak{b}\subset\mathfrak{o}\\\
\operatorname{N}\mathfrak{b}\gg
P^{d}\end{subarray}}(\operatorname{N}\mathfrak{b})^{1-n/2+m/2}(\operatorname{N}{{}^{{G}}\mathfrak{b}})^{-m/2}.$

Since
$\operatorname{N}{{}^{{G}}\mathfrak{b}}\geqslant\operatorname{N}\mathfrak{b}$,
this is acceptable provided that $n>4$, which is certainly implied by the
hypotheses in Theorems 1.3 and 1.4. Thus we can focus our remaining efforts on
establishing (4.21).

Our next goal is to analyse the integrals $K(u,P\mathbf{m})$ in (4.23) for the
case that $F$ has the shape
$F(\mathbf{x})=Q(\mathbf{x})+R(x_{1}^{\tau},x_{2}^{\tau},\ldots,x_{m}^{\tau}),$
for $\tau\in\operatorname{Gal}(K/\mathbb{Q})$ some fixed automorphism. Taking
the $l$th embedding into the real numbers gives

$F^{(l)}(\mathbf{x}^{(1)},\ldots,\mathbf{x}^{(d)})=Q^{(l)}(\mathbf{x}^{(l)})+R^{(l)}(\rho_{l}(\mathbf{x}^{(\tau)})),\quad
1\leqslant l\leqslant d$

where we write $\mathbf{x}^{(l)}=\rho_{l}(\mathbf{x})$. For each $1\leqslant
l\leqslant d$, we define $l_{\tau}$ through the relation (1.4). With this
notation we obtain

$\displaystyle\mathscr{Q}(\underline{\mathbf{x}})$
$\displaystyle=\sum_{1\leqslant l\leqslant
d}u_{l}F^{(l)}(\underline{\mathbf{x}})$ $\displaystyle=\sum_{1\leqslant
l\leqslant d}u_{l}Q^{(l)}(\mathbf{x}^{(l)})+\sum_{1\leqslant l\leqslant
d}u_{l}R^{(l)}(\rho_{l}(\mathbf{x}^{\tau}))$ $\displaystyle=\sum_{1\leqslant
l\leqslant d}u_{l}Q^{(l)}(\mathbf{x}^{(l)})+\sum_{1\leqslant l\leqslant
d}u_{l_{\tau}}R^{(l_{\tau})}(\mathbf{x}^{(l)}).$

Hence

$K(u,P\mathbf{m})=\prod_{l=1}^{d}\int_{{\mathbb{R}}^{n}}W(\mathbf{x}^{(l)})e(G^{(l)}(\mathbf{x}^{(l)})-P\mathbf{m}^{(l)}.\mathbf{x}^{(l)})\mathrm{d}\mathbf{x}^{(l)},$

with

$G^{(l)}(\mathbf{x}^{(l)})=u_{l}Q^{(l)}(\mathbf{x}^{(l)})+u_{l_{\tau}}R^{(l_{\tau})}(\mathbf{x}^{(l)}).$

Note that $G^{(l)}(\mathbf{x}^{(l)})$ is a quadratic form in
$\mathbf{x}^{(l)}$ and hence can be represented by a symmetric matrix, which
can be diagonalised using an orthogonal base change. Thus, for every tuple
$u=(u_{1},\ldots,u_{d})$, there exists a diagonal matrix
$\operatorname{Diag}(\eth_{l,i}(u))_{1\leqslant i\leqslant n}$ and an
orthogonal matrix $M_{l}(u)\in O(n)$ such that

$G^{(l)}(\mathbf{x}^{(l)})=(\mathbf{x}^{(l)})^{t}M_{l}(u)^{t}\operatorname{Diag}(\eth_{l,i}(u))M_{l}(u)\mathbf{x}^{(l)}.$

Set

$K^{(l)}(u,P\mathbf{m})=\int_{{\mathbb{R}}^{n}}W(\mathbf{x}^{(l)})e(G^{(l)}(\mathbf{x}^{(l)})-P\mathbf{m}^{(l)}.\mathbf{x}^{(l)})\mathrm{d}\mathbf{x}^{(l)},\quad\text{
for $1\leqslant l\leqslant d$}.$

With the change of coordinates $M_{l}(u)\mathbf{x}^{(l)}=\mathbf{y}^{(l)}$, we
get

$\displaystyle K^{(l)}$ $\displaystyle(u,P\mathbf{m})$
$\displaystyle=\pm\int_{{\mathbb{R}}^{n}}W(M_{l}(u)^{t}\mathbf{y}^{(l)})e((\mathbf{y}^{(l)})^{t}\operatorname{Diag}(\eth_{l,i}(u))\mathbf{y}^{(l)}-P\mathbf{m}^{(l)}.(M_{l}(u))^{t}\mathbf{y}^{(l)})\mathrm{d}\mathbf{y}^{(l)}$
$\displaystyle=\pm\int_{{\mathbb{R}}^{n}}W(M_{l}(u)^{t}\mathbf{y}^{(l)})e((\mathbf{y}^{(l)})^{t}\operatorname{Diag}(\eth_{l,i}(u))\mathbf{y}^{(l)}-PM_{l}(u)\mathbf{m}^{(l)}.\mathbf{y}^{(l)})\mathrm{d}\mathbf{y}^{(l)}.$

We are now ready to prove the following result.

######  Lemma 5.4.

For any $\varepsilon>0$, the integral $K^{(l)}(u,P\mathbf{m})$ is essentially
supported on the set of $u$ and $\mathbf{m}$ for which

$|(M_{l}(u)\mathbf{m}^{(l)})_{i}|\ll P^{-1+\varepsilon}|\eth_{l,i}(u)|,\quad
1\leqslant i\leqslant n,$

and

$|m_{i}^{(l)}|\ll P^{-1+\varepsilon}|u_{l}|,\quad m<i\leqslant n.$

Moreover, we have

$K^{(l)}(u,P\mathbf{m})\ll\prod_{i=1}^{n}\min\left(1,\frac{1}{|\eth_{l,i}(u)|^{1/2}}\right).$

###### Proof.

Recall that $M_{l}(u)\in O(n)$. In particular all entries of $M_{l}(u)$ are
bounded independently of $u$ and we obtain

$\frac{\partial^{k}}{\partial(y_{i}^{(l)})^{k}}W(M_{l}(u)^{t}\mathbf{y}^{(l)})\ll_{k}1,$

uniformly in $u$, for all $k\in\mathbb{N}$. The result now follows from Lemma
4.9. ∎

Henceforth we take $N=0$ and write $E(P;X,Y)=E(0;P;X,Y)$ in (4.23). We shall
adhere to common convention and allow the value of $\varepsilon>0$ to change
at each appearance, so that $P^{\varepsilon}\log P\ll P^{\varepsilon}$, for
example. Moreover, all implied constants are allowed to depend on
$\varepsilon$.

Applying Corollary 5.3, we deduce that

$E(P;X,Y)\ll
P^{\varepsilon}X^{1-(n-m)/2}Y^{-m/2}\sum_{\begin{subarray}{c}\mathfrak{b}\in\mathscr{B}(X,Y)\end{subarray}}\sum_{\begin{subarray}{c}\mathbf{0}\neq\mathbf{m}\in\widehat{{{}^{{G}}\mathfrak{b}}}^{n}\\\
i>m\Rightarrow
m_{i}\in\mathfrak{d}^{-1}\mathfrak{b}^{-1}\end{subarray}}\int_{\mathscr{U}}|K(u,P\mathbf{m})|\mathrm{d}u.$

Let $\delta\in{{}^{{G}}\mathfrak{b}}\mathfrak{d}$ and let $\mathfrak{p}_{1}$
be a prime ideal coprime to ${{}^{{G}}\mathfrak{b}}\mathfrak{d}$, with
$\operatorname{N}\mathfrak{p}_{1}\ll(\operatorname{N}\mathfrak{b})^{\varepsilon/d}$,
such that $(\delta)={{}^{{G}}\mathfrak{b}}\mathfrak{d}\mathfrak{p}_{1}$. On
multiplying $\delta$ by an appropriate unit, there is no loss of generality in
assuming that

$Y^{1/d}\ll|\delta^{(l)}|\ll Y^{1/d+\varepsilon},$ (5.1)

for $1\leqslant l\leqslant d$, since
$Y\leqslant\operatorname{N}{{}^{{G}}\mathfrak{b}}<2Y$. We are led to make the
change of variables

$c_{i}=\delta m_{i},$ (5.2)

for $1\leqslant i\leqslant n$, so that
$\mathbf{c}=(c_{1},\dots,c_{n})\in\mathfrak{o}^{n}$. Then

$c_{i}\in\delta\mathfrak{d}^{-1}\mathfrak{b}^{-1}=\mathfrak{p}_{1}\mathfrak{b}^{-1}{{}^{{G}}\mathfrak{b}}\subset\mathfrak{b}^{-1}{{}^{{G}}\mathfrak{b}},\quad\text{for
$m<i\leqslant n$}.$ (5.3)

We may now write

$E(P;X,Y)\ll
P^{\varepsilon}X^{1-(n-m)/2}Y^{-m/2}\sum_{\begin{subarray}{c}\mathfrak{b}\in\mathscr{B}(X,Y)\end{subarray}}\sum_{\begin{subarray}{c}\mathbf{0}\neq\mathbf{c}\in\mathfrak{o}^{n}\\\
\text{\eqref{eq:squall}
holds}\end{subarray}}\int_{\mathscr{U}}|K(u,P\delta^{-1}\mathbf{c})|\mathrm{d}u.$

Define the function

$f(u)=\prod_{1\leqslant l\leqslant d}\prod_{1\leqslant i\leqslant
n}\min\left(1,\frac{1}{|\eth_{l,i}(u)|^{1/2}}\right).$ (5.4)

Let $\mathscr{R}(\mathbf{m})$ be the set of $u\in\mathscr{U}$ such that

$|(M_{l}(u)\mathbf{m}^{(l)})_{i}|\ll P^{-1+\varepsilon}|\eth_{l,i}(u)|,\quad
1\leqslant i\leqslant n,1\leqslant l\leqslant d,$

and

$|m_{i}^{(l)}|\ll P^{-1+\varepsilon}|u_{l}|,\quad m<i\leqslant n,\,1\leqslant
l\leqslant d.$

We now have

$E(P;X,Y)\ll
P^{\varepsilon}X^{1-(n-m)/2}Y^{-m/2}\sum_{\begin{subarray}{c}\mathfrak{b}\in\mathscr{B}(X,Y)\end{subarray}}\sum_{\begin{subarray}{c}\mathbf{0}\neq\mathbf{c}\in\mathfrak{o}^{n}\\\
\text{\eqref{eq:squall}
holds}\end{subarray}}\int_{\mathscr{R}(\delta^{-1}\mathbf{c})}f(u)\mathrm{d}u.$

Let

$L(u)=\sum_{\mathfrak{b}\in\mathscr{B}(X,Y)}\sum_{\mathbf{c}\in\mathscr{C}(u,\mathfrak{b})}1,$

where $\mathscr{C}(u,\mathfrak{b})$ is the set of non-zero vectors
$\mathbf{c}\in\mathfrak{o}^{n}$ for which (5.3) holds,

$|(M_{l}(u)\mathbf{c}^{(l)})_{i}|\ll
P^{-1+\varepsilon}Y^{1/d}|\eth_{l,i}(u)|,\quad 1\leqslant i\leqslant
n,\,1\leqslant l\leqslant d,$

and

$|c_{i}^{(l)}|\ll P^{-1+\varepsilon}Y^{1/d}|u_{l}|,\quad m<i\leqslant
n,\,1\leqslant l\leqslant d.$

Then we have

$E(P;X,Y)\ll
P^{\varepsilon}X^{1-(n-m)/2}Y^{-m/2}\int_{\mathscr{U}}f(u)L(u)\mathrm{d}u.$

Our next goal is to estimate $L(u)$. For each $1\leqslant l\leqslant d$ we
sort the eigenvalues $\eth_{l,i}(u)$ in a way such that

$|\eth_{l,1}(u)|\geqslant|\eth_{l,2}(u)|\geqslant\cdots\geqslant|\eth_{l,n}(u)|.$

Note that we can always achieve this by adjusting the orthogonal matrix
$M_{l}(u)$ with suitable permutations. Moreover, for all $1\leqslant
i\leqslant n$ and $1\leqslant l\leqslant d$, we have

$|\eth_{l,i}(u)|\ll|u_{l}|+|u_{l_{\tau}}|.$ (5.5)

It will now be useful to make the observation

$\prod_{l=1}^{d}(1+|u_{l}|+|u_{l_{\tau}}|)\ll\prod_{l=1}^{d}((1+|u_{l}|)(1+|u_{l_{\tau}}|))\ll\mathfrak{H}(u)^{2}.$
(5.6)

We proceed by proving the following result.

######  Lemma 5.5.

Let $u\in V$ such that $\mathfrak{H}(u)\leqslant P^{d+\varepsilon}/X$. If
$L(u)\neq 0$ then

$P^{-d+\varepsilon}Y\mathfrak{H}(u)^{2}\gg 1.$ (5.7)

Moreover, we have $L(u)\ll P^{\varepsilon}XJ(u),$ where

$J(u)=\prod_{1\leqslant l\leqslant d}\prod_{1\leqslant i\leqslant
m}(1+P^{-1+\varepsilon}Y^{1/d}|\eth_{l,i}(u)|).$

###### Proof.

Let us write $\mathbf{c}=(\mathbf{c}^{\prime},\mathbf{c}^{\prime\prime})$,
where $\mathbf{c}^{\prime}=(c_{1},\dots,c_{m})$ and
$\mathbf{c}^{\prime\prime}=(c_{m+1},\dots,c_{n})$. Keeping in mind (5.3), we
first fix a choice of
$\mathbf{c}^{\prime\prime}\in\left(\mathfrak{b}^{-1}{{}^{{G}}\mathfrak{b}}\right)^{n-m}$
satisfying

$|c_{i}^{(l)}|\ll P^{-1+\varepsilon}Y^{1/d}|u_{l}|,$

for $m+1\leqslant i\leqslant n$ and $1\leqslant l\leqslant d$. Choose
$\lambda\in K$ such that
$(\lambda)=\mathfrak{b}^{-1}{{}^{{G}}\mathfrak{b}}\mathfrak{p}_{2}^{-1}$, for
a suitable prime ideal $\mathfrak{p}_{2}$ of norm $O(P^{\varepsilon}).$ We may
assume that $\lambda$ is well-shaped, in the sense of (5.1), on multiplying by
a suitable unit. Thus $X^{1/d}Y^{1/d}\ll|\lambda^{(l)}|\ll
X^{1/d}Y^{1/d+\varepsilon},$ for $1\leqslant l\leqslant d$. Making the change
of variables $\mathbf{c}^{\prime\prime}=\lambda\mathbf{d}^{\prime\prime}$ and
recalling that $\operatorname{N}\mathfrak{b}\asymp X$ and
$\operatorname{N}{{}^{{G}}\mathfrak{b}}\asymp Y$, we must have

$|d_{i}^{(l)}|\ll P^{-1+\varepsilon}X^{1/d}|u_{l}|,$

for $m+1\leqslant i\leqslant n$ and $1\leqslant l\leqslant d$.

We begin by showing that (5.7) holds if $L(u)\neq 0$. Thus there exists
$\mathbf{c}\neq\mathbf{0}$ counted by $L(u)$. Suppose first that
$\mathbf{c}^{\prime\prime}\neq\mathbf{0}$. Then there exists
$i\in\\{m+1,\dots,n\\}$ such that

$1\leqslant|N_{K/\mathbb{Q}}(d_{i})|\ll
P^{-d+\varepsilon}X|\operatorname{Nm}(u)|,$

whence $1\ll P^{-d+\varepsilon}X\mathfrak{H}(u)\ll
P^{-d+\varepsilon}Y\mathfrak{H}(u)^{2},$ since $X\leqslant Y$. This is
satisfactory for (5.7). Suppose next that $\mathbf{c}^{\prime}\neq\mathbf{0}$.
In particular we have

$|(M_{l}(u)\mathbf{c}^{(l)})_{i}|\ll
P^{-1+\varepsilon}Y^{1/d}|\eth_{l,i}(u)|,\quad 1\leqslant i\leqslant
n,\,1\leqslant l\leqslant d.$ (5.8)

As $M_{l}(u)$ is an orthogonal matrix, this implies that

$|c_{j}^{(l)}|\ll P^{-1+\varepsilon}Y^{1/d}\max_{1\leqslant i\leqslant
n}|\eth_{l,i}(u)|,\quad 1\leqslant j\leqslant n,\,1\leqslant l\leqslant d,$

whence (5.6) yields

$1\ll P^{-d+\varepsilon}Y\prod_{l=1}^{d}\max_{1\leqslant i\leqslant
n}|\eth_{l,i}(u)|\ll
P^{-d+\varepsilon}Y\prod_{l=1}^{d}(|u_{l}|+|u_{l_{\tau}}|)\ll
P^{-d+\varepsilon}Y\mathfrak{H}(u)^{2}.$

This completes the proof of (5.7) under the assumption that $L(u)\neq 0$.

Turning now to the estimation of $L(u)$, it readily follows from a result in
Lang [9, Thm. 0 in §V.1] that the overall number of vectors
$\mathbf{d}^{\prime\prime}$ is

$\displaystyle\ll\left(1+\prod_{l=1}^{d}P^{-1+\varepsilon}X^{1/d}|u_{l}|\right)^{n-m}\ll\left(1+P^{-d+\varepsilon}X\operatorname{Nm}(u)\right)^{n-m}\ll
P^{\varepsilon}.$

It remains to count the number of vectors $\mathbf{c}^{\prime}$ associated to
a particular choice of $\mathbf{c}^{\prime\prime}$. Let
$L(u,\mathfrak{b},\mathbf{c}^{\prime\prime})$ be the number of
$\mathbf{c}^{\prime}\in\mathfrak{o}^{m}$ such that (5.8) holds. Assume that
the matrix $M_{l}(u)$ is given by
$M_{l}(u)=(m_{l\alpha\beta})_{1\leqslant\alpha,\beta\leqslant n}$. Write

$M_{l}(u)=(M_{l}^{\prime}(u)M_{l}^{\prime\prime}(u)),$

with
$M_{l}^{\prime}(u)=(m_{l\alpha\beta})_{\begin{subarray}{c}1\leqslant\alpha\leqslant
n\\\ 1\leqslant\beta\leqslant m\end{subarray}}$ and
$M_{l}^{\prime\prime}(u)=(m_{l\alpha\beta})_{\begin{subarray}{c}1\leqslant\alpha\leqslant
n\\\ m<\beta\leqslant n\end{subarray}}$. Then we consider the system of
inequalities

$|M_{l}^{\prime}(u)\mathbf{c}^{\prime(l)}+r_{li}|\ll
P^{-1+\varepsilon}Y^{1/d}|\eth_{l,i}(u)|,\quad 1\leqslant i\leqslant
n,\,1\leqslant l\leqslant d,$

where $\mathbf{r}_{l}=(r_{li})_{1\leqslant i\leqslant
n}=M_{l}^{\prime\prime}(u)\mathbf{c}^{\prime\prime(l)}$.

Write

$c_{i}=\sum_{l=1}^{d}c_{il}{\omega}_{l},\quad c_{il}\in\mathbb{Z},\,1\leqslant
i\leqslant m.$

Then, for $1\leqslant i\leqslant n$, we can write

$(M^{\prime}_{l}(u)\mathbf{c}^{\prime(l)})_{i}=\sum_{\beta=1}^{m}m_{li\beta}c_{\beta}^{(l)}=\sum_{\beta=1}^{m}m_{li\beta}\sum_{k=1}^{d}c_{\beta
k}{\omega}_{k}^{(l)}=\sum_{\beta=1}^{m}\sum_{k=1}^{d}m_{li\beta}{\omega}_{k}^{(l)}c_{\beta
k}.$

Let $H$ be the $dn\times dm$ matrix given by

$H=(m_{li\beta}{\omega}_{k}^{(l)})_{(l,i)\times(k,\beta)},$

with $1\leqslant l\leqslant d$, $1\leqslant i\leqslant n$, $1\leqslant
k\leqslant d$, $1\leqslant\beta\leqslant m$, and consider the lattice

$\Lambda=H\mathbb{Z}^{md}\subset\mathbb{R}^{nd}.$

Then $L(u,\mathfrak{b},\mathbf{c}^{\prime\prime})$ counts lattice points in
$\Lambda$ which lie in a box of side length

$\ll P^{-1+\varepsilon}Y^{1/d}|\eth_{l,i}(u)|,\quad 1\leqslant i\leqslant
n,\,1\leqslant l\leqslant d.$

We claim that the successive minima of the lattice $\Lambda$ are bounded above
and below by constants depending only on $K$ and $n$. Taking this on faith, it
will then follow that

$L(u,\mathfrak{b},\mathbf{c}^{\prime\prime})\ll\prod_{1\leqslant l\leqslant
d}\prod_{1\leqslant i\leqslant
m}(1+P^{-1+\varepsilon}Y^{1/d}|\eth_{l,i}(u)|),$

which will settle the lemma, on summing over $O(X)$ choices for
$\mathfrak{b}\in\mathscr{B}(X,Y)$.

To check the claim, we order the index tuples $(l,i)$ and $(k,\beta)$ in the
matrix $H$ lexicographically. Write

$A_{lk}=(m_{li\beta}{\omega}_{k}^{(l)})_{1\leqslant i\leqslant
n,1\leqslant\beta\leqslant
m}={\omega}_{k}^{(l)}(m_{li\beta})_{(i,\beta)}={\omega}_{k}^{(l)}B_{l},$

with the $n\times m$ matrix $B_{l}=(m_{li\beta})_{1\leqslant i\leqslant
n,1\leqslant\beta\leqslant m}$. Note that $B_{l}$ has orthogonal and norm one
columns for $1\leqslant l\leqslant d$. We can then write $H$ as a block matrix

$H=(A_{lk})_{1\leqslant l,k\leqslant d}.$

Let $B=B(u)$ be the $nd\times md$ matrix which is a diagonal block matrix,
with the matrices $B_{1},\ldots,B_{d}$ on the diagonal. Let $W$ be the
$md\times md$ block matrix, with blocks ${\omega}_{k}^{(l)}E_{m}$ at each
place $1\leqslant l,k\leqslant d$, where $E_{m}$ is the $m$-dimensional
identity matrix. Then

$H=BW.$

Consider the lattice $\Gamma=W\mathbb{Z}^{md}\subset\mathbb{R}^{md}$ and note
that this only depends on the basis ${\omega}_{1},\ldots,{\omega}_{d}$.
Moreover, if $\mathbf{w}\in\Gamma$, then

$\langle
B\mathbf{w},B\mathbf{w}\rangle=\mathbf{w}^{t}B^{t}B\mathbf{w}=\mathbf{w}^{t}E_{md}\mathbf{w}=\langle\mathbf{w},\mathbf{w}\rangle.$

Hence the successive minima of the lattice $\Lambda$ coincide with those of
$\Gamma$, which thereby establishes the claim. ∎

It follows from the previous result that

$\begin{split}E(P;X,Y)\ll&P^{\varepsilon}X^{2-(n-m)/2}Y^{-m/2}\\\
&\times\int_{\mathscr{U}^{*}}f(u)\prod_{1\leqslant i\leqslant
m}\prod_{1\leqslant l\leqslant
d}(1+P^{-1+\varepsilon}Y^{1/d}|\eth_{l,i}(u)|)\mathrm{d}u,\end{split}$ (5.9)

where $f(u)$ is given by (5.4) and $\mathscr{U}^{*}$ is the set of
$u\in\mathscr{U}$ such that (5.7) holds.

Recall that the $\eth_{l,i}(u)$ are the eigenvalues of the matrix associated
to the quadratic form

$u_{l}Q^{(l)}(\mathbf{x}^{(l)})+u_{l_{\tau}}R^{(l_{\tau})}(\mathbf{x}^{(l)}).$

The next result collects together a number of properties concerning the size
of the eigenvalues $\eth_{l,i}(u)$.

######  Lemma 5.6.

Assume that Assumptions 1–3 hold and suppose that $\tilde{m}\geqslant m-1$ is
the degree of the polynomial appearing in Assumption 3. For each $1\leqslant
l\leqslant d$ we order the eigenvalues $\eth_{l,i}(u)$ such that

$|\eth_{l,1}(u)|\geqslant|\eth_{l,2}(u)|\geqslant\cdots\geqslant|\eth_{l,n}(u)|.$

Then there exist constants $C_{1},\dots,C_{d}>0$ such that the following
holds:

  1. (1)

If $|u_{l_{\tau}}|\leqslant C_{l}|u_{l}|$, then

$|\eth_{l,1}(u)|\ll|u_{l}|\quad\text{ and }\quad|\eth_{l,n-1}(u)|\gg|u_{l}|.$

Moreover, if $m=1$ and $\tilde{m}=0$, then

$|\eth_{l,1}(u)|\ll|u_{l}|\quad\text{ and }\quad|\eth_{l,n}(u)|\gg|u_{l}|.$

  2. (2)

If $|u_{l_{\tau}}|>C_{l}|u_{l}|$, then

$|\eth_{l,m+1}(u)\cdots\eth_{l,n}(u)|\gg\frac{|u_{l}|^{n-\tilde{m}}}{|u_{l_{\tau}}|^{m-\tilde{m}}}.$

###### Proof.

To begin with, according to Assumption 3, for each $1\leqslant l\leqslant d$
there exists a constant $C_{l}$ such that

$|\det(Q^{(l)}+tR^{(l_{\tau})})|\gg|t|^{\tilde{m}},$

for $|t|\geqslant C_{l}$.

We start by examining the case $|u_{l_{\tau}}|\leqslant C_{l}|u_{l}|$. The
first bound $|\eth_{l,1}(u)|\ll|u_{l}|$ follows directly from (5.5). Assume
now that $u_{l}\neq 0$. Note that each of the eigenvalues $\eth_{l,i}(u)$
arises by multiplication with $u_{l}$ from the eigenvalues of the matrix
corresponding to

$Q^{(l)}+\frac{u_{l_{\tau}}}{u_{l}}R^{(l_{\tau})}.$

Write $\tilde{\eth}_{l,i}(u)$ for those eigenvalues in the same ordering.
Assume that the lower bound $|\tilde{\eth}_{l,n-1}(u)|\gg 1$ is not satisfied.
Thus there exists a sequence of $t_{j}$ in the range $|t_{j}|\leqslant C_{l}$
such that $\tilde{\eth}_{l,n-1}(t_{j})\rightarrow 0$, for
$j\rightarrow\infty$, where we write $\tilde{\eth}_{l,n-1}(t)$ for the second
smallest eigenvalues of $Q^{(l)}+tR^{(l_{\tau})}$. As the set of $t$ is
compact there is a convergent subsequence, convergent to $t^{\prime}$ say,
with $\operatorname{rank}(Q^{(l)}+t^{\prime}R^{(l_{\tau})})<n-1$. This
contradicts Assumption 2.

Now we consider the case $m=1$ and $\tilde{m}=0$. By Assumptions 1 and 3 we
deduce that $\det(Q^{(l)}+tR^{(l_{\tau})})$ is a non-zero constant independent
of $t$. In particular the rank of this matrix is always $n$ and the argument
above shows that $|\eth_{l,n}(u)|\gg|u_{l}|$.

Next we consider the case $|u_{l_{\tau}}|>C_{l}|u_{l}|$ and $u_{l}\neq 0$.
Again, we write $\tilde{\eth}_{l,i}(u)$ for the eigenvalues of
$Q^{(l)}+\frac{u_{l_{\tau}}}{u_{l}}R^{(l_{\tau})}.$ Note that we have

$|\tilde{\eth}_{l,i}(u)|\ll|\frac{u_{l_{\tau}}}{u_{l}}|,\quad 1\leqslant
i\leqslant n.$

Moreover, we observe that

$|\tilde{\eth}_{l,1}(u)\cdots\tilde{\eth}_{l,n}(u)|=|\det(Q^{(l)}+\frac{u_{l_{\tau}}}{u_{l}}R^{(l_{\tau})})|\gg\left|\frac{u_{l_{\tau}}}{u_{l}}\right|^{\tilde{m}}.$

We therefore find that

$\displaystyle|\tilde{\eth}_{l,m+1}(u)\cdots\tilde{\eth}_{l,n}(u)|$
$\displaystyle\gg\left|\frac{u_{l}}{u_{l_{\tau}}}\right|^{m}\left|\frac{u_{l_{\tau}}}{u_{l}}\right|^{\tilde{m}}=\left|\frac{u_{l}}{u_{l_{\tau}}}\right|^{m-\tilde{m}}$

From this we obtain the lower bound

$|\eth_{l,m+1}(u)\cdots\eth_{l,n}(u)|\gg|u_{l}|^{n-m}\left|\frac{u_{l}}{u_{l_{\tau}}}\right|^{m-\tilde{m}}=\frac{|u_{l}|^{n-\tilde{m}}}{|u_{l_{\tau}}|^{m-\tilde{m}}},$

which completes the proof of the lemma. ∎

We now continue with our analysis of $E(P;X,Y)$ in (5.9). Let $E_{1}(P;X,Y)$
denote the overall contribution from the case $Y\geqslant P^{d}$, and let
$E_{2}(P;X,Y)$ denote the remaining contribution. The following pair of
results treats these two quantities in turn.

######  Lemma 5.7.

Assume $\tilde{m}\geqslant m-1$. Then $E_{1}(P;X,Y)\ll
P^{d(2-n/2+m/2)+\varepsilon}+P^{-dm+\varepsilon}$.

###### Proof.

On recalling the definition (5.4) of $f(u)$, we deduce from (5.9) that

$\displaystyle E_{1}(P;X,$ $\displaystyle Y)\ll
P^{\varepsilon}X^{2-(n-m)/2}Y^{-m/2}P^{-md}Y^{m}$
$\displaystyle\times\int_{\begin{subarray}{c}\mathscr{U}^{*}\end{subarray}}\left(\prod_{1\leqslant
l\leqslant d}(1+|u_{l}|+|u_{l_{\tau}}|)^{m/2}\right)\prod_{1\leqslant
l\leqslant d}\prod_{m<i\leqslant
n}\min\left(1,\frac{1}{|\eth_{l,i}(u)|^{1/2}}\right)\mathrm{d}u,$

since (5.5) implies that

$\prod_{1\leqslant i\leqslant
m}(1+|\eth_{l,i}(u)|)^{1/2}\ll(1+|u_{l}|+|u_{l_{\tau}}|)^{m/2}.$

Here, we recall that $\mathscr{U}^{*}$ is the set of $u\in\mathscr{U}$ such
that (5.7) holds. Consider for a moment a fixed value of $l$. If
$C_{l}|u_{l}|\geqslant|u_{l_{\tau}}|$, then Lemma 5.6 yields

$\prod_{m<i\leqslant
n}\min\left(1,\frac{1}{|\eth_{l,i}(u)|^{1/2}}\right)\ll\min\left(1,|u_{l}|^{-(n-m-1)/2}\right).$

If $C_{l}|u_{l}|<|u_{l_{\tau}}|$ and $\tilde{m}\geqslant m-1$, then Lemma 5.6
yields

$\displaystyle\prod_{m<i\leqslant
n}\min\left(1,\frac{1}{|\eth_{l,i}(u)|^{1/2}}\right)$
$\displaystyle\ll\min\left(1,\frac{1}{|\eth_{l,m+1}(u)\cdots\eth_{l,n}(u)|^{1/2}}\right)$
$\displaystyle\ll\min\left(1,\frac{|u_{l_{\tau}}|^{1/2}}{|u_{l}|^{(n-m+1)/2}}\right).$

In either of these two cases we therefore have

$\prod_{m<i\leqslant
n}\min\left(1,\frac{1}{|\eth_{l,i}(u)|^{1/2}}\right)\ll(1+|u_{l}|+|u_{l_{\tau}}|)^{1/2}\min\left(1,|u_{l}|^{-(n-m)/2}\right),$

whence

$\displaystyle E_{1}(P;X,Y)\ll$ $\displaystyle
P^{\varepsilon}X^{2-(n-m)/2}Y^{-m/2}P^{-md}Y^{m}$
$\displaystyle\times\int_{\mathscr{U}}\left(\prod_{1\leqslant l\leqslant
d}(1+|u_{l}|+|u_{l_{\tau}}|)^{(m+1)/2}\right)\prod_{1\leqslant l\leqslant
d}\min\left(1,\frac{1}{|u_{l}|^{(n-m)/2}}\right)\mathrm{d}u.$

It follows from (5.6) that

$\displaystyle E_{1}(P;X,Y)$ $\displaystyle\ll
P^{\varepsilon}X^{2-(n-m)/2}Y^{-m/2}P^{-md}Y^{m}\int_{\mathfrak{H}(u)\leqslant
P^{d+\varepsilon}/X}\mathfrak{H}(u)^{m+1-(n-m)/2}\mathrm{d}u$
$\displaystyle\ll
P^{\varepsilon}X^{2-(n-m)/2}Y^{-m/2}P^{-dm}Y^{m}(1+(P^{d}/X)^{2+m-(n-m)/2})$
$\displaystyle\ll
P^{\varepsilon}X^{-m}Y^{m/2}P^{d(2-n/2+m/2)}+P^{\varepsilon}X^{2-(n-m)/2}Y^{m/2}P^{-dm}.$

The contribution gets maximal for $Y\asymp X^{2}$, in which case we get the
upper bound

$\displaystyle E_{1}(P;X,Y)$ $\displaystyle\ll
P^{\varepsilon}X^{-m}X^{m}P^{d(2-n/2+m/2)}+P^{\varepsilon}X^{2-n/2+3m/2}P^{-dm}$
$\displaystyle\ll
P^{d(2-n/2+m/2)+\varepsilon}+X^{2-n/2+3m/2}P^{-dm+\varepsilon}.$

The first term is satisfactory for the lemma. If $2-n/2+3m/2\leqslant 0$, then
the second term is $O(P^{-dm+\varepsilon})$, which is satisfactory. If
$n\leqslant 3+3m$, on the other hand, then we take $X\ll P^{d}$ and get the
satisfactory upper bound $O(P^{d(2-n/2+m/2)+\varepsilon})$. ∎

######  Lemma 5.8.

Assume that $n\geqslant m+4$ and $\tilde{m}\geqslant m-1$. Let

$\kappa=\begin{cases}1&\text{ if $m=1$ and $\tilde{m}=0$,}\\\ 0&\text{
otherwise.}\end{cases}$

Then $E_{2}(P;X,Y)$ is

$\ll
P^{-d(n-m-4+\kappa)/4+\varepsilon}+P^{-1/2+\varepsilon}+P^{-2m+d(3m+4-\kappa-n)/2+\varepsilon}+P^{-2m+d(3m+4-\kappa-n)/4+\varepsilon}.$

###### Proof.

For $1\leqslant i\leqslant m$, we clearly have

$\min\left(1,\frac{1}{|\eth_{l,i}(u)|^{1/2}}\right)(1+P^{-1+\varepsilon}Y^{1/d}|\eth_{l,i}(u)|)\ll
1+P^{-1+\varepsilon}Y^{1/d}|\eth_{l,i}(u)|^{1/2}.$

Hence we find that $E_{2}(P;X,Y)$ is

$\displaystyle\ll P^{\varepsilon}X^{2-(n-m)/2}Y^{-m/2}$
$\displaystyle\times\int_{\mathscr{U}^{*}}\left(\prod_{1\leqslant l\leqslant
d}\prod_{m<i\leqslant
n}\min\left(1,\frac{1}{|\eth_{l,i}(u)|^{1/2}}\right)\right)\prod_{1\leqslant
l\leqslant d}\prod_{1\leqslant i\leqslant
m}(1+P^{-1+\varepsilon}Y^{1/d}|\eth_{l,i}(u)|^{1/2})\mathrm{d}u$

If $C_{l}|u_{l}|\geqslant|u_{l_{\tau}}|$, then Lemma 5.6 leads to the bound

$\prod_{m<i\leqslant
n}\min\left(1,\frac{1}{|\eth_{l,i}(u)|^{1/2}}\right)\ll\min\left(1,|u_{l}|^{-(n-m-1+\kappa)/2}\right).$

For the case $C_{l}|u_{l}|<|u_{l_{\tau}}|$ we still have

$\displaystyle\prod_{m<i\leqslant
n}\min\left(1,\frac{1}{|\eth_{l,i}(u)|^{1/2}}\right)$
$\displaystyle\ll\min\left(1,\frac{|u_{l_{\tau}}|^{1/2}}{|u_{l}|^{(n-m+1)/2}}\right),$

since $\tilde{m}\geqslant m-1$. We now deduce that in either case we have

$\prod_{m<i\leqslant
n}\min\left(1,\frac{1}{|\eth_{l,i}(u)|^{1/2}}\right)\ll(1+|u_{l}|+|u_{l_{\tau}}|)^{1/2}\min\left(1,|u_{l}|^{-(n-m+{\kappa})/2}\right).$

It now follows from (5.6) that $E_{2}(P;X,Y)$ is

$\displaystyle\ll~{}$ $\displaystyle P^{\varepsilon}X^{2-(n-m)/2}Y^{-m/2}$
$\displaystyle\times\int_{\mathscr{U}^{*}}\mathfrak{H}(u)^{1-(n-m+\kappa)/2}\prod_{1\leqslant
l\leqslant d}\prod_{1\leqslant i\leqslant
m}(1+P^{-1+\varepsilon}Y^{1/d}(|u_{l}|+|u_{l_{\tau}}|)^{1/2})\mathrm{d}u$
$\displaystyle\ll~{}$ $\displaystyle P^{\varepsilon}X^{2-(n-m)/2}Y^{-m/2}$
$\displaystyle\times\int_{\mathscr{U}^{*}}\mathfrak{H}(u)^{-\frac{n-m-2+\kappa}{2}}\prod_{1\leqslant
l\leqslant
d}(1+P^{-1+\varepsilon}Y^{1/d}(|u_{l}|+|u_{l_{\tau}}|)^{1/2})^{m}\mathrm{d}u.$

Let $I_{1}$ denote the contribution to the integral from those $u$ for which
there exists at least one $u_{l}$ with $|u_{l}|\gg(P/Y^{1/d})^{2}$, and let
$I_{2}$ denote the remaining contribution.

On recalling that $\mathscr{U}^{*}$ is the set of $u\in\mathscr{U}$ such that
(5.7) holds, it is clear that

$I_{2}\ll\int_{\begin{subarray}{c}P^{d/2-\varepsilon}/Y^{1/2}\ll\mathfrak{H}(u)\ll
P^{d+\varepsilon}/X\\\ |u_{l}|\ll P^{2}Y^{-2/d},~{}1\leqslant l\leqslant
d\end{subarray}}\mathfrak{H}(u)^{-\frac{n-m-2+\kappa}{2}}\mathrm{d}u.$

Turning to $I_{1}$, we see that

$\displaystyle\prod_{1\leqslant l\leqslant d}(1+P^{-1+\varepsilon}$
$\displaystyle Y^{1/d}(|u_{l}|+|u_{l_{\tau}}|)^{1/2})^{m}$ $\displaystyle\ll
P^{\varepsilon}\prod_{\begin{subarray}{c}1\leqslant l\leqslant d\\\
|u_{l}|+|u_{l_{\tau}}|\geqslant(P/Y^{1/d})^{2}\end{subarray}}(P^{-1}Y^{1/d}(|u_{l}|+|u_{l_{\tau}}|)^{1/2})^{m}$
$\displaystyle\ll
P^{\varepsilon}\mathfrak{H}(u)^{m}(P^{-1}Y^{1/d})^{m\sharp\\{1\leqslant
l\leqslant d:|u_{l}|+|u_{l_{\tau}}|\geqslant(P/Y^{1/d})^{2}\\}},$

by (5.6). But if there is one $u_{l}$ with $|u_{l}|\gg(P/Y^{1/d})^{2}$, then
clearly

$\sharp\\{1\leqslant l\leqslant
d:|u_{l}|+|u_{l_{\tau}}|\geqslant(P/Y^{1/d})^{2}\\}\geqslant 2.$

Hence, since $P^{-1}Y^{1/d}\ll 1$, it now follows that

$I_{1}\ll
P^{\varepsilon}\int_{P^{d/2-\varepsilon}/Y^{1/2}\ll\mathfrak{H}(u)\ll
P^{d+\varepsilon}/X}\mathfrak{H}(u)^{-\frac{n-m-2+\kappa}{2}+m}(P^{-1}Y^{1/d})^{2m}\mathrm{d}u.$

In summary we have shown that

$E_{2}(P;X,Y)\ll
P^{\varepsilon}X^{2-(n-m)/2}Y^{-m/2}\left(I_{1}+I_{2}\right),$

with $I_{1},I_{2}$ as above.

Since $n\geqslant m+4$, the exponent of $\mathfrak{H}(u)$ in $I_{2}$ is less
than or equal to $-1$. If $\frac{n-m-2+\kappa}{2}>1$, then it follows from
(4.12) that

$I_{2}\ll P^{\varepsilon}(P^{d/2}/Y^{1/2})^{-\frac{n-m-2+\kappa}{2}+1}.$

However, if $\frac{n-m-2+\kappa}{2}=1$, then we apply (4.13) to deduce that
the same bound holds.