Generalised quadratic forms over totally real number fields

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.


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 2 over Q and let Q(X 1 , . . ., X n ) = 1 i,j n c i,j X i X j be a non-singular quadratic form, with symmetric coefficients c i,j ∈ o K .For given N ∈ o K , it is very natural to ask about the solubility of Q(x 1 , . . ., x n ) = N, with x 1 , . . ., x n ∈ o K .If n 4, a number field version of the Hardy-Littlewood circle method is capable of establishing the Hasse principle for these equations.When n 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/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 2. A generalised quadratic form is given by for symmetric coefficients c i,j,τ,τ ′ = c j,i,τ ′ ,τ ∈ o K .
We will be interested in the set of (x 1 , . . ., x n ) ∈ o n K for which F (x 1 , . . ., x n ) = N, for given N ∈ o K , in which case x τ i should be interpreted as the conjugate of x i under τ ∈ Gal(K/Q).Definition 1.1 encompasses standard integral quadratic forms over o K and forms defined using norms and traces.For example, let Tr K/Q,H : K → K be the partial trace, defined via Tr K/Q,H (u) = τ ∈H u τ , for any subset H ⊂ Gal(K/Q).Then, a natural generalisation of the question about representing elements of o K as a sum of squares is to ask about the existence of x ∈ o n K such that for given N ∈ o K and a given subset H ⊂ Gal(K/Q).
The coefficients of a generalised quadratic form F (X 1 , . . ., X n ) form a dn × dn matrix M = (c i,j,τ,τ ′ ) (i,τ )×(j,τ ′ ) .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 M, which after reordering rows and columns, contains a n × 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 M is not much bigger than n.Let W : (K ⊗ Q R) n → R 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 ) = W (x/P ), as P → ∞, for given N ∈ 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 ∈ o K [X 1 , . . ., X n ], studying non-trivial zeros of Q over o K is equivalent to studying K-rational points on the smooth quadric X ⊂ P n−1 Q in P dn−1

Q
. Our first result shows that, after Weil restriction, the space of generalised quadratic forms is rich enough to capture the arithmetic over Q of arbitrary codimension d complete intersections of quadrics in P M −1 Q , provided that d | M .Let F (X 1 , . . ., X n ) be a generalised quadratic form and let ω 1 , . . ., ω d be a Z-basis for o K .Any element x ∈ o n K can be written x = ω 1 u 1 + • • • + ω d u d for (u 1 , . . ., u d ) ∈ Z dn .Taking the Weil restriction corresponds to writing down a set of quadratic forms Q 1 , . . ., Q d ∈ Z[U 1 , . . ., U d ], in dn variables, such that (1.2) We henceforth call {Q 1 , . . ., Q d } the descended system.We shall prove the following result in §2.
Theorem 1.2.Let K/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 o K -solubility for a generalised quadratic form therefore becomes a question of 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 P M −1 Q over 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 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 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 P M −1 Q when M 9d.The latter result is particularly significant, since it allows one to handle arbitrary generalised quadratic forms over K in n 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 ∈ o K [X 1 , . . ., X n ] and R ∈ o K [X 1 , . . ., X m ] be quadratic forms in n and m variables, respectively, for 1 m n.The generalised quadratic forms we shall treat take the shape for a fixed non-trivial automorphism τ ∈ Gal(K/Q).Let ρ 1 , . . ., ρ d be the d distinct embeddings of K into R, where we recall that K is totally real.For each 1 l d, we define l τ through the relation ρ lτ τ = ρ l .
(1.4) Suppose that A is the n × n symmetric matrix defining Q and that B is the n × n symmetric matrix given by the condition that its upper left m × m submatrix defines R, with all other entries equal to 0. For any 1 l d, we shall write A (l) and B (l) for the l-th embeddings of A and B, respectively.We make the following key hypotheses about A and B.
Assumption 1. Assume that the descended system has codimension d in P dn−1 .Furthermore, assume that det A = 0 and that the upper left m × m submatrix of B is non-singular.
Our first result deals with the special case m = 1.
Theorem 1.3.Let K/Q be a totally real Galois extension of degree d 2. Suppose that m = 1 and that Assumption 1 holds.Assume that det(A (l) + tB (lτ ) ) is a constant polynomial in t, for each 1 l d, where l τ is defined via (1.4).Let n 6 and assume that the descended system has non-singular points everywhere locally.Then there exist constants c > 0 and ∆ > 0 such that 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 τ 1 ) 2 + Q(X 3 , . . ., X n ) meets the hypotheses of the theorem, for example, where Q We are also able to prove an asymptotic formula for N W (F ; P ) for arbitrary m 1, provided we make additional assumptions about the matrices A and B. Assumption 2. For all 1 l d, assume that rank(A (l) + tB (lτ ) ) n − 1, for all t ∈ R, where l τ is defined via (1.4).Assumption 3.For all 1 l d, assume that det(A (l) + tB (lτ ) ) has degree at least m − 1, viewed as a polynomial in t.
When m = 1 and det(A (l) + tB (lτ ) ) has degree exactly 0 in Assumption 3, we see that Assumption 2 is implied by Assumption 1, since then rank(A (l) +tB (lτ ) ) = rank(A (l) ) = n.For general m 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 ∈ M n×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 λA + µB, with (λ, µ) = (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 A (l) + tB (lτ ) , since the matrix B (lτ ) 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/Q be a totally real Galois extension of degree d 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 On taking m = 1, we note that this result subsumes Theorem 1.3 when n 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 for quadratic forms Q 1 and Q 2 over 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 m + 5 in the statement of Theorem 1.4 if Q(0, . . ., 0, X m+1 , . . ., X n ) has rank at least 5.
1.2.Inhomogeneous setting.We now assume that N ∈ o K is non-zero.Then we may write Our next result demonstrates that sharper results are available if N = 0 and Q, R are both diagonal.Suppose that ) with H = {id, τ }.We will prove the following result.
Theorem 1.5.Let K/Q be a totally real Galois extension of degree d 2. Assume that N ∈ o K is non-zero and that n 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 Z p for every prime p. Then there exist constants c > 0 and 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 = Q( √ 2) in (1.5) and assume that a 1 , . . ., a n , b 1 , . . ., b m ∈ Z are all non-zero.Then it follows from Theorem 1.5 that our work treats the shifted descended system m i=1 when n m + 4 and N 1 , N 2 ∈ Z are not both zero.
1.3.Some words on the proof.Let F (X 1 , . . ., X n ) be a generalised quadratic form defined over o K and let N ∈ o K .Our analysis of N W (F, N ; P ) relies on a Fourieranalytic interpretation of the indicator function Browning and Vishe [2, Thm 1.2] have extended to arbitrary number fields the smooth δ-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 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 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 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 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 → ∞.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 = 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.

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 Q in dn variables.Let K/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,τ,τ ′ ) of a generalised quadratic form, with c i,j,τ,τ ′ = c j,i,τ ′ ,τ for all 1 i, j n and τ, τ ′ ∈ Gal(K/Q).We can write each coefficient c i,j,τ,τ ′ ∈ K with respect to the basis {ω 1 , . . ., ω d } as c i,j,τ,τ ′ = d k=1 c (k) i,j,τ,τ ′ ω k .We proceed to compute the descended system explicitly, by writing Let {ρ 1 , . . ., ρ d } be a dual basis of {ω 1 , . . ., ω d } with respect to the trace, so that (Tr K/Q (ρ i ω j )) i,j is the identity matrix and any α ∈ K can be written in the form and we arrive at our descended system (1.2), with for rational coefficients By construction, the coefficients β p,l,i,m,j satisfy β p,l,i,m,j = β p,m,j,l,i , for all 1 p, l, m d and 1 i, j n.Moreover, they depend linearly on the given set of coefficients (c (k) i,j,τ,τ ′ ).Now the space of all tuples (c (k) i,j,τ,τ ′ ) of rational numbers satisfying the symmetry relation c Similarly, the space of all symmetric rational tuples (β p,l,i,m,j ) is naturally parametrised by Q 1 2 dn(dn+1)d .We define the map 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 β p,l,i,m,j = 0 for all 1 p, l, m d and 1 i, j n.By the non-degeneracy of the trace as a bilinear form, we deduce that is of maximal rank, and hence we obtain Applying the same argument again, we finally obtain and hence c (k) i,j,τ,τ ′ = 0 for all 1 k d.

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/Q is a totally real Galois extension of degree d.We shall henceforth write o = 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 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 a in K one defines the dual ideal â = {α ∈ K : Tr K/Q (αx) ∈ Z for all x ∈ a}.
In particular â = a −1 d −1 , where d = {α ∈ K : αô ⊆ o} denotes the different ideal of K and is itself an integral ideal.One notes that ô = d −1 .Furthermore, we have â ⊆ b if and only if b ⊆ a.An additional integral ideal featuring in our work is the denominator ideal a γ = {α ∈ o : αγ ∈ o}, associated to any γ ∈ K. Recall that N a = |o/a| is the ideal norm of any integral ideal a.One important property of the ideal norm is that N a τ = N a for any τ ∈ Gal(K/Q).(This follows from the isomorphism o/a → o/a τ given by α → α τ .)Furthermore, we have N a ∈ a for any integral ideal a.
We will write (a, b) = a + b for the greatest common divisor of two integral ideals a, b ⊂ o.When these ideals are coprime, meaning that a + b = o, we shall adopt the abuse of notation (a, b) = 1.We close this section by recording the following basic result.
which would mean that σ 0 is a character modulo b 1 .Suppose that σ 0 is primitive and suppose that there is a prime ideal which is a contradiction.Thus h is coprime to b and we must have h | d.Suppose now that bd = a γ h for some h | d such that (h, b) = 1.We wish to deduce that σ 0 is primitive, for which we suppose for a contradiction that there exists a proper divisor b Finally we note that σ 0 is a trivial character if and only if γ ∈ ô, which is equivalent to a γ ⊇ d.This is clearly impossible for any primitive character σ 0 modulo a proper ideal b o, since (d/e, b) = 1 if a γ = be.
We proceed to define a particularly convenient additive character modulo b.Associated to any non-zero integral ideal b is the subset F(b) ⊂ K given by Note that condition (i) implies that α ∈ bd and condition (ii) implies that p 1 ∤ b τ d for any τ ∈ Gal(K/Q).We may now record a variant of [2, Lemma 2.3], which shows that Proof.We consider the integral ideal c = bd.Observe that d τ = d for all τ ∈ Gal(K/Q), since d = ô−1 and the trace is invariant under the action of the Galois group.Taking ε = 1 in Lemma 3.1(ii), we can find α ∈ c and a prime ideal p 1 coprime to c τ = b τ d, for every τ ∈ Gal(K/Q), with N p 1 ≪ N b, and such that (α) = cp 1 .It follows from Lemma 3.1(i) that there exists ν ∈ p 1 such that ((ν), c τ ) = 1 for any τ ∈ Gal(K/Q).But this implies that g = N K/Q (ν) is coprime to c τ , for any τ ∈ Gal(K/Q), with g ∈ p 1 .We will show that c = a γ with γ = g/α, after which an application of Lemma 3.3 with e = d will complete the proof.To check the claim we note that since p 1 | (g) and bd is coprime with (g).

3.3.
The G-invariant ideal and an important Z-module.Let F be a generalised quadratic form, as in Definition 1.1.Let G = G F ⊂ Gal(K/Q) be the subset of automorphisms τ ∈ Gal(K/Q) that actually appear in F .Note that G = {id} if and only if F is a standard quadratic form.For any integral ideal b we define the G-invariant ideal to be This is the least common multiple of the ideals b τ −1 for τ ∈ G.
Next, associated to our generalised quadratic form F is a generalised bilinear form This defines a map for any vectors x, y, u, v ∈ K n .(However, this fails to be a bilinear form on K n since B(λx; y), B(x; λy) and λB(x; y) needn't be equal for λ ∈ K.) For any ideal b ⊂ o, let

This is an additive group and it is clear that
, where G b is the Ginvariant ideal defined in (3.1).By testing the hypothesis with a ≡ 0 mod G b, we have Hence We claim that H b has the structure of a finitely generated Z-module.To see this, let e i be the ith unit vector, for 1 i n.Observe that (N b)ω j e i ∈ H b for all 1 i n and 1 j d.Hence the image of H b under the isomorphism o n ∼ = Z nd is a lattice of full rank and, thus, finitely generated as a Z-module.
The set H 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 o n .In the special case (1.3) it will be easier to calculate H 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,τ,τ ′ ) (i,τ )×(j,τ ′ ) of a generalised quadratic form as a nd × nd matrix.Lemma 3.5.There is a constant C 1 > 0, depending only on F , such that for all b we have Moreover, there is an integral ideal d 1 , such that one can take C 1 = 1 for all ideals b with (b, d 1 ) = 1.
Proof.Let ∆ = rank(c i,j,τ,τ ′ ) (i,τ )×(j,τ ′ ) .Let S ⊂ {1, . . ., n} × Gal(K/Q) be a subset of indices such that the vectors (c i,j,τ,τ ′ ) (j,τ ′ ) , (i, τ ) ∈ S , are linearly independent and |S | is maximal.Then, for any (k, σ) ∈ {1, . . ., n} × Gal(K/Q) there are numbers a Observe that H ′ b ⊂ H b .Moreover, if b and (α) are coprime, then the ideal (α) may be omitted in the definition of H ′ b .Finally, we observe that there is an injection We now wish to provide an alternative upper bound involving H b , under a suitable assumption on the generalised quadratic form.Definition 3.6.We say that F (X 1 , . . ., X n ) is admissible if there exist vectors In this language a standard quadratic form is admissible if and only if it is nonsingular.We may now prove the following result.Lemma 3.7.Assume that F is admissible.Then there exists a constant C 2 > 0, depending only on F , such that Moreover, there exists an integral ideal d 2 such that one can take We can use this result to get information about the index of H b in o n via the identity Lemma 3.7 is of the expected magnitude, which we can see by considering the case of the standard diagonal quadratic form Proof of Lemma 3.7.Let v 1 , . . ., v n be a set of vectors as in Definition 3.6.By scaling these vectors with a rational integer, we may assume that v i ∈ o n for all 1 i n.We define the auxiliary set and observe that H b ⊂ H b .Next, consider the map which is injective by the definition of admissibility in Definition 3.6.Let Γ be the image of o n under the map ϕ.Then ϕ induces an isomorphism Note that Γ only depends on B(X; Y ) and vectors v 1 , . . ., v n , and hence can be taken to be independent of the ideal b.As in (3.4), we therefore obtain where C Γ is a constant only depending on Γ.Moreover, there is an ideal This completes the proof of the lemma.

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 δ K in (1.6).Let K be a totally real Galois extension of Q of degree d.Let Q 1 and let α ∈ 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 where N b = |o/b| denotes the norm of the ideal b and the notation * σ (mod b) means that the sum is taken over primitive additive characters modulo b.Furthermore, we have h(x, y) ≪ x −1 and h(x, y) = 0 only if x 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 ρ 1 , . . ., ρ d : K ֒→ R be the distinct real embeddings of K, and let We extend the absolute value on R to give a norm on V via |v| = max 1 l d |v l |, which we extend to V n in the obvious way.
Let N ∈ o and let F (X 1 , . . ., X n ) be a generalised quadratic form defined over o.Our central concern is with the asymptotic behaviour of the sum 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 , . . ., X n ) be a generalised quadratic form and let {ω 1 , . . ., ω d } be a Z-basis for o.We have seen in (1.2) how there is a descended system {Q 1 , . . ., Q d } of quadratic forms, that is associated to F via with variables U l = (U l1 , . . ., U ln ) for 1 l d.
We will need to be able to relate the descended system to the embedded system, which amounts to how F (x) embeds in V for given x ∈ K n .We extend F : K n → K to get a map V n → V , through the identification of K with V .Associated to x is the vector (x (1) , . . ., x (d) ), with x (l) ∈ R n for 1 l d.Let l ∈ {1, . . ., d}.To any τ ∈ Gal(K/Q) may be associated a unique integer l τ ∈ {1, . . ., d} such that (1.4) holds.Then we define where c Thus ρ l (F (x)) is a real quadratic form in the dn variables x (1) , . . ., x (d) .We call {F (1) , . . ., F (d) } the embedded system.In particular, it is clear that for any v = (v 1 , . . ., v d ) ∈ V and x ∈ 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 ρ l (F (x)) = F (l) (x (l) ), for 1 l 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.
where W is the dn × dn block matrix Switching appropriate rows and columns takes W to Diag(A, . . ., A), In particular, it follows that for any 1 l 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 ξ = (ξ 1 , . . ., ξ d ) ∈ R dn such that J Q 1 ,...,Q d (ξ) has rank d, where is the associated d × dn Jacobian matrix.Define the smooth weight function and let δ > 0 be a small parameter.In this paper we shall work with the weight function W : V n → R 0 , which is given by where x is identified with (x (1) , . . ., x (d) ), and where W is the matrix in (4.5).It is clear that W is infinitely differentiable and that it is supported on the region |W −1 x− ξ| δ.Ultimately we will want to work with a value of δ that is sufficiently small, but which still satisfies 1 ≪ δ 1 for an absolute implied constant.
for any Q 1.Here the constant c , for any A > 0. Furthermore, we have h(x, y) ≪ x −1 for all y and h(x, y) = 0 only if x 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 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) = 0 only if If F were a standard quadratic form over o, we would proceed by breaking the sum over x into residue classes modulo 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 a, h ∈ o n and any primitive character σ modulo b, one may have σ(F (a + h)) = σ(F (a)) even when h ∈ b n .In this way, we see that a special role will be played by the set H b , that was introduced in §3.3.Lemma 4.1.We have where the sum over b is over non-zero integral ideals and Proof.Our approach is based on breaking the x-sum in (4.6) into residue classes modulo G b. Since Q = P and G b n ⊂ H b , it follows that this sum equals for any primitive character σ modulo b.We apply the multi-dimensional Poisson summation formula (cf.[2, §5]).Recalling that K is totally real, we find that the inner x-sum is equal to Our work hinges upon the following upper bound for this sum.
Lemma 4.2.We have where H b is given by (3.3).
Proof.For fixed a ∈ (o/b) * , we have in the notation of (3.2).We observe that the function u → ψ(2γaB(u; h)) is a character modulo G b n , and it is the trivial character precisely when We rewrite γ in the form γ = g/α with (α) = bdp 1 for some prime ideal p 1 and g ∈ p 1 ∩Z with the property that ((g), Finally, since this condition on u is invariant modulo G b n , this is equivalent to the condition 2B(u; h) ∈ b, for all u ∈ o n , which is equivalent to specifying that h ∈ H b , by (3.3).The statement of the lemma now follows.Proof.This follows from combining Lemmas 3.7 and 4.2.
It is straightforward to show that S b (N ; m) vanishes unless m satisfies additional constraints, as demonstrated in the following result.However, orthogonality of characters gives

Since we automatically have
of the lemma follows.
We shall also need to establish a multiplicativity property for the exponential sums.This is achieved in the following result.
We claim that ψ(γµb 2 2 •) defines a primitive character modulo b 1 .For this we note that ).Now µ is coprime to b 1 and so is b 2 .Hence the common divisor of these ideals most be coprime to the ideal b 1 , as claimed.Thus Lemma 3.3 establishes the claim that ψ(γµb where from which the lemma follows. 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 ρ (v) : V → C such that where In our analysis it will be useful to have the notion of a height function on V .Accordingly, we define In the closing stages of our argument we will need to estimate integrals involving powers of H(v) over various regions in V .First, it follows from [2, Lemma 5.3] that We can use this to deduce two further bounds that will play important roles.
For any A 1 and ε > 0, we claim that If α < −1 then we can clearly assume that ε < −α − 1.But then the conditions of integration imply that (H(v)/A) −α−1−ε 1, whence Next, for any B 1 and ε > 0, we claim that To see this we note that (B/H(v)) α+1+ε 1, under the conditions of the integral, if α −1.But then

by (4.11).
Returning to the function p ρ (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 ε > 0, we have p ρ (v) ≪ P ε , for any v ∈ V .Moreover, for any ε > 0 and A 1, we have Recall here that ρ > 0. The next result is a straightforward consequence of the previous result, once combined with (4.9) and the bound which follows from the fact that W is compactly supported.
for any A 1, where 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 x factors as ), where G (l) = ρ l (G) is a cubic form over 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, k) in generic situations.Define Q(x (1) , . . ., for fixed v ∈ V , where F (l) is the quadratic form (4.2).Thus Q is a quadratic form over R in dn variables.Let us write, temporarily, x = (x (1) , . . ., x (d) ) and k = (k (1) , . . ., k (d) ).Then, in the light of (4.3), we may write

.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 Q ∈ R[X 1 , . . ., X m ] be a quadratic form with coefficients of maximum modulus Q and eigenvalues ρ 1 , . . ., ρ m .Let λ ∈ R m and suppose that w : R m → R is any smooth weight function supported on Furthermore, if |λ| 4 Q then the integral is O w,A (|λ| −A ) for any A 1.
We will apply this result with λ = k and with the real quadratic form in (4.14).Note that Q ≪ |v|.Next, define where M (l) is the dn × dn matrix associated to F (l) .The function F (v) is a real form of degree dn in the variables v 1 , . . ., v d .The following estimate is a direct consequence of Lemma 4.9.
Unfortunately, it appears difficult to extract anything useful from the second bound, unless the generalised quadratic form is assumed to have extra structure.4.6.Contribution from the trivial character.In this section we study the overall contribution from the vector m = 0 in the expression for N W (F, N ; P ) in Lemma 4.1.This contribution is in the notation of that result.It will ease notation if we put t = N/P 2 ∈ R. Assuming that the descended system has codimension d, we begin by analysing the exponential integral I b (t; 0), writing where ρ = N b/P d and where, by an abuse of notation, dx is the surface measure obtained by eliminating d variables from the equation F (x) − t = v.We shall think of f (y) as a function of y = (y 1 , . . ., y d ) on R d , in which t is fixed and bounded absolutely.The following result summarises its main properties.
where τ = A −1 t and w = A −1 y.It will clearly suffice to prove the properties recorded in the lemma for the integral on the right hand side, f (w) say, regarded as a function of w.Making the change of variables s = u − ξ, we have Next, we recall that J Q 1 ,...,Q d (ξ) has rank d.We may assume without loss of generality that Let ϕ : R dn → R dn be given by The implicit function theorem implies that there exist open subsets W ′ , W ⊂ R dn with 0 ∈ W ′ and ϕ(0) ∈ W , such that ϕ : W ′ → W is a bijection and has differentiable inverse ϕ −1 on W .It is now clear that we wish to choose δ > 0 small enough to ensure that s ∈ W ′ whenever |s| δ.
We may now conclude that where s ′ = (s 1,2 , . . ., s 1,n , . . ., s d,2 , . . ., s d,n ), and s 1,1 , s 2,1 , . . ., s d,1 are implicitly given by s ′ and w, and is the associated Jacobian.Since ϕ −1 is smooth this implies that f (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 ∈ R, and ε fixed as in the corollary, we have Furthermore, in view of Lemma 4.11, it follows from (4.16) and [2, Lemma 4.1] that for any A 0, where According to (4.17), we have is the usual singular integral for the descended system.In particular, arguing as in Davenport and Lewis [3, §6], a standard argument ensures that σ ∞ (t) > 0, since ξ is a non-singular real point on the descended system.We summarise our preliminary treatment of the main term in the following result.
In order to proceed further, it is clear that one requires a good enough upper bound for S b (N ; 0), in order to show that the error term is satisfactory and the sum over 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 for any admissible F such that n 5.In the setting of Theorems 1.4 and 1.5 we shall produce better bounds for S b (N ; 0) which allow such a deduction under milder hypotheses.
We close this section with a formal analysis of the singular series ignoring issues of convergence.This is summarised in the following result.We have S(N ) > 0 if the shifted descended system has a non-singular p-adic solution for every prime p.
Proof.We may write say.It follows from Corollary 4.6 that S(k Since K is Galois we may assume that p admits a factorisation (p) = (p Then the union of I ℓ over ℓ 0 exactly matches the set of integral ideals whose norm is a power of p. Hence ) ℓe , we conclude that each such character induces a character modulo p ℓ .In order to complete the proof of the claim it remains to show that we get all p ℓd characters modulo p ℓ this way.But the number of characters is precisely as required.
We may now conclude from orthogonality of characters that from which the first part of the lemma follows.The second part is standard.Using (1.2), the solubility of F (x) − N in o/p ℓ o can be reduced to the solubility of a shifted descended system Q i (u 1 , . . ., u d ) − N i modulo primes powers, for 1 i d, where we have written 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 S(N ) > 0. The details of this will not be repeated here.The primary task is to establish conditions under which there is an absolute constant ∆ > 0 such that E(N ; P ) = O(P −∆ ).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 b such that N b and N G b are constrained to lie in dyadic intervals.Note that N b ≪ P d and N G b (N b) 2 , since #G = 2. Accordingly, we let X, Y be parameters such that We then write E(N ; P ; X, Y ) for the overall contribution to E(N ; P ) from non-zero ideals b ⊂ o for which We denote by 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 ∆ > 0 such that 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 ν 1 A + ν 2 B, for suitable n × 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 ∈ M n×n (L) such that B has only non-zero entries in the upper left m × m submatrix, which we also assume to be non-singular.Let det(A) = 0. Assume that all singular points of the intersection of the two quadratic forms associated to A and B have the shape (0, x ′′ ) with x ′′ = (x m+1 , . . ., x n ), and that the intersection has codimension 2. Then we have Proof.Assume that there is some λ ∈ L with rank(A + λB) n − 2.
Let V 0 ⊂ A n be the affine subspace given by the kernel of A + λB.Then dim V 0 2. Let P(V 0 ) = V ⊂ P n−1 and let Q B ⊂ P n−1 be the quadric given by the matrix B. Then dim V 1 and dim Q B = n − 2 as projective varieties.We deduce that the intersection We deduce that x lies on the quadric given by A and as it is in the kernel of A + λB, it is a singular point of the intersection Q A ∩ Q B .We claim that x ′ = 0, i.e. x is not of the shape (0, x ′′ ).Assume for a moment that x = (0, x ′′ ).Note that 0 = (A + λB)(0, x ′′ ) = A(0, x ′′ ).This is a contradiction to A being non-singular.Hence we found a singular point of the intersection Q A ∩ Q B which is not of the form (0, x ′′ ).
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 . ., X τ m ), as in (1.3).Suppose that A is the n × n symmetric matrix defining Q and that B is the n × n symmetric matrix given by the condition that its upper left m × 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 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 H b , we lower bound its index in o n .Lemma 5.2.There exist non-zero constants κ 1 , . . ., κ n , κ1 , . . ., κm ∈ K, depending only on F and K, such that Assume that A has symmetric entries a i,j ∈ o, for 1 i, j n, and that B has symmetric entries b i,j ∈ o, for 1 i, j m.Then the associated bilinear form takes the shape Let ω 1 , . . ., ω d be an integral basis of o with ω 1 = 1.Let l ∈ {1, . . ., d} and j ∈ {1, . . ., n} and consider a vector k such that the j-th entry is equal to ω l and all other entries are equal to zero.Then the condition As the matrix ( Thus we find that Ah ∈ (βb) n and B(h ′ ) τ ∈ (βb) m , where h ′ = (h 1 , . . ., h m ).As both matrices A and B are non-singular, this implies that Putting these together, the statement of the lemma easily follows.
Corollary 5.3.Let N ∈ o and let F be given by (1.3).Suppose that Assumption 1 holds.Then Inserting this into Lemma 4.2 yields the desired upper bound.We have already observed in Lemma 4.
Returning to Lemma 4.12, it immediately follows from this that the overall contribution from the tail N b ≫ P d is Since N G b N 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 m) in (4.23) for the case that F has the shape F (x) = Q(x) + R(x τ 1 , x τ 2 , . . ., x τ m ), for τ ∈ Gal(K/Q) some fixed automorphism.Taking the lth embedding into the real numbers gives where we write x (l) = ρ l (x).For each 1 l d, we define l τ through the relation (1.4).With this notation we obtain ) is a quadratic form in 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 , . . ., u d ), there exists a diagonal matrix Diag(ð l,i (u)) 1 i n and an orthogonal matrix M l (u) ∈ O(n) such that With the change of coordinates M l (u)x (l) = y (l) , we get l) .y (l) )dy (l) .
We are now ready to prove the following result.
Lemma 5.4.For any ε > 0, the integral K (l) (u, P m) is essentially supported on the set of u and m for which i | ≪ P −1+ε |u l |, m < i n.Moreover, we have In particular all entries of M l (u) are bounded independently of u and we obtain uniformly in u, for all k ∈ 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 ε > 0 to change at each appearance, so that P ε log P ≪ P ε , for example.Moreover, all implied constants are allowed to depend on ε.
Applying Corollary 5.3, we deduce that Let δ ∈ G bd and let p 1 be a prime ideal coprime to G bd, with N p 1 ≪ (N b) ε/d , such that (δ) = G bdp 1 .On multiplying δ by an appropriate unit, there is no loss of generality in assuming that for 1 l d, since Y N G b < 2Y .We are led to make the change of variables for 1 i n, so that c = (c 1 , . . ., c n ) ∈ o n .Then We may now write Define the function ( Let R(m) be the set of u ∈ U such that where C (u, b) is the set of non-zero vectors c ∈ o n for which (5.3) holds, Our next goal is to estimate L(u).For each 1 l d we sort the eigenvalues ð l,i (u) in a way such that u)|.Note that we can always achieve this by adjusting the orthogonal matrix M l (u) with suitable permutations.Moreover, for all 1 i n and 1 l d, we have (5.5) It will now be useful to make the observation (5.6) We proceed by proving the following result.
Lemma 5.5.Let u ∈ V such that H(u) P d+ε /X.If L(u) = 0 then (5.7) Moreover, we have L(u) ≪ P ε XJ(u), where 2 , for a suitable prime ideal p 2 of norm O(P ε ).We may assume that λ is well-shaped, in the sense of (5.1), on multiplying by a suitable unit.Thus We begin by showing that (5.7) holds if L(u) = 0. Thus there exists c = 0 counted by L(u).Suppose first that c ′′ = 0. Then there exists i ∈ {m + 1, . . ., n} such that 2 , since X Y .This is satisfactory for (5.7).Suppose next that c ′ = 0.In particular we have (5.8) As M l (u) is an orthogonal matrix, this implies that This completes the proof of (5.7) under the assumption that L(u) = 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 d ′′ is It remains to count the number of vectors c ′ associated to a particular choice of c ′′ .Let L(u, b, c ′′ ) be the number of c ′ ∈ o m such that (5.8) holds.Assume that the matrix M l (u) is given by M l (u) = (m lαβ ) 1 α,β n .Write . Then we consider the system of inequalities l) .Write Then, for 1 i n, we can write Let H be the dn × dm matrix given by Then L(u, b, c ′′ ) counts lattice points in Λ which lie in a box of side length We claim that the successive minima of the lattice Λ are bounded above and below by constants depending only on K and n.Taking this on faith, it will then follow that which will settle the lemma, on summing over O(X) choices for b ∈ B(X, Y ).
To check the claim, we order the index tuples (l, i) and (k, β) in the matrix H lexicographically. Write with the n × m matrix B l = (m liβ ) 1 i n,1 β m .Note that B l has orthogonal and norm one columns for 1 l d.We can then write H as a block matrix Let B = B(u) be the nd×md matrix which is a diagonal block matrix, with the matrices B 1 , . . ., B d on the diagonal.Let W be the md × md block matrix, with blocks ω Consider the lattice Γ = W Z md ⊂ R md and note that this only depends on the basis ω 1 , . . ., ω d .Moreover, if w ∈ Γ, then Bw, Bw = w t B t Bw = w t E md w = w, w .
Hence the successive minima of the lattice Λ coincide with those of Γ, which thereby establishes the claim.
It follows from the previous result that where f (u) is given by (5.4) and U * is the set of u ∈ U such that (5.7) holds.
Recall that the ð l,i (u) are the eigenvalues of the matrix associated to the quadratic form ).The next result collects together a number of properties concerning the size of the eigenvalues ð l,i (u).
Lemma 5.6.Assume that Assumptions 1-3 hold and suppose that m m − 1 is the degree of the polynomial appearing in Assumption 3.For each 1 l d we order the eigenvalues ð l,i (u) such that Then there exist constants C 1 , . . ., C d > 0 such that the following holds: (  (5.5).Assume now that u l = 0. Note that each of the eigenvalues ð l,i (u) arises by multiplication with u l from the eigenvalues of the matrix corresponding to Write ðl,i (u) for those eigenvalues in the same ordering.Assume that the lower bound | ðl,n−1 (u)| ≫ 1 is not satisfied.Thus there exists a sequence of t j in the range |t j | C l such that ðl,n−1 (t j ) → 0, for j → ∞, where we write ðl,n−1 (t) for the second smallest eigenvalues of Q (l) + tR (lτ ) .As the set of t is compact there is a convergent subsequence, convergent to t ′ say, with rank(Q (l) + t ′ R (lτ ) ) < n − 1.This contradicts Assumption 2. Now we consider the case m = 1 and m = 0.By Assumptions 1 and 3 we deduce that det(Q (l) + tR (lτ ) ) is a non-zero constant independent of t.In particular the rank of this matrix is always n and the argument above shows that |ð l,n (u)| ≫ |u l |.
Next we consider the case |u lτ | > C l |u l | and u l = 0. Again, we write ðl,i (u) for the eigenvalues of Q (l) + u lτ u l R (lτ ) .Note that we have Moreover, we observe that We therefore find that From this we obtain the lower bound 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 P d , and let E 2 (P ; X, Y ) denote the remaining contribution.The following pair of results treats these two quantities in turn.
Proof.On recalling the definition (5.4) of f (u), we deduce from (5.9) that Here, we recall that U * is the set of u ∈ U such that (5.7) holds.Consider for a moment a fixed value of l.
In either of these two cases we therefore have The contribution gets maximal for Y ≍ X 2 , in which case we get the upper bound The first term is satisfactory for the lemma.If 2 − n/2 + 3m/2 0, then the second term is O(P −dm+ε ), which is satisfactory.If n 3 + 3m, on the other hand, then we take X ≪ P d and get the satisfactory upper bound O(P d(2−n/2+m/2)+ε ).Then E 2 (P ; X, Y ) is Hence we find that E 2 (P ; X, Y ) is since m m − 1.We now deduce that in either case we have Let I 1 denote the contribution to the integral from those u for which there exists at least one u l with |u l | ≫ (P/Y 1/d ) 2 , and let I 2 denote the remaining contribution.
On recalling that U * is the set of u ∈ U such that (5.7) holds, it is clear that Turning to I 1 , we see that Hence, since P −1 Y 1/d ≪ 1, it now follows that In summary we have shown that with I 1 , I 2 as above.
Since n m+4, the exponent of H(u) in I 2 is less than or equal to −1.If n−m−2+κ 2 > 1, then it follows from (4.12) that However, if n−m−2+κ 2 = 1, then we apply (4.13) to deduce that the same bound holds.On the other hand, (4.12) and (4.13) also yield We conclude that We now consider these three terms separately, starting with the third and recalling that n − m 4. If −3m/4 + n/4 − (4 − κ)/4 0, then we get an upper bound In the opposite case we get the upper bound ≪ P −dm/2+ε ≪ P −m+ε , on using Y P d and d 2.
Proof.Let p be a prime ideal of residue degree 1, so that N p = p, for a rational prime p.We may assume that p is unramified in K and that since the desired estimate is trivial otherwise.Since K/Q is Galois, this means that there is a factorisation (p) = p 1 • • • p d into prime ideals, where p 1 , . . ., p d are the d conjugates of p under Gal(K/Q), satisfying N p i = p for 1 i d.
It will be convenient to write S p = S p (N ; v) and p = p τ −1 in the proof.Then p and p are distinct prime ideals, with G p = pp and N p = N p = p.Choose γ = g/α ∈ F(p) as in Lemma 3.4, so that ψ(γ•) is a primitive character modulo p. Then we can write It turns out that the remaining sum Σ 2 (a) can also be interpreted as a product of Gauss sums.First, we observe that we have the factorisation We now piece everything together in (6.2).To begin with, it follows from squaring and differencing that where G is given by (6.1).
Our next task is to analyse the oscillatory integral K(u, P m) when F is given by (1.5), based on (4.15).To the fixed automorphism τ ∈ Gal(K/Q) in (1.5), we can associated a unique integer l τ ∈ {1, . . ., d}, as in (1.4).We therefore have m , 0, . . ., 0).Then it follows that the quadratic form (4.14) has an underlying matrix which is the block diagonal matrix      If m is given coordinates m = (m (1) , . . ., m (d) ) on V n , then we have )e G (l) (x (l) ) − P m (l) .x (l) dx (l) , where G (l) has underlying matrix u l A l + u lτ B l .Since this matrix is diagonal, on assuming that the weight W is chosen suitably, we may further factorise to obtain Proof.Clearly we get exponential decay in K(u, P m) unless P |m| ≪ |u|P ε , as we now assume.However, on examining each of the factors in K(u, P m) separately, the essential support of K(u, P m) is rendered clear.Next, for each i m and 1 l d, we have ρ l (L i (u)) = a We now piece everything together in our expression (4.23) for E(N ; P ; X, Y ).We shall continue to adhere to the convention that the value of ε > 0 is allowed to change at each appearance, and that all implied constants are allowed to depend on ε.
Recall the definition of U from (4.22).Combining (4.23) and Lemma 6. and R(m) denotes the set of u ∈ U such that (6.4) holds.
We now make the exact same change of variables c = δm that we made previously in (5.2).Then, in particular, we can assume that (5.3) holds.Moreover, on dropping the information about G(m), Lemma 6.

for 1 m
n and non-zero a 1 , . . ., a n , b 1 , . . ., b m ∈ o K , and where τ ∈ Gal(K/Q) is a fixed non-trivial automorphism.Taking m = n and a i = b i = 1 for 1 i n, we are led to an instance of the partial trace problem in (1.1

. 6 )
, involving an infinite sum over non-zero integral ideals 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 b for generalised quadratic forms F .Instead we shall break into residue classes modulo a larger ideal G b, which is the least common multiple of the ideals b τ −1 , as τ ranges over the automorphisms that actually occur in F .Poisson summation then leads to the analysis of certain exponential sums S b (N ; m) and oscillatory integrals I b (N ; m), which are indexed by b ⊂ o K and suitable vectors m ∈ K n .While the treatment of S b (N ; m) is relatively standard, the main challenge is to understand I b (N ; m).

Lemma 3 . 1 .
Let ε > 0 and let b, c be integral ideals.Then (i) there exists α ∈ b such that ord p (α) = ord p (b) for every prime ideal p | c; (ii) there exists α ∈ b and an unramified prime ideal p coprime to b τ for all

any choice of b. Lemma 3 . 4 .
Let b o be a non-zero ideal.Then there exists γ ∈ F(b) such that ψ(γ•) defines a non-trivial primitive additive character modulo b.

D n/ 2 K P 2d N 4 . 4 .
where we recall that G b = G b −1 d −1 is the dual of G b. Putting everything together in (4.6), we have therefore established that N W (F, N ; P ) = c P b≪P d m∈ G b n (N G b) −n S b (N ; m) Ĩb (m), with S b (N ; m) as in the statement of the lemma and Ĩb (m) = V n W (x/P )h N b P d , | Nm(F (x) − N )| P 2d ψ (−m.x) dx.A simple change of variables yields Ĩb (m) = P dn I b (N/P 2 ; P m), as required.The exponential sum.We proceed by discussing S b (N ; m) in Lemma 4.1, for m ∈ G b n .Let γ = g/α ∈ F(b) be as in Lemma 3.4.Then we have * σ (mod b) σ(x) = a∈(o/b) * ψ(γax), for any x ∈ o.It follows that S b (N ; m) = a∈(o/b) * ψ(−γaN ) x (mod G b) ψ (γaF (x) + m.x) .(4.7)

Corollary 4 . 3 .
Assume that F is admissible, in the sense of Definition 3.6.Let b be an integral ideal and let m ∈ K n .Then S b (N ; m) ≪ (N b) 1−n/2 (N G b) n .

Lemma 4 . 4 .
We have S b (N ; m) = 0 unless m.h ∈ d −1 for all h ∈ H b .Proof.Returning to the definition of S b (N ; m) in Lemma 4.1 and noting that G b n ⊂ H b ⊂ o n , we may write S b (N ; m) = * σ (mod b) a∈o n /H b σ(−N ) h∈H b / G b n σ(F (a))ψ(m.a)ψ(m.h).

Lemma 4 . 5 .
Let b be a non-zero integral ideal and suppose that b = b 1 b 2 for integral ideals b 1 , b 2 , such that gcd(N b 1 , N b 2 ) = 1.Then, for any N ∈ o and any m ∈ G b n , we have S b (N; m) = S b 1 (N b 2 2 N ; (N b 2 )m)S b 2 (N b 1 2 N ; (N b 1 )m).Proof.According to Lemma 3.4, there exists γ = g/α ∈ F(b) such that ψ(γ•) is a primitive character modulo b.Then, (4.7) implies thatS b (N ; m) = a∈(o/b) * ψ(−γaN ) x (mod G b) ψ (γaF (x) + m.x) .Let us write N b i = b i for i = 1, 2. The assumption gcd(b 1 , b 2 ) = 1 implies that ( G b 1 , G b 2 ) = 1.Moreover, we have b 1 ∈ b 1 , b 2 ∈ b 2 and ((b 1 ), b 2 ) = ((b 2 ), b 1 ) = 1.(4.8)According to Lemma 3.1(i) we find elements λ, µ ∈ o such that ord p (λ) = ord p (b 1 ) and ord p (µ) = ord p (b 2 ) for all p | G b 1 G b 2 .It follows from the Chinese remainder theorem, in the form Lemma 3.2, that we can write a = µb + λc for b (mod b 1 ) and c (mod b 2 ).Likewise, we claim that we can write x = b 2 b+b 1 c, for b (mod G b 1 ) and c (mod G b 2 ).To prove the claim it suffices to show that there is an isomorphism o/ G b 1 × o/ G b 2 → o/ G b, given by (u, v) → b 2 u + b 1 v.This map is clearly well-defined, since b 1 ∈ G b 1 and b 2 ∈ G b 2 .Moreover, injectivity follows from the coprimality conditions ((b 2 ), G b 1 ) = ((b 1 ), G 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 ∈ G b 1 and b 2 ∈ G b 2 , it follows that

Corollary 4 . 6 . 4 . 5 .
Let b be a non-zero integral ideal and suppose that b = b 1 b 2 for integral ideals b 1 , b 2 , such that gcd(N b 1 , N b 2 ) = 1.Then S b (N ; 0) = S b 1 (N ; 0)S b 2 (N ; 0).Proof.On making an obvious change of variables to the a-sum and the x-sum in (4.7), we note that S b (c 2 N ; 0) = S b (N ; 0) for any c ∈ Z which is coprime to b.The statement now follows from an application of Lemma 4.5.The exponential integral.In this section we discuss the exponential integral I b (t; k) that appears in Lemma 4.1, for given t ∈ V and k ∈ V n .It will be convenient to set 0 < ρ = N b P d ≪ 1, with which notation we have

Lemma 4 . 11 .
Assume that the descended system has codimension d.There exist positive constants C, C 0 , C 1 , . . .such that the function f : R d → R is a smooth weight function that is supported on [−C, C] d and satisfies

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 m in the asymptotic formula for N W (F, N ; P ) in Lemma 4.1.This contribution is ≪ P (n−2)d E(N ; P ), where E(N ; P ) = 0 =b⊂o N b≪P d 0 =m∈ G b n (N G b) −n |S b (N ; m)||I b (N/P 2 ; P m)|.(4.19)

5 .
.21) for any X, Y satisfying (4.20).It follows from Corollary 4.8 that I b (N/P 2 ; P m) ≪ A P ε U |K(u, P m)|du + P −A , for any A 1, where K(u, P m) is given by (4.10) andU = u ∈ V : H(u) ; P ; X, Y ) ≪ A P −A + P ε Y −n b∈B(X,Y ) 0 =m∈ G b n |S b (N ; m)| U |K(u, P m)|du,(4.23)for any A 1, where B(X, Y ) is the set of non-zero ideals b ⊂ o for which X N b < 2X and Y N G b < 2Y .Homogeneous case: proof of Theorems 1.3 and 1.4
1 β m, and consider the lattice Λ = HZ md ⊂ R nd .
at each place 1 l, k d, where E m is the m-dimensional identity matrix.Then H = BW.

1 2 1 2 1 2
2 yieldsS b (N ; δ −1 c) ≪ (N b) −(n−m)/2+ε (N G b) n−m/2 g(b) ≪ X −(n−m)/2+ε Y n−m/2 g(b), where g(b) = p|(b,N ) N p N b N p p k N b k 2 p k .In this notation we conclude thatE(N ; P ; X, Y ) ≪ A X −(n−m)/2 P ε Y m/2 b∈B(X,Y ) 0 =c∈o n (5.3) holds g(b) R(δ −1 c) f (u)du + P −A .Let L(u) = b∈B(X,Y ) c∈C (u,b) g(b),where C (u, b) is the set of non-zero vectors c ∈ o n for which (5.3) holds and|c (l) i | ≪ P −1+ε Y 1/d |ρ l (L i (u))| if i m, P −1+ε Y 1/d |u l | if i > m, Suppose that a, a 1 , a 2 are integral ideals such that a = a 1 a 2 , with a 1 and a 2 coprime.Let α 1 , α 2 ∈ o satisfy ord p (α 1 ) = ord p (a 1 ) and ord p (α 2 ) = ord p (a 2 ), for allp | a. Then o/a = {α 1 µ + α 2 β : β ∈ o/a 1 , µ ∈ o/a 2 }.3.2.Construction of primitive characters.Let ψ(•) = exp(2πi Tr K/Q (•)) be a character on K.The following result gives a way to construct primitive characters o/b → C. We begin by showing that σ 0 is an additive character modulo b if and only if bd ⊂ a γ .But σ 0 is an additive character modulo b if and only if σ 0 (x + z) = σ 0 (x) for all x ∈ o and z ∈ b.But this happens if and only if γz ∈ ô for all z ∈ b, which is if and only if bd ⊂ a γ .This establishes the claim.Now suppose that bd ⊂ a γ , which means that a γ | bd.Thus there is an integral ideal h such that bd = a γ h.We wish to show that σ 0 is primitive if and only if h | d with (h, b) = 1.To do so we note that σ 0 is primitive if and only if a +i d for any y ∈ [−C, C] d and any i 1 , . .., i d 0. The constants C, C 0 , C 1 , . ..dependonly on the coefficients of F and the parameter δ in the definition of W .Proof.In the course of the proof it will be convenient to write s = (s 1 , . . ., s d ), u = (u 1 , . . ., u d ), ξ = (ξ 1 , . . ., ξ d ) and t = (t 1 , . . ., t d ).Recall the definition of the weight function W in §4.2 for a suitable fixed ξ ∈ R dn .Making the change of variables in (4.4), we see that 2+ε ,