An asymptotic formula for representations of integers by indefinite hermitian forms

We fix a maximal order $\mathcal O$ in $\F=\R,\C$ or $\mathbb{H}$, and an $\F$-hermitian form $Q$ of signature $(n,1)$ with coefficients in $\mathcal O$. Let $k\in\N$. By applying a lattice point theorem on the $\F$-hyperbolic space, we give an asymptotic formula with an error term, as $t\to+\infty$, for the number $N_t(Q,-k)$ of integral solutions $x\in\mathcal O^{n+1}$ of the equation $Q[x]=-k$ satisfying $|x_{n+1}|\leq t$.


Introduction
The representation theory of quadratic forms has a long history. It starts with the qualitative problem of determining which integers are represented by a given quadratic form. For example, Fermat, Legendre and Lagrange found which integral numbers are represented as sums of two, three and four squares respectively. After that, the quantitative problem was considered by Jacobi, Kloosterman and Liouville among others. For instance, Jacobi proved that the number of ways to write any positive integer number k as a sum of four squares is 8 m|k, 4∤m m, by determining the Fourier coefficients of the theta function associated to the form x 2 1 + · · · + x 2 4 . These examples are positive definite quadratic forms. For indefinite forms, the literature is much less abundant as the definite case. In the present work, we study the quantitative problem for quadratic and hermitian forms of signature (n, 1).
Let F = R, C or H, and O a maximal order in F. Thus O is the ring of integer numbers Z in the real case, the ring of integers of an imaginary quadratic extension of Q in the complex case (e.g. the Gaussian integers Z[ √ −1]), and, for instance, the ring of Hurwitz integers if F = H, though there are many other choices. We denote by C * the conjugate transpose of C and we let B[C] = C * BC, where B ∈ M(m, F) and C ∈ M(m, l; F).
We consider an F-hermitian matrix with respect to the canonical involution on F, with a ∈ N and A ∈ M n (O) a positive definite F-hermitian matrix. We also denote by Q the induced F-hermitian form where x = (x 1 , . . . , x n ) t ∈ F n . We are interested, for k ∈ N, in the solution vectors x ∈ O n+1 of the equation Since Q is an indefinite form, this set is either empty or infinite.
In this paper we establish an asymptotic formula, for large t, for the number Our main result, Theorem 5.1, is the asymptotic formula is the local density of the representation (1.2) (see (4.5)). The number τ is defined in (3.3). It depends only on F, or more precisely, on the lower bound of the first eigenvalue of the Laplace-Beltrami operator on Γ\H n F , where Γ is an arithmetic lattice and H n F is the n-dimensional F-hyperbolic space. When F = R, formula (1.5) holds with τ = n − 3/2 (note that 2ρ = n − 1).
In the particular case when F = R and Q = I n,1 = In −1 , the main theorem says us that the number N t (I n,1 , −k) of vectors x ∈ Z n+1 such that x 2 1 + · · · + x 2 n − x 2 n+1 = −k and |x n+1 | ≤ t satisfies the following asymptotic estimate, as t → +∞, This formula is due to J. Ratcliffe and S. Tschantz [RT97]. The present article was inspired by this work. Our main tool is the hyperbolic lattice point theorem of P. Lax and R. Phillips [LP82] (improved by B. M. Levitan [Le87]) in the real case, and in the general case (F = R, C and H) by R. Bruggeman, R. Miatello and N. Wallach [BMW99]. For F = R we use the best lower bound known for the first eigenvalue of the Laplace-Beltrami operator on Γ\H n R , obtained in [EGM90] and in [CLPS91]. After applying the lattice point theorem, we determine the coefficient in the main term by using the theory of Siegel [Sie44] on quadratic forms and its generalization for hermitian forms given by Raghavan [Rag62]. The formula (1.5) holds only for F = R and C since there is not a generalization of Siegel's theory in the quaternionic case.
Consider the real norm · on F n+1 given by the positive definite F-hermitian form Q = ( A a ), that is where x = (x 1 , . . . , x n ) t ∈ F n . Note that for a solution x ∈ R(Q, −k), we have Thus, we are also obtaining an asymptotic formula, as r → +∞, for the number of solutions x ∈ O n+1 of (1.2) with x lying in the ball of radius r centered at the origin with respect to · . In this context, the problem has been considered by some authors (see [Sch85], [Sa90], [BR95]). They count integer points on homogeneous spaces (hyperbolic spaces in our particular case) by using harmonic analysis on the corresponding groups and obtain similar asymptotic formulas for N t (Q, −k). The ternary real case is considered in [Bo01]. The outline of the paper is as follows. In Section 2 we introduce the geometric context, relating N t (Q, −k) with the number of elements in an arithmetic subgroup of Iso(H n F ) satisfying a geometric condition. In Section 3, we apply the lattice point theorem to count such lattice points. Section 4 uses Siegel's theory to compute the main term of the formula. We conclude with Section 5 which contains our main theorem and some examples.

F-Hyperbolic space
Throughout the paper, given R a ring with identity, we denote by M(m, n; R) the set of m × n matrices with coefficients in R. When n = m we just write M(m, R). Let R m denote the right R-module M(m, 1; R). We denote by GL(m, F) the general linear group and by SL(m, F) its derived group, the special linear group. We denote by C * the conjugate transpose of C and we let B[C] = C * BC, where B ∈ M(m, F) and C ∈ M(m, k; F).
We now introduce a model for Riemannian symmetric spaces of real rank one and negative curvature (leaving out the Cayley plane). These are the real, complex and quaternionic hyperbolic spaces. For a general reference on this subject see [BH99, II. §10] and [Mo73,§19].
Let Q be the matrix defined as in (1.1). The set will serve as the set of points for the Q-Klenian model of n-dimensional F-hyperbolic geometry. Note that condition Q[x] < 0 is well defined on the projective space. The metric in this model is given by for any x = (x 1 , . . . , x n+1 ) ∈ F n+1 − {0}, where Q(x, y) = y * Qx. It is easy to check that the distance function on H n F (Q) satisfies It is standard to use the unit ball model B n F = {u ∈ F n : |u| < 1} for the F-hyperbolic space of dimension n. Write I n,1 = In −1 . The map is an identification between H n F (I n,1 ), the Klenian model for Q = I n,1 , and the open unit ball model B n F . The metric here is This metric induces a metric on H n F (I n,1 ) under the identification (2.4), which coincides with (2.2) for Q = I n,1 . Moreover, the linear map We denote by the Q-unitary group and by SU(Q, F) = U(Q, F)∩SL(n+1, F) the special Q-unitary group. For Q = I n,1 , it is known the classical notation U(I n,1 , F) = Ø(n, 1), U(n, 1), Sp(n, 1) and SU(I n,1 , F) = SO(n, 1), SU(n, 1), Sp(n, 1), for F = R, C, H respectively. It will be useful to known who is the center of U(Q, F). This is given by Indeed, by the distance formula (2.3), its elements act by isometries. Let Iso(H n F (Q)) (resp. Iso + (H n F (Q))) denote the set of isometries (resp. orientation-preserving isometries) of H n F (Q). It is clear that the elements of Z(U(Q, F)) act as the identity map on H n F (Q). Moreover, we have that ) is an exact sequence. Furthermore, the group PU(Q, F) := U(Q, F)/Z(U(Q, F)) is, up to finite index, the full isometry group Iso(H n F (Q)). Remark 2.1. When F = R the group Ø(Q) := U(Q, R) has four connected compo- When F = C, the group Iso(H n C (Q)) is generated by PU(Q, C) and the conjuga- We fix a maximal order O in F; this means that the subset O ⊂ F satisfies the following conditions: and O is maximal among all orders (subsets of F satisfying (i)-(iii)). Here r F := dim R (F).
The elements of O will be called integers. The ring of integers Z is the only order in R. In C, the maximal orders are the rings of integers of the imaginary quadratic extensions Q( When F = H there are many orders. We refer to [MR03] for more detailed information. As a canonical example in this case, the reader may take Let Γ Q be the set of unimodular matrices in U(Q, F), that is This is a discrete subgroup of U(Q, F) with finite center. The action of Γ Q on H n F (Q) is discontinuous, not free and the quotient Γ Q \H n F (Q) is of finite volume and not compact.
On the other hand, the group Γ Q acts by left multiplication on the set R(Q, k) given in (1.3).
Lemma 2.2. The set of Γ Q -orbits in R(Q, k) is finite.
From now on, we fix k ∈ N such that Q represents −k, that is, there exists x ∈ O n+1 satisfying Q[x] = −k. Let F be a (finite) set of representantives of the Γ Q -orbits of R(Q, −k). Let Γ Q,y be the stabilizer of y in Γ Q , which is finite. We conclude this section by relating the number N t (Q, k) defined in (1.4), with the cardinality of subsets of lattice points in Γ Q .
Fix t > 0, thus cosh(s) = a 1/2 k −1/2 t. Let g ∈ Γ Q and y ∈ F such that gy = But the condition on the left ensures that x ∈ R t (Q, −k). We conclude that The proposition follows by counting the elements of these sets.

Lattice point theorem
In this section we use lattice point theorems to determine the asymptotic distribution, for t → +∞, of the number of elements in the sets We will follow the notation in [BMW99] since its lattice point theorem works for every hyperbolic spaces. Let where Z denotes the center of U 0 (Q, F). The group U(Q, F) is connected for F = C, H, thus Z is as in (2.6). When F = R, U 0 (Q, R) = PSO(Q) (see Remark 2.1) has trivial center.
The group G is a connected semisimple group of real rank one and trivial center. Let g be the Lie algebra of G and let θ be the Cartan involution on g given by θ(X) = −X * with Cartan decomposition g = k ⊕ p.
The group G acts transitively on H n Thus K ∼ = SO(n), U(n) respectively. The Lie algebra of K is k, and then it is a maximal compact subgroup of G. Let B be the bilinear form on g given by where B K denotes the Killing form on g. The map g → [ge n+1 ] from G to H n F (Q) gives rise to an G-equivariant bijection between the symmetric space G/K and H n F (Q). Under this identification, the Riemannian structure given in (2.2) on H n F (Q) corresponds to the structure induced on G/K by the bilinear form B restricted to p. This is the standard Riemannian metric on H n F (Q) which, when F = R gives H n R (Q) constant curvature −1, and pinched sectional curvature in the interval [−4, −1] to H n F (Q), if F = C, H. Fix G = N AK an Iwasawa decomposition of G and let g = k ⊕ a ⊕ n be the corresponding decomposition at the Lie algebra level. Let M be the centralizer of A in K. Let 2ρ denote the sum of the positive roots of G, which is equal to n − 1, 2n, 4n + 2 for F = R, C, H respectively.
Let Γ be a non cocompact lattice in G. Let ∆ be the Laplace operator on Γ\H n F (Q). We identify −∆ with the Casimir operator C of G, with respect to the form B. We fix a complete orthonormal set {ϕ j } of real valued eigenfunctions of C, with (exceptional) eigenvalues 0 < λ 1 ≤ · · · ≤ λ N < ρ 2 , written as λ j = ρ 2 − ν 2 j , with 0 < ν N ≤ ν N −1 ≤ · · · ≤ ν 1 < ρ. Now we can state the hyperbolic lattice point theorem. It was proved for the real hyperbolic space by P. Lax and R. Phillips [LP82], with an improved error term by B. M. Levitan [Le87] and generalized by R. Bruggeman, R. Miatello and N. Wallach [BMW99] for any symmetric space of real rank one.
Theorem 3.1. In the notation above, for [x], [y] ∈ H n F (Q), we have that Note that the summation in (3.2) can be restricted to the indices j such that ρ + ν j > 2ρ n n+1 (in the real case we can replace > by ≥). In the case when the term of ρ + ν 1 is meaningful, we will allow the error term to increase up to ρ + ν 1 . Put and F = C, H, We see that in (3.4) the error term depends on the first eigenvalue of the Laplace operator on Γ\H n F (Q). The following theorem was proved in [EGM90] and [CLPS91]. The notation PSO(Q) was introduced in Remark 2.1.
Proof. It is sufficient to prove the lemma for Q = I n,1 since, if we write A = L * L and set T = L √ a , the map g → T gT −1 induces an isomorphisms of Lie groups from U(Q, F) to U(I n,1 , F), which preserves the Killing form.
Recall that a is the Lie algebra of the group A from the Iwasawa decomposition taken at the beginning of this section. This is a maximal abelian subalgebra of p. It has dimension one. Let H 0 ∈ a such that B(H 0 , H 0 The adjoint representation restricted to K left invariant the set S and this action is transitive. The stabilizer of H 0 is M . Then, we have a K-equivariant bijection from the manifold K/M to S. Moreover, we take the Riemannian metric B on K/M and S, and we have an isometry. The assertion follows by showing that the Riemannian manifold S is isometric to S nr F −1 , the (nr F − 1)-dimensional sphere on R nr F .
We take the lattice Γ 0 Q in G given by the image of Γ Q under the projection from U(Q, F) 0 to G. In the real case, we have Γ 0 Q = Γ Q ∩ PSO(Q). Let us denote by w the number of units in O. It is clear that Let {g 1 , . . . , g κ } be a set of representatives of the Γ 0 Q -coclases of Γ Q . For example Now we can apply the lattice point theorem to our problem. For each 1 ≤ j ≤ κ, Theorem 3.1 implies that A trivial verification shows that vol(Γ 0 Q \H n R (Q)) = 2 vol(Γ Q \H n R (Q)) for F = R and vol(Γ 0 Q \H n F (Q)) = vol(Γ Q \H n F (Q)) otherwise, where vol(Γ Q \H n F (Q)) denotes the volume on any fundamental domain in H n F (Q) relative to Γ Q (the action of Γ Q on H n F (Q) is not faithful). Furthermore, Lemma 3.4 gives ζ = vol(S nr F −1 ). These considerations imply, by adding (3.5) over j, that if F = H. Applying these equations to (2.7), we obtain that where w = w if F = R, C and w = 2 otherwise. Recall that cosh(s) = a 1/2 k −1/2 t. Notice that we can replace the error term in (3.8) by O(t τ ) since e s ∼ 2 cosh(s) = 2 a 1/2 k −1/2 t as s → +∞, and furthermore e 2ρs = 2 2ρ a ρ k −ρ t 2ρ + O(t τ ).
Collecting all the information in this section, we have obtained the following formula.
The last assertion follows from Remark 3.3.

The mass of the representation
The object of this section is to obtain a formula for the term y∈F |Γ Q,y | −1 by using Siegel's theory on quadratic forms and its generalization to complex hermitian forms given by Raghavan. Our main references are [Sie67] and [Rag62]. From now on we make the assumption F = R or C, since there are no similar results to those in [Sie67] in the quaternionic hermitian case.
We denote by d O the discriminant of the quotient field of O and we assume from now on that n > 2.
We pick v ∈ F n and R ∈ M(n, F) such that the matrix has signature (n, 1). For y ∈ F n+1 such that Q[y] = −k, let U(Q, F) y denote the set of elements U ∈ U(Q, F) such that U y = y. Note that if y ∈ O n , the stabilizer of y in Γ Q is Γ Q,y = U(Q, F) y ∩ GL(n + 1, O). We fix on Ω(Q, W ) the volume element dω = det(W Q −1 ) 1/2 {dX} {dW } following the notation in [Sie67] and [Rag62]. Similarly, on Ω(Q, W ; y) we use the volume element The groups U(Q, F) and U(Q, F) y act by left multiplication on Ω(Q, W ) and Ω(Q, W ; y) respectively, and the given measures on these varieties are invariant under these actions. Siegel (and [Rag62] for the hermitian case) defines the measure of the representation of −k ∈ Z by Q as Here µ(Q) denotes the measure of the unit group Γ Q given by the volume of any fundamental domain for the action of Γ Q on Ω(Q, W ), and similarly, µ(y, Q) is the measure of the representation y, given by (r F / |d O |) n(n+1)/2 times the volume of any fundamental domain for the action of Γ Q,y on Ω(Q, W ; y). These numbers do not depend on W . We will recover from the right hand of (4.1) the term y∈F |Γ Q,y | −1 and then, by applying Siegel's main theorem, we will obtain an explicit formula for this term. By [Sie67,Thm. 7 Let F (y) be a fundamental domain of the action of Γ Q,y on Ω(Q, W ; y). By definition µ(y, Q) = | det W | 1/2 | det Q| −1/2 (r 2 F /|d O |) n(n+1)/4 F (y) dω * , but the measure dω * is invariant by Γ Q,y , then Using [Sie67, Thm. 6, Ch IV] and [Ram61, Lemma 9], we have that .
Finally, (4.2) and (4.3) imply that Now, we will recall Siegel's main theorem for indefinite quadratic and hermitian forms (see [Sie44, Thm. 1] and [Rag62, Thm. 7]). For every rational prime p, the p-adic density of representation of −k by Q is Define δ(Q, −k) = p δ p (Q, −k), the local density, where the product is over all prime numbers. See [Sie44, Thm. 1] for the real case (it is mentioned in page 580 that (4.6) holds for n = 3) and [Rag62, Thm. 7] for the complex case. See [Bo01] for the ternary quadratic case (n = 2 and F = R).
By combining equation (4.4) and Theorem 4.1, we obtain an expression for the term y∈F |Γ Q,y | −1 , the main goal of this section.
Corollary 4.2. We have that

Main theorem
We can now state the main result in this paper, which follows by combining Proposition 3.5 and Corollary 4.2. The last assertion is a consequence of Remark 3.3. We first recall some terminology: r F = dim R (F), O is a maximal order in F, d O is the discriminant of O and ρ = r F 2 (n + 1) − 1. Theorem 5.1. Let Q be an F-hermitian matrix as in (1.1), with F = R and n ≥ 3 or F = C and n ≥ 2. We fix k ∈ N such that −k is represented by Q. Then, the number N t (Q, −k) of elements x ∈ O n+1 such that Q[x] = −k and |x n+1 | ≤ t, satisfies the following asymptotic estimate as t → +∞, where τ is as in (3.3). Moreover, when F = R and n > 2, the formula (3.9) holds for τ = n − 3/2.
Remark 5.2. In order to get an explicit value of the main term in (5.1) for a fixed quadratic or hermitian form Q, one needs to determine the local density δ(Q, −k). When F = R, T. Yang [Ya98] computed this local density for any quadratic form Q. For F = C see [Hi99].
We conclude the article considering the case of the Lorentzian hermitian form over the Gaussian integers, that is Q = I n,1 for F = C and O = Z[ √ −1] (thus d O = −4). Let us assume that where λ 1 is the first nonzero eigenvalue for the Laplace operator ∆ on Γ 0 Q \H n C (Q). Jian-Shu Li [Li91] proves this for ∆ on the subspace of nondegenerate forms in L 2 (Γ 0 Q \H n C (Q)). Similar to Remark 3.3, we obtain that ν 1 ≤ n − 1, thus ν 1 > ρ n−1 n+1 , then τ given in (3.3) can be taken as τ = 2n − 1.