Hybrid bounds for automorphic forms on ellipsoids over number fields

We prove upper bounds for Hecke-Laplace eigenfunctions on certain Riemannian manifolds of arithmetic type. The manifolds under consideration are d-fold products of 2-spheres or 3-spheres, realized as adelic quotients of quaternion algebras over totally real number fields.


Introduction
Given a Riemannian manifold X, it is a classical problem to give pointwise upper bounds for L 2 -normalized Laplace eigenfunctions in terms of the eigenvalue and/or properties of X.This is particularly interesting if the manifold is of "arithmetic type", i.e. is acted on by a suitable family of Hecke operators, in which case one considers Laplace eigenfunctions that are simultaneously eigenfunctions of the Hecke algebra.The classical situation is X := Γ\H, where H = SL 2 (R)/SO 2 is the hyperbolic plane and Γ ⊆ SL 2 (R) is an arithmetic subgroup.
1.1.Bounds on 2-dimensional ellipsoids.In [BM] we considered the case where X is a finite union of 2-spheres S 2 = SO 3 /SO 2 , each of them realized as a connected component of an adelic quotient attached to a totally definite quaternion algebra over Q.The present paper is a continuation of [BM]: on the one hand, by a new treatment of the amplifier we improve significantly the main result in [BM], on the other hand we extend the argument to arbitrary totally real number fields F of degree d over Q; the underlying manifold is then a union of d-fold products of 2-spheres.An application of the number field setting is given in the next subsection.
We proceed to describe our results in more detail.Let F/Q be a totally real field and B be a totally definite quaternion algebra defined over F (i.e. at all real places B is isomorphic to the Hamilton quaternions).We denote canonical involution, the reduced trace and reduced norm on B by z → z * , tr(z) = z + z * , nr(z) = zz * respectively.Let B 0 ⊂ B denote the pure (i.e. the trace 0) quaternions.Then (B 0 , nr), (B, nr) are quadratic F -spaces whose associated inner product is denoted by Conversely any ternary quadratic space V is similar 1 to some unique quaternionic space (B 0 , nr).The corresponding quaternion algebra B is the subalgebra C(V ) + of the Clifford algebra C(V ) of elements invariant under the canonical involution.
Let O ⊂ B be an order.To O is associated a finite disjoint union of quotients of d = [F : Q] products of 2-spheres The quotients X i are called the components of X(O) and the set indexing the components is the set of classes in the genus of the ternary quadratic lattice (O 0 , nr) where O 0 = B 0 ∩ O. Fix an SO 3 (R)-invariant Riemannian metric on S 2 .This induces a volume form and a d-tuple of Laplace operators ∆ = (∆ 1 , . . ., ∆ d ) on (S 2 ) d that descent to X (2) (O).We are interested in obtaining nontrivial bounds for the L ∞ -norm of an L 2 -normalized ∆-eigenfunction ϕ on X (2) (O) in terms of the eigenvalues λ = (λ 1 , . . ., λ d ) of ∆ and of the total volume vol(X (2) (O)).One has which is essentially the number of classes in the genus of (O 0 , nr).The trivial bound in this case is (see [Sar] for a general result) |λ| 1/4 , with |λ| := j=1...d (1 + |λ j |).
Our objective is to improve over this bound simultaneously in the λ and the volume aspect; such non-trivial bounds are called "hybrid".In this generality this is hopeless: the previous bound is indeed sharp both in the volume and in the λ-aspect.The possibility of constructing Laplace eigenfunctions with large sup-norm comes from the fact that ∆-eigenfunctions have very large multiplicities (roughly ≈ V 2 |λ| 1/2 ).As explained in [BM], a way to resolve this issue is to require ϕ to be also an eigenfunction of a family of "Hecke" operators, indexed by the complement of a finite, fixed subset of the prime ideals of F , {T p } p disc(O) .The Hecke operators {T p , p disc(O)} together with ∆ generate a commutative algebra of self-adjoint operators on L 2 (X(O)); in particular this space admits an orthonormal basis made of Laplace-Hecke eigenfunctions.
Theorem 1.Let O be an Eichler order and let ϕ be an L 2 -normalized Hecke-Laplace eigenfunction.Then one has ϕ ∞ |λ| 1 4 (V 2 |λ| 1 2 ) − 1 20 .Individually, we obtain the following bounds in the λ and in the volume aspect The first non-trival bound of this sort was obtained for F = Q, B indefinite and O fixed by Iwaniec and Sarnak [IS95]; for O varying (of square-free level) a bound simultaneously non-trivial in vol(X(O)) and in |λ| was obtained by the first named author and R. Holowinsky [BH10].This result was extended by Templier [Tem10] to the case of a totally real number field and for B indefinite at one place.In the definite case, the first non-trivial result we are aware of is due to Vanderkam [Van97]: for F = Q, B the Hamilton quaternions and O the maximal order, he obtained 24 +ε .Unaware of his work, we obtained in [BM] a hybrid bound for general B and any Eichler order O of the shape . Observe that (1.1) is stronger in both aspects.The improvement in the volume aspect comes from a new way to deal with the amplifier (occurring from the amplification method) which may be of general interest.In the λ aspect, the improvement comes from the use of Vanderkam's method.Our bound in (1.1), however, is marginally weaker than (1.2) because of some technical obstacles in the number field case.We remark that the best possible result in the situation of Theorem 1 should be ϕ ∞ |λ| 1/4 (V 2 |λ| 1/2 ) −1/2+ε .In particular, in the level aspect, we arrive at 33% of the conjectured bound which is similar to Weyl's bound vs. the Lindelöf Hypothesis for Riemann's zeta function.Gergely Harcos and Nicolas Templier kindly informed us that for the (indefinite) discriminant quadratic form b 2 − 4ac over Q, they obtained in [HT] the exponent −1/12 which they recently improved to the same exponent −1/6.This convergence of exponents obtained independently and in fairly different contexts makes it therefore likely that this result will be hard to improve with the present technology.
1.2.Application to 3-dimensional ellipsoids.We illustrate the extension of [BM] to general totally number fields by providing non-trivial sup-norm bounds for Hecke-Laplace eigenfunctions on manifolds X that are finite unions of d-fold products of 3-spheres S 3 = SO 4 /SO 3 (i.e.bounds for automorphic forms of orthogonal groups in 4 variables).The main point here is that there is a close relationship between automorphic forms on orthogonal groups in 4 variables over F and automorphic forms on orthogonal groups in 3-variables over suitable (possibly split) quadratic extension E of F ; this is an extension of the well known fact that SO 4 (R) is a double cover of SO 3 (R) × SO 3 (R).
We now briefly describe the set-up for the sup-norm problem on SO 4 , more details will be given later.We have seen that ternary quadratic spaces can be realized in terms of quaternion algebras, and a similar correspondence holds for quaternary spaces that we now describe: given a totally definite quaternary quadratic space (V, Q) over F of discriminant ∆, let E be the (totally real) quadratic where the involution * is extended E-linearly to B E in the obvious way.Clearly nr is F -valued on B and (B , nr) defines a non-degenerate quaternary quadratic space over F .Then for any totally definite quaternary quadratic space (V, Q) as above there is a unique quaternion algebra B such that (V, q) is similar 2 to (B , nr).The group SO(B ) embeds naturally into B × E .We consider the following simple example of an integral quadratic lattice in (B , nr): let O be an Eichler order in a quaternion algebra B over a fixed totally real field F and let E be a fixed quadratic algebra over F , either a field or F × F .Constructing B as above, the Eichler order O induces the quadratic lattice (O , nr) where indexed over the set of classes in the genus of the quaternary quadratic lattice (O , nr), hence Let ϕ be an L 2 -normalized Hecke-Laplace eigenfunction on X (3) (O).The trivial bound is now ϕ ∞ |λ| 1/2 , and we obtain the following improvement in the volume aspect: Theorem 2. In the situation described above, one has Remark 1.1.The present bound is a direct application of the arguments of the proof of Theorem 1; yet it seems to be the first instance of a non-trivial arithmetic sup-norm bound for a manifold 2 Even isometric if q is positive at every archimedean place by Eichler's norm theorem which does not factor into surfaces.Several improvements are possible which will be discussed in a future work: (1) We have considered here only the volume aspect.The diophantine counting Lemma 5 of §4 of this paper would yield quite directly some non-trivial hybrid bounds for some SO Qautomorphic forms, namely those, which at each archimedean place of F correspond (via the identification SO Q (F σ ) SO 4 (R)) to pure weight vectors with respect to the action of the maximal torus SO 2 (R) × SO 2 (R) < SO 4 (R).Laplace-Hecke eigenfunctions on 3-dimensional ellipsoids on the other hand, correspond to SO 3 (R)-invariant vectors; these are potentially long linear combinations of pure weight vectors, and Lemma 5 in its present form is not sufficient to obtain hybrid bounds for such functions.
(2) The present bound depends on the quadratic extension E. Making it explicit and non-trivial in this aspect requires a more precise description of the local structure of the quaternary quadratic lattices considered at the places where E is ramified and versions of the counting Lemmata 2 -5 taking this aspect into account.Observe that in the present case, the amplification method does not a priori require that E splits at many small places (as is the case in [DFI95] or [Ven10, §7]), for the group SO Q (F v ) has rank at least 1 for almost all places of F (the places at which B is unramified).
In the next section we introduce general notations and describe how the problem translates in the adelic setting.Section 3 discusses reduction theory for totally definite quadratic forms over totally real number fields, and we discuss general results about the representation of algebraic integers by such quadratic forms.In Section 5 we apply the pretrace formula and the amplification method in a by now standard way and reduce the problem of bounding the sup norm of Hecke-Laplace eigenforms to the diophantine problems of the previous section.In fact, as in previous treatments of sup-norm problems on arithmetic manifolds, the heart of the matter is the solution of a certain diophantine problem, in our case bounding representation numbers of quadratic forms of large discriminant of F -integral vectors that are almost parallel or almost orthogonal to a given vector.The first bound in (1.1) and the bound (1.3) follow only from Lemma 2 which is at least in principle not much more than a generalized Lipschitz principle.The second bound in (1.1) is more complicated and requires Lemmata 3 -5.

Preliminaries
2.1.Notation.Let F/Q be a totally real number field of degree d, O F its ring of integers, U = O × F its group of units and U + the subgroup of totally positive units.For a place v of F , we denote by F v the associated local field; for v non-archimedean, q v will denote the order of the residue field.A typical real place of F will be denoted as an embedding σ : F → R and the list of real places will be denoted by σ 1 , . . ., σ d ; for x ∈ F , we write x σ = σ(x) ∈ R for the corresponding conjugate.
We denote by and archimedean components of F .We denote the norm on F by N and use the same notation for natural extension of the norm to the F -ideals or to various F -algebras related to F (F v , F ∞ , A, etc...).Let B be a quaternion algebra defined over F , let B × be the group of units, Z its center (the subgroup of scalars), B 1 the subgroup of norm 1; we denote by PB × = Z\B × the projective quaternions.All these are considered as F -algebraic groups in the evident way; we denote the trace-0 quaternions and trace-0 quaternions of norm 1 by B 0 and B 0,1 respectively (considered as algebraic varieties over F ).We write B(A), B × (A), B 1 (A), B × (A f ), ..., B(F v ), B × (F v ) etc. for the sets of rational points over the corresponding F -algebras.
Let us recall finally that the conjugation action of group of units B × on the ternary quadratic space (B 0 , nr) is isometric (i.e.preserves the norm form) and that the map where Spin(B 0 ) denotes the spin group (the simply connected covering group of SO(B 0 )).

Representations at the archimedean place.
Recall that for any integer m ≥ 0 there is a unique irreducible (unitary) representation of SU 2 (C) Spin 3 (R) of degree d m = m + 1, denoted π m , and any irreducible representation of SU 2 (C) is isomorphic to some π m .The representation π m may be realized concretely as the space of complex homogeneous polynomials of degree m in two variables on which SU 2 (C) ⊂ GL 2 (C) acts by linear change of variables.The Casimir element (say with respect to the inner product on the Lie algebra (X, C acts on any realization by multiplication by the scalar be the stabilizer of (say) the north pole of S 2 under the natural projection SU 2 (C) → SO 3 (R).This is a maximal torus of SU 2 (C) which we parametrize as Let e : T(R) → C 1 be the character e(κ(θ)) = exp(ιθ).If V m is any vector space realizing π m and l ∈ Z, let V l m be the subspace of vectors "of weight l", that is, the vectors satisfying (2) and zero otherwise.Remark 2.1.The representation π m occurs in the right regular representation L 2 (SU 2 (C)) with multiplicity d m .When is m even, π m descents via the natural projection SU 2 (C) → SO 3 (R) to an irreducible representation of SO 3 (R).The direct sum of the weight zero vectors of each such copy of π m therefore injects into L 2 (SO 3 (R)) SO2(R)) = L 2 (S 2 ) and the image is the space harmonic homogenous polynomials of degree m/2 in R 3 (i.e.polynomials P such that ∆ R 3 P = 0).The action of Casimir element on that space corresponds to that of a fixed multiple of the Laplace operator ∆ S 2 .
Given a non-zero ϕ ∈ V l m and g ∈ SU 2 (C), we write for the corresponding normalized matrix coefficient, where ( , ) m is some SU 2 (C)-invariant inner product on V m .By definition g → |p m,l (g)| is bi-T(R)-invariant, and therefore depends only on the inner product of the north pole on S 2 with its image by the corresponding rotation.The following decay estimate holds as g ∈ SU 2 (C) gets "away" from T(R) (i.e.t gets away from ±1): Proof.By symmetry we may assume 0 m−l (t)| where (for α, β ≥ 0 integers) is the Jacobi polynomial.Let us recall that P (α,β) n has degree n and that {P (α,β) n | n ≥ 0} is orthogonal with respect to the inner product and that In particular, Remark 2.2.The above bound exhibits significant decay as t gets away from ±1 uniformly for |l| ≤ (1 + m) 1−δ for any fixed δ > 0. It is plausible that this holds also for very large values of l m: for instance in the extreme case l = m one has: More generally it is conjectured in [EMN94] that in (2.1), the term m+1 |l|+1 can be replaced by m+1 be the irreducible representation obtained from the corresponding normalized matrix coefficient.

Adelic interpretation of ellipsoids.
As in [BM,§4], it is useful to realize X (2) (O) as an adelic quotient: for instance, in this realization the Hecke operators admit a simple and natural description.The identification at the archimedean places (S 2 ) d Z(F ∞ )\B(F ∞ )/K ∞ extends readily to the adelic realization Therefore our original problem is equivalent to bounding a certain automorphic function ϕ on the adelic quotient (3) an eigenfunction of a certain Hecke algebra H(O), which is a commutative algebra of normal operators commuting with the Casimir operators C. We will recall the definition of the Hecke algebra H(O) in section §5; our assumptions imply that the B × (A)-translates of ϕ generate an automorphic representation 3), and π v is an unramified principal series representation for every finite place v disc(O).
If π is finite dimensional then it is one-dimensional and ϕ is proportional to the function for some (quadratic) character χ on F × \A × .In this case ϕ is constant on the various components of X(O) with value equal to ±V −1/2 2 ; in particular their Laplace eigenvalues are (0, • • • , 0), and the bounds of Theorem 1 are a fortiori satisfied.Therefore we can restrict ourselves to infinite dimensional representations for the rest of the paper.

A slight generalization.
With no extra effort we can consider a slightly more general setting: let χ : F × \A × → C 1 be a unitary Hecke character, and L 2 (Z(A)B × (F )\B × (A), χ) the space of square-integrable functions on B × (F )\B × (A) satisfying with respect to the inner product be the eigenvalues of the Casimir operator (C B 1 (Fσ) ) σ and let Let O ⊂ B be an Eichler order (the intersection of two maximal orders) and let ϕ ∈ π be a smooth, O × -invariant (thus π v is an unramified principal series representation at every place not dividing disc(O)) corresponding to some fixed weight l = (l σ ) σ ∈ Z d with respect to action of the maximal torus K ∞ SO 2 (R) d .In particular, π v is an unramified principal series representation at every place not dividing disc(O).
We normalize the Haar measure dg This implies [Vig80] that .
Under these conditions, we prove the following slightly more general version of Theorem 1: Theorem 3.Under the above assumptions one has In particular we obtain the following bounds in the λ and in the volume aspect Remark 2.3.Here we have assumed that the weight l = (l σ ) σ of ϕ is fixed.This is merely to simplify exposition (and also because the lowest weight case is arguably the most interesting one): the proof of Theorem 3 together with the bound (2.1) yields immediately a non-trivial bound for ϕ ∞ as long as for any fixed δ > 0. We expect that such non-trivial bound hold for all l, and this would follow from good enough bounds for Jacobi polynomials (cf.Remark 2.2).

Reduction of definite quadratic forms
Let Note that the determinants of two equivalent forms over O F may differ by the square of a unit.The quadratic form defines a bilinear form For any real embedding σ : F → R denote by Q σ the conjugated form.We assume that Q is totally positive definite, that is, Q σ is positive definite for all σ.
Minkowski developed a reduction theory for rational positive definite quadratic forms (see e.g.[Cas78, Chapter 12]) that has been extended to arbitrary number fields by Humbert [Hum40].We summarize some basic facts.Every quadratic form is equivalent (over O F ) to some form of the shape for all 1 ≤ i, j ≤ n and all embeddings σ, and Here and henceforth all implied constants depend only on n and F .Clearly, We denote the eigenvalues of the matrix ), hence by Cramer's rule the eigenvalues of (A σ ) −1 are O(1), and therefore Let Q be the quadratic form in n−1 variables that is derived from Q by setting x n = 0. Let Ã be the corresponding (n − 1) × (n − 1)-submatrix of A, and denote by ∆ its determinant.The (n, n)th-entry of A −1 is by Cramer's rule ∆/∆ (up to sign); hence N ∆/∆ is integral for all N ∈ n.Therefore the ideal n ( ∆)/(∆) is integral, and we obtain

Representation numbers of quadratic forms
In this section we establish several lemmata to bound certain averages of representation numbers of O F -integers by some totally definite quadratic form Q. To perform the counting we will frequently use the following consequence of Dirichlet's unit theorem: let A 1 , . . ., A d > 0 be any positive real numbers and write As a consequence we find Indeed, (4.1) implies that for each principal ideal (x) of norm N x ≤ A there are O(log(2+A/N x) d−1 ) generators satisfying the size constraints in (4.2), hence the left hand side of (4.2) is at most We remark that the estimate (4.2) is a trivial lattice point count if all A j 1.It is a little less trivial if some A j are very large and others are very small.
We use the notation r Q ( ) to denote the number of integral representations of by Q.
Lemma 2. Let Q be a totally positive-definite integral quaternary quadratic form of determinant ∆ and level n.Let y > 1.Then for any ε > 0, the implied constants depending on ε alone.
Proof.All of these bounds are proved in a similar way.We start with (4.3).We use the representation (3.2) together with the bounds (3.4).By (4.2) we have (y/N h j ) 1/2 non-zero and hence (y/N h j ) 1/2 + 1 choices in total for x j getting a bound In order to prove (4.5), we choose as before x 4 , x 3 , x 2 in ways.Here we used again (3.4), (3.5) and (3.7).Note that Q(x) = 2 implies x σ j y 1/d for all σ.Once we have fixed x 2 , x 3 , x 4 we are left with counting pairs (x 1 , ) satisfying (a 1i a 1j − a 11 a ij )x i x j .
Note that ξ σ ∆ σ y 1/d and D σ (∆ σ ) 2 y 2/d .It follows that x σ 1 ∆ σ y 1/d .Let us first assume that 2a 11 = b 2 , say, is a square in F and hence in O F .If D = 0, then there are (N D) ε pairs of principal ideals (b − a 11 x 1 − ξ), (b + a 11 x 1 + ξ) whose product equals (D), and by (4.1) each of these has (yN ∆) ε generators g ∈ O F satisfying g σ (∆ σ ) 2 y 2/d .If D = 0, we choose freely in O(y) ways (by (4.2)), and then there are at most two choices for x 1 .We determine how often the case D = 0 happens.The quantity D is a ternary positive definite quadratic form in x 2 , . . ., x n whose determinants of its upper left k × k submatrices (1 ≤ k ≤ 3) are precisely the determinants of the (k + 1) × (k + 1) upper left submatrices of A. In particular we see that D = 0 if and only if x 2 = x 3 = x 4 = 0.
Let us now assume that 2a 11 is not a square in F .Then D = 0, and we need to solve a Pell-type equation.There are (N D) ε ideals ( √ 2a 11 − a 11 x 1 − ξ) in the totally real field E = F ( √ 2a 11 ) of relative norm D, and again by (4.1) each of these yield (yN ∆) ε solutions.This establishes (4.5).
Finally we prove (4.4).Again we fix x 4 , x 3 , x 2 as above, and we fix 1 .This gives a total count of and we are left with counting pairs (x 1 , 2 ) satisfying 2a 11 1 2 − (a 11 x 1 + ξ) 2 = D with ξ and D as above.Now we argue exactly as in the previous case.
For a polynomial Lemma 3. a) Let P (x, y) ∈ O F [x, y] be a quadratic polynomial and assume that its quadratic homogeneous part is a totally positive definite quadratic form.Let ∈ O F .Then there (H(P )N ) ε solutions to P (x, y) = .
b) Let Q be a totally positive definite integral ternary quadratic form over F of discriminant ∆ and let ∈ O F .Then r Q ( ) N 1/2 N (∆ ) ε .c) Let Q be a totally positive definite integral quaternary quadratic form over F of discriminant ∆ and let ∈ O F .Then r Q ( ) N N (∆ ) ε .
Remark 4.1.The proof gives slightly stronger bounds for parts b) and c); for instance in the situation of part b) we obtain r Q ( ) (N 1/2 N ∆ −1/6 + 1)(N (∆ )) ε , but we do not need these refinements.
Proof.a) Without loss of generality we can write P (x, y) = Q(x, y) + L(x, y) = where Q is a totally positive quadratic form over O F of discriminant ∆ and L(x, y) = α 1 x + α 2 y is a linear form over O F .By a linear change of variables this is equivalent to Q(x, y) = |∆| 2 + Q(−α 1 , α 2 )|∆|, where x and y satisfy certain congruence conditions modulo |∆| 2 .We are now left with a norm form equation of the totally imaginary field E = F ( −|∆|) over the totally real field F , and hence there are at most which case there is exactly one solution).
b) We use the representation (3.2) with n = 3 together with the bounds (3.3), (3.4), (3.5).By (4.2) we can choose x 3 in (N /N h 3 + 1) 1/2 (N /N ∆ 1/3 + 1) 1/2 ways, and are left with an inhomogeneous binary problem for which part (a) applies.c) This is proved in the same way.We choose x 4 and x 3 and are left with a binary problem.
In the following lemma we denote by . 2 the usual Euclidean norm on R n , which is (in general) not induced by the inner product (3.1).
Lemma 4. Let Q(x) = 1 2 x t Ax be a positive definite ternary quadratic form with real coefficients and eigenvalues Proof.The assertions are clear if Q(y) = y 2 1 + y 3 2 + y 2 3 and x = (0, 0, 1) t is the north pole.In the general case, we write 1 2 A = B t B for some unique positive symmetric matrix B ∈ GL 3 (R), so that Bx 2 = 1.Let S ∈ O 3 (R) be any orthogonal matrix with SBx = (0, 0, 1) t .Since A has eigenvalues 1, the same holds for B and hence for SB.For the proof of a) in the general case we conclude y − x 2 SB(y − x) 2 , and for the two vectors SBy, SBx the above special case applies.The other two parts are proved in the same way.
The rather complicated proof of the next lemma follows to some extent the argument in [Van97, Lemma 2.1].
We start with the proof of (4.6).By (4.2) there are (N η N ) 1/2 choices for y 0 = 0, and for each of them there are by Lemma 3b at most N 1/2 (N ( ∆)) ε choices for ỹ.This gives the first term in (4.6).
We proceed to count the solutions with y 0 = 0.There are at most 2 linearly dependent solutions to Q(ỹ) = (namely ỹ and −ỹ), hence after adding 1 to the count of (4.6) we can assume that there are at least two linearly independent solutions ỹ1 = (y 11 , y 12 , y 13 ) t , ỹ2 = (y 21 , y 22 , y 23 ) t , say, satisfying Recall that by Lemma 4c any solution y to (4.8) satisfies y σ i ( σ ) 1/2 for 1 ≤ i ≤ 3 and all σ.Now any other solution ỹ3 satisfies det(ỹ by Lemma 4b, as well as Q(ỹ 3 ) = .By (4.2) there are 1 + N 3/2 N η 1/2 choices for the determinant (including 0).For a fixed value of the determinant and some z3 ∈ O 3 12 is a binary problem in α, β with (N (∆ )) ε solutions by Lemma 3a.Alternatively, by Lemma 3b we have the trivial bound N 1/2 (N ( ∆)) ε for the number of solutions with y 0 = 0. Combining these two counts gives the last term in (4.6).
We proceed to prove (4.7).Let δ 1,j δ 2,j = η j .We will fix δ 1,j , δ 2,j later and assume for the moment only We start with the latter.There are at most two solutions with ỹ = 0. From now on we consider only solutions y with ỹ = 0.Among these we define an equivalence relation: we call y = (y 0 , ỹ), z = (z 0 , z) ∈ O 4 F with Q(y) = Q(z) = equivalent if z = cỹ for some c ∈ K.We claim that the cardinality of each equivalence class [y] is small: Clearly (c) ⊆ (y 1 , y 2 , y 3 ) −1 .Fix a fractional ideal a ⊇ O F in the ideal class of (y 1 , y 2 , y 3 ).Then (c) ⊆ (y 1 , y 2 , y 3 ) −1 a = (α), say, where N α N (y 1 , y 2 , y 3 ) −1 N −1/2 , and we choose a generator such that |α σj | N α 1/d , say.Hence we can write c = dα with d ∈ O F .After multiplication with 1/α, the equation Q(z) = becomes an integral binary problem in d and z 0 which by Lemma 3a has (N ( ∆)) ε solutions.It is therefore enough to count the number of equivalence classes, and to this end we pick a set of representatives y; then by construction the corresponding vectors ỹ are pairwise not collinear.
5. Application of the pre-trace formula 5.1.The general set-up.As in [BM], the proof of Theorem 3 (which implies Theorem 1) follows from an application of a pre-trace formula which we recall now.Fix weights l ∈ Z d and a Hecke character χ as in Theorem 3.
Let H(O) be the convolution (spherical) Hecke algebra generated by these bi-O × -invariant functions f α on B × (A).This algebra is commutative, and it follows from our assumptions (by O × -invariance) that ϕ is an eigenfunction for the action of H(O) by convolution: for f ∈ H(O) one has Given such an f , we consider more generally the convolution operator on L 2 (B × (K)\B × (A), χ) where we use as before the notation h = h f h ∞ ∈ PB × (A) and p m,l was defined in (2.2).This is an integral operator with kernel given by It decomposes into an orthonormal (finite) basis of O × K ∞ -invariant H(O)-eigenfunctions of weight l with respect to K ∞ containing ϕ, and from of the normalization (2.4) of Haar measures one finds that Choosing f appropriately, one can assume that λ f (ψ) ≥ 0 for any such ψ with λ f (ϕ) positive (and large): this is the principle of the amplification method.Taking g = g , we obtain We construct the amplifier λ f (ϕ) by a slight generalization of [BM, §5.2] to the number field F : We need to address the fact that the group of units We fix (once and for all) a fundamental domain D 0 for the action of U + on the hyperboloid {y ∈ Then the cone D := F diag ∞,+ D 0 is a fundamental domain for the action of U + on F × ∞,+ .Given some parameter L ≥ 1, consider the four sets where , 1 , 2 denote generators contained in D of principal integral prime ideals p ⊆ O F coprime with disc(O).For any is then a suitable linear combination of the f α( ) , cf. [BM,p. 25].Observe that for γ ∈ PB × (F ) we have the implication , where O g is the order f Og f .This remark along with our choice of amplifier yields the bound Remark 5.1.As in [BM], we have now reduced the problem to a counting problem for the number of representations of integers by a quaternary quadratic form.However, in the present paper we use explicitly the fact that this quadratic form is associated to an order and in particular represents 1; this is one of the reasons for the present improvement over the bounds in [BM].Another reason is that the bounds (4.4) and (4.5) exploit the average over the amplifier and treat the quadratic part of essentially as a new variable of the quadratic form.
to (5.1).If η σ > 2 −C for all σ, then N η (LV |m|) −100d , and we are left with (5.5) We choose The discussion of (5.5) as a function of 0 ≤ η ≤ 1 with this choice of L is elementary, but a bit tedious.The easiest way is to write N η = |m| β with β ≤ 0. Then (5.5) becomes For each i ∈ {1, 2, 3, 4}, the exponent is a piecewise linear function in β that is elementary to discuss. Figure 1 displays the four functions.
This completes the proof of Theorem 3.

Bounds for automorphic forms on 3-dimensional ellipsoids
One of our main reasons for extending [BM] from Q to general totally real number fields is that this will allow us to obtain bounds for automorphic forms associated to quaternary quadratic spaces.There is a close relationship between automorphic functions associated to quaternary quadratic spaces and automorphic functions attached to ternary quadratic spaces but over a quadratic extension of the base field.In the present section we review this connection (see also [Pon76]).6.1.Quaternary quadratic spaces and quaternion algebras.Until said otherwise, F can be any number field.Let (V, Q) be a non-degenerate quaternary quadratic space defined over F , then there is a unique quaternion algebra B defined over F such that (V, Q) is similar to the following quaternary quadratic space (B , nr): let E be the quadratic F -algebra equipped with either the F -invariant involution σ(x, y) = (y, x) if E = F × F or the canonical Finvariant involution if E is a field.In the split case we view F as embedded diagonally into F × F .
Let B E := B ⊗ F E. Slightly abusing notations, we denote by • * the extension to B E of the canonical involution of B, by nr(z) = zz * the associated norm form on B E and by σ Then nr is F -valued on B and (B nr) defines a non-degenerate quaternary quadratic space over F such that disc(B ) is a square if E = F × F and E = F ( disc(B )) otherwise.
We now proceed to describe the orthogonal group SO(B ) along these lines: for any w ∈ B × E , the map z → wzσ(w) * leaves B invariant and defines a proper similitude with factor λ(w) = nr E/F (nr(w)).In particular, if w is such that nr(w) ∈ F (i.e.nr(w) = σ(nr(w))), the map is a special orthogonal transformation of (B , nr); moreover the map w → ρ w induces an isomorphism of F -algebraic groups SO(B ) {w ∈ B × E , nr(w) = σ(nr(w))}/Z F .Here Z denotes the group of scalars of B × and we view B × E as a F -algebraic group (of dimension 8) and nr : B × E → G m,E and σ : B × E → B × E as algebraic maps.We also note that the stabilizer in SO(B ) of the vector 1 ∈ B is precisely where B 0 = B 0 E ∩ B is the orthogonal subspace to 1.In the split case we have B In the sequel, we denote by G < G the F -algebraic groups defined (at the level of their F -points) by 6.2.Automorphic forms associated to orthogonal groups in four variables.From now on we restrict to the following situation: F is a fixed totally real number field, E a fixed totally real quadratic extension of F (possibly F × F ) and B is a totally definite quaternion algebra over F (as explained above, any totally definite quaternary quadratic space is similar to some (B , nr)).In the following all implied constants may depend on F and E.
In that space we consider the following quadratic lattice: Let O ⊂ B be an Eichler order, O E a maximal order in E It is easy to see that where the disjoint union is indexed by the set of classes in the genus of (O , nr), and for a given representative of a genus class (Λ i , nr), Γ i is isomorphic to the subgroup of SO(B )(F ) which preserves the lattice Λ i .As above we are interested in a supnorm bound of functions ϕ on X (3) (O) which are eigenfunctions of the Laplace operator ∆ = (∆ σ ) σ and of a suitable algebra of Hecke operators H(K O ).The Laplace operator is normalized so that the eigenvalues have the shape (6.1) λ = (λ σ ) σ , where λ σ = −m σ (m σ + 2), for m σ ∈ N ≥0 .
These requirements imply that ϕ is identified with a smooth, K O K ∞ -invariant function, contained in the subspace V π ⊂ L 2 (G(F )\G(A)) of an automorphic representation π = ⊗ v π v (as before we may assume that π is infinite dimensional).By [HS,Thm. 4.13], there exists an automorphic representation π ⊗ v πv of G(A) (with unitary central character) such that V π ⊂ V 1 π|G(A) where V 1 π|G(A) denotes the restriction to G(A) of a certain subspace of V π (in most cases V 1 π = V π ).In other words, π is an automorphic representation on B × (A E ) whose central character on Z E (A) A × E is trivial when restricted to Z F (A) A × .In particular, in order to bound ϕ is it sufficient to bound φ: obviously, where m σ is defined by (6.1).For such a representation, the space of K σ SO 3 (R) invariant vectors is one-dimensional.Let E mσ be such a non-zero vector, and for g = (g 1 , g 2 ) ∈ B 1 σ × B 1 σ let P mσ (g σ ) = g σ .E mσ , E mσ E mσ , E mσ be the corresponding matrix coefficient.For g = (g σ be the product of these matrix coefficients. 6.4.The pre-trace formula.Let us (for notational simplicity) first assume that E is a field.By the amplified pretrace formula for B × (A E ) we have for g ∈ B × (A E ) and a suitable parameter L to be chosen in a moment This is the same expression as (5.1) except that the underlying field is now called E rather than F (this applies also to the definition of the sets L i ), and the matrix coefficient is different.In particular, O B E ,g is locally everywhere conjugated to O B E , and its level n B E satisfies as in Section 5.In view of (6.1) and (6.2) this may be rewritten if ϕ 2 = 1, which matches the generic bound in the λ-aspect and improves it in the volume aspect.The split case E = F × F is very similar.In this case the sums over ∈ L i and over γ in (6.3) factor, and we can apply the argument of Section 5.2 for each factor.This completes the proof of Theorem 2. Remark 6.1.The function P m is bi-∆B 1 σ invariant (i.e.spherical) and may be expressed in terms of the character χ m of the representation π m , namely P m (g) = P m (g 1 /g 2 , 1) = χ m (t) = 1 m + 1 sin((m + 1)θ) sin(θ) , t = cos(θ) = g 1 /g 2 , 1 B = 1 2 tr(g 1 /g 2 ).

d
i=1 π mi ∈ Irr(SU 2 (C) d ) from the above identification and d m = σ (m σ + 1) =: |m| its dimension.Given a realization V m = σ V mσ of π m and a d-tuple l = (l σ ) σ of integers, we denote by V l m = σ V lσ mσ the tensor product of weight l σ vectors, and for g y by (3.4), (3.5) and (3.7).