On sup-norm bounds part I: ramified Maa{\ss} newforms over number fields

We prove new upper bounds for the sup-norm of Hecke Maa{\ss} newforms on $GL(2)$ over a number field. Our newforms are more general than those considered in a recent paper by Blomer, Harcos, Maga, and Mili\`cevi\`c: we do not require square free level. Furthermore, we allow for non-trivial central character. Over the rationals we cover the best bounds as obtained by Saha.


Introduction
In this paper we prove bounds for the L ∞ -norm of automorphic forms on GL 2 which improve upon the local bounds.The problem of establishing such bounds is commonly referred to as sup-norm problem for GL 2 .Just recently in the paper [2] this problem was solved over number fields for square free level and trivial central character.Previously, in [15], this problem was solved over Q with arbitrary level and central character.In the present paper we go beyond these ideas.Containing each of them as special cases, our results recover the strongest bounds from both.
The true size of the sup-norm of automorphic forms is still quite mysterious.For conjectures and results concerning the sup-norm problem we refer to [2,15] and the references within.Let us just say that there are many results towards improved upper bounds for the sup-norm in various settings.However, there are also some results giving lower bounds.On non-compact surfaces these lower bounds can have two sources.First they can come from the transition region of Whittaker functions.In this case the peak appears far from the so-called bulk of the manifold.Such lower bounds were considered for example in [16] and [6].Second, they can appear in the bulk of the manifold, which also appears in the compact setting.In this case we refer to [5] for more details.The reason for mentioning these two sources for lower bounds is that we will encounter them in some sense in our arguments.Roughly speaking our proof will consist of two main parts.First we will obtain bounds by estimating Whittaker expansions.Second we use the amplification method to obtain good bounds in the bulk.Now we dive straight into a more detailed set-up of the scene, so that we can give precise statements of our main theorems.
Acknowledgements.I would like to thank A. Saha and A. Booker helpful discussions and valuable feedback.

Set-up and basic definitions
Let F be a number field of degree n = r 1 + 2r 2 , where r 1 is the number of real embeddings and 2r 2 is the number of complex embeddings.By O F we denote the ring of integers in F .Prime ideals in O F are typically denoted by p.Each prime ideal gives rise to a non-archimedean place of F which we also denote by p.The corresponding (canonically normalized) valuation (used for field elements and ideals interchangeably) will be called v p (•) and gives rise to the absolute value , where q p = N (p).In a similar spirit we use ν for an archimedean place and at the same time for the corresponding embedding ν : F → F ν .We put Here and in the following |•| always denotes the standard absolute value on F, R ⊂ C. We equip the archimedean fields with the standard Lebesgue measure coming from R or C, respectively.
If F p is the local non-archimedean field associated to p then we write o p for its ring of integers and ̟ p for its uniformizer.These fields are equipped with two measures.First the Haar measure µ p on (F, +), which we normalize such that µ p (o p ) = 1.Further we have the Haar measure µ × p on (F × , •).This will be normalized to satisfy µ × p (o × p ) = 1.We define F ∞ = ν F ν and equip it with the modulus coming from the real (respectively complex) embeddings only.Let A fin denote the finite adéles equipped with the absolute value |•| fin being the product of all the local absolute values.The usual adéle ring is then given by A F = F ∞ × A fin and equipped with |•| A and µ in the usual manner.We also define the set of totally positive field elements F + to contain all x ∈ F such that x ν > 0 for all real ν.Furthermore put F 0 (A F ) = {a ∈ A F : |a| AF = 1} and F + ∞ = R + ⊂ F ∞ diagonally.Further let us choose ideal representatives θ 1 , . . ., θ hF ∈ ÔF , where h F denotes the class number of F .We write d F for the discriminant of F and d for the different ideal of F .Then by [ .
Here we use the classical (analytic) notation which separates the archimedean and non-archimedean parts.A complete description of the local factors can be found in [13].If χ is the trivial character this leads to the Dedekind zeta function.In this case the local factors reduce to ζ p (s) = L p (1, s) = (1 − q −s p ) −1 and we write ζ n (s) = p|n ζ p (s).At the archimedean places we have Let R be a commutative ring with 1. Typically this will be one of the objects introduced above.Then we set G(R) = GL 2 (R).We will also need the subgroups We use the following compact subgroups of G(R) which depend on the underlying ring R. Define At the non-archimedean places we additionally need the smaller groups Further let be the long Weyl element.Let us briefly describe the measures on the groups in use.Locally we will stick to the measure convention from [15,16].This means, we use the identifications N (R) = (R, +), A(R) = R × , and Z(R) = R × to transport the measures defined on the local fields to the corresponding groups.Further we take µ Kp to be the probability Haar measure on K p .Globally we choose the product measure on K and A(A F ) coming from the previously defined local measures.On the group N (A F ) = A F we put the measure This corresponds to the normalization Vol(N (F ) \ N (A F )) = 1, as can be seen from strong approximation together with [14, Chapter I, Proposition 5.2].Here it is important to be aware of the convention in [14, p. 211] while identifying the Minkowski space with F ∞ .Finally we define as in [10].
In this text we are interested in bounding the sup-norm of Maaß newforms over F which are spherical at infinity.More precisely we will study functions which are right K 1 (n)-invariant, and eigenfunctions of the Casimir element (C ν ) ν ∈ U(q ∞ ) with eigenvalues (λ ν ) ν .These are automorphic forms in the sense of [3, 4.2].Thus, it is standard procedure to associate an cuspidal automorphic representation1 π φ to φ.As explained in [3, 4.6] each cuspidal automorphic representation with central character ω can be (uniquely) realized as a closed invariant subspace of L 2 0 (G(F ) \ G(A F ), ω).In this way the problem of estimating the sup-norm of the Maaß newform φ is closely linked to the underlying cuspidal automorphic representation π φ .However, the sup-norm itself is only defined for smooth elements in L 2 0 (G(F ) \ G(A F ), ω) and it does not make sense in different realizations of π φ .Therefore we will make the following convention.
Convention 1.Let (π, V π ) be a cuspidal automorphic representation with central character ω π .Then there is an intertwiner σ : Let us make some remarks concerning this convention.
• First note that this is indeed well defined.First we observe that by multiplicity one for GL 2 the intertwiner σ is unique up to scaling.However, the scaling does not matter since we L 2 -normalize the image.Secondly K-finiteness ensures that the L ∞ -norm of σ(v) is defined.
• This convention may seem unnecessary at first.But it gives us the flexibility to realize π in arbitrary models without changing the fixed cusp form whose sup-norm we want to bound.
• The restriction to K-finite vectors shows that we should actually work with the G(A F )module underlying π.
Let us now describe the structure of the cuspidal automorphic representation π, keeping in mind that we are mainly interested in spherical Maaß newforms.We write V π for the representation space of π.First note that since (π, V π ) is an cuspidal automorphic representation it is in particular unitary and admissible.For convenience we assume throughout the text that the central character This can be achieved without loss of generality by twisting by an unramified character.
By the tensor product theorem [9, Theorem 4] we may assume that Where (π p , V π,p ) (respectively (π ν , V π,ν ) ) is an irreducible representation of G(F p ) (reps G(F ν )) with central character ω π,p (respectively ω π,ν ).Note that this decomposition also preserves the subspaces of K-finite vectors.Since we are only interested in automorphic forms which are spherical eigenfunctions of the Casimir operator we can restrict ourselves to a very particular situation at the archimedean places.Indeed we will always assume that These are principal series representations and the representation space is denoted by We define the invariants In particular we have ν is an eigen vector of the Casimir operator with eigenvalue This justifies calling t ν the spectral parameter of π.For convenience we will exclude exceptional Maaß forms and will therefore assume t ν ∈ R.
Note that a representation π featuring these types of representations at the archimedean places are spherical.In other words, each representation (π ν , V π contains a K ν -invariant vector v • ν which is unique up to scaling.At the non-archimedean places we define n p to be the log-conductor of π p .Then there exists a unique up to scaling vector v • p ∈ V π,p which is K 1,p (n p )-invariant.Globally we define the conductor of π to be the ideal n = p p np .Thus, V π contains a unique (up to scaling) vector which is K 1 (n)-invariant.The vector does the job and we will call it the (global) new vector.
With this restrictions on π in place we observe that is a Hecke-Maaß newform over F with central character ω π which is K 1 (n)-invariant and has Casimir eigenvalue (λ ν ) ν .Furthermore, by our convention v • ∞ = φ • ∞ .This is exactly the setting in which we will study sup-norm problem.It is the natural generalization of classical Maaß wave forms on the upper half plane H.

Statement of results
The theorems we state deal with newforms coming from cuspidal automorphic representations (π, V π ).Assume that at the infinite places π features spherical principal series representations with spectral data (t ν ) ν .We associate the numbers T = (T ν ) ν defined by To capture the arithmetic properties of π we define the following ideals.First let m be the conductor of the central character ω π .Further let n = n 2 n 2 0 be the conductor of π, where n 2 denotes the square-free part of n.
Remark 1.1.Note that if we take F = Q this theorem reduces to [15,Theorem 3.2].On the other hand if F is an arbitrary number field but n is square-free then we recover [2,Theorem 1].
This theorem fails to meet our expectations for non-totally real fields F .Therefore, we will prove a second theorem generalizing [2, Theorem 2].Theorem 1.2.Let F be number field with maximal totally real subfield F R such that [F : F R ] = m ≥ 2. For a cuspidal automorphic representation (π, σ) with conductor n and spectral parameter (t ν ) ν we have where v • is a new vector.
At this point let us briefly add some speculations concerning the correct local bound.For compact locally symmetric spaces it seems quite clear what is correct local bound for the supnorm of eigenfunctions.See for example [?].The non compact situation seems to be much more complicated.While in a fixed compact set the same local bounds as for compact manifolds remain true, it has been shown in [6] that globally these upper bounds are false.Thus, it still remains unclear what one should consider to be the correct local bound.We suggest that the right answer might come from the analysis of elliptic PDE's with boundary condition on manifolds with corners.This is due to the fact that one my compactify the underlying locally symmetric space (see [4]).The resulting compact space will have a boundary.Then the fact that we are usually dealing with cusp forms suggests that one should look at solutions to elliptic operators with Dirichlet boundary conditions.

Guide to the rest of the paper
Let us now briefly give an overview over the rest of the paper.In Section 2 we find a nice generating domain for G(A F ) which is tailor made for the transformation behaviour of φ • .
We then move on towards the study of Whittaker functions associated to newforms.This will take up most of Section 3 and culminate in the first upper bounds which are good towards the cusps.
The next step is to define an integral operator which will serve as an approximate spectral projector.This operator will then lead to what is usually called an amplified pre-trace formula.The geometric side of this pre-trace inequality can then be estimated as in [2].
Finally, in Section 5, we will give complete proofs for the theorems stated above.

The reduction step
In this section we follow [15, Section 3.2] to derive a generating domain for We then continue from there and show that in order to solve the sup-norm problem for the automorphic forms under consideration we only have to bound our functions (and possibly their twists) on very special elements in G(A F ).The central results of this section are Corollary 2.2 below.

Local preliminaries
Several steps that are necessary to deal with powerful level rely on local methods.In this section we briefly recall the ingredients needed from [15].Let p be a finite place and let (π p , V π,p ) denote an admissible, irreducible representation of (F p ). Define n p = a(π p ) to be the log-conductor of π p .Let ω π,p be the central character of π p and let m p = a(ω π,p ) be its log-conductor.
Most local computations rely on the decomposition This is [15, (3)] or originally [16,Lemma 2.13].This decomposition suggests to define the invariants t p (g), l p (g) and n 0,p (g) in the obvious way by writing Let us collect some simple results capturing the behavior of this invariants in crucial situations.
If n p is even then Proof.The first part is a consequence of [15, Lemma 2.2,( 2)].The second part holds since for even n p one has n 0,p = n 1,p .
Proof.The case ǫ p = 1 is very simple.One writes It is a straight forward calculation to check k ′ ∈ K 0 p (1).In the remaining case we write As before we have k ′ ∈ K 0 p (1).To verify the equality one only needs the observation that since n p is odd we have n 0,p = n 1,p − 1.

The global generating domain
Our goal is to recreate the argument from [15, Section 3.2] coupled with the results from [2, Section 5] to deal with arbitrary number fields.As one expects this general setting brings the class group and the unit group into the picture.We start with several definitions.For any ideal L in O F we define Before we proceed let us make the following little observation.
) for all p |L with n p odd.
Proof.The proof proceeds by applying Lemma 2.1 for each p |L.
Corollary 2.1.For g p ∈ J p and v ∈ o × p we have a(v)g ∈ J p .
Proof.Obviously a(v)g p ∈ J p .One then concludes using Lemma 2.3 and the fact In terms of the local invariants we can write Note that n 2 is square free and that we have n = n 2 0 n 2 .Now we want to use the generating domain from [2] for the square free ideal n 2 .Recall the group , where |y| ∞ is maximal and θ i ∈ Ô× F is an ideal representative.We will call such matrices special.Define We can write down a generating domain in the spirit of [15,Proposition 3.6].
The proof follows exactly the steps in [15] exploiting that the fundamental domain F (n 2 ) from [2] is already given adélically.
Proof.Let ω n be the diagonal embedding of ω in K n .Then the determinant map is surjective.Thus we can apply strong approximation to the element gh −1 n ω −1 n and find Using the properties of F (n 2 ) we write By construction of K * we can assume We can write Let us treat each product appearing above separately.First we include the product over p ∤ n into K 1 (n).Next we notice that if p | n but p ∤ n 2 then n p must be even.Since obviously k * p ω ∈ K p we apply Lemma 2.1 to absorb the second product into J n .In the two remaining cases, namely p | n 2 , n p must be odd.First for p ∤ L we apply Lemma 2.2 to obtain a(̟ n1,p p ).
Now it follows from Lemma 2.3 that also the third product is contained in J n .Finally we use Lemma 2.2 and Lemma 2.3 again to get One concludes the proof by writing f = pa(θ i ) for a special matrix p ∈ F n2 and some

The action of η L
The next step is to understand how the matrix η L acts on the automorphic functions under consideration.
Let us define the character We also impose that ω L π,∞ | F∞ = 1 and ω π,p (̟ p ) = 1 for all p.Strong approximation for A × F shows that this defines an unique character which is F × invariant.
Let us make some observations.Locally one has Now let (π, V π ) be a cuspidal automorphic representation.Then we define the twisted representation (π L , V π ) by 2 and looks locally like In particular the log-conductor of the new central character coincides with the log-conductor of Further we note that this twist does not change the spectral data at ∞. Concerning the conductor of π L we have the following statement.
Lemma 2.4.The log-conductor of π L is n and This corresponds to [15,Lemma 3.4].
Proof.Note that for p ∤ L one simply has π L p = π p .However at the places p | L the representation π L p is equivalent to πp up to some unramified twist.Here πp denotes the contragient representation of π p .Since a(π p ) = a(π p ) it suffices to show that the vector given in (2.4) has the correct transformation behavior under K 1 (n).
We proceed place by place.For p ∤ L and ν there is nothing to do.For p | L we calculate It is easy to verify that k ′ p z(det(k p )) −1 ∈ K 1,p (n p ).Therefore, using (2.2) and (2.3) we have Observe that (π L , V π ) is also a cuspidal automorphic representation.Furthermore, an intertwiner, σ L , to This leads us to the definition of the twisted newform φ L Giving us exactly the ingredient we needed to understand the action of η L on φ • .We derive the following corollary.
Therefore, we reduced the sup-norm problem for the newform φ • to bounding the newforms φ L • on very special matrices.In the following we will fix an arbitrary L | n 2 and write φ = φ L • .We can therefore focus on bounding φ at a(θ i )(J n × F n 2 ).

Bounds via Whittaker expansions
In this section we consider the Whittaker expansion of cusp forms.This will lead to first upper bounds for the newform φ • which is sufficient close to the cusps.The main result is Proposition 3.1 below.
Throughout this section let (π, V π ) be a cuspidal automorphic representation with new vector v • ∈ V π and associated newform φ • = σ(v • ).Without loss of generality we assume that φ • is L 2 -normalized.Further we fix g ∈ J n and n(x)a(y) ∈ F n 2 .

The Whittaker expansion of cusp forms
Let ψ = v ψ v p ψ p be the standard additive character of A F as defined in [16].Recall Having fixed the additive character we define the corresponding global Whittaker function We want to factor this global function into a product of local functions each of which matches the ones studied in [15].Therefore, we define the shifted local characters p π p .The purpose of this twist is that the central character ω ′ πp of π ′ p is trivial on the uniformizer.Now let W p be the Whittaker new vector associated to the representation π ′ p with respect to the character ψ ′ p normalized by W p (1) = 1.These are exactly the Whittaker functions studied in [15,16].At infinity we take local Whittaker function W ν to be the Whittaker vector associated to v • ν normalized by W ν , W ν = 1.This matches the situation in [2].Having defined these local functions we achieve the factorization The translation in the finite part comes from the shift in the local additive characters as explained in [16,Remark 2.11].The constant c ψ comes from our re-normalization of the local functions.
For 1 ≤ i ≤ h F and g ∈ J n we have the well known Whittaker expansion For convenience we split the local terms in the archimedean part W ∞ , the unramified part and the ramified part We also collect all the unramified twists together and write η(q . Since |η| = 1 this factor does not influence any of the upcoming estimates.
Let us continue by gathering some properties of λ n and λ ur .First we recall the following standard result.Lemma 3.1.If p ∤ n then there are unramified characters χ 1,p and χ 2,p such that π ′ p = χ 1,p ⊞ χ 2,p .In this case we have Proof.This follows from [7, Theorem 4.6.4]and [7, Theorem 4.6.5].
We can extract the following fact about the support of unramified coefficients.
In order to describe the unramified coefficients in terms of more ore less well known terms we quickly introduce the Hecke operators.For p ∤ n and k ∈ N define

The local new vector v •
p is an eigenvector of the operator π p (½ X p,k ) and we denote its eigenvalue by λ(p k ).For any ideal a co-prime to n we define the global Hecke operator by T (a) = p |a π p (½ X p,vp(a) ).It is clear that the global new vector v • and therefore also the newform φ • , is an eigenvector of this operator with eigenvalue λ(a) = p|a λ(p vp(a) ).We can now make a connection between λ ur and the Hecke eigenvalues λ(•).At this point let us remark that we follow the normalization of [7,Section 4.6] which differs from the one used in [15] and [2].Lemma 3.2.We have Proof.The proof proceeds locally by showing This can be done by induction using [7, Proposition 4.6.4,Proposition 4.6.6]and Lemma 3.1.
Next let us inspect the support of λ n .
Proof.Since g ∈ J n we have g p ∈ K p a(̟ n1,p p ) and n 1,p (g p ) = n 0,p .But W p (a(̟ Note that we used Corollary 2.1 to include a(v ′ ) into g p for v ′ ∈ o × p where Later on it will make sense to view λ n as a locally constant function on the adéles in an obvious way.It will then be crucial to determine sets on which this function is constant.
Proof.The proof of this little lemma goes back to the decomposition (2.1) and the fact that W πp is well defined by its values on g t,l,v .
First let us write g p = zng t,l,v k.
Then one observes that a(̟ k p u 1 )g p = zn ′ g t+k,l,vu −1 By doing the same for u 2 we observe, that the claimed equality follows when The last condition leads to Combining the support properties from Lemma 3.3 and Corollary 3.1 we derive Where It is easy to deal with the constant c φ• .
Lemma 3.5.We have Proof.As in [16] we observe It is a well known fact that L(1, π, Ad) Before continuing we fix a parameter R = (R ν ) ν and define the box This box will be used to truncate the Whittaker expansion.We will mostly use R ν ≍ Tν yν except in Section 3.4 below, where we allow arbitrary R.
We will estimate each one of these 3 quantities in the upcoming subsections.

Counting field elements in boxes
This subsection is concerned with estimating the number of field elements in different adélic boxes.These estimates will be needed in order to estimate S 1 , S 2 (R), and E. We start by considering some archimedean boxes.The following argument is almost completely taken from [2].Take R ν = Tν +T Let us start by establishing a simple but crucial property of these sets.
We also need good estimates for ♯(I(k) ∩ aı −1 ).These are obtained by a standard volume argument.Let us start with some pre-requests.
Choose a fundamental set P for the lattice aı −1 ⊂ F ∞ .Without loss of generality we can assume 0 ∈ P. Let D be the diameter of P. It is an elementary fact (see [14]) that Further we define Proof.The proof is an elementary volume calculation.Its much easier when one draws a picture.
As consequence of Minkowski-theory we can choose P such that D ≪ N (aı −1 ) We are now ready to count points in our boxes.
Lemma 3.8.One has Proof.By the construction of P we have Vol(P) ≤ Vol(J(k)) Vol(P) .
We conclude by the calculations above.
For the estimation of S 2 (R) we need to count field elements with strong non-archimedean restrictions.We will be able to reduce this problem to [2, Lemma 6].
Define the sets It will be useful to know the volumes of these sets.
Lemma 3.9.We have Proof.This is a standard adélic volume computation done place by place.The key facts we use are µ p (o × p ) = ζ p (1) −1 , µ p (̟ r p o p ) = q −r p , and that both µ p and µ × p are Haar measures for o × p .
Finally we are ready to prove the following counting result.
Lemma 3.10.We have Proof.Let us call the set we want to count S. If S is empty we have nothing to show.Thus, take q 0 ∈ S. Now define the shifted set Define the idéle s by s ν = 2 1/[Fν :R] R q0 and After noting that 0 ∈ S ′ we conclude that To estimate the last set we use [2,Lemma 6].This yields We are left with calculating the adélic norm of s.This is done using To prove the second part we suppose p|n q Let us define the ideal m = p|n p kp .Then in order to have q ∈ C ı (k) one needs N ((q)) ≥ N (m)ı[ı] −1 n .But for q ∈ B(R) we require |q| ∞ ≤ |R| ∞ .Now we conclude by Roughly the same reasoning applies to elements of ı −1 ∩ B(R).

The sum S 1
In this section we will treat the sum S 1 .Due to the transition region of the archimedean Whittaker function this argument requires Note that in view of Corollary 3.2 the sum S 1 is empty if |R| ∞ < N (ı) −1 .Therefore we assume throughout this section.Let us fix a ∈ ı such that (3.4) holds.
The proof will be in the spirit of [2].We start by expressing the archimedean Whittaker functions explicitly in terms of the K-Bessel function.We have else.This holds as in [2, p. 19 ν e −πtν .Thus using [17, (3.1)] one derives We define and observe that for k ν ≤ 0 and q ∈ I ν (k ν ) we have But for q ∈ B(R) we also have qa Finally we are ready to estimate S 1 .
Proof of Lemma 3.11.First we shift the sum by a.This gives Then we partition B(|a| R) using the boxes defined in (3.5).In each box we exploit (3.9) to get Inserting the result from Lemma 3.8 yields To estimate the remaining sums we use ideas of [2].Let us treat each place at its own.We start with ν real and obtain Similarly one treats the complex places: Putting everything together gives for all ν then we obtain ∞ + N (n 0 m 1 (g)) T y Proof.We consider two cases.First assume |y| ∞ ≤ a 3 T 1 3

∞ . Then the balancing assumption implies |y
ν for all ν.Therefore, we have ∞ , one argues analogously to obtain Recalling that |a| ∞ ≪ F N (n 0 m 1 (g)) completes the proof.

The sum S 2 (R)
In this section we will estimate the sum S 2 (R) by reducing it to well known averages of Hecke eigenvalues and local Whittaker functions.Lemma 3.12.We have .
Proof.We start by defining Using [2, Corollary 1] we observe that In particular, if N (m) ≫ |R| ∞ then I(m) must be empty.By Lemma 3.2 we have S 2 (R) At this stage we apply Hölder to the m 2 -sum.This yields S 2 (R) . Here Before we continue it is important to recall that our Hecke operators are differently normalized than the ones in [2], [15] and [12].It is well known that This was proved in [12] over Q.
We now use Jensen's inequality exploiting that the q-sum is short by (3.11).This yields . (3.12) We will first continue to analyze the m 2 -sum.For the sake of notation we define In order to use the notation from Section 3.2 we set By the local definition of λ n we can view it as a function on A ı f in .Lemma 3.4 then implies that this function is constant on the sets C ı (k, [u]).Therefore, we have Using (3.7) and Lemma 3.10 reveals The integral appearing here can be estimated using the local result [15,Proposition 2.11].This is done as follows: Note that here we crucially rely on g p ∈ J n in order to apply the upper bounds for the local integrals.Inserting this estimate in our expression for S ram we get The result from Lemma 3.10 yields .
In the end we note that by the Rankin-trick we have

The error E
For R as in (3.8) we will roughly prove that the error is always absorbed in the main-contribution.More precisely we have the following lemma.
Lemma 3.13.Under the balancing assumption for all ν we have ∞ N (n 0 ) ∞ N (n 0 m 1 (g)) ∞ N (n 0 m 1 (g)) Proof.For S ⊂ {ν} we define By the exponential decay of the K-Bessel function we have the bound for q ∈ I ν (k ν ) and k ν ≥ 1.We now decompose E as follows Again we included the shift by a only in the archimedean part.Note that by Corollary 3.2 below the sum ∞ N (ı) −1 to the sum over k.First note that Lemma 3.12 is good enough to deal with the non-archimedean part of the sum.To deal with the archimedean part we use the same approach as in Section 3.3.In particular with (3.9) and (3.15) we have We obtain N (m 1 (g)) ∞ N (n 0 m 1 (g)) .
∞ concludes the proof.It is obvious from Lemma 3.13 that the error gets absorbed.

Bounds in the bulk
After having obtained estimates for automorphic forms near the cusps we now have to determine their size in the bulk.To do so we will use a variant of the so called amplification method.More precisely we will define an integral operator which approximates an spectral projector on a certain subspace of L 2 (G(F ) \ G(A F )) related to the automorphic form under consideration.An geometric estimation of the kernel then yields the desired estimate.
Let (π, V π ) be a cuspidal automorphic representation with new vector v • and associated newform φ • = σ(v • ).Throughout this section we fix an square free ideal q such that all the units that are quadratic residues modulo q are indeed contained in (O × F ) 2 .We will further assume that (q, n) = 1.Later on we will see that one can actually construct such an ideal.

Amplification and the spectral expansion
Therefore, we further define n invariant and can be considered as an element of the Hilbert space n and n(x)a(y) ∈ F n 2 .We will now define the kernel function which serves as an approximate spectral projector mentioned in the introduction to this section.We do this place by place and immediately give some basic properties.
Let ν be an archimedean place.Then we define for k ν as in [2, Lemma 9].By uniqueness of the spherical vector we have The number c ν (π ν ) depends only on the equivalence class of π ν and is given by the spherical transform of f ν at π ν .By a suitable parametrization of spherical representations of G(F ν ) one relates this to the classical Selberg/Harish-Chandra transform of k ν .Therefore, we have p is the chopped of matrix coefficient as defined in [15,Section 2.4].By construction (see [15,Proposition 2.13]) there is δ π Let us remark that Then we put ω πp (z) −1 π p (zk)w • p dµ p (zk) = w • p .
We also have the estimate ] ≪ q 2+ǫ p .We will treat the remaining places at once.To do so we set S ur = {p : (p, q n) = 1}.And we define the unramified Hecke algebra This is a commutative algebra by [7,Theorem 4.6.1].To an integral ideal c we associate the special element κ c = ⊗ p∈Sur κ p,vp(c) ∈ H ur where This is well defined since the central character is unramified at the places under consideration.This function is constructed such that π(½ X p,k ) = R(κ p,k ).Therefore, we have for w • ur = ⊗ p∈Sur w • p that R(κ c )w • ur = λ(c)w • ur .Now let us fix a large parameter L such that N (q) ≪ (log L) A for some constant A. We then define the sets P q = {a : a = (α) for α ∈ F × + ∩ (1 + q)}, J (q) = {a : (a, q) = 1} and P(L) = {α ∈ O F : (α) ∈ P q is a prime ideal with N (α) ∈ [L, 2L] and ((α), n) = 1}/ ∼ .
In the last definition we say α ∼ β if (α) = (β).We identify P(L) with a suitable fundamental domain for ∼.We can arrange that α v ≍ L [F :Q] for all ν and all α ∈ P(L).
We now need a lower bound for ♯P(L).Since we can not assume q to be fixed (it might depend on n) we need a stronger argument than the one given in [2].The following variation of the generalized Siegel-Walfisz theorem does the job.
This is a very lazy estimate but it uses some heavy machinery.So we will sketch the proof.
Proof.Let Cl q F = J (q)/P q be the ray class group.The explicit formula [13, VI, Theorem 1] for the cardinality of Cl q F implies ♯Cl q F ≪ N (q).For our purposes this is enough.
Remark 4.1.We could also work with the weaker assumption N (q) ≪ ǫ N (n) ǫ .In this case we can still obtain a good lower bound for ♯P(L) using a version of Linnik's theorem over number fields.
To α ∈ O F we associate the numbers We finally define the unramified test function to be This defines an operator R(f ur ) such that Using [7, Proposition 4.6.4/4.6.6] and (4.2) and arguing as in [2, (9.17)] one gets On the other hand we can linearise f ur as usual to obtain The coefficients y α are very similar in spirit to the coefficients w m in [2, (9.16)].Indeed Thus, most importantly we have 2 for some j = 1, 2 and α 1 , α 2 ∈ P(L), 0 else.
Combining everything we define Associated to this function there is the integral operator f (g)φ(gx)dg .
In particular we have The corresponding automorphic kernel is given by The spectral expansion of K f will enable us to bound the sup-norm of φ ′ in terms of the geometric definition of K f .Let us work out the spectral expansion in detail.
We decompose Let us first deal with the cuspidal part.
Lemma 4.2.For any g ∈ G(A F ) we have Proof.We begin by fixing a basis B cusp for L 2 0 (X) containing φ ′ and consisting of R(F ) eigenfunctions.This is possible by a standard multiplicity one argument.For Ψ ∈ B cusp let c Ψ be the associated R(f )-eigenvalue.Then we obtain We can choose B cusp in such a way that for each Ψ there is a cuspidal automorphic representation (π Ψ , V Ψ ) and Ψ = σ Ψ (v) for some pure tensor v ∈ V Ψ .Then we have By [15,Corollary 2.16] we have δ Ψ ∈ {0, δ π }.In particular δ Ψ ≥ 0. At the archimedean places positivity of c ν (π Ψ,ν ) is ensured by the definition of k ν .Finally also c ur must obviously be positive since R(f ur ) is a positive operator.Therefore c Ψ ≥ 0 for all Ψ ∈ B cusp .An explicit lower bound for c φ ′ follows from (4.3), (4.1), and [15,Proposition 2.13].We then conclude by dropping all unnecessary terms.
The argument for the continuous part is quite similar.We obtain Lemma 4.3.For g ∈ G(A F ) one has K cont (g, g) ≥ 0.
Proof.Using the theory of Eisenstein series we have the expansion This is [10, (5.21)].Let us briefly recall the notation.We define the space We then have a representation (π s , H(s)) of G(A F ) where G(A F ) acts by right translation.For s ∈ i R we have the inner product We can also view H(s) as a trivial holomorphic fibre bundle over H = H(0).For φ ∈ H we then define Ψ(s) = Ψ • H(•) s ∈ H(s).Where, is naturally defined via the Iwasawa decomposition of G(A F ). Further, to Ψ ∈ H we associate the Eisenstein series The sum in (4.6) is taken over an orthonormal basis B H for H.As earlier it is no problem to choose this basis to consist of R(F ) eigenfunctions.For Ψ ∈ B H we denote the corresponding R(f )-eigenvalue by c Ψ (0).Note that then also Ψ(s) is an R(f )eigenfunction but the eigenvalue may depend on s.Thus by putting h = g we obtain We can now argue as before using the construction of f to show that c Ψ (s) ≥ 0 for all Ψ.This concludes the proof.
Finally we also treat the residual part of the spectrum.
Lemma 4.4.As long as n 0 = O F we have for any g, h ∈ G(A F ). Otherwise we still have K sp (g, g) ≥ 0 and the only contribution comes from characters χ 2 = ω π with a(χ p ) ≤ 1 at p |q n 1 [n 1 ] −1 n0 and a(χ p ) = 0 otherwise.
Proof.We start from the spectral expansion of K sp .This reads Since the character χ factors and also f is almost a pure tensor the last integral factors in the local integrals and the unramified part I ur (χ ur ).By Lemma A.1 it is clear that I ur (χ ur ) ≥ 0. The lemma then follows from the evaluation of the itnegrals I p (χ p ) given in Lemma A.2 and A.3.By combining the last three lemmata with the definition of K f we conclude This gives an upper bound for φ ′ in terms of the geometry of G(F ) and the test function f .We will estimate this further in the next section.

Estimating the geometric expansion
In this subsection we prove an upper bound for φ ′ which is good in the bulk.This will be done by estimating the right hand side of (4.7).
Proposition 4.1.Assume we have (q, n) = 1 and N (q) ≪ log(N (n)) A .Then for The only thing we will have to do is to exploit the support properties of f to reduce the estimate to the counting problem solved in [2].Comparing this result to [2, Theorem 1] and [16,Theorem 3.2] shows that the exponents here are indeed as expected.
Proof.To save chalk we put By inserting the linearization of f ur given in (4.4) into (4.7)yields Let us analyze the support of f p and κ (α) place by place.At this point we will also have to exploit the special structure of g.
It is now straight forward to choose a suitable representative for γ ∈ Z(F ) \ GL 2 (F ) such that we arrive at the analogue of [2, (9.20)].In our case this reads Since our coefficients y α have the same properties as the corresponding w m in [2] we can replicate the argument from [2, p. 26].One quickly sees, that this argument does not produce any new q dependence.We arrive at |φ ′ (g)| 2 ≪ ǫ N (q) 2+ǫ L ǫ N (n 1 m 1 ) |T | .
∞ (N n 2 ) 13 N (n) ǫ and noting that n 2 n 0 = n 1 leads to (4.8).Note that we include factor N (n) ǫ in the definition of L to make sure that N (q) ≪ ǫ log(L) A .
As in [2] we can give another estimate for non-totally-real number fields.Proposition 4.2.Let C ≤ N (q) ≪ log(N (n)) A , where C is an explicitly computable constant depending only on the field F .Further let F R be the maximal totally real subfield of F and let m = [F : F R ] ≥ 2. Then we have Proof.To see this one uses the second list in [2, p.37] together with (4.9).This yields .
completes the proof. 4

The endgame
In this section we put all the pieces together to prove the theorems stated in the beginning.

Constructing the ideal q
The section on amplification is dependent on the existence of a square-free ideal q which eliminates certain technicalities coming from the unit group of F .Here we will show that one can actually construct q with the desired properties.Lemma 5.1.There is an absolute constant A > 0 depending only on F such that for any n there is an ideal q satisfying the following two properties.
• We have C ≤ N (q) ≪ log(N (n)) A , where C is the absolute constant from Proposition 4.2.
• If x is a quadratic residue modulo q then x ∈ (O × F ) 2 .
Next we consider fields that are not totally real.Therefore, we find a maximal, totally real subfield F R .Put m = [F : F R ] ≥ 2.
Proof of Theorem 1.2.We start by choosing q accordingly and using Corollary 2.2 to reduce the problem as far as we can.Now observe that for |y| ∞ > |T | .
Now one concludes by interpolation as in [2].

A Evaluation of some integrals
In this appendix we will evaluate local integrals that appear in the residual part ot the spectral expansion.More precisely we will calculate the integral This concludes the proof.

For p |q
p (1) : a − d ∈ ̟ p o p .
].One notes, that the Gamma factors are due to the L 2 -normalization in the archimedean Whittaker model.By Stirling's approximation one finds Γ( 1 2 + it ν ) ≫ e − π 2 tν and |Γ(1 + 2it)| ≫ T If p | n then we use Lemma 2.3 to see that g ′ p ∈ ωK 0 p (1) if p | n 2 and g ′ p K p otherwise.Using the support property of f p we conclude that First note that if p ∤ n we have g ′ p = 1.This case consists of two sub cases Namelya(θ −1 i )γa(θ i ) ∈ Z(F p ) K0,p (1) if p | q, Z(F p )K p a(̟ vp(α) p)K p else.