aa r X i v : . [ m a t h . N T ] J u l HYBRID SUP-NORM BOUNDS FOR MAASS NEWFORMS OF POWERFULLEVEL
ABHISHEK SAHA
Abstract.
Let f be an L -normalized Hecke–Maass cuspidal newform of level N , character χ andLaplace eigenvalue λ . Let N denote the smallest integer such that N | N and N denote the largestinteger such that N | N . Let M denote the conductor of χ and define M = M/ gcd( M, N ). Weprove the bound k f k ∞ ≪ ε N / ε N / ε M / λ / ε , which generalizes and strengthens previ-ously known upper bounds for k f k ∞ . This is the first time a hybrid bound (i.e., involving both N and λ ) has been established for k f k ∞ in the case of non-squarefree N . The only previously known bound in the non-squarefree casewas in the N -aspect; it had been shown by the author that k f k ∞ ≪ λ,ε N / ε provided M = 1.The present result significantly improves the exponent of N in the above case. If N is a squarefreeinteger, our bound reduces to k f k ∞ ≪ ε N / ε λ / ε , which was previously proved by Templier.The key new feature of the present work is a systematic use of p -adic representation theoretictechniques and in particular a detailed study of Whittaker newforms and matrix coefficients forGL ( F ) where F is a local field. Introduction
The main result.
Let f be a Hecke–Maass cuspidal newform on the upper half plane ofweight 0, level N , character χ , and Laplace eigenvalue λ . We normalize the volume of Y ( N ) to beequal to 1 and assume that h f, f i := R Y ( N ) | f ( z ) | dz = 1. The problem of bounding the sup-norm k f k ∞ := sup z ∈ Y ( N ) | f ( z ) | in terms of the parameters N and λ is interesting from several points ofview (quantum chaos, spectral geometry, subconvexity of L -functions, diophantine analysis) andhas been much studied recently. For squarefree levels N , there were several results, culminating inthe best currently known bound due to Templier [16], which states that k f k ∞ ≪ ε λ / ε N / ε . The exponent 5 /
24 above for λ has stayed stable since the pioneering work of Iwaniec and Sarnak[7] (who proved k f k ∞ ≪ ε λ / ε in the case N = 1), and it will likely require some key new ideato improve it. The exponent 1 / N in the above bound also appears difficult to improve, atleast for squarefree levels, as it seems that the method used so far, primarily due to Harcos andTemplier [3, 4, 14, 16] has been pushed to its limit. The purpose of the present paper is to showthat the situation is very different for powerful (non-squarefree) levels.To state our result, we introduce a bit of notation. Let N denote the smallest integer such that N | N . Let N be the largest integer such that N | N . Thus N divides N and N = N N . Notethat if N is squarefree, then N = N and N = 1. On the other hand, if N is a perfect square or The author is supported by EPSRC grant EP/L025515/1. If N has the prime factorization N = Q p p n p , then N has the prime factorization N = Q p p ⌊ n p / ⌋ and N hasthe prime factorization N = Q p p ⌈ n p / ⌉ . if N is highly powerful (a product of high powers of primes) then N ≍ N ≍ √ N . Also, let M be the conductor of χ (so M divides N ) and put M = M/ gcd( M, N ). Note that M divides N ,and in fact M equals 1 if and only if M divides N . We will refer to the complementary situationof M > M ∤ N ) as the case when the character χ is highly ramified .We prove the following result, which generalizes and strengthens previously known upper boundsfor k f k ∞ . Theorem. (Theorem 3.2) We have k f k ∞ ≪ ε N / ε N / ε M / λ / ε . Thus in the squarefree case, our result reduces to that of Templier. However when N ≍ N ≍√ N , and M | N (i.e., χ is not highly ramified), our result gives k f k ∞ ≪ ε N / ε λ / ε . Theexponent 1 / N is due to the present author, and was proved only very recently [12]. It was shown that k f k ∞ ≪ λ,ε N / ε when χ is trivial (no dependence on λ was proved). The results of this paper not only substantiallyimprove those of [12] but also use quite different methods. We believe that the approach we take inthis paper, characterized by a systematic use of adelic language and local representation-theoretictechniques that separate the difficulties place by place, is the right one to take for powerful levels.As for the optimum upper bound for k f k ∞ , it is reasonable to conjecture that(1) k f k ∞ ≪ ε N ε λ / ε , if M = 1 (i.e., provided χ is not highly ramified). If true, (1) is optimal as one can prove lowerbounds of a similar strength. If χ is highly ramified, we cannot expect (1) to hold, for reasonsexplained in [13]. Roughly speaking, in the highly ramified case, the corresponding local Whittakernewforms can have large peaks due to a conspiracy of additive and multiplicative characters. Thisleads to a lower bound for k f k ∞ that is larger than N ε in the N -aspect. This phenomenon wasfirst observed by Templier [15] in the case when χ is maximally ramified ( M = N ) and extended(to cover a much bigger range of M ) in our paper [13]. The fact that the factor M / is present inour main theorem above (giving worse upper bounds in the highly ramified case) fits nicely withthis theme.In the table below we compare the upper bound provided by this paper with the lower boundprovided by our paper [13]. We consider newforms of level N = p n , 1 ≤ n ≤
5. The secondcolumn gives the possible values of M in each case. The third, fourth and fifth columns give thecorresponding values of N , N and M , respectively. The sixth column gives the upper boundprovided by the main theorem of this paper and should serve as a nice numerical illustration of ourresult in the depth aspect ( N = p n , n → ∞ ). The seventh and final column gives the correspondinglower bound proved in Theorem 3.3 of [13]. The difference between these last two columns reflectsthe gap in the state of our current knowledge. As the table makes clear, the larger upper boundsfor highly ramified χ are often matched by larger lower bounds. Added in proof:
Recent work of the author with Yueke Hu suggests that (1) may not hold in general in thecase of powerful levels. This is due to the failure of Conjecture 1 in certain cases, which we have recently discovered.
YBRID SUP-NORM BOUNDS FOR MAASS NEWFORMS OF POWERFUL LEVEL 3
Finally, we have coloured blue all the quantities on the last column that we (optimistically)conjecture to be in fact the true size of k f k ∞ (up to a factor of ( N λ ) ε ) in those cases. N M N N M k f k ∞ ≪ ε N ε λ / ε × . . . k f k ∞ ≫ ε N − ε λ / − ε × . . .p p p N / p p p p N / p p p p N / N / p p or p p p N / p p p p N / N / p p or p p p N / p p p p N / p p p p N / N / p p or p or p p p N / p p p p N / N / p p p p N / N / Organization of this paper.
The remainder of Section 1 is an extended introduction thatexplains some of the main features of our work. In Section 2, which is the technical heart of thispaper, we undertake a detailed analytic study of p -adic Whittaker newforms and matrix coefficientsfor representations of GL ( F ) where F is a nonarchimedean local field of characteristic 0. The twomain results we prove are related to a) the support and average size of p -adic Whittaker newforms,b) the size of eigenvalues of certain matrix coefficients. These might be of independent interest. InSection 3, we prove the main result above. Perhaps surprisingly, and in contrast to our previouswork [12], no counting arguments are needed in this paper beyond those supplied by Templier forthe squarefree case. Also, in contrast to [12], we do not need any powerful version of the “gapprinciple”. Instead, we rely almost entirely on the p -adic results of Section 2.1.3. Squarefree versus powerful levels.
The first bound for k f k ∞ in the N -aspect was provedby Blomer and Holowinsky [1] who showed that k f k ∞ ≪ λ,ǫ N + ǫ . They also proved the hybridbound k f k ∞ ≪ ( λ / N ) − . These results were only valid under the assumption that N issquarefree. After that, there was fairly rapid progress (again only assuming N squarefree) byHarcos and Templier [3, 4, 14, 16], culminating in the hybrid bound due to Templier describedearlier. Note that the N -exponent in Templier’s case is 1 /
3, which may be viewed as the “Weylexponent”, as it is a third of the way from the trivial bound of N / ε towards the expectedoptimum bound of N ε .For a long time, there was no result at all when N is not squarefree. Indeed, all the papersof Harcos and Templier rely crucially on using Atkin-Lehner operators to move any point of H toa point of imaginary part ≥ N (which is essentially equivalent to using a suitable Atkin–Lehner As mentioned earlier, this optimum bound is only expected to hold when χ is not highly ramified. ABHISHEK SAHA operator to move any cusp to infinity). This only works if N is squarefree. In [12], the first (andonly previous) result for Maass forms of non-squarefree level was proved; assuming that M = 1 weshowed that k f k ∞ ≪ λ,ε N / ε . A key new idea in [12] was to look at the behavior of f around cusps of width 1 and to formulate all the geometric and diophantine results around such a cusp.Apart from this, the overall strategy was not that different from the works of Harcos and Templierand the exponent of 5 /
12 obtained was weaker than the exponent 1/3 for the squarefree case.An initial indication that the exponent 1 / N -aspect might be beaten for powerful levelswas given by Marshall [8], who showed recently that for a newform g of level N and trivial characteron a compact arithmetic surface (i.e., coming from a quaternion division algebra) the bound k g k ∞ ≪ ε λ / ε N / ε holds true. In particular, when N is sufficiently powerful, this gives a “sub-Weyl” exponent of 1 / N -aspect. Marshall’s proof does not work for the usual Hecke–Maass newforms f on theupper-half plane of level N that we consider in this paper (though it does work for certain shifts ofthese f when restricted to a fixed compact set). Finally, the main result of the present paper gives(when χ is not highly ramified) the bound k f k ∞ ≪ ε N / ε N / ε λ / ε which may be viewedas a strengthened analogue of Marshall’s result for cusp forms on the upper-half plane.As indicated already, the powerful level case has been historically more difficult than the square-free case. It may thus seem surprising that in the powerful case, we succeed in obtaining betterexponents than in the squarefree case. However this seems to be a relatively common phenomenon.For example, for the related problem of quantum unique ergodicity in the level aspect, the knownresults in the squarefree case [10] give mass equidistribution with no power-savings but for powerfullevels one obtains mass equidistribution with power savings [11]. Again, for the problem of provingstrong subconvexity bounds in the conductor aspect for Dirichlet L -functions, one only has a Weylexponent 1 / . .. < / Fourier expansions and efficient generating domains.
It seems worth noting explicitlythe following interesting technical aspect of our work: the method of Fourier (Whittaker) expansion,once one chooses a good (adelic) generating domain , leads to the rather strong bound k f k ∞ ≪ ε M / N / ε λ / ε . Note that this bound reduces to the “trivial bound” when N is squarefree,but is almost of the same strength (in the N -aspect) as our main theorem when N is sufficientlypowerful. In this subsection, we briefly explain the ideas behind this.It is best to work adelically here. Let φ be the automorphic form associated to f , and let g = g f g ∞ ∈ G ( A ) , where g f denotes the finite part of g and g ∞ denotes the infinite component.Then k f k ∞ = sup g ∈ G ( A ) | φ ( g ) | . Because of the invariance properties for φ , it suffices to restrict g to a suitable generating domain D ⊂ G ( A ). Roughly speaking, D can be any subset of G ( A ) suchthat the natural map from D to Z ( A ) G ( Q ) \ G ( A ) / ¯ K is a surjection where ¯ K is a subgroup of G ( A )generated by a set of elements under which | φ | is right-invariant.The Whittaker expansion for φ , which we want to exploit to bound | φ ( g ) | , looks as follows: φ ( g ) = X q ∈ Q =0 W φ ( (cid:20) q (cid:21) g ) . YBRID SUP-NORM BOUNDS FOR MAASS NEWFORMS OF POWERFUL LEVEL 5
The above is an infinite sum, but two things make it tractable. First of all there is an integer Q ( g f ),depending on g f , such that the sum is supported only on those q whose denominator divides Q ( g f ).Secondly, the sum decays very quickly after a certain point | q | > T ( g ∞ ) due to the exponentialdecay of the Bessel function. The upshot is that(2) | φ ( g ) | ≪ X n ∈ Z =0 | n | 2. In this case Q ( g f ) = 1 and T ( g ∞ ) = λ / /y ,leading to the bound | φ ( g ) | ≪ ε λ / ε as expected. Next, suppose φ corresponds to a newform oflevel N where N is squarefree. In this case one can include the Atkin-Lehner operators inside thesymmetry group ¯ K above. Harcos and Templier showed that one can take D to be the subset of G ( A ) consisting of the elements with g f = 1 and for which g ∞ = (cid:20) y x (cid:21) such that y ≥ √ / (2 N ) (andsome additional properties). For such an element, one again has Q ( g f ) = 1 and T ( g ∞ ) = λ / /y leading to the bound | φ ( g ) | ≪ ε ( λ / /y ) / ε ≪ N / ε λ / ε .When N is non-squarefree, it is not possible to construct a generating domain D with a finitevalue of sup g ∈ D T ( g ∞ ) for which all points have g f = 1. Classically, this means that any fundamentaldomain (for the full symmetry group generated by Γ ( N ) and the Atkin-Lehner operators) musttouch the real line. The idea we used in our previous paper [12] was to take the infinite part of D essentially the same as in the squarefree case and take the finite part to be a certain nice subsetof Q p | N GL ( Z p ). Classically, our choice of generating domain in [12] corresponded to taking discsaround cusps of width 1. Assuming χ = 1, this choice again gave Q ( g f ) = 1 and T ( g ∞ ) = λ / /y leading to the same bound | φ ( g ) | ≪ ε N / ε λ / ε as earlier. Thus, in all the above papers, theworst case bound obtained by the Whittaker expansion (i.e., for smallest y ) was just the trivialbound N / ε λ / ε (also, all these papers restricted to χ = 1.)In this paper we choose a somewhat different generating domain from that of [12]. For simplicity,we describe this domain here in the special case when N = p n , M = p m , for some prime p andsome non-negative integers n , m . Take D to consist of the elements g p g ∞ where g ∞ = (cid:20) y x (cid:21) with y ≥ √ / g p ∈ GL ( Z p ) (cid:20) p n (cid:21) . It is easy to prove this is a generating domain. The difficultieslie in computing Q ( g f ) and in proving that the required Ramanujan type bounds on average hold.These key technical local results involve intricate calculations that take up a good part of Section2. We are able to prove that sup g ∈ D Q ( g f ) = p max( m,n ) = M √ N . Also, T ( g ∞ ) = λ / /y as usual. Strictly speaking, this inequality is not completely accurate as one has to add an (usually smaller) error termcoming from peaks of the local Whittaker and K -Bessel functions. ABHISHEK SAHA This leads to the surprisingly strong Whittaker expansion bound of | φ ( g ) | ≪ ε M / N / ε λ / ε in this case. Classically, the generating domain described above corresponds to taking discs aroundthe cusps of the group Γ ( p n , p n ) = { (cid:20) a bc d (cid:21) ∈ SL ( Z ) : p n | b, p n | c } . We remark here that the function f ′ ( z ) := f ( z/p n ) is a Maass form for Γ ( p n , p n ).When N is not a perfect square, the generating domain we actually use is slightly different thandescribed above. Roughly speaking, we exploit the existence of Atkin-Lehner operators at primesthat divide N to an odd power. This does not change the value of sup g ∈ D Q ( g f ) T ( g ∞ ) and so doesnot really affect the Whittaker expansion analysis; however it makes it easier to count lattice pointsfor amplification (described in the next subsection). In any case, the Whittaker expansion boundwe prove ultimately (see Section 3.4) is | φ ( g ) | ≪ ε ( N M λ / /y ) / ε where y ≥ N /N , leading tothe worst case bound of | φ ( g ) | ≪ ε M / N / ε λ / ε . This, as mentioned earlier, is essentially ofthe same strength (in the N -aspect) as our main theorem when N is sufficiently powerful.It bears repeating that the main tools used for the above bound are local, relating to the rep-resentation theory of p -adic Whittaker functions. This supports the assertion of Marshall [8] that N / ε should be viewed as the correct local bound in the level aspect (when χ is not highly rami-fied). Our analysis of these p -adic Whittaker functions also lead to other interesting questions. Forexample, one can ask for a sup-norm bound for these local Whittaker newforms, and in Conjecture1, we predict a Lindel¨of type bound when χ is not highly ramified (this conjecture was originallymade in our paper [13]). One of the key technical results in Section 2 essentially proves an averaged version of this conjecture (this is the Ramanujan type bound on average alluded to earlier).1.5. The pre-trace formula and amplification. Recall that our main theorem states that k f k ∞ ≪ ε N / ε N / ε M / λ / ε . As we have seen above, the method of Fourier (Whittaker)expansion gives us the bound k f k ∞ ≪ ε N / ε M / λ / ε (with even better bounds when the rel-evant point on our generating domain has a large value for y ) so we need to save a further factor of( N /N ) / λ / . This is done by amplification , whereby we choose suitable test functions at eachprime to obtain a pre-trace formula and then estimate its geometric side via some point countingresults due to Harcos–Templier [4] and Templier [16]. The basic idea is that by choosing these localtest functions carefully (constructing an amplifier) one should be able to boost the contribution ofthe newform f to the resulting pre-trace formula. The details for this are given (in a fairly flexibleadelic framework) in Sections 3.5 – 3.7.The unramified local test functions that we use in this paper are standard and essentially goback to Iwaniec–Sarnak (the key point is to exploit a simple identity relating the eigenvalues for theHecke operators T ( ℓ ) and T ( ℓ )). However, our ramified local test functions are very different fromthe papers of Harcos–Templier or our previous paper [12]. In all those past papers, the ramifiedtest functions had been simply chosen to be the characteristic functions of the relevant congruencesubgroups. In contrast, we use a variant of the local test function used by Marshall in [8]. Themain results about this test function are proved in Sections 2.6 – 2.8. Roughly speaking, it is (therestriction to a large compact subgroup of) the matrix coefficient for a local vector v ′ obtained bytranslating the local newform. The key property of this test function is that its unique non-zeroeigenvalue is fairly large (and v ′ is an eigenvector with this eigenvalue). YBRID SUP-NORM BOUNDS FOR MAASS NEWFORMS OF POWERFUL LEVEL 7 Our choice of test functions at ramified primes ensures that any pre-trace formula involving themaverages over relatively few representations of level N . It may be useful to view this as a ramifiedanalogue of the classical (unramified) amplifier. Indeed, the resulting “trivial bound” obtainedvia the pre-trace formula (by choosing the unramified test functions trivially) matches exactly(on compact subsets) with the strong local bounds obtained via Whittaker expansion. This is animportant point because it means that we only need to save a further factor of ( N /N ) / λ / byputting in the unramified amplifier and counting lattice points. This is carried out in Section 3.7.It is worth noting that we do not need any new counting results in this paper beyond thoseproved by Harcos and Templier. This is because the counting part of our paper is only concernedwith the squarefree integer N /N . In particular, the role of amplification in this paper to improvethe N exponent is relatively minor when N is highly powerful (note that N /N approaches anegligible power of N as N gets more powerful). Indeed, when N is a perfect square ( N = N ), allour savings in the N -aspect come from Whittaker expansion and we do not gain anything furtherby amplification. In contrast, our paper [12] had a relatively poor bound coming from Whittakerexpansion but we then saved a non-trivial power of N via amplification.The technical reason why the method of amplification does not improve the N -aspect too muchbeyond our strong local bounds is that our ramified test functions have relatively large support.Consequently, we do not have many global congruences related to N , and congruences are essentialfor savings via counting. More precisely, our ramified test functions are supported on the maximalcompact subgroup at primes that divide N to an even power, and supported on a (slightly) smallersubgroup at primes that divide N to an odd power (it is the latter case that leads to the savingsof ( N /N ) / ). If we were to reduce the support of our test functions further and thus forcenew congruences, the resulting savings via counting would be eclipsed by the resulting loss due tothe fact that our pre-trace formula would now be averaging over more representations of level N .Somehow the ramified and unramified parts of the amplifier seem to work against each other andthe key point is to strike the right balance.It would be an interesting and challenging problem to detect any additional cancellation on thegeometric side of our pre-trace formula by going beyond counting lattice points and perhaps takinginto account the phases of the matrix coefficient used to construct the ramified test function. Sucha result could potentially push the upper-bound for k f k ∞ below N / .1.6. Notations. We collect here some general notations that will be used throughout this paper.Additional notations will be defined where they first appear in the paper.Given two integers a and b , we use a | b to denote that a divides b , and we use a | b ∞ to denotethat a | b n for some positive integer n . For any real number α , we let ⌊ α ⌋ denote the greatest integerless than or equal to α and we let ⌈ α ⌉ denote the smallest integer greater than or equal to α . Thesymbol A denotes the ring of adeles of Q and A f denotes the subset of finite adeles. For any twocomplex numbers α, z , we let K α ( z ) denote the modified Bessel function of the second kind.The groups GL , SL , PSL , Γ ( N ) and Γ ( N ) have their usual meanings. The letter G alwaysstands for the group GL . If H is any subgroup of G , and R is any subring of R , then H ( R ) + denotes the subgroup of H ( R ) consisting of matrices with positive determinant. However, we always gain a non-trivial savings in the λ aspect via amplification. ABHISHEK SAHA We let H = { x + iy : x ∈ R , y ∈ R , y > } denote the upper half plane. For any γ = (cid:18) a bc d (cid:19) in GL ( R ) + , and any z ∈ H , we define γ ( z ) or γz to equal az + bcz + d . This action of GL ( R ) + on H extends naturally to the boundary of H .We say that a function f on H is a Hecke–Maass cuspidal newform of weight 0, level N , character χ and Laplace eigenvalue λ if it has the following properties: • f is a smooth real analytic function on H . • f satisfies (∆ + λ ) f = 0 where ∆ := y − ( ∂ x + ∂ y ). • For all γ = (cid:20) a bc d (cid:21) ∈ Γ ( N ), f ( γz ) = χ ( d ) f ( z ). • f decays rapidly at the cusps of Γ ( N ). • f is orthogonal to all oldforms. • f is an eigenfunction of all the Hecke and Atkin-Lehner operators. The study of newforms f as above is equivalent to the study of corresponding adelic newforms φ which are certain functions on G ( A ). For the details of this correspondence, see Remark 3.1.We use the notation A ≪ x,y,z B to signify that there exists a positive constant C , depending atmost upon x, y, z , so that | A | ≤ C | B | . The symbol ǫ will denote a small positive quantity. Thevalues of ǫ and that of the constant implicit in ≪ ǫ,... may change from line to line.1.7. Acknowledgements. I would like to thank Edgar Assing, Simon Marshall, Paul Nelson, andNicolas Templier for useful discussions and feedback.2. Local calculations Preliminaries. We begin with fixing some notations that will be used throughout this section.Let F be a non-archimedean local field of characteristic zero whose residue field has cardinality q .Let o be its ring of integers, and p its maximal ideal. Fix a generator ̟ of p . Let | . | denote theabsolute value on F normalized so that | ̟ | = q − . For each x ∈ F × , let v ( x ) denote the integersuch that | x | = q − v ( x ) . For a non-negative integer m , we define the subgroup U m of o × to be theset of elements x ∈ o × such that v ( x − ≥ m .Let G = GL ( F ) and K = GL ( o ). For each integral ideal a of o , let K ( a ) = K ∩ (cid:20) o oa o (cid:21) , K ( a ) = K ∩ (cid:20) a oa o (cid:21) , K ( a ) = K ∩ (cid:20) o ao o (cid:21) . Write w = (cid:20) − (cid:21) , a ( y ) = (cid:20) y (cid:21) , n ( x ) = (cid:20) x (cid:21) , z ( t ) = (cid:20) t t (cid:21) for x ∈ F, y ∈ F × , t ∈ F × . Define subgroups N = { n ( x ) : x ∈ F } , A = { a ( y ) : y ∈ F × } , Z = { z ( t ) : t ∈ F × } , B = N A , and B = ZN A = G ∩ [ ∗ ∗∗ ] of G .We normalize Haar measures as follows. The measure dx on the additive group F assigns volume1 to o , and transports to a measure on N . The measure d × y on the multiplicative group F × assignsvolume 1 to o × , and transports to measures on A and Z . We obtain a left Haar measure d L b on Assuming the previous properties, this last property is equivalent to the weaker condition that f is an eigenfunc-tion of almost all Hecke operators. YBRID SUP-NORM BOUNDS FOR MAASS NEWFORMS OF POWERFUL LEVEL 9 B via d L ( z ( u ) n ( x ) a ( y )) = | y | − d × u dx d × y. Let dk be the probability Haar measure on K . TheIwasawa decomposition G = BK gives a left Haar measure dg = d L b dk on G .For each irreducible admissible representation σ of G (resp. of F × ), we define a ( σ ) to be thesmallest non-negative integer such that σ has a K ( p a ( σ ) )-fixed (resp. U a ( σ ) -fixed) vector.2.2. Some matrix invariants. From now on, fix π to be a generic irreducible admissible unitaryrepresentation of G . Let n = a ( π ) , and let ω π denote the central character of π . It is convenient now to introduce some notation. Define • n := ⌈ n ⌉ , • n := n − n = ⌊ n ⌋ , • m = a ( ω π ), • m = max(0 , m − n ).Note that m = 0 if and only if m ≤ n ; this can be viewed as the case when ω π is not highlyramified.Next, for any g ∈ G , we define two integers t ( g ) and l ( g ) which depend on g and n . Recall thedisjoint double coset decomposition [13, Lemma 2.13]:(3) G = G t ∈ Z G ≤ l ≤ n G v ∈ o × /U min( l,n − l ) ZN a ( ̟ t ) wn ( ̟ − l v ) K ( p n ) . Accordingly, given any matrix g ∈ G , we define t ( g ) and l ( g ) to be the unique integers such that • ≤ l ( g ) ≤ n , • g ∈ ZN a ( ̟ t ( g ) ) wn ( ̟ − l ( g ) v ) K ( p n ) for some v ∈ o × . Remark 2.1. It is illuminating to restate these matrix invariants slightly differently. Let g in G .The Iwasawa decomposition tells us that g ∈ ZN a ( y ) k where k = (cid:20) a bc d (cid:21) ∈ K . Then one can checkthat l ( g ) = min( v ( c ) , n ) , and t ( g ) = v ( y ) − l ( g ) . In the sequel, we will often consider matrices g lying in the set Ka ( ̟ n ). The next few lemmasexplicate some key properties of this set. Lemma 2.2. Suppose that k ∈ K and n is odd (so n = n + 1 ). Then (1) l ( ka ( ̟ n )) ≥ n if and only if k ∈ N ( o ) K ( p ) . (2) l ( ka ( ̟ n )) ≤ n if and only if k ∈ wK ( p ) .Proof. We first assume that l ( ka ( ̟ n )) ≥ n and prove that k ∈ BK ( p ). For brevity, put l = l ( ka ( ̟ n )). So we can write ka ( ̟ n ) = bwn ( ̟ − l v ) k ′ where b ∈ B , k ′ ∈ K ( p n ) and n ≥ l ≥ n .Therefore k = b ′ wn ( ̟ n − l v ) k where k = a ( ̟ n ) k ′ a ( ̟ − n ) ∈ K ( p n ) and b ′ ∈ B . So to completethe proof that k ∈ BK ( p ), it suffices to check that there exists a matrix b ∈ B such that b wn ( ̟ n − l v ) ∈ K ( p ). By explicit verification, b = h ̟ n − l v ̟ l − n v − i works. Having provedthat k ∈ BK ( p ), it follows immediately that k ∈ B ( o ) K ( p ) = N ( o ) K ( p ).The proof that l ( ka ( ̟ n )) ≤ n implies k ∈ wK ( p ) is similar. The reverse implications followusing N ( o ) K ( p ) ∩ wK ( p ) = ∅ . (cid:3) Lemma 2.3. Suppose that k ∈ K ( p ) , n is odd, and g ∈ { , (cid:20) ̟ (cid:21) } . Then kgwa ( ̟ n ) = k ′ a ( ̟ n ) g ′ z where k ′ ∈ K , l ( k ′ a ( ̟ n )) ≤ n , g ′ ∈ { , (cid:20) ̟ n (cid:21) } , and z ∈ Z .Proof. If g = 1, then kgwa ( ̟ n ) = w ( w − kw ) a ( ̟ n ). If g = (cid:20) ̟ (cid:21) , then kgwa ( ̟ n ) = w ( w − kw ) a ( ̟ n ) (cid:20) ̟ n (cid:21) z ( − ̟ n − n ). The result follows from Lemma 2.2 as ( w − kw ) ∈ K ( p ). (cid:3) Lemma 2.4. Suppose that g ∈ Ka ( ̟ n ) . Then t ( g ) = min( n − l ( g ) , − n ) .Proof. This follows by an explicit computation similar to the proof of Lemma 2.2. We omit thedetails. (cid:3) Our goal. It may be worthwhile to declare at this point the output from the rest of Section2 that will be needed for our main theorem.In Sections 2.4 – 2.5, we will study the local Whittaker newform W π , which is a certain functionon G . Given a compact subset J of G , we are interested in the following questions:(1) For each g ∈ J , provide a good upper bound for the quantity sup {| y | : W π ( a ( y ) g ) = 0 } . (2) Prove an average Ramanujan-type bound for the function | W π ( a ( y ) g ) | whenever g ∈ J and W π ( a ( y ) g ) = 0.For our global applications, it will be useful to have the set J to be relatively large (so that wecan create a generating domain out of it with a relatively small archimedean component) while alsomaking sure that the supremum of the upper bound above (as g varies in J ) is fairly small (soas to optimize the Whittaker expansion bound). We will choose J to equal the set Ka ( ̟ n/ ) if n is even and equal to { g ∈ Ka ( ̟ n ) : l ( g ) ≤ n } if n is odd. For this set J we will answer thetwo questions above in Proposition 2.11 below. This proposition will be of key importance for ourglobal Whittaker expansion bound.Next, in Sections 2.6 – 2.8, we will study a certain test function Φ ′ π . This test function, viewed asa convolution operator, is essentially idempotent, and therefore has exactly one non-zero positiveeigenvalue. In Proposition 2.13, we determine the size of this non-zero eigenvalue, and we also provethat a ( ̟ n ) · W π is an eigenvector with this eigenvalue. This proposition will be of key importancefor our global bound coming from the amplified trace formula.In view of the technical material coming up, it is worth emphasizing that Propositions 2.11 and2.13 are the only results from the rest of Section 2 that will be used in Section 3.2.4. The Whittaker newform. Fix an additive character ψ : F → S with conductor o . Then π can be realized as a unique subrepresentation of the space of functions W on G satisfying W ( n ( x ) g ) = ψ ( x ) W ( g ). This is the Whittaker model of π and will be denoted W ( π, ψ ). Definition 2.5. The normalized Whittaker newform W π is the unique function in W ( π, ψ ) invariantunder K ( p n ) that satisfies W π (1) = 1.The following lemma is well-known and so we omit its proof. Lemma 2.6. Suppose that W π ( a ( y )) = 0 . Then | y | ≤ , i.e., y ∈ o . This is essentially the local analogue of the quantity Q ( g f ) described in the introduction. YBRID SUP-NORM BOUNDS FOR MAASS NEWFORMS OF POWERFUL LEVEL 11 Lemma 2.7 (Proposition 2.28 of [13]) . Let ˜ π denote the contragredient representation of π . Let t ∈ Z , ≤ l ≤ n , v ∈ o × , and assume that ω π ( ̟ ) = 1 . We have W ˜ π ( a ( ̟ t ) wn ( ̟ − l v )) = ε (1 / , π ) ω π ( v ) ψ ( − ̟ t + l v − ) W π ( a ( ̟ t +2 l − n ) wn ( − ̟ l − n v )) . Define g t,l,v := a ( ̟ t ) wn ( ̟ − l v ). Let ˜ X denote the group of characters µ of F × such that µ ( ̟ ) = 1. For each µ ∈ ˜ X , and each x ∈ F , define the Gauss sum G ( x, µ ) = R o × ψ ( xy ) µ ( y ) d × y. We will need two additional results for the results of the next subsection. The first one is a keyformula from [13]. Lemma 2.8 (Prop. 2.23 of [13]) . Assume that ω π ∈ ˜ X . We have W π ( g t,l,v ) = X µ : a ( µ ) ≤ l,µ ∈ ˜ X c t,l ( µ ) µ ( v ) , where the coefficients c t,l ( µ ) can be read off from the following identity ε ( 12 , µπ ) ∞ X t = −∞ q ( t + a ( µπ ))( − s ) c t,l ( µ ) ! L ( s, µπ ) − = ω π ( − ∞ X a =0 W π ( a ( ̟ a )) q − a ( − s ) G ( ̟ a − l , µ − ) ! L (1 − s, µ − ω − π π ) − . (4)The next result deals with conductors of character twists. While the proof is quite easy, itinvolves a question that comes up frequently in such problems, see, e.g. Remark 1.9 of [11]. Lemma 2.9. Let l ≤ n be a non-negative integer. For each character µ with a ( µ ) = l , we have a ( µπ ) ≤ max( n, l + m ) . Furthermore, for each r ≥ , (cid:12)(cid:12)(cid:12) { µ ∈ ˜ X : a ( µ ) = l, a ( µπ ) = max( n, l + m ) − r } (cid:12)(cid:12)(cid:12) ≤ q l − r . Proof. If π is supercuspidal we have l + m ≤ n . Writing π as a twist of a minimal supercuspidal,the result follows from Tunnell’s theorem [17, Prop. 3.4] on conductors of twists of supercuspidalrepresentations. If π is principal series, then it follows from the well-known formula a ( χ ⊞ χ ) = a ( χ ) + a ( χ ) . If π is a twist of the Steinberg representations, it follows from the formula a ( χ St) =max(2a( χ ) , (cid:3) The support and average size of W π . In this subsection, we will prove an importanttechnical result (Proposition 2.10) about the size and support of W π . This will then be combinedwith the results of the previous subsection to deduce Proposition 2.11 which will be needed for ourglobal application. To motivate all these results, we first recall a certain conjecture made in [13]. Conjecture 1 (Local Lindelof hypothesis for Whittaker newforms) . Suppose that a ( ω π ) ≤ n (i.e., m = 0 ). Then ≪ sup g ∈ G | W π ( g ) | ≪ ε q nε . There is no loss of generality in this assumption as we can always twist π by a character of the form | | ir to ensurethis. This conjecture (originally stated as [13, Conjecture 2]) seems to be quite hard as it impliessquare-root cancelation in sums of twisted GL − ε -factors. However, for the purpose of this paper,we can prove a bound that is (at least) as strong as the above conjecture on average . This isachieved by the second part of the next proposition, which generalizes some results obtained in [11,Section 2], which considered the special case ω π = 1. Proposition 2.10. The following hold. (1) If W π ( g ) = 0 , then t ( g ) ≥ − max(2 l ( g ) , l ( g ) + m, n ) . (2) Suppose t ( g ) = − max(2 l ( g ) , l ( g ) + m, n ) + r where r ≥ . Then we have (cid:18)Z v ∈ o × | W π ( a ( v ) g ) | d × v (cid:19) / ≪ q − r/ . Proof. By twisting π with a character of the form | | ir if necessary (which does not change | W π | ),we may assume ω π ∈ ˜ X . Also assume n ≥ n = 0 is trivial. Because of the cosetdecomposition from earlier, we may further assume that g = g t,l,v := a ( ̟ t ) wn ( ̟ − l v )). Finally,because of Lemma 2.7, we can assume (by replacing π by ˜ π if necessary) that 0 ≤ l ≤ n . Thedesired result then is the following: • Let 0 ≤ l ≤ n . If W π ( g t,l,v ) = 0, then t ≥ − max( n, l + m ). Further if t = − max( n, l + m )+ r where r ≥ (cid:18)Z v ∈ o × | W π ( g t,l,v ) | d × v (cid:19) / ≪ q − r/ . In the notation of (4), the above is equivalent to: Claim 1: Let 0 ≤ l ≤ n . If there exists µ ∈ ˜ X such that a ( µ ) ≤ l and c t,l ( µ ) = 0 then t ≥ − max( n, l + m ). Further if t = − max( n, l + m ) + r where r ≥ P µ ∈ ˜ Xa ( µ ) ≤ l | c t,l ( µ ) | ≪ q − r/ . Define the quantities d t,l ( µ ) via the following identity (of polynomials in q ± s ).(5) ε ( 12 , µπ ) ∞ X t = −∞ q ( t + a ( µπ ))( − s ) c t,l ( µ ) ! L ( s, µπ ) − = ∞ X t = −∞ q ( t + a ( µπ ))( − s ) d t,l ( µ ) ! Note that (for fixed l and µ ) d t,l ( µ ) is non-zero for only finitely many t . Furthermore, c t,l ( µ ) = P ∞ i =0 α i d t − i,l ( µ ) where | α | = 1 and | α i | ≪ q − i/ . (In fact, if π is supercuspidal, α i = 0 for all i > d t,l ( µ ) rather than c t,l ( µ ).Therefore using (4) it suffices to prove the following: Claim 2: Let ≤ l ≤ n . Define the quantities d t,l ( µ ) via the identity ∞ X t = −∞ q ( t + a ( µπ ))( − s ) d t,l ( µ ) ! = ω π ( − ∞ X a =0 W π ( a ( ̟ a )) q − a ( − s ) G ( ̟ a − l , µ − ) ! L (1 − s, µ − ω − π π ) − . (6) If there exists µ ∈ ˜ X such that a ( µ ) ≤ l and d t,l ( µ ) = 0 then t ≥ − max( n, l + m ) . Further if t = − max( n, l + m ) + r where r ≥ then P µ ∈ ˜ Xa ( µ ) ≤ l | d t,l ( µ ) | ≪ q − r/ . YBRID SUP-NORM BOUNDS FOR MAASS NEWFORMS OF POWERFUL LEVEL 13 We only consider the case L ( s, π ) = 1, as the case L ( s, π ) = 1 is similar but easier. (Note that L ( s, π ) = 1 iff either m = n or n = 1.)Let µ ∈ ˜ X be such that a ( µ ) ≤ l . As L ( s, π ) = 1, we can use the well-known formulas statedin [13, Equation (6) and Lemma 2.5] to deduce that the quantity on the RHS of (6) lying insidethe bracket is a constant of absolute value ≪ q − l/ if a ( µ ) = l or a ( µ ) = 0 , l = 1; and isequal to 0 otherwise. Furthermore, there are at most 2 characters µ ∈ ˜ X with a ( µ ) ≤ n and L ( s, µ − ω − π π ) = 1 (this can be checked, for example, using the classification written down in [13,Sec. 2.2]).We henceforth assume that a ( µ ) = l or a ( µ ) = 0 , l = 1; else there is nothing to prove as d t,l ( µ ) = 0. Suppose first that L ( s, µ − ω − π π ) = 1. Then by equating coefficients on both sidesof (6), we see that d t,l ( µ ) = 0 ⇒ t = − a ( µπ ) ≥ − max( n, a ( µ ) + m ) ≥ − max( n, l + m ) , usingLemma 2.9. Furthermore if t = − max( n, l + m ) + r , then X µ ∈ ˜ Xa ( µ ) ∈{ l, } L ( s,µπ )=1 | d t,l ( µ ) | ≪ X µ ∈ ˜ Xa ( µ ) ∈{ l, } a ( µπ )=max( n,l + m ) − r q − l ≪ q − r/ , again using Lemma 2.9.Suppose next that L ( s, µ − ω − π π ) = 1. In this case µ = 1 so if d t,l ( µ ) = 0 we must have a ( µ ) = l . Also, the right side of (6) is of the form α + α q − − s ) + α q − − s ) with α i ≪ q − ( l + i ) / . Furthermore if α = 0 then a ( µπ ) = 0 ≤ n − α = 0 then α = 0 and a ( µπ ) ≤ max( n , m ) ≤ n − 1. So again equating coefficients and using Lemma 2.9, we see that d t,l ( µ ) = 0 ⇒ t ≥ − n ≥ − max( n, l + m ) , and furthermore if t = − max( n, l + m ) + r , then X µ ∈ ˜ Xa ( µ )= lL ( s,µπ ) =1 | d t,l ( µ ) | ≪ X i =0 X µ ∈ ˜ Xa ( µ )= la ( µπ )=max( n,l + m ) − r − i q − l − i ≪ q − r/ , again using Lemma 2.9. Putting everything together, the proof of Claim 2 is complete. (cid:3) Next, for any g ∈ G , define n ( g ) = min( l ( g ) , n − l ( g )) , q ( g ) = max( n , n ( g ) − n + m ) . We note the useful bounds 0 ≤ n ( g ) ≤ n and n ≤ q ( g ) ≤ n + m . Proposition 2.11. Suppose that g ∈ Ka ( ̟ n ) . Assume further that either n is even or l ( g ) ≤ n .Then the following hold (1) If for some y ∈ F × , we have W π ( a ( y ) g ) = 0 , then v ( y ) ≥ − q ( g ) . (2) Suppose b = − q ( g ) + r where r ≥ . Then we have (cid:18)Z v ∈ o × (cid:12)(cid:12)(cid:12) W π ( a ( ̟ b v ) g ) (cid:12)(cid:12)(cid:12) d × v (cid:19) / ≪ q − r/ . Proof. This follows immediately by putting together Lemma 2.4 and Proposition 2.10. (cid:3) Remark 2.12. Note that the map on o × given by v 7→ | W π ( a ( vy ) g ) | is U n ( g ) invariant for all y ∈ F × , g ∈ G . Hence the second part of the above proposition is equivalent to | o × /U n ( g ) | X v ∈ o × /U n g ) (cid:12)(cid:12)(cid:12) W π ( a ( ̟ b v ) g ) (cid:12)(cid:12)(cid:12) d × v ≪ q − r/ . Test functions. We now change gears and start looking at certain local test functions (relatedto matrix coefficients) that will be used later by us in the trace formula. We begin with somedefinitions. Let C ∞ c ( G, ω − π ) be the space of functions κ on G with the following properties:(1) κ ( z ( y ) g ) = ω − π ( y ) κ ( g ).(2) κ is locally constant.(3) | κ | is compactly supported on Z \ G .Given κ , κ in C ∞ c ( G, ω − π ) we define the convolution κ ∗ κ ∈ C ∞ c ( G, ω − π ) via(7) ( κ ∗ κ )( h ) = Z Z \ G κ ( g − ) κ ( gh ) dg, which turns C ∞ c ( G, ω − π ) into an associative algebra.Next let σ be a representation of G with central character equal to ω π . Then, for any κ ∈ C ∞ c ( G, ω − π ), and any vector v ∈ σ , we define R ( κ ) v to be the vector in σ given by(8) R ( κ ) v = Z Z \ G κ ( g )( σ ( g ) v ) dg. Let v π be any newform in the space of π , i.e., any non-zero vector fixed by K ( p n ). Equivalently v π can be any vector in π corresponding to W π under some isomorphism π ≃ W ( π, ψ ). Thus v π isunique up to multiples. Put v ′ π = π ( a ( ̟ n )) v π . Note that v ′ π is up to multiples the unique non-zerovector in π that is invariant under the subgroup a ( ̟ n ) K ( p n ) a ( ̟ − n ).Let h , i be any G -invariant inner product on π (this is also unique up to multiples). Define amatrix coefficient Φ π on G as follows: Φ π ( g ) = h v π , π ( g ) v π ih v π , v π i which is clearly independent of the choice of v π or the normalization of inner product.Let K := K ( p n − n ) = ( K if n is even, K ( p ) if n is odd.Put Φ ′ π ( g ) = ( Φ π ( a ( ̟ − n ) ga ( ̟ n )) = h v ′ π ,π ( g ) v ′ π ih v ′ π ,v ′ π i if g ∈ ZK , g / ∈ ZK . Then it follows that Φ ′ π ∈ C ∞ c ( G, ω − π ) and Φ ′ π ( g − ) = Φ ′ π ( g ). In particular, the operator R (Φ ′ π )is self-adjoint. Proposition 2.13. There exists a positive real constant δ π depending only on π and satisfying δ π ≫ q − n − m such that the following hold. (1) R (Φ ′ π ) v ′ π = δ π v ′ π , (2) Φ ′ π ∗ Φ ′ π = δ π Φ ′ π . YBRID SUP-NORM BOUNDS FOR MAASS NEWFORMS OF POWERFUL LEVEL 15 Remark 2.14. The above Proposition is a refinement of a result of Marshall [8] , who proved asimilar result in the special case ω π = 1 , using a slightly different test function which does notdifferentiate between n odd and even. Remark 2.15. In fact with some additional work one can prove δ π ≍ q − n − m . The rest of this section will be devoted to proving this proposition. We note a useful corollary. Corollary 2.16. Let σ be a generic irreducible admissible unitarizable representation of G suchthat ω σ = ω π and let v σ be any vector in the space of σ . Suppose that R (Φ ′ π ) v σ = δv σ for somecomplex number δ . Then δ ∈ { , δ π } ; in particular, δ is a non-negative real number.Proof. We have δδ π v σ = δ π R (Φ ′ π ) v σ = R (Φ ′ π ∗ Φ ′ π ) v σ = R (Φ ′ π ) R (Φ ′ π ) v σ = δ v σ , implying that δ ∈ { , δ π } . (cid:3) Some preparatory lemmas.Lemma 2.17. Consider the representation π | K of K and let π ′ be the subrepresentation of π | K generated by v ′ π . Then π ′ is a finite dimensional irreducible representation of K .Proof. We know that π ′ is isomorphic to a direct sum of irreducible representations of K . Howeverif there were more than one summand in the decomposition of π ′ , then the representation π | K (and hence the representation π ) would contain a a ( ̟ n ) K ( p n ) a ( ̟ − n )-fixed subspace of dimen-sion greater than one; by newform theory this is impossible. Hence π ′ is irreducible. The finitedimensionality of π ′ follows from the admissibility of π . (cid:3) Lemma 2.18. Let π ′ be as in the above Lemma. Then both the claims of Proposition 2.13 holdwith the quantity δ π defined as follows: δ π = Z Z \ G | Φ ′ π ( g ) | dg = Z K | Φ ′ π ( g ) | dg = 1[ K : K ] dim( π ′ ) . Proof. Note that h , i is an invariant inner product for π ′ . It follows immediately (from the orthonor-mality of matrix coefficients) that the last two quantities are equal. The equality of the middle twoquantities is immediate from our normalization of Haar measures.We now show that this quantity satisfies the claims of Proposition 2.13. First of all, R (Φ ′ π ) v ′ π is avector in π that is invariant under the subgroup a ( ̟ n ) K ( p n ) a ( ̟ − n ). It follows that R (Φ ′ π ) v ′ π = δv ′ π for some constant δ. Taking inner products with v ′ π immediately shows that δ = δ π . This provesthe first assertion of the Proposition. The second assertion is a standard property of convolutionsof matrix coefficients. (cid:3) Proof of Proposition 2.13 in the case of non-supercuspidal representations. We now prove Proposi-tion 2.13 for all non-supercuspidal representations π . It suffices to show thatdim( π ′ ) ≪ q n + m , where π ′ is as in Lemma 2.17.We can embed π inside a representation χ ⊞ χ , consisting of smooth functions f on G satisfying f (cid:18)(cid:20) a b d (cid:21) g (cid:19) = | a/d | χ ( a ) χ ( d ) f ( g ) . Here χ and χ are two (not necessarily unitary) characters. Let f ′ be the function in χ ⊞ χ thatcorresponds to v ′ π . Let K ′ be the (normal) subgroup of K consisting of matrices (cid:20) a bc d (cid:21) such that a ≡ d ≡ p n + m ), b ≡ p n + m ), c ≡ p n + m ). Let V K ′ be the subspace of χ ⊞ χ consisting of the functions f that satisfy f ( gk ) = ω π ( a ) f ( g ) for all k = (cid:20) a bc d (cid:21) ∈ K ′ . Then f ′ ∈ V K ′ . Moreover (and this is the key fact!) if k ∈ K and k ′ ∈ K ′ , then the top left entries of k ′ and kk ′ k − (both these matrices are elements of K ′ ) are equal modulo p m . Hence the space V K ′ isstable under the action of K . So it suffices to prove that dim( V K ′ ) ≪ q n + m .Using the Iwasawa decomposition, it follows easily that | B ( F ) \ G ( F ) /K ′ | ≍ q n + m . Fix a set ofdouble coset representatives S for B ( F ) \ G ( F ) /K ′ . Since any element of V K ′ is uniquely determinedby its values on S , it follows that dim( V K ′ ) ≪ q n + m . The proof is complete.2.8. Proof of Proposition 2.13 in the case of supercuspidal representations. We nowassume that π is supercuspidal. In this case, m ≤ n , hence m = 0. So, we suffices to prove that(9) Z K | Φ π ( a ( ̟ − n ) ga ( ̟ n )) | dg ≫ q − n . The next Proposition gives a formula for Φ π , which may be of independent interest. Proposition 2.19. For ≤ l < n , we have Φ π ( n ( x ) g t,l,v ) = G ( − ̟ l − n , G ( ̟ t + l v − − x, ω π ( − v ) δ t, − l + ε ( 12 , π ) ω π ( v ) X µ ∈ ˜ Xa ( µ )= n − la ( µ ˜ π )= n − l − t G ( ̟ l − n , µ ) G ( vx − ̟ t + l , µ ) ε ( 12 , µ ˜ π ) . (10) Proof. Using the usual inner product in the Whittaker model, and the fact that W π ( a ( t )) is sup-ported on t ∈ o × , (as π is supercuspidal) it follows that(11) Φ π ( n ( x ) g t,l,v ) = Z o × ψ ( − ux ) ω π ( u ) W π ( g t,l,vu − ) d × u. On the other hand, by the formula [13, Prop. 2.30] for W π , and using Proposition 2.7 we have W π ( a ( ̟ t ) wn ( ̟ − l v )) = ω π ( − v − ) ψ ( − ̟ t + l v − ) G ( ̟ l − n , δ t, − l + ε (1 / , ˜ π ) ω π ( v − ) ψ ( − ̟ t + l v − ) X µ ∈ ˜ Xa ( µ )= n − la ( µπ )= n − t − l G ( ̟ l − n , µ − ) ε (1 / , µ − π ) µ ( − v ) . Substituting this into (11), we immediately get the required result. (cid:3) To obtain (9), we will need to substitute the formula from the above proposition and integrate.The following elementary lemma (which is similar to Lemma 2.6 of [6]) will be useful; we omit itsproof. YBRID SUP-NORM BOUNDS FOR MAASS NEWFORMS OF POWERFUL LEVEL 17 Lemma 2.20. Let f be a function on G that is right K ( p n ) -invariant. Then Z G f ( g ) dg = n X k =0 A k Z B f ( bwn ( ̟ − k )) db, where A = (1 + q − ) − , A n = q n (1 + q − ) − , and for < k < n , A k = q k (1 − q − )(1 + q − ) − . We now complete the proof of (9). Using Lemma 2.20, it suffices to prove that(12) Z b ∈ Bbwn ( ̟ − n ) ∈ a ( ̟ − n ) K a ( ̟ n ) | Φ π ( bwn ( ̟ − n )) | db ≫ q − n . Now, note that the quantity z ( u ) n ( x ) a ( y ) wn ( ̟ − n ) lies in a ( ̟ − n ) K a ( ̟ n ) if and only if : u = ̟ n u ′ , y = ̟ − n y ′ , x = ̟ − n x ′ , y ′ ∈ o × , u ′ ∈ o × , x ′ ∈ o , y ′ − x ′ ∈ p n − n . Hence the left side of (12) is equal to(13) q − n Z y ′ ∈ o × , x ′ ∈ o x ′ ∈ y ′ + p n − n | Φ π ( n ( ̟ − n x ′ ) g − n , − n ,y ′− ) | dx ′ d × y ′ Now, we can exactly evaluate the integral in (13) using Proposition 2.19. We expand out | Φ π ( n ( ̟ − n x ′ ) g − n ,n ,y ′− ) | and observe that the main (diagonal) terms are simple to evaluate aswe know the modulus-squared of Gauss sums. Indeed, the contribution to (13) from the diagonalterms is simply q − n Z y ′ ∈ o × , x ′ ∈ o x ′ ∈ y ′ + p n − n X µ ∈ ˜ Xa ( µ )= n a ( µ ˜ π )= n q − n ≍ q − n . On the other hand, the contribution from the cross terms is zero. Indeed, each cross term in-volves an integral like R y ′ ∈ o × , x ′ ∈ o y ′− x ′ − ∈ p n − n o × µ − µ (( y ′− x ′ − ̟ n − n ) which equals 0 because of theorthogonality of characters. This completes the proof of (12).3. Supnorms of global newforms From now on, we move to a global setup and consider newforms on GL ( A ) where A is thering of adeles over Q . For any place v of Q , we will use the notation X v for each local object X introduced in the previous section. The corresponding global objects will be typically denotedwithout the subscript v . The archimedean place will be denoted by v = ∞ . We will usually denotea non-archimedean place v by p where p is a rational prime. The set of all non-archimedean places(primes) will be denoted by f .We fix measures on all our adelic groups (like A , GL ( A ), etc.) by taking the product of the localmeasures over all places (for the non-archimedean places, these local measures were normalized inSection 2.1; at the archimedean place we fix once and for all a suitable Haar measure). We normalizethe Haar measure on R to be the usual Lebesgue measure. We give all discrete groups the countingmeasure and thus obtain a measure on the appropriate quotient groups. Statement of result. As usual, let G = GL . Let π = ⊗ v π v be an irreducible, unitary,cuspidal automorphic representation of G ( A ) with central character ω π = Q v ω π v . For each prime p , let the integers n p , n ,p , n ,p , m p , m ,p be defined as in Section 2.2. We put N = Q p p n p , N = Q p p n ,p , N = Q p p n ,p , M = Q p p m p , M = Q p p m ,p . Thus, N is the conductor of π , M is the conductor of ω π , N is the largest integer such that N | N , and N = N/N is the smallestinteger such that N | N . Let N = N /N = N/N . Note that N is a squarefree integer and isthe product of all the primes p such that p divides N to an odd power. If N is squarefree, then N = N = N and N = 1 while if N is a perfect square then N = N = √ N and N = 1. Notealso that M = M/ gcd( M, N ).We assume that π ∞ is a spherical principal series representation whose central character istrivial on R + . This means that π ∞ ≃ χ ⊞ χ , where for i = 1 , 2, we have χ = | y | it sgn( y ) m , χ = | y | − it sgn( y ) m , with m ∈ { , } , t ∈ R ∪ ( − i , i ).Let K ( N ) = Q p ∈ f K ,p ( p n p ) = Q p ∤ N G ( Z p ) Q p | N K ,p ( p n p ) be the standard congruence sub-group of G (ˆ Z ) = Q p ∈ f G ( Z p ); note that K ( N ) G ( R ) + ∩ G ( Q ) is equal to the standard congruencesubgroup Γ ( N ) of SL ( Z ). Let K ∞ = SO ( R ) be the maximal connected compact subgroup of G ( R ) (equivalently, the maximal compact subgroup of G ( R ) + ). We say that a non-zero automor-phic form φ ∈ V π is a newform if φ is K ( N ) K ∞ -invariant. It is well-known that a newform φ exists and is unique up to multiples, and corresponds to a factorizable vector φ = ⊗ v φ v . We define k φ k = R Z ( A ) G ( F ) \ G ( A ) | φ ( g ) | dg. Remark 3.1. If φ is a newform, then the function f on H defined by f ( g ( i )) = φ ( g ) for each g ∈ SL ( R ) is a Hecke-Maass cuspidal newform of level N (and character ω π ). Precisely, it satisfiesthe relation (14) f (cid:18)(cid:20) a bc d (cid:21) z (cid:19) = Y p | N ω π,p ( d ) f ( z ) for all (cid:20) a bc d (cid:21) ∈ Γ ( N ) . The Laplace eigenvalue λ for f is given by λ = + t where t is as above. (Note that λ ≍ (1 + | t | ) . )Furthermore, any Hecke-Maass cuspidal newform f is obtained in the above manner from anewform φ in a suitable automorphic representation π . The newform φ can be directly constructedfrom f via strong approximation. It is clear that sup g ∈ G ( A ) | φ ( g ) | = sup z ∈ Γ ( N ) \ H | f ( z ) | . Our main result is as follows. Theorem 3.2. Let π be an irreducible, unitary, cuspidal automorphic representation of G ( A ) suchthat π ∞ ≃ χ ⊞ χ , where for i = 1 , , we have χ = | y | it sgn( y ) m , χ = | y | − it sgn( y ) m , with m ∈ { , } , t ∈ R ∪ ( − i , i ) . Let the integers N , N , M be defined as above and let φ ∈ V π be anewform satisfying k φ k = 1 . Then sup g ∈ G ( A ) | φ ( g ) | ≪ ε N / ε N / ε M / (1 + | t | ) / ε . For two characters χ , χ on R × , we let χ ⊞ χ denote the principal series representation on G ( R ) that is unitarilyinduced from the corresponding representation of B ( R ); this consists of smooth functions f on G ( R ) satisfying f (cid:18)(cid:20) a b d (cid:21) g (cid:19) = | a/d | χ ( a ) χ ( d ) f ( g ) . YBRID SUP-NORM BOUNDS FOR MAASS NEWFORMS OF POWERFUL LEVEL 19 Remark 3.3. Assume that π has trivial central character and N ≍ √ N (this is the case whenever N is sufficiently “powerful”). Then we get sup g ∈ G ( A ) | φ ( g ) | ≪ t,ε N + ε , which is a considerableimprovement of the best previously known result sup g ∈ G ( A ) | φ ( g ) | ≪ t,ε N + ε due to the author [12] . Atkin-Lehner operators and a generating domain. Let π be as in Section 3.1 and φ ∈ V π a newform. In order to prove Theorem 3.2 we will restrict the variable g to a carefullychosen generating domain inside G ( A ). In order to do this, we will have to consider the newform φ along with some of its Atkin-Lehner translates. The object of this section is to explain these ideasand describe our generating domain. The main result in this context is Proposition 3.6 below.We begin with some definitions. For any integer L , let P ( L ) denote the set of distinct primesdividing L . For any subset S of P ( N ), let η S , h S ∈ G ( A f ) be defined as follows: η S,p = (cid:20) p n p (cid:21) if p ∈ S , η S,p = 1 otherwise; h S,p = a ( p n ,p ) if p ∈ S , h S,p = 1 otherwise. Define K S = Y p ∈ S G ( Z p ) ⊂ G ( A f ) , J S = K S h S ⊂ G ( A f ) . Finally, define J S = { g ∈ J S : l ( g p ) ≤ n ,p for all p ∈ S such that n p is odd } . Using Lemma 2.2, we see that g ∈ Q p ∈ S G ( Q p ) belongs to J S iff g p ∈ wK p ( p ) a ( p n ,p ) for all p ∈ S for which n p is odd. If L divides N , we abuse notation by denoting h L = h P ( L ) , K L = K P ( L ) , J L = J P ( L ) , J L = J P ( L ) . For any 0 < c < ∞ , let D c be the subset of B ( R ) + ≃ H defined by D c := { n ( x ) a ( y ) : x ∈ R , y ≥ c } . Finally, for L > 0, define F L = { n ( x ) a ( y ) ∈ D √ / (2 L ) : z = x + iy satisfies | cz + d | ≥ /L ∀ (0 , = ( c, d ) ∈ Z } . Next, for any subset S of P ( N ), let ω Sπ = Q v ω Sπ,v be the unique character on Q × \ A × with thefollowing properties:(1) ω Sπ, ∞ is trivial on R + .(2) ω Sπ,p | Z × p is trivial if p ∈ S and equals ω π,p | Z × p if p / ∈ S .Note that ω P ( N ) π = 1, ω ∅ π = ω π , and for each S , ω Sπ has conductor Q p / ∈ S p m p . Define theirreducible, unitary, cuspidal, automorphic representation π S by π S = ˜ π ⊗ ω Sπ = π ⊗ ( ω − π ω Sπ ) . Akey observation is that for every S , the representation π S has conductor N and its central character ω π S = ω − π ( ω Sπ ) has conductor M . We have π ∅ = π and π P ( N ) = ˜ π. Lemma 3.4. The function φ S on G ( A ) given by φ S ( g ) := ( ω − π ω Sπ )(det( g )) φ ( gη S ) is a newform in π S .Proof. It is clear that φ S is a vector in π S , and one can easily check from the defining relation thatit is K ( N ) K ∞ invariant. (cid:3) The existence, as well as uniqueness, of the character ω Sπ follows from the identity A × = Q × R + Q p Z × p . Remark 3.5. In the special case ω π = 1 , one has π S = π for every subset S of P ( N ) . In thiscase, for each S , the involution π ( η S ) on V π corresponds to a classical Atkin-Lehner operator, and φ S = ± φ with the sign equal to the Atkin-Lehner eigenvalue. We will call the natural map on Z ( A ) G ( Q ) \ G ( A ) /K ( N ) K ∞ induced by g gη S the adelic Atkin-Lehner operator associated to S . Recall that J N = Q p | N wK p ( p ) a ( p n ,p ) Q p | N, p ∤ N G ( Z p ) a ( p n ,p ) ⊂ G ( A f ). The next Propositiontells us that any point in Z ( A ) G ( Q ) \ G ( A ) /K ( N ) K ∞ can be moved by an adelic Atkin-Lehneroperator to a point whose finite part lies in J N and whose infinite component lies in F N . Proposition 3.6. Suppose that g ∈ G ( A ) . Then there exists a subset S of P ( N ) such that g ∈ Z ( A ) G ( Q ) ( J N × F N ) η S K ( N ) K ∞ . Proof. Let w N be the diagonal embedding of w = (cid:20) − (cid:21) into K N . The determinant map from w N h N K ( N ) h − N w − N is surjective onto Q p Z × p . Hence by strong approximation for gh − N w − N , we canwrite gh − N w − N = zg Q g + ∞ ( w N h N kh − N w − N ) where z ∈ Z ( A ), g Q ∈ G ( Q ), g + ∞ ∈ G ( R ) + , k ∈ K ( N ).In other words,(15) g ∈ Z ( A ) G ( Q ) g + ∞ w N h N K ( N ) . Using Lemma 1 from [3], we can find a divisor N of N , and a matrix W ∈ M ( Z ) such that W ≡ (cid:20) ∗ (cid:21) mod N , W ≡ (cid:20) ∗ ∗ ∗ (cid:21) mod N , det( W ) = N , W ∞ g + ∞ ∈ F N K ∞ . Above, W ∞ denotes the element W considered as an element of G ( R ) + . Let S be the set ofprimes dividing N . Note that W p ∈ K ,p ( p ) (cid:20) p (cid:21) if p ∈ S , W p ∈ K ,p ( p ) if p | N but p / ∈ S , and W p ∈ G ( Z p ) if p ∤ N . Since W ∈ G ( Q ), it follows from the above and from (15) that g ∈ Z ( A ) G ( Q ) F N K ∞ Y p ∈ S K ,p ( p ) (cid:20) p (cid:21) w Y p | N p / ∈ S K ,p ( p ) w Y p | Np ∤ N G ( Z p ) h N K ( N )= Z ( A ) G ( Q ) ( J N × F N ) η S K ( N ) K ∞ , where in the last step we have used Lemma 2.3. (cid:3) Corollary 3.7. Let π , φ be as in Theorem 3.2. Suppose that for all subsets S of P ( N ) and all g ∈ J N , n ( x ) a ( y ) ∈ F N , we have | φ S ( gn ( x ) a ( y )) | ≪ ε N / ε M / N − / (1 + | t | ) / ε . Then the conclusion of Theorem 3.2 is true.Proof. This follows from the above Proposition and the fact | φ S ( gn ( x ) a ( y )) | = | φ ( gn ( x ) a ( y ) η S ) | . (cid:3) YBRID SUP-NORM BOUNDS FOR MAASS NEWFORMS OF POWERFUL LEVEL 21 Sketch of proof modulo technicalities. In this subsection, we prove Theorem 3.2 assumingsome key bounds whose proofs will take the rest of this paper. For brevity we put T = 1 + | t | .Also, recall that N = N /N . We need to show that for each g ∈ G ( A ), | φ ( g ) | ≪ ε N / ε M / N − / T / ε . By letting φ run over all its various Atkin-Lehner translates φ S , S ⊆ P ( N ), we may assume (byCorollary 3.7) that g ∈ J N F N . Therefore in what follows, we will not explicitly keep track ofthe set S , but instead prove the following: Given an automorphic representation π as in Section3.1 (with associated quantities N , N , T , M as defined earlier), a newform φ ∈ V π satisfying k φ k = 1 , and elements g ∈ J N , n ( x ) a ( y ) ∈ F N , we have (16) | φ ( gn ( x ) a ( y )) | ≪ ε N / ε M / N − / T / ε . As noted, the above statement implies Theorem 3.2. Implicit here is the fact that we are letting π vary among the various π S , which all have exactly the same values of N, N , N , M , T as π does,and moreover the corresponding newforms φ S all satisfy k φ S k = k φ k .We will prove (16) by a combination of two methods. First, in Proposition 3.8, we will use theWhittaker expansion to bound this quantity. Precisely, we will prove the following bound:(17) | φ ( gn ( x ) a ( y )) | ≪ ε ( N T ) ε (cid:18) N M TN y (cid:19) / + N T / N ! / . To prove the above bound, we will rely on Proposition 2.11. Next, in Proposition 3.16, we will usethe amplification method to bound this quantity. We will prove that for each Λ ≥ 1, we have(18) | φ ( gn ( x ) a ( y )) | ≪ ε ( N T Λ) ε N M " T + N / T / y Λ + Λ / T / ( N − / + y ) + Λ T / N − . The proof of this bound will rely on Proposition 2.13 and some counting arguments due to Harcosand Templier. Combining the two bounds will lead to Theorem 3.2, as we explain now.Choose Λ = T / N / . Then (18) becomes | φ ( gn ( x ) a ( y )) | ≪ ε ( N T ) ε N M (cid:20) T / N − / + T / N − / y (cid:21) . (19)If y ≤ T / N − / , then we use (19) to immediately deduce (16). If y ≥ T / N − / , then weuse (17) to obtain the bound(20) | φ ( gn ( x ) a ( y )) | ≪ ε ( N T ) ε M / N / N − / T / which is much stronger than (16)! This completes the proof.3.4. The bound via the Whittaker expansion. Let π , φ be as in Section 3.1 with k φ k = 1.The object of this section is to prove the following result. Proposition 3.8. Let x ∈ R , y ∈ R + , g ∈ J N . Then | φ ( gn ( x ) a ( y )) | ≪ ε ( N T ) ε (cid:18) N M Ty (cid:19) / + (cid:16) N T / (cid:17) / ! . Remark 3.9. If we assume Conjecture 1 stated earlier, then we can improve the bound in Propo-sition 3.8 to ( N T ) ε (cid:18)(cid:16) N M Ty (cid:17) / + (cid:0) T / (cid:1) / (cid:19) . We now begin the proof of Proposition 3.8. One has the usual Fourier expansion at infinity(21) φ ( n ( x ) a ( y )) = y / X n ∈ Z =0 ρ φ ( n ) K it (2 π | n | y ) e ( nx ) . The next Lemma notes some key properties about the Fourier coefficients appearing in the aboveexpansion. Lemma 3.10. The Fourier coefficients ρ φ ( n ) satisfy the following properties. (1) | ρ φ ( n ) | = | ρ φ (1) λ π ( n ) | where λ π ( n ) are the coefficients of the L -function of π . (2) | ρ φ (1) | ≪ ε ( N T ) ε e πt/ . (3) P ≤| n |≤ X | λ π ( n ) | ≪ X ( N T X ) ε .Proof. All the parts are standard. The first part is a basic well-known relation between the Fouriercoefficients and Hecke eigenvalues. The second part is due to Hoffstein-Lockhart [5]. The last partfollows from the analytic properties of the Rankin-Selberg L -function (e.g., see [2]). (cid:3) The Fourier expansion (21) is a special case of the more general Whittaker expansion that wedescribe now. Let g f ∈ G ( A f ). Then the Whittaker expansion for φ says that(22) φ ( g f n ( x ) a ( y )) = X q ∈ Q =0 W φ ( a ( q ) g f n ( x ) a ( y ))where W φ is a global Whittaker newform corresponding to φ given explicitly by W φ ( g ) = Z x ∈ A / Q φ ( n ( x ) g ) ψ ( − x ) dx. Putting g f = 1 in (22) gives us the expansion (21). On the other hand, the function W φ factors as W φ ( g ) = c Q v W v ( g v ) where(1) W p = W π p at all finite primes p .(2) | W ∞ ( a ( q ) n ( x ) a ( y )) | = | qy | / | K it (2 π | q | y ) | The constant c is related to L (1 , π, Ad); for further details on this constant, see [13, Sec. 3.4].For any g = Q p | N g p ∈ J N , define N g = Y p | N p n ( g p ) , Q g = Y p | N p q ( g p ) where the integers n ( g p ), q ( g p ) are as defined just before Proposition 2.11. Note that the “usefulbounds” stated there imply that N g | N and Q g | N M . Lemma 3.11. Suppose that g ∈ J N and W φ ( a ( q ) gn ( x ) a ( y )) = 0 for some q ∈ Q . Then we have q = nQ g for some n ∈ Z .Proof. We have W π p ( a ( q ) g p ) = 0 for each p | N and W π p ( a ( q )) = 0 for each p ∤ N . Now the resultfollows from Proposition 2.11 and Lemma 2.6. (cid:3) YBRID SUP-NORM BOUNDS FOR MAASS NEWFORMS OF POWERFUL LEVEL 23 Henceforth we fix some g ∈ J N . By comparing the expansion (22) for g f = g with the trivialcase g f = 1, we conclude that φ ( gn ( x ) a ( y )) = X n ∈ Z =0 W φ ( a ( n/Q g ) gn ( x ) a ( y ))= X n ∈ Z =0 Y p | N W π p ( a ( n/Q g ) g ) c Y p ∤ N W π p ( a ( n )) W ∞ ( a ( n/Q g ) n ( x ) a ( y ))= (cid:18) yQ g (cid:19) / X n ∈ Z =0 ( | n | , N ∞ ) / ρ φ (cid:18) n ( | n | , N ∞ ) (cid:19) λ π N ( n ; g ) K it (cid:18) π | n | yQ g (cid:19) χ n (23)where χ n is some complex number of absolute value 1, and for each non-negative integer n wedefine λ π N ( n ; g ) := Y p | N W π p (cid:16) a ( np − q ( g p ) ) g p (cid:17) . The tail of the sum (23) consisting of the terms with 2 π | n | y/Q g > T + T / ε is negligiblebecause of the exponential decay of the Bessel function. Put R = Q g ( T + T / ε πy ) ≍ Q g Ty . Using the Cauchy-Schwarz inequality and Lemma 3.10, we therefore have | φ ( gn ( x ) a ( y )) | ≪ ε ( N T ) ε e πt (cid:18) yQ g (cid:19) × X The function λ π N ( n ; g ) satisfies the following properties. (1) Suppose that n is a positive integer such that n | N ∞ , and n , n ′ are two integers coprimeto N such that n ≡ n ′ (mod N g ) . Then | λ π N ( n n ; g ) | = | λ π N ( n ′ n ; g ) | . (2) For any integer r , and any n | N ∞ , X rN g ≤| n | < ( r +1) N g ( n ,N )=1 | λ π N ( n n ; g ) | ≪ N g n − / . Proof. Let p | N and u , u ∈ Z × p . Then by (3), it follows that for all w ∈ Q × p , | W π p ( a ( wu ) g p ) | = | W π p ( a ( wu ) g p ) | whenever u ≡ u mod ( p n ( g p ) ) . It follows that if n | N ∞ , then(25) | λ π N ( n n ; g ) | = | λ π N ( n ′ n ; g ) | if n ≡ n ′ (mod N g ) . Furthermore using the above and the Chinese remainder theorem,1 N g X n mod N g ( n ,N )=1 | λ π N ( n n ; g ) | = Y p | N Z Z × p (cid:12)(cid:12)(cid:12) W π p (cid:16) a ( n vp − q ( g p ) ) g p (cid:17)(cid:12)(cid:12)(cid:12) d × v ! and hence by Proposition 2.11, N g P n mod N g ( n ,N )=1 | λ π N ( n n ; g ) | ≪ n − / . (cid:3) Lemma 3.13. We have X If we assume Conjecture 1, then the bound on the right side can be improved to ( N T ) ε (cid:16) T / + Q g Ty (cid:17) . Proof. Let f ( y ) = min( T / , (cid:12)(cid:12) yT − (cid:12)(cid:12) − / ) . Then it is known that te πt | K it ( y ) | ≪ f ( y ); see, e.g.,[16, (3.1)]. Using the previous lemma, we may write te πt X For all X > , we have X Proof. This follows from the last part of Lemma 3.10 using a similar (but simpler) argument as inthe above Lemma. (cid:3) Finally, by combining (24), Lemma 3.13, and Lemma 3.15, we get the bound(26) | φ ( gn ( x ) a ( y )) | ≪ ε ( N T ) ε (cid:18) Q g Ty + N g T / (cid:19) . Taking square roots, and using that Q g ≤ N M , N g ≤ N , we get the conclusion of Proposition 3.8.3.5. Preliminaries on amplification. Our aim for the rest of this paper is to prove the followingproposition. As explained in Section 3.3, this will complete the proof of our main result. Proposition 3.16. Let Λ ≥ be a real number. Let n ( x ) a ( y ) ∈ F N , g ∈ J N . Then (27) | φ ( gn ( x ) a ( y )) | ≪ ε (Λ N T ) ε N M " T + N / T / y Λ + Λ / T / ( N − / + y ) + Λ T / N − . Recall that h N = Q p | N a ( p n ,p ) . Define the vector φ ′ ∈ V π by φ ′ ( g ) = φ ( gh N ) . Then the problem becomes equivalent to bound the quantity φ ′ ( k N n ( x ) a ( y )) where k N ∈ K N = Q p | N G ( Z p ), k N h N ∈ J N . Note that φ ′ is K ′ ( N ) K ∞ -invariant where K ′ ( N ) := h N K ( N ) h − N .Define the function Φ ′ N on Q p | N G ( Q p ) by Φ ′ N = Q p | N Φ ′ π p , with the functions Φ ′ π p defined inSection 2.6. By Proposition 2.13, it follows that R (Φ ′ N ) φ ′ := Z ( Z \ G )( Q p | N Q p ) Φ ′ N ( g )( π ( g ) φ ′ ) dg = δ N φ ′ where δ N ≫ N − M − . Note also that if g ∈ Q p | N G ( Q p ) and Φ ′ N ( g ) = 0, then g ∈ Z ( Q p ) G ( Z p )for each prime p dividing N and g ∈ Z ( Q p ) K p ( p ) for each prime p dividing N . Also, recall that R (Φ ′ N ) is a self-adjoint, essentially idempotent operator.Next, we consider the primes not dividing N . Let H ur be the usual global (unramified) convolu-tion Hecke algebra; it is generated by the set of all functions κ ur on Q p ∤ N G ( Q p ) such that for eachfinite prime p not dividing N ,(1) κ p ∈ C ∞ c ( G ( Q p ) , ω − π p ) ,(2) κ p is bi- G ( Z p ) invariant.It is well-known that H ur is a commutative algebra and is generated by the various functions κ ℓ (as ℓ varies over integers coprime to N ) where κ ℓ = Q p ∤ N κ ℓ,p and the function κ ℓ,p in C ∞ c ( G ( Q p ) , ω − π p )is defined as follows:(1) κ ℓ,p ( zka ( ℓ ) k ) = | ℓ | − / ω − π p ( z ) for all z ∈ Z ( Q p ), k ∈ G ( Z p ).(2) κ ℓ,p ( g ) = 0 if g / ∈ Z ( Q p ) G ( Z p ) a ( ℓ ) G ( Z p ).Then, it follows that for each κ ur ∈ H ur , R ( κ ur ) φ ′ := Z Q p ∤ N ( Z \ G )( Q p ) κ ur ( g )( π ( g ) φ ′ ) dg = δ ur φ ′ where δ ur is a complex number (depending linearly on κ ur ). Furthermore, R ( κ ℓ ) φ ′ = λ π ( ℓ ) φ ′ where the Hecke eigenvalues λ π ( ℓ ) were defined earlier in Lemma 3.10. Moreover, we note thatas κ ur varies over H ur , the corresponding operators R ( κ ur ) form a commuting system of normaloperators. Indeed, if we define κ ∗ ℓ = (cid:16)Q p | ℓ ω − π p ( ℓ ) (cid:17) κ ℓ , and extend this via multiplicativity andanti-linearity to all of H ur , then we have an involution κ κ ∗ on all of H ur . It is well-known that κ ∗ ( g ) = κ ( g − ) and hence R ( κ ∗ ) is precisely the adjoint of R ( κ ).Finally, we consider the infinite place. For g ∈ G ( R ) + , let u ( g ) = | g ( i ) − i | g ( i )) denote the hyperbolicdistance from g ( i ) to i . Each bi- Z ( R ) K ∞ -invariant function κ ∞ in C ∞ c ( Z ( R ) \ G ( R ) + ), can beviewed as a function on R + via κ ∞ ( g ) = κ ∞ ( u ( g )). For each irreducible spherical unitary principalseries representation σ of G ( R ), we define the Harish-Chandra–Selberg transform ˆ κ ∞ ( σ ) viaˆ κ ∞ ( σ ) = Z Z ( R ) \ G ( R ) + κ ∞ ( g ) h σ ( g ) v σ , v σ ih v σ , v σ i dg where v σ is the unique (up to multiples) spherical vector in the representation σ . It is known thatfor all such σ , R ( κ ∞ ) v σ = ˆ κ ∞ ( σ ) v σ ; in particular, R ( κ ∞ ) φ ′ = ˆ κ ∞ ( π ∞ ) φ ′ . By [16, Lemma 2.1] there exists such a function κ ∞ on G ( R ) with the following properties:(1) κ ∞ ( g ) = 0 unless g ∈ G ( R ) + and u ( g ) ≤ κ ∞ ( σ ) ≥ σ of G ( R ).(3) ˆ κ ∞ ( π ∞ ) ≫ g ∈ G ( R ) + , | κ ∞ ( g ) | ≤ T and moreover, if u ( g ) ≥ T − , then | κ ∞ ( g ) | ≤ T / u ( g ) / .Henceforth, we fix a function κ ∞ as above.3.6. The amplified pre-trace formula. In this subsection, we will use L ( X ) as a shorthand for L ( G ( Q ) \ G ( A ) /K ′ ( N ) K ∞ , ω π ).Let the functions Φ ′ N , κ ∞ be as defined in the previous subsection. Consider the space of functions κ on G ( A ) such that κ = Φ ′ N κ ur κ ∞ with κ ur in H ur . We fix an orthonormal basis B = { ψ } of thespace L ( X ) with the following properties: • φ ′ ∈ B , • Each element of B is an eigenfunction for all the operators R ( κ ) with κ as above, i.e., forall ψ ∈ B , there exists a complex number λ ψ satisfying R (Φ ′ N ) R ( κ ur ) R ( κ ∞ ) ψ = R ( κ ) ψ := Z Z ( A ) \ G ( A ) κ ( g )( π ( g ) ψ ) dg = λ ψ ψ. Such a basis exists because the set of all R ( κ ) as above form a commuting system of normaloperators. The basis B naturally splits into a discrete and continuous part, with the continuouspart consisting of Eisenstein series and the discrete part consisting of cusp forms and residualfunctions.Given a κ = Φ ′ N κ ur κ ∞ as above, we define the automorphic kernel K κ ( g , g ) for g , g ∈ G ( A )via K κ ( g , g ) = X γ ∈ Z ( Q ) \ G ( Q ) κ ( g − γg ) . YBRID SUP-NORM BOUNDS FOR MAASS NEWFORMS OF POWERFUL LEVEL 27 A standard calculation tells us that if ψ = ⊗ v ψ v is an element of L ( X ) such that for each place v , ψ v is an eigenfunction for R ( κ v ) with eigenvalue λ v , then one has(28) Z Z ( A ) G ( Q ) \ G ( A ) K κ ( g , g ) ψ ( g ) dg = ( Y v λ v ) ψ ( g ) Lemma 3.17. Suppose that κ ur = κ ′ ur ∗ ( κ ′ ur ) ∗ for some κ ′ ur ∈ H ur . Put κ = Φ ′ N κ ur κ ∞ . Then if ψ ∈ B then Z Z ( A ) G ( Q ) \ G ( A ) K κ ( g , g ) ψ ( g ) dg = λ ψ ψ ( g ) for some λ ψ ≥ . Moreover λ φ ′ ≥ M − N − | λ ′ ur | ˆ κ ∞ ( π ∞ ) where the quantity λ ′ ur is defined by R ( κ ′ ur ) φ ′ = λ ′ ur φ ′ .Proof. By our assumption that ψ ∈ B , a complex number λ ψ as above exists. We can write λ ψ = λ ψ,N λ ψ, ur λ ψ, ∞ using the decomposition R ( κ ) = R (Φ ′ N ) R ( κ ur ) R ( κ ∞ ). We have λ ψ, ∞ ≥ κ ∞ ( σ ) ≥ σ of G ( R ). We have λ ψ,N ≥ R ( κ ′ ur ) ψ = λ ′ ψ ψ then λ ψ, ur = | λ ′ ψ | ≥ λ ψ ≥ 0. The last assertion is immediate from the results of the previous subsection. (cid:3) Henceforth we assume that κ ur = κ ′ ur ∗ ( κ ′ ur ) ∗ for some κ ′ ur ∈ H ur and we put κ = Φ ′ N κ ur κ ∞ .Then spectrally expanding K κ ( g, g ) along B and using the above Lemma, we get for all g ∈ G ( A ), M − N − ˆ κ ∞ ( π ∞ ) | λ ′ ur φ ′ ( g ) | ≤ K κ ( g, g ) . Note that ˆ κ ∞ ( π ∞ ) ≥ 1. Next we look at the quantity K κ ( g, g ). Assume that g = k N n ( x ) a ( y )with k N = Q p | N k p ∈ K N , k N h N ∈ J N . The second condition means that k p ∈ wK p ( p ) for all p | N . We have K κ ( g, g ) = X γ ∈ Z ( Q ) \ G ( Q ) Φ ′ N ( k − N γk N ) κ ur ( γ ) κ ∞ (( n ( x ) a ( y )) − γn ( x ) a ( y )) . Above we have Φ ′ N ( k − N γk N ) ≤ 1, and moreover if Φ ′ N ( k − N γk N ) = 0 then we must have a) k − p γk p ∈ Z ( Q p ) G ( Z p ) for all primes p dividing N , and b) k − p γk p ∈ Z ( Q p ) K p ( p ) for all primes p dividing N . The condition a) implies that γ ∈ Z ( Q p ) G ( Z p ) for all primes p dividing N . The condition b),together with the fact that k p ∈ wK p ( p ) for all p | N , implies that γ ∈ Z ( Q p ) K ,p ( p ) for all primes p dividing N .Finally we have κ ∞ ( g ) = 0 if det( g ) < 0, and if det( g ) > κ ∞ ( g ) = κ ∞ ( u ( g )) asexplained earlier, whence κ ∞ (( n ( x ) a ( y )) − γn ( x ) a ( y )) = κ ∞ ( u ( z, γz )) , z = x + iy where for any two points z , z on the upper-half plane, u ( z , z ) denotes the hyperbolic distancebetween them, i.e., u ( z , z ) = | z − z | z )Im( z ) . Putting everything together, we get the followingProposition. Proposition 3.18. Let κ ′ ur ∈ H ur and suppose that R ( κ ′ ur ) φ ′ = λ ′ ur φ ′ . Let κ ur = κ ′ ur ∗ ( κ ′ ur ) ∗ and κ = Φ ′ N κ ur κ ∞ . Then for all z = x + iy and all k ∈ K N such that kh N ∈ J N , we have | φ ′ ( kn ( x ) a ( y )) | ≤ M N | λ ′ ur | X γ ∈ Z ( Q ) \ G ( Q ) + ,γ ∈ Z ( Q p ) K ,p ( p ) ∀ p | N γ ∈ Z ( Q p ) G ( Z p ) ∀ p | N | κ ur ( γ ) κ ∞ ( u ( z, γz )) | Conclusion. We now make a specific choice for κ ur . Let Λ ≥ S = { ℓ : ℓ prime, ( ℓ, N ) = 1 , Λ ≤ ℓ ≤ } . Define for each integer r , c r = ( | λ π ( r ) | λ π ( r ) if r = ℓ or r = ℓ , ℓ ∈ S, κ ′ ur = P r c r κ r , and κ ur = κ ′ ur ∗ ( κ ′ ur ) ∗ . Given this, let us estimate the quantities appearingin Proposition 3.18.First of all, we have λ ′ ur = P ℓ ∈ S ( | λ π ( ℓ ) | + | λ π ( ℓ ) | ) . By the well-known relation λ π ( ℓ ) − λ π ( ℓ ) = ω π ℓ ( ℓ ), it follows that | λ π ( ℓ ) | + | λ π ( ℓ ) | ≥ 1. Hence λ ′ ur ≫ ε Λ − ε .Next, using the well-known relation κ m ∗ κ ∗ n = X t | gcd( m,n ) Y p | t ω π p ( t ) Y p | n ω − π p ( n ) κ mn/t , we see that κ ur = X ≤ l ≤ y l κ l where the complex numbers y l satisfy: | y l | ≪ Λ , l = 1 , , l = ℓ or l = ℓ ℓ or l = ℓ ℓ or l = ℓ ℓ with ℓ , ℓ ∈ S , otherwise.We have | κ l ( γ ) | ≤ l − / . Moreover κ l ( γ ) = 0 unless γ ∈ Z ( Q p ) G ( Z p ) a ( ℓ ) G ( Z p ) for all p ∤ N . Wededuce the following bound,(29) | φ ′ ( kn ( x ) a ( y )) | ≪ ε Λ − ε M N X ≤ l ≤ y l √ l X γ ∈ Z ( Q ) \ G ( Q ) + ,γ ∈ Z ( Q p ) K ,p ( p ) ∀ p | N γ ∈ Z ( Q p ) G ( Z p ) a ( ℓ ) G ( Z p ) ∀ p ∤ N | κ ∞ ( u ( z, γz )) | . Define M ( ℓ, N ) = { (cid:20) a bc d (cid:21) , a, b, c, d ∈ Z , a > , N | c, ad − bc = ℓ } . The following lemma follows immediately from strong approximation. Lemma 3.19. Let γ ∈ G ( Q ) + and ℓ be a positive integer coprime to N . Suppose that for eachprime p , γ ∈ Z ( Q p ) G ( Z p ) a ( ℓ ) G ( Z p ) . Suppose also that for each prime p | N , γ ∈ Z ( Q p ) K ,p ( p ) .Then there exists z ∈ Z ( Q ) such zγ ∈ M ( ℓ, N ) . YBRID SUP-NORM BOUNDS FOR MAASS NEWFORMS OF POWERFUL LEVEL 29 Proof. Omitted. (cid:3) Let us take another look at (29) in view of the above Lemma. The sum in (29) is over allmatrices γ in Z ( Q ) \ G ( Q ) + such that γ ∈ Z ( Q p ) K ,p ( p ) for p | N and γ ∈ Z ( Q p ) G ( Z p ) a ( ℓ ) G ( Z p )for p ∤ N . The latter condition can equally well be taken to over all p as Z ( Q p ) G ( Z p ) a ( ℓ ) G ( Z p ) = Z ( Q p ) G ( Z p ) a ( ℓ ) G ( Z p ) = Z ( Q p ) G ( Z p ) if ℓ and p are coprime, which is certainly the case when p | N . Therefore the above Lemma, together with the fact that the natural map from M ( ℓ, N )to Z ( Q ) \ G ( Q ) + is an injection, implies that the sum in (29) can replaced by a sum over the set M ( ℓ, N ). Hence, writing g = kh N ∈ J N as before, we get(30) | φ ( gn ( x ) a ( y )) | = | φ ′ ( kn ( x ) a ( y )) | ≪ ε Λ − ε M N X ≤ l ≤ y l √ l X γ ∈ M ( ℓ,N ) | κ ∞ ( u ( z, γz )) | . For any δ > 0, we define N ( z, ℓ, δ, N ) = |{ γ ∈ M ( ℓ, N ) : u ( z, γz ) ≤ δ }| . We have the following counting result due to Templier [16, Proposition 6.1]. Proposition 3.20. Let z = x + iy ∈ F N . For any < δ < and a positive integer ℓ coprime to N , let the number N ( z, ℓ, δ, N ) be defined as above.For Λ ≥ , define A ( z, Λ , δ, N ) = X ≤ l ≤ y l √ l N ( z, ℓ, δ, N ) . Then A ( z, Λ , δ, N ) ≪ ε Λ ε N ε (cid:20) Λ + Λ N / δ / y + Λ / δ / N − / + Λ / δ / y + Λ δN − (cid:21) . Proof. This is just Prop. 6.1 of [16]. (cid:3) Now (30) gives us(31) | φ ( gn ( x ) a ( y )) | ≪ ε Λ − ε M N Z | κ ∞ ( δ ) | dA ( z, Λ , δ, N ) . Using Proposition 3.20 and the property | κ ∞ ( δ ) | ≤ min( T, T / δ / ), we immediately deduce Propo-sition 3.16 after a simple integration, as in [16, 6.2]. 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