On the sup-norm of Maass cusp forms of large level. III
aa r X i v : . [ m a t h . N T ] J u l ON THE SUP-NORM OF MAASS CUSP FORMS OF LARGELEVEL. III
GERGELY HARCOS AND NICOLAS TEMPLIER
Abstract.
Let f be a Hecke–Maass cuspidal newform of square-free level N and Laplacian eigenvalue λ . It is shown that k f k ∞ ≪ λ,ǫ N − + ǫ k f k for any ǫ > Introduction
This note deals with the problem of bounding the sup-norm of eigenfunctionson arithmetic hyperbolic surfaces. It is natural to restrict this problem to Hecke–Maass cuspidal newforms which are square-integrable joint eigenfunctions of theLaplacian and Hecke operators. We consider the noncompact modular surfaceΓ ( N ) \ H equipped with its hyperbolic metric and associated measure; the totalvolume is then asymptotically equal to N o (1) . We shall L -normalize all Hecke–Maass cuspidal newforms f with respect to that measure, namely(1.1) Z Γ ( N ) \ H | f ( z ) | dxdyy = 1 . It is interesting to bound the sup-norm k f k ∞ in terms of the two basic parameters:the Laplacian eigenvalue λ and the level N .In the λ -aspect, the first nontrivial bound is due to Iwaniec and Sarnak [6] whoestablished k f k ∞ ≪ N,ǫ λ + ǫ for any ǫ >
0. Their key idea was to make use ofthe Hecke operators, through the method of amplification, in order to go beyond k f k ∞ ≪ N λ which is valid on any Riemannian surface by [9].In the N -aspect, the “trivial” bound is k f k ∞ ≪ λ,ǫ N ǫ see [1, 3, 7]. Here andlater the dependence on λ is continuous. The first nontrivial bound in the N -aspect is due to Blomer–Holowinsky [3, p. 673] who proved k f k ∞ ≪ λ,ǫ N − + ǫ ,at least for square-free N . In [11] the second named author revisited the proofby making a systematic use of geometric arguments, and derived a stronger expo-nent: k f k ∞ ≪ λ,ǫ N − + ǫ . Helfgott–Ricotta (unpublished) improved some of theestimates in [11] and obtained k f k ∞ ≪ λ,ǫ N − + ǫ . In [5] we introduced a moreefficient treatment of the counting problem at the heart of the argument andderived the estimate k f k ∞ ≪ λ,ǫ N − + ǫ . We shall improve this estimate further. Date : Sep 2011.2010
Mathematics Subject Classification.
Key words and phrases. automorphic forms, trace formula, amplification, diophantineapproximation.This work was partially supported by a grant Theorem 1.1.
Let f be an L -normalized Hecke–Maass cuspidal newform ofsquare-free level N . Then for any ǫ > we have a bound k f k ∞ ≪ λ,ǫ N − + ǫ , where the implied constant depends continuously on λ . Remarks. (i) It seems that − is the natural exponent for the sup-norm prob-lem in the level aspect. Examples of such exponents are the Weyl exponent (resp. Burgess exponent ) in the subconvexity problem for GL in thearchimedean (resp. nonarchimedean) aspect, or their doubles in the GL -setting.(ii) Independently, Blomer–Michel [2] obtain a bound of the same quality forHecke eigenforms on unions of arithmetic ellipsoids. In this paper we areconcerned in (2.2) with solutions of an indefinite quadratic equation det( γ ) = l , whereas arithmetic ellipsoids involve definite quadratic forms.(iii) From Atkin–Lehner theory we may assume that Im z ≫ N − when investi-gating the sup-norm. The critical range is actually when Im z ≤ N − + o (1) .Otherwise the details of the proof below show that | f ( z ) | is significantly lessthan N − .The present note is derived from [10] which is motivated by the comparison ofthe method in [6] for the λ -aspect with our method in [5, 11] for the N -aspect. Theadvantage of the new argument in [10] is that it can be adapted to the λ -aspectto reproduce the bound k f k ∞ ≪ N,ǫ λ + ǫ , which is key for establishing hybridbounds simultaneously in the λ and N -aspects. Compared to [5, 11] the readerwill find below two improvements coming from a Pell equation and a uniformcount of lattice points [8].2. Counting lattice points
Notation.
To make this section self-contained we recall the definitions from [5,11]. Let GL ( R ) + act on the upper-half plane H = { x + iy, y > } by fractionallinear transformations. Denote by u ( , ) the following function of the hyperbolicdistance:(2.1) u ( w, z ) = | w − z | Im w Im z . For z ∈ H and l, N ≥ M ∗ ( z, l, N ) be the number of matrices γ = (cid:18) a bc d (cid:19) in M ( Z ) such that(2.2) det( γ ) = l, c ≡ N ) , u ( γz, z ) ≤ N ǫ , c = 0 , ( a + d ) = 4 l. We write f g meaning that for all ǫ > C ( ǫ ) > f ( N ) ≤ C ( ǫ ) N ǫ g ( N ) for all N ≥
1. To simplify notation we omit the dependencein λ . For example Theorem 1.1 says k f k ∞ N − .Let F ( N ) be the set of z ∈ H such that Im z ≥ Im δz for all Atkin–Lehneroperators δ of level N . In this section we shall only use the fact ([5, Lemma 2.2])that for all z = x + iy ∈ F ( N ), we have N y ≫ a, b ) ∈ Z distinct from (0 ,
0) we have(2.3) | az + b | ≥ N .
Lattice points.
We have the following uniform estimate for the number oflattice points in a disc ([8, Lemma 2]):
Lemma 2.1.
Let M be a euclidean lattice of rank and D be a disc of radius R > in M ⊗ Z R (not necessarily centered at ). If λ ≤ λ are the successiveminima of M , then (2.4) M ∩ D ≪ Rλ + R λ λ Remarks. (i) Let d ( M ) > M . Minkowski’s second The-orem asserts that λ λ ≍ d ( M ). When R → ∞ , the leading term of (2.4) is R d ( M ) as expected.(ii) It is easier to establish the upper-bound ≪ R λ (which also has theadvantage of having only two terms). One can verify that(2.5) 1 + Rλ + R λ λ ≪ R λ . Thus the estimate in (2.4) is always better.(iii) We have equality (up to a constant) in (2.5) if and only if R ≪ λ . In theapplications below it is often the case that R ≪ λ . However this is notalways the case, and then the improvement of (2.4) on the easier bound issignificant.2.3. Counting.
The following is an improvement on [5, Lemma 4.2]:
Lemma 2.2.
Let z = x + iy ∈ F ( N ) and ≤ L ≤ N O (1) . Then (2.6) X ≤ l ≤ L M ∗ ( z, l, N ) LN y + L N + L N .
If we restrict to l being a perfect square, then one can improve by a factor L : (2.7) X ≤ l ≤ L,l is a square M ∗ ( z, l, N ) L N y + LN + L N .
Proof.
We briefly recall the beginning of the argument in [5, Lemma 4.2]. Let γ = (cid:18) a bc d (cid:19) satisfy (2.2). In coordinates we have(2.8) (cid:12)(cid:12) − cz + ( a − d ) z + b (cid:12)(cid:12) ≤ Ly N ǫ . As in [5, 6] we verify that | c | L /y , so there are L / ( N y ) possible valuesof c .Consider the lattice h , z i inside C . Its covolume equals y and its shortest lengthis at least N − / by (2.3). In the inequality (2.8) we are counting lattice points ( a − d, b ) in a disc of volume Ly centered at cz . Hence by Lemma (2.1), thereare L yN − + Ly y possible pairs ( a − d, b ) for each value of c .As in [5, 6] one can deduce from (2.8) that | a + d | L . This concludes theproof of (2.6).For (2.7) we instead use the identity(2.9) ( a − d ) + 4 bc = ( a + d ) − l. The left-hand side is non-zero by assumption (2.2). Since l is a perfect square, foreach given triple ( a − d, b, c ) the number of pairs ( a + d, l ) satisfying (2.9) is (cid:3) The following is a refinement of (2.7).
Lemma 2.3.
Let z = x + iy ∈ F ( N ) and ≤ l ≤ Λ ≤ N O (1) . Then (2.10) X ≤ l ≤ Λ M ∗ ( z, l l , N ) Λ N y + Λ N + Λ N .
Proof.
Let γ = (cid:18) a bc d (cid:19) satisfy (2.2). We have(2.11) (cid:12)(cid:12) − cz + ( a − d ) z + b (cid:12)(cid:12) ≤ l l y N ǫ . This implies | c | Λ /y , so there are Λ / ( N y ) possible values of c .For each value of c , we again apply Lemma 2.1 to the lattice h , z i of covolume y and shortest length at least N − . In the inequality (2.11) we are countinglattice points ( a − d, b ) in a disc of volume Λ y . This implies that there are Λ yN − + Λ y y possible pairs ( a − d, b ) satisfying (2.11).Further, since det( γ ) = l l , we have:(2.12) ( a − d ) + 4 bc = ( a + d ) − l l . The left-hand side is already determined by the values of c and ( a − d, b ). It isnonzero by assumption (2.2). This is a generalized Pell equation in the remainingvariables a + d and l .Without loss of generality we can assume that l is square-free. One can deducefrom (2.8) that | a + d | Λ . If l = 1 then we are done with a divisor bound asin the proof of (2.7).If l > √ = 1 . · · · , which isbounded away from 1 (for better estimates, see [4] and the references herein). Wededuce that the number of pairs ( a + d, l ) of solutions of (2.12) is ≪ Λ o (1) γ ’s is(2.13) Λ N y · (1 + Λ N y + Λ y ) . This concludes the proof of the lemma. (cid:3)
Special matrices.
We let M u ( z, l, N ) be the number of matrices satisfy-ing (2.2) but with the condition c = 0 instead of c = 0 (upper-triangular). Lemma 2.4.
Let z = x + iy ∈ F ( N ) and ≤ Λ ≤ N O (1) . Then the followingestimates hold, where l , l run through prime numbers: (2.14) X ≤ l ,l ≤ Λ M u ( z, l l , N ) Λ + Λ N y + Λ y, (2.15) X ≤ l ,l ≤ Λ M u ( z, l l , N ) Λ + Λ N y + Λ y, (2.16) X ≤ l ,l ≤ Λ M u ( z, l l , N ) N y + Λ y. Proof.
We need to count the number of matrices γ = (cid:18) a b d (cid:19) ∈ M ( Z ) such that(2.17) | ( a − d ) z + b | ≤ ady N ǫ and ad = l l (resp. ad = l l , and ad = l l ).We again consider the lattice h , z i of covolume y and shortest length at least N − . In the inequality (2.17) we are counting lattice points ( a − d, b ) in a disc ofvolume ady .We consider (2.14) first. There are N y + Λ y possible pairs of integers( a − d, b ) satisfying (2.17). Each pair gives rise to O (Λ) matrices γ (this is because ad = l l ).Next we consider (2.15). There are N y + Λ y pairs of integers( a − d, b ) satisfying (2.17). Each pair gives rise to O (Λ) matrices γ (this is because ad = l l ).Finally we consider (2.16). There are N y + Λ y pairs of integers( a − d, b ) satisfying (2.17). Since l and l are primes, we have either ( a = 1 , d = l l ) or ( a = l , d = l l ) or ( a = l , d = l ), or equivalent configurations. In eachconfiguration, and for a given value of a − d , there are Λ o (1) pairs ( a, d ). Thuseach pair ( a − d, b ) gives rise to Λ o (1) matrices γ . (cid:3) We note that a similar proof also yields:(2.18) X ≤ l ≤ Ll prime M u ( z, l, N ) L N y + Ly.
Finally let M p ( z, l, N ) be the number of matrices satisfying (2.2) but insteadwith the condition ( a + d ) = 4 l (parabolic) and with no restriction on c ≡ N ).Then [5, Lemma 4.1] gives(2.19) M p ( z, l, N ) = 2 δ (cid:3) ( l ) , ≤ l < y − N − ǫ . Here δ (cid:3) ( l ) = 1 , l is a perfect square or not.We let M ( z, l, N ) := M ∗ ( z, l, N )+ M u ( z, l, N )+ M p ( z, l, N ) which is the numberof matrices satisfying the first three conditions in (2.2). Proof of Theorem 1.1
Applying the amplification method of Friedlander–Iwaniec as in [6] and [5, § | f ( z ) | X l ≥ y l √ l M ( z, l, N ) . Here Λ > y l ∈ R ≥ satisfies: y l := Λ , l = 1 , , l = l or l l or l l or l l with Λ < l , l <
2Λ primes,0 , otherwise . By [11, § | f ( x + iy ) | ( N y ) − . Thus we may assume that y < N − when establishing Theorem 1.1. Without loss of generality we can also assume z = x + iy ∈ F ( N ).We shall choose Λ = N − ǫ . This implies Λ < y − N − ǫ , thus the conditionin (2.19) is satisfied. Therefore the contribution in (3.1) of the parabolic ma-trices is ≪ Λ using (2.19).The contribution in (3.1) of the upper-triangular matrices with l = 1 is Λ(1 + N y + y ) using (2.18). For Λ < l <
2Λ it is Λ − + N y + Λ y using (2.18) again. For Λ < l < it is N y + Λ y using (2.14) ofLemma 2.4. For Λ < l < it is Λ N y + Λ y using (2.15) of Lemma 2.4.For l > Λ it is N y + Λ y using (2.16) of Lemma 2.4.It now remains to consider the matrices in (2.2) counted by M ∗ . The contribu-tion in (3.1) of l = 1 is Λ( Ny + N − ) using (2.6) in Lemma 2.2. For Λ < l < Λ Ny + Λ N + Λ N using (2.6) again. For Λ < l < it is Λ Ny + Λ N + Λ N using (2.6) again. For Λ < l < it is Λ Ny + Λ N + Λ N using Lemma 2.3. For l > Λ it is Ny + Λ N + Λ N using (2.7).Altogether we obtain that(3.2) Λ | f ( z ) | Λ + Λ N + Λ N .
Choosing Λ := N − ǫ , all three terms above are equal to N + o (1) . This concludesthe proof of Theorem 1.1. (cid:3) Remark.
The conclusion of Theorem 1.1 holds true for Hecke–Maass cuspidalnewforms of an arbitrary nebentypus and also for holomorphic modular forms.Indeed the amplification method again yields the inequality (3.1) above and therest of the proof goes through without change. Also the assumption that f be anewform is not necessary since Atkin–Lehner theory reduces the general case tothe case of newforms. For an oldform f , the bound would be in terms of the levelfrom which f was induced. Acknowledgements.
We thank the referee for his helpful comments.
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