Hybrid sup-norm bounds for Maass newforms of powerful level

Abstract

Let f be an L 2 -normalized Hecke-Maass cuspidal newform of level N , character χ and Laplace eigenvalue λ. Let N1 denote the smallest integer such that N |N 2 1 and N0 denote the largest integer such that N 2 0 |N . Let M denote the conductor of χ and define M1 = M/ gcd(M, N1). We prove the bound, which generalizes and strengthens previously known upper bounds for f ∞ . This is the first time a hybrid bound (i.e., involving both N and λ) has been established for f ∞ in the case of non-squarefree N . The only previously known bound in the non-squarefree case was in the N -aspect; it had been shown by the author that f ∞ ≪ λ,ε N 5/12+ε provided M = 1. The present result significantly improves the exponent of N in the above case. If N is a squarefree integer, our bound reduces to f ∞ ≪ε N 1/3+ε λ 5/24+ε , which was previously proved by Templier.The key new feature of the present work is a systematic use of p-adic representation theoretic techniques and in particular a detailed study of Whittaker newforms and matrix coefficients for GL2(F ) where F is a local field.

Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces

“…It is clear that R(κ ur ) is a normal, non-negative operator. Moreover, by a standard argument (see (5.6-5.8) of [3] and Section 3.7 of [17]) we get that Λ, l = 1, 1, l = ℓ 1 ℓ 2 or l = ℓ 2 1 ℓ 2 2 with ℓ 1 , ℓ 2 ∈ P, 0, otherwise.…”

“…Next, we apply the theorem with O = O 0 (C ′ ) where C ′ = C 2 1 /C is the squarefree integer obtained by taking the product of all primes which divide C to an odd power. Then, it was shown in [17,Sec. 2.7] that the action of K O on (a suitable right-translate of) φ generates a representation of dimension ≪ lcm(M,C 1 ) C ′ .…”

“…Let H ur be the set of all compactly supported functions on p∈ur GL 2 (Q p ) that are bi-GL 2 (Z p ) invariant for each p ∈ ur and transform under the action of the centre by ω −1 π . For each positive integer ℓ satisfying (ℓ, S) = 1, define the functions κ ℓ in H ur as in Section 3.5 of [17]; these correspond to the usual Hecke operators T ℓ .…”

“…As we are not looking for a bound in the archimedean aspect, the choice of κ ∞ is unimportant. However for definiteness, let us fix the function κ ∞ on G ∞ as in [17] (see end of Section 3.5). In particular this choice has the property that κ ∞ (g) = 0 ⇒ det(ι ∞ (g ∞ )) > 0, u(ι ∞ (g ∞ )) ≤ 1 27 where for g ∈ GL 2 (R) + , we let u(g) = |g(i)−i| 2 4Im(g(i)) denote the hyperbolic distance from g(i) to i.…”

“…For an explanation why we get such different exponents for squarefree and squarefull levels, see Section 1.5 of this introduction. An interesting fact is that this squarefree/squarefull dichotomy in the sup-norm bound is also present in the case of newforms on GL 2 (see Section 1.3 of [17]), but for utterly different reasons! Remark 1.1.…”