On the Sup-Norm of Maass Cusp Forms of Large Level: II

Harcos, Gergely; Templier, Nicolas

doi:10.1093/imrn/rnr202

Cited by 29 publications

(53 citation statements)

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“…We consider an amplified second moment which we transform by a pre-trace formula into a sum over a sort of automorphic kernel, see (5.3). This starting point is similar in most investigations of sup-norms of eigenfunction on arithmetically defined manifolds, see, for example, [2,9,11,16,21]. In all cases one encounters eventually an interesting diophantine problem the solution of which is at the heart of the problem.…”

Section: Principle Of Proofmentioning

confidence: 67%

Sup-norms of Eigenfunctions on Arithmetic Ellipsoids

Blomer

Michel

2011

International Mathematics Research Notices

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Section: Principle Of Proofmentioning

confidence: 67%

Sup-norms of Eigenfunctions on Arithmetic Ellipsoids

Blomer

Michel

2011

International Mathematics Research Notices

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“…We also obtain a bound for newforms that generalizes and strengthens previously known results; see Theorem 3. 7 Finally, we remark that the results of this paper appear to be the first time that the local bound in the conductor aspect has been improved upon for squarefull conductors, for any kind of automorphic form on a compact domain. (In the noncompact case, this had been achieved in our previous paper [11].)…”

Section: A Classical Reformulationmentioning

confidence: 76%

“…Hence (6) allows us to define a character on Γ O which (by abusing notation) we also denote by χ. Let N χ be an integer such that (7) O…”

Section: A Classical Reformulationmentioning

confidence: 99%

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

Saha

2019

Math. Ann.

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Let D be an indefinite quaternion division algebra over Q. We approach the problem of bounding the sup-norms of automorphic forms φ on D × (A) that belong to irreducible automorphic representations and transform via characters of unit groups of orders of D. We obtain a non-trivial upper bound for φ ∞ in the level aspect that is valid for arbitrary orders. This generalizes and strengthens previously known upper bounds for φ ∞ in the setting of newforms for Eichler orders. In the special case when the index of the order in a maximal order is a squarefull integer N , our result specializes to φ ∞ ≪π ∞,ǫ N 1/3+ǫ φ 2.A key application of our result is to automorphic forms φ which correspond at the ramified primes to either minimal vectors (in the sense of [11]), or p-adic microlocal lifts (in the sense of [15]). For such forms, our bound specializes to φ ∞ ≪ǫ C 1 6 +ǫ φ 2 where C is the conductor of the representation π generated by φ. This improves upon the previously known local bound φ ∞ ≪ λ,ǫ C 1 4 +ǫ φ 2 in these cases.

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“…The corresponding forms f are (Hecke-Maass or holomorphic) newforms with respect to the group Γ 1 (C). For such newforms and for squarefree C there were several results [5,10,11,39,41] culiminating in the bound φ ∞ ≪ k/λ,ǫ C 1/3+ǫ due to Harcos and Templier. (Here, for simplicity, we have quoted the bound only in the conductor-aspect, noting that a hybrid result was proved by Templier in [41].) This bound was generalized to the case of powerful (non-squarefree) C by the third author [35].…”

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confidence: 99%

Some analytic aspects of automorphic forms on GL(2) of minimal type

Nelson

Saha

2019

Comment. Math. Helv.

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Let π be a cuspidal automorphic representation of PGL 2 (A Q ) of arithmetic conductor C and archimedean parameter T , and let φ be an L 2 -normalized automorphic form in the space of π. The sup-norm problem asks for bounds on φ ∞ in terms of C and T . The quantum unique ergodicity (QUE) problem concerns the limiting behavior of the L 2 -mass |φ| 2 (g) dg of φ. All previous work on these problems in the conductor-aspect has focused on the case that φ is a newform.In this work, we study these problems for a class of automorphic forms that are not newforms. Precisely, we assume that for each prime divisor p of C, the local component πp is supercuspidal (and satisfies some additional technical hypotheses), and consider automorphic forms φ for which the local components φp ∈ πp are "minimal" vectors. Such vectors may be understood as non-archimedean analogues of lowest weight vectors in holomorphic discrete series representations of PGL 2 (R).For automorphic forms as above, we prove a sup-norm bound that is sharper than what is known in the newform case. In particular, if π∞ is a holomorphic discrete series of lowest weight k, we obtain the optimal bound C 1/8−ǫ k 1/4−ǫ ≪ǫ |φ|∞ ≪ǫ C 1/8+ǫ k 1/4+ǫ . We prove also that these forms give analytic test vectors for the QUE period, thereby demonstrating the equivalence between the strong QUE and the subconvexity problems for this class of vectors. This finding contrasts the known failure of this equivalence [31] for newforms of powerful level.

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On the Sup-Norm of Maass Cusp Forms of Large Level: II

Abstract: Abstract. Let f be a Hecke-Maass cuspidal newform of square-free level N and Laplacian eigenvalue λ. It is shown that f ∞ λ, N −1/12+ f 2 for any > 0, with an implied constant depending continuously on λ. The proof is short and selfcontained.

Cited by 29 publications

References 7 publications

Sup-norms of Eigenfunctions on Arithmetic Ellipsoids

Sup-norms of Eigenfunctions on Arithmetic Ellipsoids

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

Some analytic aspects of automorphic forms on GL(2) of minimal type

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