Algebra & Number Theory

Hybrid sup-norm bounds for Maass newforms of powerful level

Abhishek Saha

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Let f be an L2-normalized Hecke–Maass cuspidal newform of level N, character χ and Laplace eigenvalue λ. Let N1 denote the smallest integer such that N|N12 and N0 denote the largest integer such that N02|N. Let M denote the conductor of χ and define M1 = Mgcd(M,N1). We prove the bound fεN016+εN113+εM112λ524+ε, 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 nonsquarefree N. The only previously known bound in the nonsquarefree case was in the N-aspect; it had been shown by the author that fλ,εN512+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 fεN13+ελ524+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.

Article information

Algebra Number Theory, Volume 11, Number 5 (2017), 1009-1045.

Received: 13 October 2015
Revised: 25 October 2016
Accepted: 16 December 2016
First available in Project Euclid: 12 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F03: Modular and automorphic functions
Secondary: 11F41: Automorphic forms on GL(2); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces [See also 14J20] 11F60: Hecke-Petersson operators, differential operators (several variables) 11F72: Spectral theory; Selberg trace formula 11F85: $p$-adic theory, local fields [See also 14G20, 22E50] 35P20: Asymptotic distribution of eigenvalues and eigenfunctions

Maass form sup-norm automorphic form newform amplification


Saha, Abhishek. Hybrid sup-norm bounds for Maass newforms of powerful level. Algebra Number Theory 11 (2017), no. 5, 1009--1045. doi:10.2140/ant.2017.11.1009.

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  • V. Blomer and R. Holowinsky, “Bounding sup-norms of cusp forms of large level”, Invent. Math. 179:3 (2010), 645–681.
  • G. Harcos and P. Michel, “The subconvexity problem for Rankin–Selberg $L$-functions and equidistribution of Heegner points II”, Invent. Math. 163:3 (2006), 581–655.
  • G. Harcos and N. Templier, “On the sup-norm of Maass cusp forms of large level: II”, Int. Math. Res. Not. 2012:20 (2012), 4764–4774.
  • G. Harcos and N. Templier, “On the sup-norm of Maass cusp forms of large level III”, Math. Ann. 356:1 (2013), 209–216.
  • J. Hoffstein and P. Lockhart, “Coefficients of Maass forms and the Siegel zero”, Ann. of Math. $(2)$ 140:1 (1994), 161–181.
  • Y. Hu, “Triple product formula and mass equidistribution on modular curves of level ${N}$”, Int. Math. Res. Not. (2017), art. id. rnw322.
  • H. Iwaniec and P. Sarnak, “$L^\infty$ norms of eigenfunctions of arithmetic surfaces”, Ann. of Math. $(2)$ 141:2 (1995), 301–320.
  • S. Marshall, “Local bounds for $L^p$ norms of Maass forms in the level aspect”, Algebra Number Theory 10:4 (2016), 803–812.
  • D. Milićević, “Sub-Weyl subconvexity for Dirichlet $L$-functions to prime power moduli”, Compos. Math. 152:4 (2016), 825–875.
  • P. D. Nelson, “Equidistribution of cusp forms in the level aspect”, Duke Math. J. 160:3 (2011), 467–501.
  • P. D. Nelson, A. Pitale, and A. Saha, “Bounds for Rankin–Selberg integrals and quantum unique ergodicity for powerful levels”, J. Amer. Math. Soc. 27:1 (2014), 147–191.
  • A. Saha, “On sup-norms of cusp forms of powerful level”, Preprint, 2014.
  • A. Saha, “Large values of newforms on GL(2) with highly ramified central character”, Int. Math. Res. Not. 2016:13 (2016), 4103–4131.
  • N. Templier, “On the sup-norm of Maass cusp forms of large level”, Selecta Math. $($N.S.$)$ 16:3 (2010), 501–531.
  • N. Templier, “Large values of modular forms”, Camb. J. Math. 2:1 (2014), 91–116.
  • N. Templier, “Hybrid sup-norm bounds for Hecke–Maass cusp forms”, J. Eur. Math. Soc. $($JEMS$)$ 17:8 (2015), 2069–2082.
  • J. B. Tunnell, “On the local Langlands conjecture for $GL(2)$”, Invent. Math. 46:2 (1978), 179–200.