## Algebra & Number Theory

### Hybrid sup-norm bounds for Maass newforms of powerful level

Abhishek Saha

#### Abstract

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 $M1 = M∕gcd(M,N1)$. We prove the bound $∥f∥∞≪εN01∕6+εN11∕3+εM11∕2λ5∕24+ε$, 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 $∥f∥∞≪λ,εN5∕12+ε$ 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 $∥f∥∞≪εN1∕3+ελ5∕24+ε$, 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.

#### Article information

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

Dates
Revised: 25 October 2016
Accepted: 16 December 2016
First available in Project Euclid: 12 December 2017

https://projecteuclid.org/euclid.ant/1513090721

Digital Object Identifier
doi:10.2140/ant.2017.11.1009

Mathematical Reviews number (MathSciNet)
MR3671430

Zentralblatt MATH identifier
06748165

#### Citation

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. https://projecteuclid.org/euclid.ant/1513090721

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