On Exceptional Maass Forms

TL;DR

This paper establishes relations between Satake parameters of Maass forms and shows that the Ramanujan-Petersson conjecture at a single prime implies it for all primes and forms, improving bounds for exceptional cases.

Contribution

It proves relations between Satake parameters at different places and links conjectures for non-exceptional and exceptional Maass forms, extending known results.

Findings

Ramanujan-Petersson conjecture at one prime implies it at all primes for Maass forms.

Improved Kim and Sarnak's bound to 7/64 for exceptional Maass forms.

Established relations between Satake parameters at finite and archimedean places.

Abstract

We prove certain relations between Satake parameters of cuspidal representations of $\GL_2(\mathbb{A}_{\mathbb{Q}})$ at finite and archimedean places. Consequently, we show that the Ramanujan-Petersson conjecture at a fixed prime $p\nmid N$ for \textit{non-exceptional} Maass forms of level $N$ implies the conjecture at $p$ for \textit{all} Maass forms of level $N$ and the Selberg's $1/4$-eigenvalue conjecture simultaneously. As an application, we improve Kim and Sarnak's $7/64$-bound towards the Satake parameters at all $p\nmid N$ for exceptional Maass forms.