On Exceptional Maass Forms

2020/11/18 by Liyang Yang, Yang, Liyang
Mathematics · #Advanced Algebra and Geometry #Analytic Number Theory Research #Advanced Mathematical Identities

paper · pdf · doi:10.48550/arxiv.2011.09054

Abstract

We prove certain relations between Satake parameters of cuspidal representations of \GL₂(\mathbbA_ℚ) at finite and archimedean places. Consequently, we show that the Ramanujan-Petersson conjecture at a fixed prime p\nmid N for non-exceptional Maass forms of level N implies the conjecture at p for 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.

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