Lifting to GL(2) over a quaternion division algebra and an explicit construction of CAP representations

2014/05/19 by Masanori Muto, Muto, Masanori, Hiro-aki Narita +3
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1405.4770

openalex publication_date 2014/05/19 · openalex created_date 2016/07/22 · openalex updated_date 2026/07/28

Abstract

The aim of this paper is to carry out an explicit construction of CAP representations of GL(2) over a division quaternion algebra with discriminant two. We first construct cusp forms on such group explicitly by lifting from Maass cusp forms for the congruence subgroup of level 2. We show that this lifting is non-zero and Hecke-equivariant. This allows us to determine each local component of such a cuspidal representation. We then know that our cuspidal representations provide examples of CAP representations, and in fact, counterexamples of the Generalized Ramanujan conjecture.

Citations

• On the Saito-Kurokawa lifting

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