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

TL;DR

This paper explicitly constructs CAP representations of GL(2) over a quaternion division algebra by lifting Maass cusp forms, providing concrete examples and counterexamples to the Ramanujan conjecture.

Contribution

It offers an explicit lifting method for constructing CAP representations of GL(2) over a quaternion algebra, detailing their local components and properties.

Findings

Constructed explicit cusp forms via lifting from Maass forms

Demonstrated non-zero and Hecke-equivariant liftings

Provided examples of CAP representations that counter the Ramanujan conjecture

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.