# M. Muto et al: LIFTING TO GL(2) OVER A DIVISION QUATERNION ALGEBRA, AND AN EXPLICIT CONSTRUCTION OF CAP REPRESENTATIONS

MASANORI MUTO, HIRO-AKI NARITA and AMEYA PITALE, LIFTING TO GL(2) OVER A DIVISION QUATERNION ALGEBRA, AND AN EXPLICIT CONSTRUCTION OF CAP REPRESENTATIONS, Nagoya Mathematical Journal, Volume 222 – Issue 1 – June 2016, DOI: http://dx.doi.org/10.1017/nmj.2016.15, Published online: 07 June 2016, pp. 137-185

Abstract: The aim of this paper is to carry out an explicit construction of CAP representations of $\text{GL}(2)$ over a division quaternion algebra with discriminant two. We first construct cusp forms on such a group explicitly by lifting from Maass cusp forms for the congruence subgroup ${\rm\Gamma}_{0}(2)$. We show that this lifting is nonzero and Hecke-equivariant. This allows us to determine each local component of a cuspidal representation generated by such a lifting. We then show that our cuspidal representations provide examples of CAP (cuspidal representation associated to a parabolic subgroup) representations, and, in fact, counterexamples to the Ramanujan conjecture.