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

Masanori Muto, Hiroaki Narita, Ameya Pitale

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The aim of this paper is to carry out an explicit construction of CAP representations of 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 . 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.

Original languageEnglish
Pages (from-to)137-185
Number of pages49
JournalNagoya Mathematical Journal
Publication statusPublished - 2016 Jun 1
Externally publishedYes



  • 2010 Mathematics subject classification 11F55 11F70

ASJC Scopus subject areas

  • Mathematics(all)

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