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

Masanori Muto; Hiroaki Narita; Ameya Pitale

doi:10.1017/nmj.2016.15

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

Masanori Muto, Hiroaki Narita, Ameya Pitale

Research output: Contribution to journal › Article

1 Citation (Scopus)

Abstract

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 language	English
Pages (from-to)	137-185
Number of pages	49
Journal	Nagoya Mathematical Journal
Volume	222
DOIs	https://doi.org/10.1017/nmj.2016.15
Publication status	Published - 2016 Jun 1
Externally published	Yes

Fingerprint

Quaternion Algebra

Division Algebra

Cusp Form

Congruence Subgroups

Parabolic Subgroup

Ramanujan

Discriminant

Equivariant

Counterexample

Cap

Keywords

2010 Mathematics subject classification 11F55 11F70

ASJC Scopus subject areas

Mathematics(all)

Cite this

Muto, M., Narita, H., & Pitale, A. (2016). Lifting to GL(2) over a division quaternion algebra, and an explicit construction of cap representations. Nagoya Mathematical Journal, 222, 137-185. https://doi.org/10.1017/nmj.2016.15

Lifting to GL(2) over a division quaternion algebra, and an explicit construction of cap representations. / Muto, Masanori; Narita, Hiroaki; Pitale, Ameya.

In: Nagoya Mathematical Journal, Vol. 222, 01.06.2016, p. 137-185.

Research output: Contribution to journal › Article

Muto, M, Narita, H & Pitale, A 2016, 'Lifting to GL(2) over a division quaternion algebra, and an explicit construction of cap representations', Nagoya Mathematical Journal, vol. 222, pp. 137-185. https://doi.org/10.1017/nmj.2016.15

Muto M, Narita H, Pitale A. Lifting to GL(2) over a division quaternion algebra, and an explicit construction of cap representations. Nagoya Mathematical Journal. 2016 Jun 1;222:137-185. https://doi.org/10.1017/nmj.2016.15

Muto, Masanori ; Narita, Hiroaki ; Pitale, Ameya. / Lifting to GL(2) over a division quaternion algebra, and an explicit construction of cap representations. In: Nagoya Mathematical Journal. 2016 ; Vol. 222. pp. 137-185.

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10.1017/nmj.2016.15