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

Masanori Muto; Hiro Aki 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, Hiro Aki Narita, Ameya Pitale

School of Fundamental Science and Engineering

Research output: Contribution to journal › Article › peer-review

3 Citations (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

Keywords

2010 Mathematics subject classification 11F55 11F70

ASJC Scopus subject areas

Mathematics(all)

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10.1017/nmj.2016.15

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

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.",

keywords = "2010 Mathematics subject classification 11F55 11F70",

author = "Masanori Muto and Narita, {Hiro Aki} and Ameya Pitale",

note = "Funding Information: Received June 21, 2014. Accepted June 11, 2015. 2010 Mathematics subject classification. 11F55, 11F70. The second author was partly supported by Grant-in-Aid for Scientific Research (C) 24540025, Japan Society for the Promotion of Science. The third author was partly supported by National Science Foundation grant DMS-1100541.",

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