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

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
Externally published	Yes

Keywords

2010 Mathematics subject classification 11F55 11F70

ASJC Scopus subject areas

Mathematics(all)

Access to Document

10.1017/nmj.2016.15

Fingerprint Dive into the research topics of 'Lifting to GL(2) over a division quaternion algebra, and an explicit construction of cap representations'. Together they form a unique fingerprint.

Cite this

@article{db4b601ec4b24d84908985786c77e6fd,

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. Publisher Copyright: {\textcopyright} 2016 by The Editorial Board of the Nagoya Mathematical Journal.",

year = "2016",

month = jun,

day = "1",

doi = "10.1017/nmj.2016.15",

language = "English",

volume = "222",

pages = "137--185",

journal = "Nagoya Mathematical Journal",

issn = "0027-7630",

publisher = "Nagoya University",

}

TY - JOUR

T1 - Lifting to GL(2) over a division quaternion algebra, and an explicit construction of cap representations

AU - Muto, Masanori

AU - Narita, Hiro Aki

AU - Pitale, Ameya

N1 - 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. Publisher Copyright: © 2016 by The Editorial Board of the Nagoya Mathematical Journal.

PY - 2016/6/1

Y1 - 2016/6/1

N2 - 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.

AB - 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.

KW - 2010 Mathematics subject classification 11F55 11F70

UR - http://www.scopus.com/inward/record.url?scp=85047190565&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85047190565&partnerID=8YFLogxK

U2 - 10.1017/nmj.2016.15

DO - 10.1017/nmj.2016.15

M3 - Article

AN - SCOPUS:85047190565

VL - 222

SP - 137

EP - 185

JO - Nagoya Mathematical Journal

JF - Nagoya Mathematical Journal

SN - 0027-7630

ER -

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

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Cite this