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 | |
| Publication status | Published - 2016 Jun 1 |
| Externally published | Yes |
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Keywords
- 2010 Mathematics subject classification 11F55 11F70
ASJC Scopus subject areas
- Mathematics(all)
Cite this
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
}
TY - JOUR
T1 - Lifting to GL(2) over a division quaternion algebra, and an explicit construction of cap representations
AU - Muto, Masanori
AU - Narita, Hiroaki
AU - Pitale, Ameya
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 -