Abstract
As was first essentially pointed out by Tomoyoshi Ibukiyama, Hecke eigenforms on the indefinite symplectic group GSp(1, 1) or the definite symplectic group GSp∗(2) over ℚ right invariant by a (global) maximal open compact subgroup are conjectured to have the same spinor L-functions as those of paramodular new forms of some specified level on the symplectic group GSp(2) (or GSp(4)). This can be viewed as a generalization of the Jacquet-Langlands-Shimizu correspondence to the case of GSp(2) and its inner forms GSp(1,1) and GSp∗(2). In this paper we provide evidence of the conjecture on this explicit functorial correspondence with theta lifts: a theta lift from GL(2)×B× to GSp(1, 1) or GSp∗(2) and a theta lift from GL(2) × GL(2) (or GO(2, 2)) to GSp(2). Here B denotes a definite quaternion algebra over ℚ. Our explicit functorial correspondence given by these theta lifts are proved to be compatible with archimedean and non-archimedean local Jacquet-Langlands correspondences. Regarding the non-archimedean local theory we need some explicit functorial correspondence for spherical representations of the inner form and non-supercuspidal representations of GSp(2), which is studied in the appendix by Ralf Schmidt.
| Original language | English |
|---|---|
| Pages (from-to) | 1443-1474 |
| Number of pages | 32 |
| Journal | Journal of the Mathematical Society of Japan |
| Volume | 69 |
| Issue number | 4 |
| DOIs | |
| Publication status | Published - 2017 Jan 1 |
| Externally published | Yes |
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Keywords
- Jacquet-Langlands correspondence
- Spinor L-functions
- Theta lifts
ASJC Scopus subject areas
- Mathematics(all)
Cite this
Jacquet-langlands-Shimizu correspondence for theta lifts to GSp(2) and its inner forms I : An explicit functorial correspondence. / Narita, Hiroaki; Schmidt, Ralf.
In: Journal of the Mathematical Society of Japan, Vol. 69, No. 4, 01.01.2017, p. 1443-1474.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Jacquet-langlands-Shimizu correspondence for theta lifts to GSp(2) and its inner forms I
T2 - An explicit functorial correspondence
AU - Narita, Hiroaki
AU - Schmidt, Ralf
PY - 2017/1/1
Y1 - 2017/1/1
N2 - As was first essentially pointed out by Tomoyoshi Ibukiyama, Hecke eigenforms on the indefinite symplectic group GSp(1, 1) or the definite symplectic group GSp∗(2) over ℚ right invariant by a (global) maximal open compact subgroup are conjectured to have the same spinor L-functions as those of paramodular new forms of some specified level on the symplectic group GSp(2) (or GSp(4)). This can be viewed as a generalization of the Jacquet-Langlands-Shimizu correspondence to the case of GSp(2) and its inner forms GSp(1,1) and GSp∗(2). In this paper we provide evidence of the conjecture on this explicit functorial correspondence with theta lifts: a theta lift from GL(2)×B× to GSp(1, 1) or GSp∗(2) and a theta lift from GL(2) × GL(2) (or GO(2, 2)) to GSp(2). Here B denotes a definite quaternion algebra over ℚ. Our explicit functorial correspondence given by these theta lifts are proved to be compatible with archimedean and non-archimedean local Jacquet-Langlands correspondences. Regarding the non-archimedean local theory we need some explicit functorial correspondence for spherical representations of the inner form and non-supercuspidal representations of GSp(2), which is studied in the appendix by Ralf Schmidt.
AB - As was first essentially pointed out by Tomoyoshi Ibukiyama, Hecke eigenforms on the indefinite symplectic group GSp(1, 1) or the definite symplectic group GSp∗(2) over ℚ right invariant by a (global) maximal open compact subgroup are conjectured to have the same spinor L-functions as those of paramodular new forms of some specified level on the symplectic group GSp(2) (or GSp(4)). This can be viewed as a generalization of the Jacquet-Langlands-Shimizu correspondence to the case of GSp(2) and its inner forms GSp(1,1) and GSp∗(2). In this paper we provide evidence of the conjecture on this explicit functorial correspondence with theta lifts: a theta lift from GL(2)×B× to GSp(1, 1) or GSp∗(2) and a theta lift from GL(2) × GL(2) (or GO(2, 2)) to GSp(2). Here B denotes a definite quaternion algebra over ℚ. Our explicit functorial correspondence given by these theta lifts are proved to be compatible with archimedean and non-archimedean local Jacquet-Langlands correspondences. Regarding the non-archimedean local theory we need some explicit functorial correspondence for spherical representations of the inner form and non-supercuspidal representations of GSp(2), which is studied in the appendix by Ralf Schmidt.
KW - Jacquet-Langlands correspondence
KW - Spinor L-functions
KW - Theta lifts
UR - http://www.scopus.com/inward/record.url?scp=85032787184&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85032787184&partnerID=8YFLogxK
U2 - 10.2969/jmsj/06941443
DO - 10.2969/jmsj/06941443
M3 - Article
AN - SCOPUS:85032787184
VL - 69
SP - 1443
EP - 1474
JO - Journal of the Mathematical Society of Japan
JF - Journal of the Mathematical Society of Japan
SN - 0025-5645
IS - 4
ER -