Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

Shalika periods on $ \mathrm{GU}(2,2)$


Authors: Masaaki Furusawa and Kazuki Morimoto
Journal: Proc. Amer. Math. Soc. 141 (2013), 4125-4137
MSC (2010): Primary 11F67; Secondary 11F66
DOI: https://doi.org/10.1090/S0002-9939-2013-11690-4
Published electronically: August 15, 2013
MathSciNet review: 3105856
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider a certain Rankin-Selberg integral on a quasi-split similitude unitary group $ \mathrm {GU}\left (2,2\right )$, which is an analogue of Jacquet-Shalika's integral for the exterior square $ L$-function for $ \mathrm {GL}(2n)$ when $ n=2$. It indeed represents the twisted exterior square $ L$-function, and we study the relationship between the existence of a pole at $ s=1$ and the non-vanishing of a unitary analogue of the Shalika period.


References [Enhancements On Off] (What's this?)

  • [1] Daniel Bump, The Rankin-Selberg method: a survey, Number theory, trace formulas and discrete groups (Oslo, 1987) Academic Press, Boston, MA, 1989, pp. 49-109. MR 993311 (90m:11079)
  • [2] W. Casselman and J. Shalika, The unramified principal series of $ p$-adic groups. II. The Whittaker function, Compositio Math. 41 (1980), no. 2, 207-231. MR 581582 (83i:22027)
  • [3] Wee Teck Gan and Joseph Hundley, The spin $ L$-function of quasi-split $ D_4$, IMRP Int. Math. Res. Pap. (2006), Art. ID 68213, 74 pp. MR 2268487 (2008k:11053)
  • [4] Wee Teck Gan and Shuichiro Takeda, On Shalika periods and a theorem of Jacquet-Martin, Amer. J. Math. 132 (2010), no. 2, 475-528. MR 2654780 (2011e:11091), https://doi.org/10.1353/ajm.0.0109
  • [5] Hervé Jacquet, Automorphic forms on $ {\rm GL}(2)$. Part II, Lecture Notes in Mathematics, Vol. 278, Springer-Verlag, Berlin, 1972. MR 0562503 (58 #27778)
  • [6] H. Jacquet and R. P. Langlands, Automorphic forms on $ {\rm GL}(2)$, Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin, 1970. MR 0401654 (53 #5481)
  • [7] Hervé Jacquet and Kimball Martin, Shalika periods on $ {\rm GL}_2(D)$ and $ {\rm GL}_4$, Pacific J. Math. 233 (2007), no. 2, 341-370. MR 2366380 (2008j:11060), https://doi.org/10.2140/pjm.2007.233.341
  • [8] Hervé Jacquet and Joseph Shalika, Exterior square $ L$-functions, Automorphic forms, Shimura varieties, and $ L$-functions, Vol.II (Ann Arbor, MI, 1988), Perspect. Math., vol. 11, Academic Press, Boston, MA, 1990, pp. 143-226. MR 1044830 (91g:11050)
  • [9] Dihua Jiang, Chufeng Nien, and Yujun Qin, Local Shalika models and functoriality, Manuscripta Math. 127 (2008), no. 2, 187-217. MR 2442895 (2010b:11057), https://doi.org/10.1007/s00229-008-0200-0
  • [10] Henry H. Kim and Muthukrishnan Krishnamurthy, Twisted exterior square lift from $ {\rm GU}(2,2)_{E/F}$ to $ {\rm GL}_6/F$, J. Ramanujan Math. Soc. 23 (2008), no. 4, 381-412. MR 2492574 (2010h:22027)
  • [11] Brooks Roberts and Ralf Schmidt, Local newforms for $ {\rm GSp}(4)$, Lecture Notes in Mathematics, vol. 1918, Springer, Berlin, 2007. MR 2344630 (2008g:11080)
  • [12] Jonathan D. Rogawski, Analytic expression for the number of points mod $ p$, The zeta functions of Picard modular surfaces, Univ. Montréal, Montreal, QC, 1992, pp. 65-109. MR 1155227 (93i:11140)
  • [13] Freydoon Shahidi, On the Ramanujan conjecture and finiteness of poles for certain $ L$-functions, Ann. of Math. (2) 127 (1988), no. 3, 547-584. MR 942520 (89h:11021), https://doi.org/10.2307/2007005
  • [14] Takashi Sugano, On the $ L$-functions associated with Hermitian modular forms of genus $ 2$. Bull. Fac. Educ. Mie Univ. 42 (Natur. Sci.), 1-28 (1991).
  • [15] Boaz Tamir, On $ L$-functions and intertwining operators for unitary groups, Israel J. Math. 73 (1991), no. 2, 161-188. MR 1135210 (92j:11047), https://doi.org/10.1007/BF02772947

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11F67, 11F66

Retrieve articles in all journals with MSC (2010): 11F67, 11F66


Additional Information

Masaaki Furusawa
Affiliation: Department of Mathematics, Graduate School of Science, Osaka City University, Sugimoto 3–3–138, Sumiyoshi-ku, Osaka 558-8585, Japan
Email: furusawa@sci.osaka-cu.ac.jp

Kazuki Morimoto
Affiliation: Department of Mathematics, Graduate School of Science, Osaka City University, Sugimoto 3–3–138, Sumiyoshi-ku, Osaka 558-8585, Japan
Email: kazukimorimo@gmail.com

DOI: https://doi.org/10.1090/S0002-9939-2013-11690-4
Received by editor(s): September 26, 2011
Received by editor(s) in revised form: February 7, 2012
Published electronically: August 15, 2013
Additional Notes: The research of the first author was supported in part by JSPS Grant-in-Aid for Scientific Research (C) 22540029
The research of the second author was supported in part by Grant-in-Aid for JSPS Fellows (23-6883) and JSPS Institutional Program for Young Researcher Overseas Visits project: Promoting international young researchers in mathematics and mathematical sciences led by OCAMI
Communicated by: Kathrin Bringmann
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society