Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Spinor $ L$-functions for generic cusp forms on $ GSp(2)$ belonging to principal series representations

Authors: Taku Ishii and Tomonori Moriyama
Journal: Trans. Amer. Math. Soc. 360 (2008), 5683-5709
MSC (2000): Primary 11F70; Secondary 11F41, 11F46.
Published electronically: June 19, 2008
MathSciNet review: 2425688
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \mathsf{G}=GSp(2)$ be the symplectic group with similitude of degree two, which is defined over $ \mathbf{Q}$. For a generic cusp form $ F$ on the adelized group $ \mathsf{G}_{\mathbf{A}}$ whose archimedean type is a principal series representation, we show that its spinor $ L$-function is continued to an entire function and satisfies the functional equation.

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

  • [An] ANDRIANOV, A. N., Dirichlet series with Euler product in the theory of Siegel modular forms of genus two, Trudy Mat. Inst. Steklov. 112 (1971), 73-94. MR 0340178 (49:4934)
  • [A-S] ASGARI, M. AND SHAHIDI, F., Generic transfer from $ GSp(4)$ to $ GL(4)$, Compos. Math. 142 (2006), 541-550. MR 2231191 (2007d:11055b)
  • [Bo] BOREL, A., Automorphic $ L$-functions, Proc. Sympos. Pure Math. 33, Part 2 (1979), 27-61. MR 546608 (81m:10056)
  • [Bu] BUMP, D., The Rankin-Selberg method: A survey. In: Number theory, trace formulas and discrete groups, Academic Press (1989), 49-109. MR 993311 (90m:11079)
  • [Er] ERDELYI, A., MAGNUS, W., OBERHETTINGER, F., and TRICOMI, F. G., Tables of integral transforms I, II, McGraw-Hill, (1954). MR 0061695 (15:868a); MR 0065685 (16:468c)
  • [G-W] GOODMAN, R, AND WALLACH, N., Whittaker vectors and conical vectors. J. Funct. Anal. 39 (1980), 199-279. MR 597811 (82i:22018)
  • [HC] HARISH-CHANDRA, Automorphic forms on semisimple Lie groups, Lecture Notes in Math. 62 (1968), Springer. MR 0232893 (38:1216)
  • [H-M] HOFFSTEIN, J. AND MURTY, M. R., $ L$-series of automorphic forms on $ {\rm GL}(3,R)$, In: Théorie des nombres (Quebec, PQ, 1987), de Gruyter, Berlin, (1989), 398-408. MR 1024578 (90m:11075)
  • [Ho] HORI, A., Andrianov's $ L$-functions associated to Siegel wave forms of degree two, Math. Ann. 303 (1995), 195-226. MR 1348797 (96g:11045)
  • [I] ISHII, T., On principal series Whittaker functions on $ Sp(2,\mathbf{R})$, J. Funct. Anal. 225 (2005), 1-32. MR 2149916 (2007k:22011)
  • [J] JACQUET, H., Fonctions de Whittaker associées aux groupes de Chevalley, Bull. Soc. Math. France 95 (1967), 243-309. MR 0271275 (42:6158)
  • [Kn] KNAPP, A. W., Representation theory of semisimple groups. An overview based on examples. Princeton Mathematical Series, 36. Princeton University Press (1986). MR 855239 (87j:22022)
  • [K-S] KOHNEN, W. AND SKORUPPA, N. P., A certain Dirichlet series attached to Siegel modular forms of degree two, Invent. Math. 95(1989), 541-558. MR 979364 (90b:11050)
  • [Ko] KOSTANT, B., On Whittaker vectors and representation theory, Invent. Math. 48 (1978), 101-184. MR 507800 (80b:22020)
  • [Mi] MIYAZAKI, T., The generalized Whittaker functions for $ Sp(2,\mathbf{R})$ and the gamma factor of the Andrianov $ L$-functions, J. Math. Sci. Univ. Tokyo. 7 (2000), 241-295. MR 1768466 (2001f:22047)
  • [Mi-O1] MIYAZAKI, T. AND ODA, T., Principal series Whittaker functions on $ Sp(2;\mathbf{R})$. Explicit formulae of differential equations, Automorphic forms and related topics (Seoul, 1993), 59-92, Pyungsan Inst. Math. Sci., Seoul, 1997. MR 1342321 (96e:22032)
  • [Mi-O2] MIYAZAKI, T. AND ODA, T., Principal series Whittaker functions on $ Sp(2,\mathbf{R})$ II, Tôhoku Math. J. 50 (1998), 243-260. MR 1622070 (99d:22027)
  • [Mo1] MORIYAMA, T., Entireness of the spinor $ L$-functions for certain generic cusp forms on $ GSp(2)$, Amer. J. Math 126 (2004), 899-920. MR 2075487 (2005d:11076)
  • [Mo2] MORIYAMA, T., Bessel functions on $ GSp(2, \mathbf{R})$ and Fourier expansion of automorphic forms on $ GSp(2)$, in preparation.
  • [M-S] MURASE, A. AND SUGANO, T., Shintani function and its application to automorphic $ L$-functions for classical groups I, Math. Ann 299 (1994), 17-56. MR 1273075 (96c:11054)
  • [Ni] NIWA, S., Commutation relations of differential operators and Whittaker functions on $ Sp_2(\mathbf{R})$, Proc. Japan Acad. Ser. A Math. Sci. 71, (1995), 189-191. MR 1362994 (96m:11042)
  • [No] NOVODVORSKY, M. E., Automorphic $ L$-functions for symplectic group $ GSp(4)$, Proc. Sympos. Pure Math. 33, Part 2 (1979), 87-95. MR 546610 (81c:10032)
  • [PS] PIATETSKI-SHAPIRO, I. I., $ L$-functions for $ {\rm GSp}\sb 4$. Olga Taussky-Todd: in memoriam. Pacific J. Math. (1997), Special Issue, 259-275. MR 1610879 (99a:11058)
  • [TB1] TAKLOO-BIGHASH, R., $ L$-functions for the $ p$-adic group $ {\rm GSp}(4)$, Amer. J. Math. 122 (2000), 1085-1120. MR 1797657 (2001k:11090)
  • [TB2] TAKLOO-BIGHASH, R., Spinor $ L$-functions, theta correspondence, and Bessel coefficients, Forum Math. 19 (2007), no. 3, 487-554. MR 2328118 (2008g:11081)
  • [V] VOGAN, D., Gel$ '$fand-Kirillov dimension for Harish-Chandra modules, Invent. Math. 48 (1978), 75-98. MR 0506503 (58:22205)
  • [Wa] WALLACH, N., Asymptotic expansions of generalized matrix entries of representations of real reductive groups, Lecture Notes in Mathematics 1024, Springer-Verlag (1983), 287-369. MR 727854 (85g:22029)
  • [W-W] WHITTAKER, E. T. AND WATSON, G. N., A course of modern analysis, Reprint of the fourth (1927) edition, Cambridge University Press, (1996). MR 1424469 (97k:01072)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 11F70, 11F41, 11F46.

Retrieve articles in all journals with MSC (2000): 11F70, 11F41, 11F46.

Additional Information

Taku Ishii
Affiliation: Department of Mathematics, Chiba Institute of Technology, 2-1-1 Shibazono, Narashino, Chiba, 275-0023, Japan

Tomonori Moriyama
Affiliation: Department of Mathematics, Sophia University, 7-1 Kioi-cho, Chiyoda-ku, Tokyo, 102-8554 Japan
Address at time of publication: Department of Mathematics, Graduate School of Science, Osaka University, Machikaneyama-cho 1-1, Toyonaka, Osaka, 560-0043, Japan

Keywords: Spinor $L$-functions, Novodvorsky's zeta integrals, Whittaker functions, principal series
Received by editor(s): June 15, 2005
Published electronically: June 19, 2008
Additional Notes: The first author was supported by JSPS Research Fellowships for Young Scientists.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society