On Euler products associated with noncuspidal metaplectic forms

Authors:
Shamita Dutta Gupta and Xiaotie She

Journal:
Proc. Amer. Math. Soc. **128** (2000), 2203-2212

MSC (1991):
Primary 11F55; Secondary 11F30

Published electronically:
February 25, 2000

MathSciNet review:
1707145

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we obtain an Euler product with functional equation associated to a noncuspidal metaplectic form on the double cover of . Zagier's idea of Rankin-Selberg method is used to define the convolution of and the -function.

**[1]**Daniel Bump and Jeffrey Hoffstein,*Some Euler products associated with cubic metaplectic forms on 𝐺𝐿(3)*, Duke Math. J.**53**(1986), no. 4, 1047–1072. MR**874680**, 10.1215/S0012-7094-86-05351-2**[2]**Daniel Bump and Jeffrey Hoffstein,*On Shimura’s correspondence*, Duke Math. J.**55**(1987), no. 3, 661–691. MR**904946**, 10.1215/S0012-7094-87-05533-5**[3]**Dutta Gupta S.*The Rankin-Selberg method on congruence subgroups*, Illinois J. Math. (to appear).**[4]**Shamita Dutta Gupta and Xiaotie She,*The 𝐺𝐿(2) Rankin-Selberg method for functions not of rapid decay*, J. Number Theory**71**(1998), no. 2, 159–165. MR**1633793**, 10.1006/jnth.1998.2249**[5]**Yuval Z. Flicker,*Automorphic forms on covering groups of 𝐺𝐿(2)*, Invent. Math.**57**(1980), no. 2, 119–182. MR**567194**, 10.1007/BF01390092**[6]**Yuval Z. Flicker and David A. Kazhdan,*Metaplectic correspondence*, Inst. Hautes Études Sci. Publ. Math.**64**(1986), 53–110. MR**876160****[7]**Solomon Friedberg,*Theta series correspondences and modular forms for number fields*, Modular forms (Durham, 1983) Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, 1984, pp. 75–86. MR**803363****[8]**Solomon Friedberg and Shek-Tung Wong,*On the Shimura correspondence for 𝐺𝑆𝑝(4)*, Math. Ann.**290**(1991), no. 1, 183–207. MR**1107667**, 10.1007/BF01459242**[9]**S. Gelbart and I. Piatetski-Shapiro,*On Shimura’s correspondence for modular forms of half-integral weight*, Automorphic forms, representation theory and arithmetic (Bombay, 1979), Tata Inst. Fund. Res. Studies in Math., vol. 10, Tata Inst. Fundamental Res., Bombay, 1981, pp. 1–39. MR**633657****[10]**Thomas Goetze,*Euler products associated to metaplectic automorphic forms on the 3-fold cover of 𝐺𝑆𝑝(4)*, Trans. Amer. Math. Soc.**350**(1998), no. 3, 975–1011. MR**1401521**, 10.1090/S0002-9947-98-01817-0**[11]**Jeffrey Hoffstein,*Theta functions on the 𝑛-fold metaplectic cover of 𝑆𝐿(2)—the function field case*, Invent. Math.**107**(1992), no. 1, 61–86. MR**1135464**, 10.1007/BF01231881**[12]**Tomio Kubota,*Ein arithmetischer Satz über eine Matrizengruppe*, J. Reine Angew. Math.**222**(1966), 55–57 (German). MR**0188194****[13]**Daniel B. Lieman,*The 𝐺𝐿(3) Rankin-Selberg convolution for functions not of rapid decay*, Duke Math. J.**69**(1993), no. 1, 219–242. MR**1201699**, 10.1215/S0012-7094-93-06911-6**[14]**Shinji Niwa,*Modular forms of half integral weight and the integral of certain theta-functions*, Nagoya Math. J.**56**(1975), 147–161. MR**0364106****[15]**S. J. Patterson,*A cubic analogue of the theta series*, J. Reine Angew. Math.**296**(1977), 125–161. MR**0563068****[16]**Shimura G.*On modular forms of half integral weight*, Annals of Math.**97**440-481 (1973).**[17]**Takuro Shintani,*On construction of holomorphic cusp forms of half integral weight*, Nagoya Math. J.**58**(1975), 83–126. MR**0389772****[18]**Don Zagier,*The Rankin-Selberg method for automorphic functions which are not of rapid decay*, J. Fac. Sci. Univ. Tokyo Sect. IA Math.**28**(1981), no. 3, 415–437 (1982). MR**656029**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
11F55,
11F30

Retrieve articles in all journals with MSC (1991): 11F55, 11F30

Additional Information

**Shamita Dutta Gupta**

Affiliation:
Department of Mathematics, Florida International University, Miami, Florida 33199

Email:
duttagus@fiu.edu

**Xiaotie She**

Affiliation:
Financial Data Planning Corporation, 2140 S. Dixie Highway, Miami, Florida 33133

Email:
xiaoties@fdpcorp.com

DOI:
http://dx.doi.org/10.1090/S0002-9939-00-05593-3

Keywords:
Eisenstein series,
automorphic forms,
Fourier expansions,
Euler products,
functional equations

Received by editor(s):
September 8, 1998

Published electronically:
February 25, 2000

Communicated by:
Dennis A. Hejhal

Article copyright:
© Copyright 2000
American Mathematical Society