On Euler products associated with noncuspidal metaplectic forms
Authors:
Shamita Dutta Gupta and Xiaotie She
Journal:
Proc. Amer. Math. Soc. 128 (2000), 22032212
MSC (1991):
Primary 11F55; Secondary 11F30
Published electronically:
February 25, 2000
MathSciNet review:
1707145
Fulltext PDF Free Access
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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 RankinSelberg 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
(88d:11044), http://dx.doi.org/10.1215/S0012709486053512
 [2]
Daniel
Bump and Jeffrey
Hoffstein, On Shimura’s correspondence, Duke Math. J.
55 (1987), no. 3, 661–691. MR 904946
(89c:11072), http://dx.doi.org/10.1215/S0012709487055335
 [3]
Dutta Gupta S. The RankinSelberg method on congruence subgroups, Illinois J. Math. (to appear).
 [4]
Shamita
Dutta Gupta and Xiaotie
She, The 𝐺𝐿(2) RankinSelberg method for functions
not of rapid decay, J. Number Theory 71 (1998),
no. 2, 159–165. MR 1633793
(99h:11049), http://dx.doi.org/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
(81m:10057), http://dx.doi.org/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
(88d:11049)
 [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
(87c:11043)
 [8]
Solomon
Friedberg and ShekTung
Wong, On the Shimura correspondence for
𝐺𝑆𝑝(4), Math. Ann. 290
(1991), no. 1, 183–207. MR 1107667
(92f:11063), http://dx.doi.org/10.1007/BF01459242
 [9]
S.
Gelbart and I.
PiatetskiShapiro, On Shimura’s correspondence for modular
forms of halfintegral 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
(83b:10032)
 [10]
Thomas
Goetze, Euler products associated to
metaplectic automorphic forms on the 3fold cover of
𝐺𝑆𝑝(4), Trans. Amer.
Math. Soc. 350 (1998), no. 3, 975–1011. MR 1401521
(98e:11061), http://dx.doi.org/10.1090/S0002994798018170
 [11]
Jeffrey
Hoffstein, Theta functions on the 𝑛fold metaplectic cover
of 𝑆𝐿(2)—the function field case, Invent. Math.
107 (1992), no. 1, 61–86. MR 1135464
(92k:11049), http://dx.doi.org/10.1007/BF01231881
 [12]
Tomio
Kubota, Ein arithmetischer Satz über eine Matrizengruppe,
J. Reine Angew. Math. 222 (1966), 55–57 (German). MR 0188194
(32 #5633)
 [13]
Daniel
B. Lieman, The 𝐺𝐿(3) RankinSelberg convolution for
functions not of rapid decay, Duke Math. J. 69
(1993), no. 1, 219–242. MR 1201699
(94a:11070), http://dx.doi.org/10.1215/S0012709493069116
 [14]
Shinji
Niwa, Modular forms of half integral weight and the integral of
certain thetafunctions, Nagoya Math. J. 56 (1975),
147–161. MR 0364106
(51 #361)
 [15]
S.
J. Patterson, A cubic analogue of the theta series, J. Reine
Angew. Math. 296 (1977), 125–161. MR 0563068
(58 #27795a)
 [16]
Shimura G. On modular forms of half integral weight, Annals of Math. 97 440481 (1973).
 [17]
Takuro
Shintani, On construction of holomorphic cusp forms of half
integral weight, Nagoya Math. J. 58 (1975),
83–126. MR
0389772 (52 #10603)
 [18]
Don
Zagier, The RankinSelberg 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
(83k:10056)
 [1]
 Bump D. and Hoffstein J. Some Euler products associated with cubic metaplectic forms on GL(3), Duke. Math. J. 53 10471072 (1986). MR 88d:11044
 [2]
 Bump D. and Hoffstein J. On Shimura's correspondence, Duke. Math. J. 55 661691 (1987). MR 89c:11072
 [3]
 Dutta Gupta S. The RankinSelberg method on congruence subgroups, Illinois J. Math. (to appear).
 [4]
 Dutta Gupta S. and She X. The GL(2) RankinSelberg method for functions not of rapid decay, J. Num. Th. 71 159165 (1998). MR 99h:11049
 [5]
 Flicker Y. Automorphic forms on covering groups of GL(2), Inv. Math. 57 119182 (1980). MR 81m:10057
 [6]
 Flicker Y. and Kazhdan D. Metaplectic Correspondence, Publ. Math., IHES 64 53110 (1986). MR 88d:11049
 [7]
 Friedberg S. Theta series correspondence and modular forms for number fields, In: Modular Forms. Chichester: Horwood, 7586 (1984). MR 87c:11043
 [8]
 Friedberg S. and Wong S. On the Shimura correspondence for GSp(4), Math. Ann. 290 183207 (1991). MR 92f:11063
 [9]
 Gelbart S. and PiatetskiShapiro I. On Shimura's correspondence for modular forms of halfintegral weight, Automorphic forms, Representation Theory and Arithmetic, Bombay, (1979). MR 83b:10032
 [10]
 Goetze T. Euler products associated to metaplectic automorphic forms on the 3fold cover of GSp(4), Trans. Amer. Math. Soc. 350 No. 3 9751011 (1998). MR 98e:11061
 [11]
 Hoffstein J. Theta Functions on the nfold metaplectic cover of GL(2), Invent. Math. 107 6186 (1992). MR 92k:11049
 [12]
 Kubota T. Ein arithmetischer Satz uber eine Matrizengruppe, J. Reine Angew. Math. 222 5557 (1966). MR 32:5633
 [13]
 Lieman D. The GL(3) RankinSelberg convolution for functions not of rapid decay, Duke Math. J. 69 219242 (1993). MR 94a:11070
 [14]
 Niwa S. Modular forms of half integral weight and the integral of certain thetafunctions, Nagoya Math. J. 56 147161 (1974). MR 51:361
 [15]
 Patterson S. A cubic analogue of the theta series.I, J. Reine Angew. Math. 296 125161 (1977). MR 58:27795a
 [16]
 Shimura G. On modular forms of half integral weight, Annals of Math. 97 440481 (1973).
 [17]
 Shintani T. On construction of holomorphic cusp forms of half integral weight, Nagoya Math. J. 58 83126 (1975). MR 52:10603
 [18]
 Zagier D. The RankinSelberg method for automorphic functions which are not of rapid decay, J. Fac. Sci. Univ of Tokyo, Sect. IA. 28 415437 (1981). MR 83k:10056
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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/S0002993900055933
PII:
S 00029939(00)055933
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
