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A Kloosterman sum in a relative trace formula for $GL_4$

Author(s): Yangbo Ye
Journal: Represent. Theory 2 (1998), 370-392.
MSC (1991): Primary 11L05; Secondary 11F70, 22E55
Posted: September 16, 1998
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Abstract: We study a Kloosterman sum for $GL_4$ and prove that it is equal to an exponential sum over a quadratic number field. This identity has applications in a relative trace formula for $GL_4$ which might be used to give a new proof of quadratic base change and characterize its image.

References:

1.
J. Arthur and L. Clozel, Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, Ann. Math. Studies, no. 120, Princeton Univ. Press, Princeton, 1989. MR 90m:22041

2.
T. Asai, On certain Dirichlet series associated with Hilbert modular forms and Rankin's method, Math. Ann. 226 (1977), 81-94. MR 55:276

3.
W. Duke and H. Iwaniec, A relation between cubic exponential and Kloosterman sums, Contemporary Math. 143 (1993), 255-258. MR 93m:11082

4.
S. Friedberg, Poincaré series for $GL(n)$: Fourier expansion, Kloosterman sums, and algebreo-geometric estimates, Math. Zeit. 196 (1987), 165-188. MR 88m:11032

5.
P. Gérardin and J.P. Labesse, The solution of a base change problem for $GL(2)$ (following Langlands, Saito, Shintani), Proc. Symp. Pure Math. 33 (1979), part 2, 115-133. MR 82e:10047

6.
G. Harder, R. Langlands and M. Rapoport, Algebraische Zyklen auf Hilbert-Blumenthal-Flächen, J. Reine Angew. Math. 366 (1986), 53-120. MR 87k:11066

7.
H. Jacquet, The continuous spectrum of the relative trace formula for $GL(3)$ over a quadratic extension, Israel J. Math. 89 (1995), 1-59. MR 96a:22029

8.
H. Jacquet and Y. Ye, Une remargue sur le changement de base quadratique, C.R. Acad. Sci. Paris, 311 Série I, (1990), 671-676. MR 92j:11046

9.
H. Jacquet and Y. Ye, Relative Kloosterman integrals for $GL(3)$, Bull. Soc. Math. France, 120 (1992), 263-295. MR 94c:11047

10.
H. Jacquet and Y. Ye, Distinguished representations and quadratic base change for $GL(3)$, Trans. Amer. Math. Soc., 348 (1996), 913-939. MR 96h:11041

11.
H. Jacquet and Y. Ye, Germs of Kloosterman integrals for $GL(3)$, Trans. Amer. Math. Soc., to appear. CMP 97:11

12.
N.M. Katz, Gauss Sums, Kloosterman Sums, and Monodromy Groups, Ann. Math. Studies, no. 116, Princeton Univ. Press, Princeton, 1988. MR 91a:11028

13.
Z. Mao and S. Rallis, A trace formula for dual pairs, Duke Math. J. 87 (1997), 321-341. CMP 97:11

14.
G. Stevens, Poincaré series on $GL(r)$ and Kloosterman sums, Math. Ann. 277 (1987), 25-51. MR 88m:11031

15.
J. Tate, Number theoretic background, Proc. Symp. Pure Math. 33 (1979), part 2, 3-26. MR 80m:12009

16.
Y. Ye, Kloosterman integrals and base change. In: Number Theory and its Applications in China, Contemporary Math. 77 (1988), 163-170. MR 90b:11058

17.
Y. Ye, Kloosterman integrals and base change for $GL(2)$, J. Reine Angew. Math. 400 (1989), 57-121. MR 90i:11134

18.
Y. Ye, The fundamental lemma of a relative trace formula for $GL(3)$, Comp. Math. 89 (1993), 121-162. MR 95b:22023

19.
Y. Ye, Local orbital integrals of a relative trace formula, Chinese Sci. Bull. 38 (1993), 969-971.

20.
Y. Ye, An integral transform and its applications, Math. Ann. 300 (1994), 405-417. MR 95j:11045

21.
Y. Ye, The lifting of Kloosterman sums, J. Number Theory 51 (1995), 275-287. MR 97a:11126

22.
Y. Ye, Exponential sums for $GL(n)$ and their applications to base change, J. Number Theory 68 (1998), 112-130. CMP 98:07

23.
Y. Ye, The lifting of an exponential sum to a cyclic algebraic number field of a prime degree, Trans. Amer. Math. Soc. 350 (1998), 5003-5015. CMP 97:08

24.
D. Zagier, Modular forms associated to real quadratic fields, Invent. Math. 30 (1975), 1-46. MR 52:3062

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Additional Information:

Yangbo Ye
Affiliation: Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242-1419
Email: yey@math.uiowa.edu

DOI: 10.1090/S1088-4165-98-00049-1
PII: S 1088-4165(98)00049-1
Received by editor(s): April 9, 1997
Received by editor(s) in revised form: August 27, 1998
Posted: September 16, 1998
Copyright of article: Copyright 1998, American Mathematical Society

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