A Kloosterman sum in a relative trace formula for 𝐺𝐿₄

Ye, Yangbo

doi:10.1090/S1088-4165-98-00049-1

A Kloosterman sum in a relative trace formula for $GL_4$
HTML articles powered by AMS MathViewer

by Yangbo Ye

Represent. Theory 2 (1998), 370-392

DOI: https://doi.org/10.1090/S1088-4165-98-00049-1

Published electronically: September 16, 1998

PDF | Request permission

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

James Arthur and Laurent Clozel, Simple algebras, base change, and the advanced theory of the trace formula, Annals of Mathematics Studies, vol. 120, Princeton University Press, Princeton, NJ, 1989. MR 1007299, DOI 10.1515/9781400882403
David W. Boyd, Pisot sequences which satisfy no linear recurrence, Acta Arith. 32 (1977), no. 1, 89–98. MR 427241, DOI 10.4064/aa-32-1-89-98
W. Duke and H. Iwaniec, A relation between cubic exponential and Kloosterman sums, A tribute to Emil Grosswald: number theory and related analysis, Contemp. Math., vol. 143, Amer. Math. Soc., Providence, RI, 1993, pp. 255–258. MR 1210520, DOI 10.1090/conm/143/00999
Solomon Friedberg, Poincaré series for $\textrm {GL}(n)$: Fourier expansion, Kloosterman sums, and algebreo-geometric estimates, Math. Z. 196 (1987), no. 2, 165–188. MR 910824, DOI 10.1007/BF01163653
P. Gérardin and J.-P. Labesse, The solution of a base change problem for $\textrm {GL}(2)$ (following Langlands, Saito, Shintani), Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 115–133. MR 546613, DOI 10.1090/pspum/033.2/546613
G. Harder, R. P. Langlands, and M. Rapoport, Algebraische Zyklen auf Hilbert-Blumenthal-Flächen, J. Reine Angew. Math. 366 (1986), 53–120 (German). MR 833013, DOI 10.1515/crll.1986.366.53
Hervé Jacquet, The continuous spectrum of the relative trace formula for $\textrm {GL}(3)$ over a quadratic extension, Israel J. Math. 89 (1995), no. 1-3, 1–59. MR 1324453, DOI 10.1007/BF02808192
Hervé Jacquet and Yangbo Ye, Une remarque sur le changement de base quadratique, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 11, 671–676 (French, with English summary). MR 1081622
Hervé Jacquet and Yangbo Ye, Relative Kloosterman integrals for $\textrm {GL}(3)$, Bull. Soc. Math. France 120 (1992), no. 3, 263–295 (English, with English and French summaries). MR 1180831, DOI 10.24033/bsmf.2187
Hervé Jacquet and Yangbo Ye, Distinguished representations and quadratic base change for $\textrm {GL}(3)$, Trans. Amer. Math. Soc. 348 (1996), no. 3, 913–939. MR 1340178, DOI 10.1090/S0002-9947-96-01549-8
H. Jacquet and Y. Ye, Germs of Kloosterman integrals for $GL(3)$, Trans. Amer. Math. Soc., to appear.
Nicholas M. Katz, Gauss sums, Kloosterman sums, and monodromy groups, Annals of Mathematics Studies, vol. 116, Princeton University Press, Princeton, NJ, 1988. MR 955052, DOI 10.1515/9781400882120
Z. Mao and S. Rallis, A trace formula for dual pairs, Duke Math. J. 87 (1997), 321-341.
Glenn Stevens, Poincaré series on $\textrm {GL}(r)$ and Kloostermann sums, Math. Ann. 277 (1987), no. 1, 25–51. MR 884644, DOI 10.1007/BF01457276
J. Tate, Number theoretic background, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 3–26. MR 546607, DOI 10.1090/pspum/033.2/546607
Yangbo Ye, Kloosterman integrals and base change, Number theory and its applications in China, Contemp. Math., vol. 77, Amer. Math. Soc., Providence, RI, 1988, pp. 163–170. MR 973234, DOI 10.1090/conm/077/973234
Yangbo Ye, Kloosterman integrals and base change for $\textrm {GL}(2)$, J. Reine Angew. Math. 400 (1989), 57–121. MR 1013725, DOI 10.1515/crll.1989.400.57
Yangbo Ye, The fundamental lemma of a relative trace formula for $\textrm {GL}(3)$, Compositio Math. 89 (1993), no. 2, 121–162. MR 1255692
Y. Ye, Local orbital integrals of a relative trace formula, Chinese Sci. Bull. 38 (1993), 969-971.
Yangbo Ye, An integral transform and its applications, Math. Ann. 300 (1994), no. 3, 405–417. MR 1304430, DOI 10.1007/BF01450494
Yangbo Ye, The lifting of Kloosterman sums, J. Number Theory 51 (1995), no. 2, 275–287. MR 1326749, DOI 10.1006/jnth.1995.1047
Y. Ye, Exponential sums for $GL(n)$ and their applications to base change, J. Number Theory 68 (1998), 112-130.
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.
Don Zagier, Modular forms associated to real quadratic fields, Invent. Math. 30 (1975), no. 1, 1–46. MR 382174, DOI 10.1007/BF01389846