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

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- 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
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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

- 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

## Bibliographic Information

**Yangbo Ye**- Affiliation: Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242-1419
- MR Author ID: 261621
- Email: yey@math.uiowa.edu
- Received by editor(s): April 9, 1997
- Received by editor(s) in revised form: August 27, 1998
- Published electronically: September 16, 1998
- © Copyright 1998 American Mathematical Society
- Journal: Represent. Theory
**2**(1998), 370-392 - MSC (1991): Primary 11L05; Secondary 11F70, 22E55
- DOI: https://doi.org/10.1090/S1088-4165-98-00049-1
- MathSciNet review: 1641835