Distinguished representations and quadratic base change for 𝐺𝐿(3)

Jacquet, Herve; Ye, Yangbo

doi:10.1090/S0002-9947-96-01549-8

Distinguished representations and quadratic base change for $GL(3)$
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by Herve Jacquet and Yangbo Ye

Trans. Amer. Math. Soc. 348 (1996), 913-939

DOI: https://doi.org/10.1090/S0002-9947-96-01549-8

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Abstract:

Let $E/F$ be a quadratic extension of number fields. Suppose that every real place of $F$ splits in $E$ and let $H$ be the unitary group in 3 variables. Suppose that $\Pi$ is an automorphic cuspidal representation of $GL(3,E_{\mathbb {A}})$. We prove that there is a form $\phi$ in the space of $\Pi$ such that the integral of $\phi$ over $H(F)\setminus H(F_{\mathbb {A}})$ is non zero. Our proof is based on earlier results and the notion, discussed in this paper, of Shalika germs for certain Kloosterman integrals.

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
A. Ash and D. Ginzburg, $p$-adic $L$-functions for $GL(2n)$, preprint.
Joseph N. Bernstein, $P$-invariant distributions on $\textrm {GL}(N)$ and the classification of unitary representations of $\textrm {GL}(N)$ (non-Archimedean case), Lie group representations, II (College Park, Md., 1982/1983) Lecture Notes in Math., vol. 1041, Springer, Berlin, 1984, pp. 50–102. MR 748505, DOI 10.1007/BFb0073145
Yuval Z. Flicker, On distinguished representations, J. Reine Angew. Math. 418 (1991), 139–172. MR 1111204, DOI 10.1515/crll.1991.418.139
Yuval Z. Flicker, Distinguished representations and a Fourier summation formula, Bull. Soc. Math. France 120 (1992), no. 4, 413–465 (English, with English and French summaries). MR 1194271, DOI 10.24033/bsmf.2193
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
Solomon Friedberg and Hervé Jacquet, Linear periods, J. Reine Angew. Math. 443 (1993), 91–139. MR 1241129, DOI 10.1515/crll.1993.443.91
D. Goldfeld, Kloosterman zeta functions for $GL(n,\mathbb {Z})$, Proc. Internat. Congr. Math., vol. 1, Univ. of California Press, Berkeley, 1986, pp. 417–424.
Benedict H. Gross, Some applications of Gel′fand pairs to number theory, Bull. Amer. Math. Soc. (N.S.) 24 (1991), no. 2, 277–301. MR 1074028, DOI 10.1090/S0273-0979-1991-16017-9
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
H. Iwaniec, On Waldspurger’s Theorem, Acta Arith. 49 (1987), 205–212.
Hervé Jacquet, Sur un résultat de Waldspurger, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 2, 185–229 (French). MR 868299, DOI 10.24033/asens.1506
Hervé Jacquet, Sur un résultat de Waldspurger. II, Compositio Math. 63 (1987), no. 3, 315–389 (French). MR 909385
Hervé Jacquet, On the nonvanishing of some $L$-functions, Proc. Indian Acad. Sci. Math. Sci. 97 (1987), no. 1-3, 117–155 (1988). MR 983610, DOI 10.1007/BF02837819
G. A. Lomadze, Some arithmetic applications of the theory of modular forms, Trudy Mat. Inst. Steklov. 200 (1991), 236–244 (Russian); English transl., Proc. Steklov Inst. Math. 2(200) (1993), 261–270. MR 1143372
Hervé Jacquet, Relative Kloosterman integrals for $\textrm {GL}(3)$. II, Canad. J. Math. 44 (1992), no. 6, 1220–1240. MR 1192415, DOI 10.4153/CJM-1992-073-6
—, The continuous spectrum of the relative trace formula for $GL(3)$ over a quadratic extension, Israel J. Math. (to appear).
Hervé Jacquet and Stephen Rallis, Kloosterman integrals for skew symmetric matrices, Pacific J. Math. 154 (1992), no. 2, 265–283. MR 1159511, DOI 10.2140/pjm.1992.154.265
Hervé Jacquet and Stephen Rallis, Symplectic periods, J. Reine Angew. Math. 423 (1992), 175–197. MR 1142486
Hervé Jacquet, King F. Lai, and Stephen Rallis, A trace formula for symmetric spaces, Duke Math. J. 70 (1993), no. 2, 305–372. MR 1219816, DOI 10.1215/S0012-7094-93-07006-8
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
Zhengyu Mao, Relative Kloosterman integrals for the unitary group, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 4, 381–386 (English, with English and French summaries). MR 1179042
Zhengyu Mao, Sur les sommes de Salié relatives, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), no. 12, 1257–1262 (French, with English and French summaries). MR 1226111
Hidegorô Nakano, Über Abelsche Ringe von Projektionsoperatoren, Proc. Phys.-Math. Soc. Japan (3) 21 (1939), 357–375 (German). MR 94
T. A. Springer, Some results on algebraic groups with involutions, Algebraic groups and related topics (Kyoto/Nagoya, 1983) Adv. Stud. Pure Math., vol. 6, North-Holland, Amsterdam, 1985, pp. 525–543. MR 803346, DOI 10.2969/aspm/00610525
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
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
—, Orbital integrals of a relative trace formula for $GL(3)$, 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
Don Zagier, Modular forms associated to real quadratic fields, Invent. Math. 30 (1975), no. 1, 1–46. MR 382174, DOI 10.1007/BF01389846