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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

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Periods of automorphic forms
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by Hervé Jacquet, Erez Lapid and Jonathan Rogawski
J. Amer. Math. Soc. 12 (1999), 173-240


Let $E/F$ be a quadratic extension of number fields and $G= \operatorname {Res}_{E/F}H$, where $H$ is a reductive group over $F$. We define the integral (in general, non-convergent) of an automorphic form on $G$ over $H(F)\backslash H(\mathbb A)^1$ via regularization. This regularized integral is used to derive a formula for the integral over $H(F)\backslash H(\mathbb A)^1$ of a truncated Eisenstein series on $G$. More explicit results are obtained in the case $H=GL(n)$. These results will find applications in the expansion of the spectral side of the relative trace formula.
  • James G. Arthur, A trace formula for reductive groups. I. Terms associated to classes in $G(\textbf {Q})$, Duke Math. J. 45 (1978), no. 4, 911–952. MR 518111
  • James Arthur, A trace formula for reductive groups. II. Applications of a truncation operator, Compositio Math. 40 (1980), no. 1, 87–121. MR 558260
  • James Arthur, The trace formula in invariant form, Ann. of Math. (2) 114 (1981), no. 1, 1–74. MR 625344, DOI 10.2307/1971376
  • James Arthur, On the inner product of truncated Eisenstein series, Duke Math. J. 49 (1982), no. 1, 35–70. MR 650368
  • Tetsuya Asai, On certain Dirichlet series associated with Hilbert modular forms and Rankin’s method, Math. Ann. 226 (1977), no. 1, 81–94. MR 429751, DOI 10.1007/BF01391220
  • I. N. Bernstein and A. V. Zelevinsky, Induced representations of reductive ${\mathfrak {p}}$-adic groups. I, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 4, 441–472. MR 579172, DOI 10.24033/asens.1333
  • D. Blasius and J. Rogawski, Cohomology of congruence subgroups of $SU(2,1)^{p}$ and Hodge cycles on some special complex hyperbolic surfaces, to appear in: Proc. of conference on Regulators, Jerusalem 1996 (Ed. A. Resnikov and N. Schappacher).
  • W. Casselman, Extended automorphic forms on the upper half plane, Math. Ann. 296 (1993), no. 4, 755–762. MR 1233497, DOI 10.1007/BF01445135
  • Yuval Z. Flicker, Twisted tensors and Euler products, Bull. Soc. Math. France 116 (1988), no. 3, 295–313 (English, with French summary). MR 984899, DOI 10.24033/bsmf.2099
  • 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
  • Y. Flicker, Cyclic automorphic forms on a unitary group, J. Math. Kyoto Univ. 37 (1997), 367–439.
  • Yuval Z. Flicker and Jeffrey L. Hakim, Quaternionic distinguished representations, Amer. J. Math. 116 (1994), no. 3, 683–736. MR 1277452, DOI 10.2307/2374997
  • Solomon Friedberg and Hervé Jacquet, Linear periods, J. Reine Angew. Math. 443 (1993), 91–139. MR 1241129, DOI 10.1515/crll.1993.443.91
  • S. Friedberg, S. Gelbart, H. Jacquet, and J. Rogawski, On genericity for $U(3)$ $L$-packets, pre-print.
  • S. Gelbart, H. Jacquet, and J. Rogawski, The relative trace formula and genericity of L-packets for $U(3).$ In preparation.
  • S. Gelbart, D. Soudry, and J. Rogawski, Endoscopy, theta-liftings, and period integrals for the unitary group in three variables$.$ Annals of Math. 145 (1997), 1–58.
  • Jiandong Guo, On the positivity of the central critical values of automorphic $L$-functions for $\textrm {GL}(2)$, Duke Math. J. 83 (1996), no. 1, 157–190. MR 1388847, DOI 10.1215/S0012-7094-96-08307-6
  • N. Gurevitch, Unpublished notes.
  • 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
  • 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, 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
  • H. Jacquet, Automorphic spectrum of symmetric spaces, In: Representation Theory and Automorphic Forms, (Eds. T. N. Bailey and A. W. Knapp) Proc. Symposia in Pure Math. 61, AMS, Providence, 1997.
  • 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
  • H. Jacquet and K. F. Lai, A relative trace formula, Compositio Math. 54 (1985), no. 2, 243–310. MR 783512
  • Hervé Jacquet and Stephen Rallis, Symplectic periods, J. Reine Angew. Math. 423 (1992), 175–197. MR 1142486
  • H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic representations. I, Amer. J. Math. 103 (1981), no. 3, 499–558. MR 618323, DOI 10.2307/2374103
  • 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
  • D. Jiang, Periods of residual representations of $G_{2},$, pre-print.
  • K. F. Lai, Algebraic cycles on compact Shimura surface, Math. Z. 189 (1985), no. 4, 593–602. MR 786286, DOI 10.1007/BF01168162
  • K. F. Lai, On Arthur’s class expansion of the relative trace formula, Duke Math. J. 64 (1991), no. 1, 111–117. MR 1131395, DOI 10.1215/S0012-7094-91-06405-7
  • R. P. Langlands, Eisenstein series, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 235–252. MR 0249539
  • E. Lapid, Thesis, Weizmann Institute of Science, 1998.
  • Jason Levy, A note on the relative trace formula, Canad. Math. Bull. 38 (1995), no. 4, 450–461. MR 1360595, DOI 10.4153/CMB-1995-066-x
  • Gérard Laumon and Michael Rapoport, The Langlands lemma and the Betti numbers of stacks of $G$-bundles on a curve, Internat. J. Math. 7 (1996), no. 1, 29–45. MR 1369904, DOI 10.1142/S0129167X96000049
  • 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
  • Colette Mœglin and Jean-Loup Waldspurger, Décomposition spectrale et séries d’Eisenstein, Progress in Mathematics, vol. 113, Birkhäuser Verlag, Basel, 1994 (French, with English summary). Une paraphrase de l’Écriture. [A paraphrase of Scripture]. MR 1261867
  • Peter C. Sarnak, Diophantine problems and linear groups, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 459–471. MR 1159234
  • 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
  • Don Zagier, The Rankin-Selberg 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
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Bibliographic Information
  • Hervé Jacquet
  • Affiliation: Department of Mathematics, Columbia University, New York, New York 10027
  • Email:
  • Erez Lapid
  • Affiliation: Department of Mathematics, Weizmann Institute, Rehovot, Israel
  • MR Author ID: 631395
  • Email:
  • Jonathan Rogawski
  • Affiliation: Department of Mathematics, Hebrew University, Jerusalem, Israel
  • Address at time of publication: Department of Mathematics, University of California, Los Angeles, California 90095
  • Email:
  • Received by editor(s): October 14, 1997
  • Received by editor(s) in revised form: May 14, 1998
  • Additional Notes: The first author was partially supported by NSF Grant 9619766.
    The third author was partially supported by NSF Grant 9401466.
  • © Copyright 1999 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 12 (1999), 173-240
  • MSC (1991): Primary 11F55, 11F70, 11F72
  • DOI:
  • MathSciNet review: 1625060