## Periods of automorphic forms

HTML articles powered by AMS MathViewer

- by Hervé Jacquet, Erez Lapid and Jonathan Rogawski PDF
- J. Amer. Math. Soc.
**12**(1999), 173-240 Request permission

## Abstract:

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.## References

- 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**

## Additional Information

**Hervé Jacquet**- Affiliation: Department of Mathematics, Columbia University, New York, New York 10027
- Email: hj@math.columbia.edu
**Erez Lapid**- Affiliation: Department of Mathematics, Weizmann Institute, Rehovot, Israel
- MR Author ID: 631395
- Email: erezl@wisdom.weizmann.ac.il
**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: jonr@math.huji.ac.il
- 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: https://doi.org/10.1090/S0894-0347-99-00279-9
- MathSciNet review: 1625060