Automorphic period and the central value of Rankin-Selberg L-function

Zhang, Wei

doi:10.1090/S0894-0347-2014-00784-0

Automorphic period and the central value of Rankin-Selberg L-function
HTML articles powered by AMS MathViewer

by Wei Zhang

J. Amer. Math. Soc. 27 (2014), 541-612

DOI: https://doi.org/10.1090/S0894-0347-2014-00784-0

Published electronically: January 27, 2014

PDF | Request permission

Abstract:

Using the relative trace formula of Jacquet and Rallis, under some local conditions we prove a refinement of the global Gan-Gross-Prasad conjecture proposed by Ichino-Ikeda and N. Harris for unitary groups. We need to assume some expected properties of L-packets and some part of the local Gan-Gross-Prasad conjecture.

References

Avraham Aizenbud, Dmitry Gourevitch, Stephen Rallis, and Gérard Schiffmann, Multiplicity one theorems, Ann. of Math. (2) 172 (2010), no. 2, 1407–1434. MR 2680495, DOI 10.4007/annals.2010.172.1413
R. Beuzart-Plessis, La conjecture locale de Gross-Prasad pour les représentations tempérés des groupes unitaires, arXiv:1205.2987.
M. Cowling, U. Haagerup, and R. Howe, Almost $L^2$ matrix coefficients, J. Reine Angew. Math. 387 (1988), 97–110. MR 946351
Hervé Jacquet and Nan Chen, Positivity of quadratic base change $L$-functions, Bull. Soc. Math. France 129 (2001), no. 1, 33–90 (English, with English and French summaries). MR 1871978, DOI 10.24033/bsmf.2386
Brooke Feigon, Erez Lapid, and Omer Offen, On representations distinguished by unitary groups, Publ. Math. Inst. Hautes Études Sci. 115 (2012), 185–323. MR 2930996, DOI 10.1007/s10240-012-0040-z
Yuval Z. Flicker, Twisted tensors and Euler products, Bull. Soc. Math. France 116 (1988), no. 3, 295–313 (English, with French summary). MR 984899
Wee Teck Gan, Benedict H. Gross, and Dipendra Prasad, Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups. Astérisque 346, 1–110.
Wee Teck Gan and Atsushi Ichino, On endoscopy and the refined Gross-Prasad conjecture for $(\rm SO_5,SO_4)$, J. Inst. Math. Jussieu 10 (2011), no. 2, 235–324. MR 2787690, DOI 10.1017/S1474748010000198
Paul B. Garrett, Decomposition of Eisenstein series: Rankin triple products, Ann. of Math. (2) 125 (1987), no. 2, 209–235. MR 881269, DOI 10.2307/1971310
Stephen Gelbart, Hervé Jacquet, and Jonathan Rogawski, Generic representations for the unitary group in three variables, Israel J. Math. 126 (2001), 173–237. MR 1882037, DOI 10.1007/BF02784154
David Ginzburg, Dihua Jiang, and Stephen Rallis, On the nonvanishing of the central value of the Rankin-Selberg $L$-functions, J. Amer. Math. Soc. 17 (2004), no. 3, 679–722. MR 2053953, DOI 10.1090/S0894-0347-04-00455-2
David Ginzburg, Dihua Jiang, and Stephen Rallis, Models for certain residual representations of unitary groups, Automorphic forms and $L$-functions I. Global aspects, Contemp. Math., vol. 488, Amer. Math. Soc., Providence, RI, 2009, pp. 125–146. MR 2522030, DOI 10.1090/conm/488/09567
Loukas Grafakos, Classical Fourier analysis, 2nd ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2008. MR 2445437
Benedict H. Gross, Heights and the special values of $L$-series, Number theory (Montreal, Que., 1985) CMS Conf. Proc., vol. 7, Amer. Math. Soc., Providence, RI, 1987, pp. 115–187. MR 894322
Benedict H. Gross, On the motive of a reductive group, Invent. Math. 130 (1997), no. 2, 287–313. MR 1474159, DOI 10.1007/s002220050186
Benedict H. Gross and Don B. Zagier, Heegner points and derivatives of $L$-series, Invent. Math. 84 (1986), no. 2, 225–320. MR 833192, DOI 10.1007/BF01388809
Harish-Chandra, Harmonic analysis on reductive $p$-adic groups, Lecture Notes in Mathematics, Vol. 162, Springer-Verlag, Berlin-New York, 1970. Notes by G. van Dijk. MR 0414797
Harish-Chandra, Admissible invariant distributions on reductive $p$-adic groups, Lie theories and their applications (Proc. Ann. Sem. Canad. Math. Congr., Queen’s Univ., Kingston, Ont., 1977) Queen’s Papers in Pure and Appl. Math., No. 48, Queen’s Univ., Kingston, Ont., 1978, pp. 281–347. MR 0579175
Michael Harris and Stephen S. Kudla, On a conjecture of Jacquet, Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ. Press, Baltimore, MD, 2004, pp. 355–371. MR 2058614
Michael Harris, $L$-functions and periods of adjoint motives, Algebra Number Theory 7 (2013), no. 1, 117–155. MR 3037892, DOI 10.2140/ant.2013.7.117
Richard Neal Harris, A Refined Gross-Prasad Conjecture for Unitary Groups, ProQuest LLC, Ann Arbor, MI, 2011. Thesis (Ph.D.)–University of California, San Diego. MR 2890098
Atsushi Ichino, Trilinear forms and the central values of triple product $L$-functions, Duke Math. J. 145 (2008), no. 2, 281–307. MR 2449948, DOI 10.1215/00127094-2008-052
Atsushi Ichino and Tamutsu Ikeda, On the periods of automorphic forms on special orthogonal groups and the Gross-Prasad conjecture, Geom. Funct. Anal. 19 (2010), no. 5, 1378–1425. MR 2585578, DOI 10.1007/s00039-009-0040-4
A. Ichino and W. Zhang, Spherical characters for a strongly tempered pair. Appendix to \cite{Zh2}.
Hervé Jacquet, Archimedean Rankin-Selberg integrals, Automorphic forms and $L$-functions II. Local aspects, Contemp. Math., vol. 489, Amer. Math. Soc., Providence, RI, 2009, pp. 57–172. MR 2533003, DOI 10.1090/conm/489/09547
Hervé Jacquet, Distinction by the quasi-split unitary group, Israel J. Math. 178 (2010), 269–324. MR 2733072, DOI 10.1007/s11856-010-0066-1
H. Jacquet, I. I. Piatetskii-Shapiro, and J. A. Shalika, Rankin-Selberg convolutions, Amer. J. Math. 105 (1983), no. 2, 367–464. MR 701565, DOI 10.2307/2374264
Hervé Jacquet and Stephen Rallis, On the Gross-Prasad conjecture for unitary groups, On certain $L$-functions, Clay Math. Proc., vol. 13, Amer. Math. Soc., Providence, RI, 2011, pp. 205–264. MR 2767518
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
H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic forms. II, Amer. J. Math. 103 (1981), no. 4, 777–815. MR 623137, DOI 10.2307/2374050
Erez Lapid and Stephen Rallis, On the nonnegativity of $L({1\over 2},\pi )$ for $\textrm {SO}_{2n+1}$, Ann. of Math. (2) 157 (2003), no. 3, 891–917. MR 1983784
Erez M. Lapid, On the nonnegativity of Rankin-Selberg $L$-functions at the center of symmetry, Int. Math. Res. Not. 2 (2003), 65–75. MR 1936579, DOI 10.1155/S1073792803204013
E. Lapid and Z. Mao, On a new functional equation for local integrals, preprint..
C-P. Mok, Endoscopic classification of representations of quasi-split unitary groups, arXiv:1206.0882v1.
Cary Rader and Steve Rallis, Spherical characters on $\mathfrak {p}$-adic symmetric spaces, Amer. J. Math. 118 (1996), no. 1, 91–178. MR 1375304
Y. Sakellaridis and A. Venkatesh, Periods and harmonic analysis on spherical varieties, arXiv:1203.0039..
Binyong Sun, Bounding matrix coefficients for smooth vectors of tempered representations, Proc. Amer. Math. Soc. 137 (2009), no. 1, 353–357. MR 2439460, DOI 10.1090/S0002-9939-08-09598-1
I. I. Piatetski-Shapiro, Multiplicity one theorems, 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. 209–212. MR 546599
Ye Tian, Congruent numbers and Heegner points, preprint.
J.-L. Waldspurger, Sur les valeurs de certaines fonctions $L$ automorphes en leur centre de symétrie, Compositio Math. 54 (1985), no. 2, 173–242 (French). MR 783511
J. Waldspurger, Une formule intégrale reli la conjecture locale de Gross-Prasad, me partie: extension aux représentations tempérés. arXiv:0904.0314..
P.-J. White, Tempered automorphic representations of the unitary group, arXiv: 1106.1127.
Thomas Crawford Watson, Rankin triple products and quantum chaos, ProQuest LLC, Ann Arbor, MI, 2002. Thesis (Ph.D.)–Princeton University. MR 2703041
Xinyi Yuan, Shou-Wu Zhang, and Wei Zhang, The Gross-Zagier formula on Shimura curves. Ann. of Math. Studies #184, Princeton University Press, 2012.
Zhiwei Yun, The fundamental lemma of Jacquet and Rallis, Duke Math. J. 156 (2011), no. 2, 167–227. With an appendix by Julia Gordon. MR 2769216, DOI 10.1215/00127094-2010-210
Shouwu Zhang, Heights of Heegner points on Shimura curves, Ann. of Math. (2) 153 (2001), no. 1, 27–147. MR 1826411, DOI 10.2307/2661372
Shou-Wu Zhang, Gross-Zagier formula for $\textrm {GL}_2$, Asian J. Math. 5 (2001), no. 2, 183–290. MR 1868935, DOI 10.4310/AJM.2001.v5.n2.a1
Shou-Wu Zhang, Gross-Zagier formula for $\rm GL(2)$. II, Heegner points and Rankin $L$-series, Math. Sci. Res. Inst. Publ., vol. 49, Cambridge Univ. Press, Cambridge, 2004, pp. 191–214. MR 2083213, DOI 10.1017/CBO9780511756375.008
Shouwu Zhang, Linear forms, algebraic cycles, and derivatives of L-series, preprint..
Wei Zhang, On arithmetic fundamental lemmas, Invent. Math. 188 (2012), no. 1, 197–252. MR 2897697, DOI 10.1007/s00222-011-0348-1
Wei Zhang, Fourier transform and the global Gan-Gross-Prasad conjecture for unitary groups, Ann. Math., in press..