On the lifting of Hilbert cusp forms to Hilbert-Hermitian cusp forms

Yamana, Shunsuke

doi:10.1090/tran/8096

On the lifting of Hilbert cusp forms to Hilbert-Hermitian cusp forms
HTML articles powered by AMS MathViewer

by Shunsuke Yamana PDF

Trans. Amer. Math. Soc. 373 (2020), 5395-5438 Request permission

Abstract:

We construct a lifting that associates to a Hilbert cusp form a Hilbert-Hermitian cusp form. This is a generalization of the lifting of elliptic cusp forms constructed by Ikeda to arbitrary Hilbert cusp forms.

References

Jeffrey Adams and Joseph F. Johnson, Endoscopic groups and packets of nontempered representations, Compositio Math. 64 (1987), no. 3, 271–309. MR 918414
James Arthur, Unipotent automorphic representations: conjectures, Astérisque 171-172 (1989), 13–71. Orbites unipotentes et représentations, II. MR 1021499
James Arthur, The endoscopic classification of representations, American Mathematical Society Colloquium Publications, vol. 61, American Mathematical Society, Providence, RI, 2013. Orthogonal and symplectic groups. MR 3135650, DOI 10.1090/coll/061
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
Hiraku Atobe and Hisashi Kojima, On the Miyawaki lifts of hermitian modular forms, J. Number Theory 185 (2018), 281–318. MR 3734351, DOI 10.1016/j.jnt.2017.09.005
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
Martin Eichler and Don Zagier, The theory of Jacobi forms, Progress in Mathematics, vol. 55, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 781735, DOI 10.1007/978-1-4684-9162-3
Wee Teck Gan, Benedict H. Gross, and Dipendra Prasad, Restrictions of representations of classical groups: examples, Astérisque 346 (2012), 111–170 (English, with English and French summaries). Sur les conjectures de Gross et Prasad. I. MR 3202557
David Ginzburg, Endoscopic lifting in classical groups and poles of tensor $L$-functions, Duke Math. J. 141 (2008), no. 3, 447–503. MR 2387428, DOI 10.1215/00127094-2007-002
David Ginzburg, Stephen Rallis, and David Soudry, On a correspondence between cuspidal representations of $\textrm {GL}_{2n}$ and $\widetilde \textrm {Sp}_{2n}$, J. Amer. Math. Soc. 12 (1999), no. 3, 849–907. MR 1671452, DOI 10.1090/S0894-0347-99-00300-8
David Ginzburg, Stephen Rallis, and David Soudry, The descent map from automorphic representations of $\textrm {GL}(n)$ to classical groups, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. MR 2848523, DOI 10.1142/9789814304993
Benedict H. Gross and Kevin Keating, On the intersection of modular correspondences, Invent. Math. 112 (1993), no. 2, 225–245. MR 1213101, DOI 10.1007/BF01232433
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
Guy Henniart, La conjecture de Langlands locale pour $\textrm {GL}(3)$, Mém. Soc. Math. France (N.S.) 11-12 (1984), 186 (French, with English summary). MR 743063
Tamotsu Ikeda, On the theory of Jacobi forms and Fourier-Jacobi coefficients of Eisenstein series, J. Math. Kyoto Univ. 34 (1994), no. 3, 615–636. MR 1295945, DOI 10.1215/kjm/1250518935
Tamotsu Ikeda, On the lifting of elliptic cusp forms to Siegel cusp forms of degree $2n$, Ann. of Math. (2) 154 (2001), no. 3, 641–681. MR 1884618, DOI 10.2307/3062143
Tamotsu Ikeda, On the lifting of Hermitian modular forms, Compos. Math. 144 (2008), no. 5, 1107–1154. MR 2457521, DOI 10.1112/S0010437X08003643
Tamotsu Ikeda, On the lifting of elliptic cusp forms to Siegel cusp forms of degree $2n$, Ann. of Math. (2) 154 (2001), no. 3, 641–681. MR 1884618, DOI 10.2307/3062143
Dihua Jiang, Baiying Liu, and Lei Zhang, Poles of certain residual Eisenstein series of classical groups, Pacific J. Math. 264 (2013), no. 1, 83–123. MR 3079762, DOI 10.2140/pjm.2013.264.83
Eyal Kaplan, The characterization of theta-distinguished representations of $\textrm {GL}(n)$, Israel J. Math. 222 (2017), no. 2, 551–598. MR 3722261, DOI 10.1007/s11856-017-1600-1
Martin L. Karel, Functional equations of Whittaker functions on $p$-adic groups, Amer. J. Math. 101 (1979), no. 6, 1303–1325. MR 548883, DOI 10.2307/2374142
Henry H. Kim and Takuya Yamauchi, Cusp forms on the exceptional group of type $E_7$, Compos. Math. 152 (2016), no. 2, 223–254. MR 3462552, DOI 10.1112/S0010437X15007538
Henry H. Kim and Takuya Yamauchi, Cusp forms on the exceptional group of type $E_7$, Compos. Math. 152 (2016), no. 2, 223–254. MR 3462552, DOI 10.1112/S0010437X15007538
Kazuhiro Konno, Even surfaces with $p_g=7$, $q=0$ and $K^2=16$, Math. Rep. Kyushu Univ. 18 (1991), no. 1, 15–41. MR 1157326
Stephen S. Kudla, Splitting metaplectic covers of dual reductive pairs, Israel J. Math. 87 (1994), no. 1-3, 361–401. MR 1286835, DOI 10.1007/BF02773003
Stephen Kudla and Michael Rapoport, Special cycles on unitary Shimura varieties I. Unramified local theory, Invent. Math. 184 (2011), no. 3, 629–682. MR 2800697, DOI 10.1007/s00222-010-0298-z
Stephen S. Kudla and W. Jay Sweet Jr., Degenerate principal series representations for $\textrm {U}(n,n)$, Israel J. Math. 98 (1997), 253–306. MR 1459856, DOI 10.1007/BF02937337
Erez M. Lapid and Stephen Rallis, On the local factors of representations of classical groups, Automorphic representations, $L$-functions and applications: progress and prospects, Ohio State Univ. Math. Res. Inst. Publ., vol. 11, de Gruyter, Berlin, 2005, pp. 309–359. MR 2192828, DOI 10.1515/9783110892703.309
Chung Pang Mok, Endoscopic classification of representations of quasi-split unitary groups, Mem. Amer. Math. Soc. 235 (2015), no. 1108, vi+248. MR 3338302, DOI 10.1090/memo/1108
C. Mœglin and J.-L. Waldspurger, Le spectre résiduel de $\textrm {GL}(n)$, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 4, 605–674 (French). MR 1026752, DOI 10.24033/asens.1595
C. Mœglin and J.-L. Waldspurger, Spectral decomposition and Eisenstein series, Cambridge Tracts in Mathematics, vol. 113, Cambridge University Press, Cambridge, 1995. Une paraphrase de l’Écriture [A paraphrase of Scripture]. MR 1361168, DOI 10.1017/CBO9780511470905
Omer Offen and Eitan Sayag, Global mixed periods and local Klyachko models for the general linear group, Int. Math. Res. Not. IMRN 1 (2008), Art. ID rnm 136, 25. MR 2417789, DOI 10.1093/imrn/rnm136
Goro Shimura, Euler products and Eisenstein series, CBMS Regional Conference Series in Mathematics, vol. 93, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1997. MR 1450866, DOI 10.1090/cbms/093
Shunsuke Yamana, On the lifting of elliptic cusp forms to cusp forms on quaternionic unitary groups, J. Number Theory 130 (2010), no. 11, 2480–2527. MR 2678859, DOI 10.1016/j.jnt.2010.03.020
Shunsuke Yamana, L-functions and theta correspondence for classical groups, Invent. Math. 196 (2014), no. 3, 651–732. MR 3211043, DOI 10.1007/s00222-013-0476-x
Shunsuke Yamana, Siegel series for skew Hermitian forms over quaternion algebras, Abh. Math. Semin. Univ. Hambg. 87 (2017), no. 1, 43–59. MR 3623824, DOI 10.1007/s12188-016-0127-4
Shunsuke Yamana, Degenerate principal series and Langlands classification, Automorphic forms and related topics, Contemp. Math., vol. 732, Amer. Math. Soc., [Providence], RI, [2019] ©2019, pp. 275–286. MR 3973304, DOI 10.1090/conm/732/14799