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
Additional Information
- Shunsuke Yamana
- Affiliation: Advanced Mathematical Institute, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585 Japan
- MR Author ID: 871756
- Email: yamana@sci.osaka-cu.ac.jp
- Received by editor(s): May 7, 2019
- Received by editor(s) in revised form: August 25, 2019
- Published electronically: May 28, 2020
- Additional Notes: The author is partially supported by JSPS Grant-in-Aid for Scientific Research (C) 18K03210 and (B) 19H01778
The author was partially supported by Osaka City University Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics). - © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 5395-5438
- MSC (2010): Primary 11F11, 11F30, 11F55; Secondary 11F27, 11F70
- DOI: https://doi.org/10.1090/tran/8096
- MathSciNet review: 4127881