Ordinary families of Klingen Eisenstein series on symplectic groups
Author:
Zheng Liu
Journal:
Trans. Amer. Math. Soc. 374 (2021), 3331-3395
MSC (2020):
Primary 11F33; Secondary 11F30, 11F46, 11G18
DOI:
https://doi.org/10.1090/tran/8270
Published electronically:
February 24, 2021
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We construct -variable Hida families of Klingen Eisenstein series on
for
-variable Hida families on
, and relate their images under the Siegel operator to
-adic
-functions. We also carry out some preliminary calculations of the non-degenerate Fourier coefficients of the constructed Klingen Eisenstein families.
- [AI17]
F. Andreatta and .A Iovita, Triple product
-adic
-functions associated to finite slope
-adic families of modular forms: with an appendix by Eric Urban, 1708.02785, 2017.
- [Art13] James Arthur, The endoscopic classification of representations, American Mathematical Society Colloquium Publications, vol. 61, American Mathematical Society, Providence, RI, 2013. Orthogonal and symplectic groups. {3135650}
- [Böc85a] Siegfried Böcherer, Über die Fourier-Jacobi-Entwicklung Siegelscher Eisensteinreihen. II, Math. Z. 189 (1985), no. 1, 81–110 (German). {776540}, https://doi.org/10.1007/BF01246946
- [Böc85b] Siegfried Böcherer, Über die Funktionalgleichung automorpher 𝐿-Funktionen zur Siegelschen Modulgruppe, J. Reine Angew. Math. 362 (1985), 146–168 (German). {809972}, https://doi.org/10.1515/crll.1985.362.146
- [BS00] S. Böcherer and C.-G. Schmidt, 𝑝-adic measures attached to Siegel modular forms, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 5, 1375–1443 (English, with English and French summaries). {1800123}
- [CH13] Gaëtan Chenevier and Michael Harris, Construction of automorphic Galois representations, II, Camb. J. Math. 1 (2013), no. 1, 53–73. {3272052}, https://doi.org/10.4310/CJM.2013.v1.n1.a2
- [CHLN11] Laurent Clozel, Michael Harris, Jean-Pierre Labesse, and Bao-Châu Ngô (eds.), On the stabilization of the trace formula, Stabilization of the Trace Formula, Shimura Varieties, and Arithmetic Applications, vol. 1, International Press, Somerville, MA, 2011. {2742611}
- [Coa91] John Coates, Motivic 𝑝-adic 𝐿-functions, 𝐿-functions and arithmetic (Durham, 1989) London Math. Soc. Lecture Note Ser., vol. 153, Cambridge Univ. Press, Cambridge, 1991, pp. 141–172. {1110392}, https://doi.org/10.1017/CBO9780511526053.006
- [EHLS20] Ellen Eischen, Michael Harris, Jianshu Li, and Christopher Skinner, 𝑝-adic 𝐿-functions for unitary groups, Forum Math. Pi 8 (2020), e9, 160. {4096618}, https://doi.org/10.1017/fmp.2020.4
- [Eis12] Ellen E. Eischen, 𝑝-adic differential operators on automorphic forms on unitary groups, Ann. Inst. Fourier (Grenoble) 62 (2012), no. 1, 177–243 (English, with English and French summaries). {2986270}
- [EW16] Ellen Eischen and Xin Wan, 𝑝-adic Eisenstein series and 𝐿-functions of certain cusp forms on definite unitary groups, J. Inst. Math. Jussieu 15 (2016), no. 3, 471–510. {3505656}, https://doi.org/10.1017/S1474748014000395
- [FC90] Gerd Faltings and Ching-Li Chai, Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 22, Springer-Verlag, Berlin, 1990. With an appendix by David Mumford. {1083353}
- [Fin06] Tobias Finis, Divisibility of anticyclotomic 𝐿-functions and theta functions with complex multiplication, Ann. of Math. (2) 163 (2006), no. 3, 767–807. {2215134}, https://doi.org/10.4007/annals.2006.163.767
- [Gar84] Paul B. Garrett, Pullbacks of Eisenstein series; applications, Automorphic forms of several variables (Katata, 1983) Progr. Math., vol. 46, Birkhäuser Boston, Boston, MA, 1984, pp. 114–137. {763012}
- [Gar89] Paul B. Garrett, Integral representations of Eisenstein series and 𝐿-functions, Number theory, trace formulas and discrete groups (Oslo, 1987) Academic Press, Boston, MA, 1989, pp. 241–264. {993320}
- [Har85] Michael Harris, Arithmetic vector bundles and automorphic forms on Shimura varieties. I, Invent. Math. 82 (1985), no. 1, 151–189. {808114}, https://doi.org/10.1007/BF01394784
- [Har86] Michael Harris, Arithmetic vector bundles and automorphic forms on Shimura varieties. II, Compositio Math. 60 (1986), no. 3, 323–378. {869106}
- [Hid93] Haruzo Hida, Elementary theory of 𝐿-functions and Eisenstein series, London Mathematical Society Student Texts, vol. 26, Cambridge University Press, Cambridge, 1993. {1216135}
- [Hid02] Haruzo Hida, Control theorems of coherent sheaves on Shimura varieties of PEL type, J. Inst. Math. Jussieu 1 (2002), no. 1, 1–76. {1954939}, https://doi.org/10.1017/S1474748002000014
- [Hsi14] Ming-Lun Hsieh, Eisenstein congruence on unitary groups and Iwasawa main conjectures for CM fields, J. Amer. Math. Soc. 27 (2014), no. 3, 753–862. {3194494}, https://doi.org/10.1090/S0894-0347-2014-00786-4
- [HT94] H. Hida and J. Tilouine, On the anticyclotomic main conjecture for CM fields, Invent. Math. 117 (1994), no. 1, 89–147. {1269427}, https://doi.org/10.1007/BF01232236
- [JV79] Hans Plesner Jakobsen and Michèle Vergne, Restrictions and expansions of holomorphic representations, J. Functional Analysis 34 (1979), no. 1, 29–53. {551108}, https://doi.org/10.1016/0022-1236(79)90023-5
- [Kat78] Nicholas M. Katz, 𝑝-adic 𝐿-functions for CM fields, Invent. Math. 49 (1978), no. 3, 199–297. {513095}, https://doi.org/10.1007/BF01390187
- [KR92] Stephen S. Kudla and Stephen Rallis, Ramified degenerate principal series representations for 𝑆𝑝(𝑛), Israel J. Math. 78 (1992), no. 2-3, 209–256. {1194967}, https://doi.org/10.1007/BF02808058
- [KV78] M. Kashiwara and M. Vergne, On the Segal-Shale-Weil representations and harmonic polynomials, Invent. Math. 44 (1978), no. 1, 1–47. {463359}, https://doi.org/10.1007/BF01389900
- [Liu16] Zheng Liu, 𝑝-adic 𝐿-functions for ordinary families on symplectic groups, J. Inst. Math. Jussieu 19 (2020), no. 4, 1287–1347. {4120810}, https://doi.org/10.1017/s1474748018000415
- [Liu19a]
Zheng Liu, The doubling Archimedean zeta integrals for
-adic interpolation, To appear in Math. Res. Lett., 2019.
- [Liu19b] Zheng Liu, Nearly overconvergent Siegel modular forms, Ann. Inst. Fourier (Grenoble) 69 (2019), no. 6, 2439–2506 (English, with English and French summaries). {4033924}
- [LR05] 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. {2192828}, https://doi.org/10.1515/9783110892703.309
- [LR18] Zheng Liu and Giovanni Rosso, Non-cuspidal Hida theory for Siegel modular forms and trivial zeros of 𝑝-adic 𝐿-functions, Math. Ann. 378 (2020), no. 1-2, 153–231. {4150915}, https://doi.org/10.1007/s00208-020-01966-x
- [MgW94] 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]. {1261867}
- [MW84] B. Mazur and A. Wiles, Class fields of abelian extensions of 𝑄, Invent. Math. 76 (1984), no. 2, 179–330. {742853}, https://doi.org/10.1007/BF01388599
- [Pil12] Vincent Pilloni, Sur la théorie de Hida pour le groupe 𝐺𝑆𝑝_{2𝑔}, Bull. Soc. Math. France 140 (2012), no. 3, 335–400 (French, with English and French summaries). {3059119}, https://doi.org/10.24033/bsmf.2630
- [PSR87]
I. Piatetski-Shapiro and Stephen Rallis,
-functions for the classical groups, volume 1254 of Lecture Notes in Mathematics, pages 1-52. Springer-Verlag, Berlin, 1987.
- [Sha10] Freydoon Shahidi, Eisenstein series and automorphic 𝐿-functions, American Mathematical Society Colloquium Publications, vol. 58, American Mathematical Society, Providence, RI, 2010. {2683009}
- [Shi82] Goro Shimura, Confluent hypergeometric functions on tube domains, Math. Ann. 260 (1982), no. 3, 269–302. {669297}, https://doi.org/10.1007/BF01461465
- [Shi97] 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. {1450866}
- [Shi00] Goro Shimura, Arithmeticity in the theory of automorphic forms, Mathematical Surveys and Monographs, vol. 82, American Mathematical Society, Providence, RI, 2000. {1780262}
- [Shi11] Sug Woo Shin, Galois representations arising from some compact Shimura varieties, Ann. of Math. (2) 173 (2011), no. 3, 1645–1741. {2800722}, https://doi.org/10.4007/annals.2011.173.3.9
- [SU14] Christopher Skinner and Eric Urban, The Iwasawa main conjectures for 𝐺𝐿₂, Invent. Math. 195 (2014), no. 1, 1–277. {3148103}, https://doi.org/10.1007/s00222-013-0448-1
- [Swe]
W. Jay Sweet, Jr. Functional equations of
-adic zeta integrals and representations of the metaplectic group. preprint.
- [Swe95] W. Jay Sweet Jr., A computation of the gamma matrix of a family of 𝑝-adic zeta integrals, J. Number Theory 55 (1995), no. 2, 222–260. {1366572}, https://doi.org/10.1006/jnth.1995.1139
- [Urb01] Eric Urban, Selmer groups and the Eisenstein-Klingen ideal, Duke Math. J. 106 (2001), no. 3, 485–525. {1813234}, https://doi.org/10.1215/S0012-7094-01-10633-9
- [Urb06]
Eric Urban, Groupes de Selmer et fonctions
-adiques pour les représentations modulaires adjointes, preprint, 2006.
- [Urb14] Eric Urban, Nearly overconvergent modular forms, Iwasawa theory 2012, Contrib. Math. Comput. Sci., vol. 7, Springer, Heidelberg, 2014, pp. 401–441. {3586822}
- [Vat03] V. Vatsal, Special values of anticyclotomic 𝐿-functions, Duke Math. J. 116 (2003), no. 2, 219–261. {1953292}, https://doi.org/10.1215/S0012-7094-03-11622-1
- [Wan15] Xin Wan, Families of nearly ordinary Eisenstein series on unitary groups, Algebra Number Theory 9 (2015), no. 9, 1955–2054. With an appendix by Kai-Wen Lan. {3435811}, https://doi.org/10.2140/ant.2015.9.1955
- [Wan20]
Xin Wan, Iwasawa main conjecture for Rankin-Selberg
-adic
-functions, Algebra Number Theory 14 (2020), no. 2, 383-483. MR 4104413, https://doi.org/10.2140/ant.2020.14.383
- [Wil90] A. Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. (2) 131 (1990), no. 3, 493–540. {1053488}, https://doi.org/10.2307/1971468
Retrieve articles in Transactions of the American Mathematical Society with MSC (2020): 11F33, 11F30, 11F46, 11G18
Retrieve articles in all journals with MSC (2020): 11F33, 11F30, 11F46, 11G18
Additional Information
Zheng Liu
Affiliation:
Department of Mathematics, University of California Santa Barbara, South Hall, Room 6512, Santa Barbara, California 93106
DOI:
https://doi.org/10.1090/tran/8270
Received by editor(s):
July 8, 2019
Received by editor(s) in revised form:
February 24, 2020, July 10, 2020, and July 27, 2020
Published electronically:
February 24, 2021
Additional Notes:
During her time at IAS, the author was supported by the NSF under Grant No. DMS-1638352.
Article copyright:
© Copyright 2021
American Mathematical Society