Ordinary families of Klingen Eisenstein series on symplectic groups

Liu, Zheng

doi:10.1090/tran/8270

Ordinary families of Klingen Eisenstein series on symplectic groups
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Trans. Amer. Math. Soc. 374 (2021), 3331-3395 Request permission

Abstract:

We construct $(n+1)$-variable Hida families of Klingen Eisenstein series on $\operatorname {Sp}(2n+2)$ for $n$-variable Hida families on $\operatorname {Sp}(2n)$, and relate their images under the Siegel operator to $p$-adic $L$-functions. We also carry out some preliminary calculations of the non-degenerate Fourier coefficients of the constructed Klingen Eisenstein families.

References

F. Andreatta and .A Iovita, Triple product $p$-adic $L$-functions associated to finite slope $p$-adic families of modular forms: with an appendix by Eric Urban, arXiv:1708.02785, 2017.
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
Siegfried Böcherer, Über die Fourier-Jacobi-Entwicklung Siegelscher Eisensteinreihen. II, Math. Z. 189 (1985), no. 1, 81–110 (German). MR 776540, DOI 10.1007/BF01246946
Siegfried Böcherer, Über die Funktionalgleichung automorpher $L$-Funktionen zur Siegelschen Modulgruppe, J. Reine Angew. Math. 362 (1985), 146–168 (German). MR 809972, DOI 10.1515/crll.1985.362.146
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). MR 1800123, DOI 10.5802/aif.1796
Gaëtan Chenevier and Michael Harris, Construction of automorphic Galois representations, II, Camb. J. Math. 1 (2013), no. 1, 53–73. MR 3272052, DOI 10.4310/CJM.2013.v1.n1.a2
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. MR 2742611
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. MR 1110392, DOI 10.1017/CBO9780511526053.006
Ellen Eischen, Michael Harris, Jianshu Li, and Christopher Skinner, $p$-adic $L$-functions for unitary groups, Forum Math. Pi 8 (2020), e9, 160. MR 4096618, DOI 10.1017/fmp.2020.4
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). MR 2986270, DOI 10.5802/aif.2704
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. MR 3505656, DOI 10.1017/S1474748014000395
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. MR 1083353, DOI 10.1007/978-3-662-02632-8
Tobias Finis, Divisibility of anticyclotomic $L$-functions and theta functions with complex multiplication, Ann. of Math. (2) 163 (2006), no. 3, 767–807. MR 2215134, DOI 10.4007/annals.2006.163.767
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. MR 763012
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. MR 993320
Michael Harris, Arithmetic vector bundles and automorphic forms on Shimura varieties. I, Invent. Math. 82 (1985), no. 1, 151–189. MR 808114, DOI 10.1007/BF01394784
Michael Harris, Arithmetic vector bundles and automorphic forms on Shimura varieties. II, Compositio Math. 60 (1986), no. 3, 323–378. MR 869106
Haruzo Hida, Elementary theory of $L$-functions and Eisenstein series, London Mathematical Society Student Texts, vol. 26, Cambridge University Press, Cambridge, 1993. MR 1216135, DOI 10.1017/CBO9780511623691
Haruzo Hida, Control theorems of coherent sheaves on Shimura varieties of PEL type, J. Inst. Math. Jussieu 1 (2002), no. 1, 1–76. MR 1954939, DOI 10.1017/S1474748002000014
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. MR 3194494, DOI 10.1090/S0894-0347-2014-00786-4
H. Hida and J. Tilouine, On the anticyclotomic main conjecture for CM fields, Invent. Math. 117 (1994), no. 1, 89–147. MR 1269427, DOI 10.1007/BF01232236
Hans Plesner Jakobsen and Michèle Vergne, Restrictions and expansions of holomorphic representations, J. Functional Analysis 34 (1979), no. 1, 29–53. MR 551108, DOI 10.1016/0022-1236(79)90023-5
Nicholas M. Katz, $p$-adic $L$-functions for CM fields, Invent. Math. 49 (1978), no. 3, 199–297. MR 513095, DOI 10.1007/BF01390187
Stephen S. Kudla and Stephen Rallis, Ramified degenerate principal series representations for $\textrm {Sp}(n)$, Israel J. Math. 78 (1992), no. 2-3, 209–256. MR 1194967, DOI 10.1007/BF02808058
M. Kashiwara and M. Vergne, On the Segal-Shale-Weil representations and harmonic polynomials, Invent. Math. 44 (1978), no. 1, 1–47. MR 463359, DOI 10.1007/BF01389900
Zheng Liu, $p$-adic $L$-functions for ordinary families on symplectic groups, J. Inst. Math. Jussieu 19 (2020), no. 4, 1287–1347. MR 4120810, DOI 10.1017/s1474748018000415
Zheng Liu, The doubling Archimedean zeta integrals for $p$-adic interpolation, To appear in Math. Res. Lett., 2019.
Zheng Liu, Nearly overconvergent Siegel modular forms, Ann. Inst. Fourier (Grenoble) 69 (2019), no. 6, 2439–2506 (English, with English and French summaries). MR 4033924, DOI 10.5802/aif.3299
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
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. MR 4150915, DOI 10.1007/s00208-020-01966-x
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
B. Mazur and A. Wiles, Class fields of abelian extensions of $\textbf {Q}$, Invent. Math. 76 (1984), no. 2, 179–330. MR 742853, DOI 10.1007/BF01388599
Vincent Pilloni, Sur la théorie de Hida pour le groupe $\textrm {GSp}_{2g}$, Bull. Soc. Math. France 140 (2012), no. 3, 335–400 (French, with English and French summaries). MR 3059119, DOI 10.24033/bsmf.2630
I. Piatetski-Shapiro and Stephen Rallis, $L$-functions for the classical groups, volume 1254 of Lecture Notes in Mathematics, pages 1–52. Springer-Verlag, Berlin, 1987.
Freydoon Shahidi, Eisenstein series and automorphic $L$-functions, American Mathematical Society Colloquium Publications, vol. 58, American Mathematical Society, Providence, RI, 2010. MR 2683009, DOI 10.1090/coll/058
Goro Shimura, Confluent hypergeometric functions on tube domains, Math. Ann. 260 (1982), no. 3, 269–302. MR 669297, DOI 10.1007/BF01461465
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
Goro Shimura, Arithmeticity in the theory of automorphic forms, Mathematical Surveys and Monographs, vol. 82, American Mathematical Society, Providence, RI, 2000. MR 1780262, DOI 10.1090/surv/082
Sug Woo Shin, Galois representations arising from some compact Shimura varieties, Ann. of Math. (2) 173 (2011), no. 3, 1645–1741. MR 2800722, DOI 10.4007/annals.2011.173.3.9
Christopher Skinner and Eric Urban, The Iwasawa main conjectures for $\rm GL_2$, Invent. Math. 195 (2014), no. 1, 1–277. MR 3148103, DOI 10.1007/s00222-013-0448-1
W. Jay Sweet, Jr. Functional equations of $p$-adic zeta integrals and representations of the metaplectic group. preprint.
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. MR 1366572, DOI 10.1006/jnth.1995.1139
Eric Urban, Selmer groups and the Eisenstein-Klingen ideal, Duke Math. J. 106 (2001), no. 3, 485–525. MR 1813234, DOI 10.1215/S0012-7094-01-10633-9
Eric Urban, Groupes de Selmer et fonctions $L$ $p$-adiques pour les représentations modulaires adjointes, preprint, 2006.
Eric Urban, Nearly overconvergent modular forms, Iwasawa theory 2012, Contrib. Math. Comput. Sci., vol. 7, Springer, Heidelberg, 2014, pp. 401–441. MR 3586822
V. Vatsal, Special values of anticyclotomic $L$-functions, Duke Math. J. 116 (2003), no. 2, 219–261. MR 1953292, DOI 10.1215/S0012-7094-03-11622-1
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. MR 3435811, DOI 10.2140/ant.2015.9.1955
Xin Wan, Iwasawa main conjecture for Rankin-Selberg $p$-adic $L$-functions, Algebra Number Theory 14 (2020), no. 2, 383–483. MR 4104413, DOI 10.2140/ant.2020.14.383
A. Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. (2) 131 (1990), no. 3, 493–540. MR 1053488, DOI 10.2307/1971468