Eisenstein congruence on unitary groups and Iwasawa main conjectures for CM fields

Hsieh, Ming-Lun¹

¹ Department of Mathematics, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan

Journal of the American Mathematical Society, Tome 27 (2014) no. 3, pp. 753-862

Résumé

The purpose of this article is to prove the Iwasawa main conjecture for CM fields in certain cases through a detailed study on the divisibility relation between $p$-adic $L$-functions for CM fields and Eisenstein ideals of unitary groups of degree three.

Détail

DOI : 10.1090/S0894-0347-2014-00786-4

Bibliographie

[1] Bellaã¯Che, Joã«L, Chenevier, Gaã«Tan Formes non tempérées pour 𝑈(3) et conjectures de Bloch-Kato Ann. Sci. École Norm. Sup. (4) 2004 611 662

[2] Bellaã¯Che, Joã«L, Chenevier, Gaã«Tan Families of Galois representations and Selmer groups Astérisque 2009

[3] Blasius, Don, Rogawski, Jonathan D. Motives for Hilbert modular forms Invent. Math. 1993 55 87

[4] Burungale, A. On the 𝜇-invariant of cyclotomic derivative of Katz 𝑝-adic 𝐿-functions J. Inst. Math. Jussieu 2014

[5] Casselman, W. The unramified principal series of 𝔭-adic groups. I. The spherical function Compositio Math. 1980 387 406

[6] Coates, John, Goldstein, Catherine Some remarks on the main conjecture for elliptic curves with complex multiplication Amer. J. Math. 1983 337 366

[7] Chai, Ching-Li Compactification of Siegel moduli schemes 1985

[8] Chai, Ching-Li Methods for 𝑝-adic monodromy J. Inst. Math. Jussieu 2008 247 268

[9] Deligne, Pierre, Ribet, Kenneth A. Values of abelian 𝐿-functions at negative integers over totally real fields Invent. Math. 1980 227 286

[10] Faltings, G., Chai, C.-L. Degeneration of abelian varieties 1990

[11] Finis, Tobias Divisibility of anticyclotomic 𝐿-functions and theta functions with complex multiplication Ann. of Math. (2) 2006 767 807

[12] Fujiwara, K. Arithmetic compactifications of Shimura varieties (I) 1989

[13] Gelbart, Stephen, Piatetski-Shapiro, Ilya, Rallis, Stephen Explicit constructions of automorphic 𝐿-functions 1987

[14] Greenberg, Ralph Iwasawa theory and 𝑝-adic deformations of motives 1994 193 223

[15] Greenberg, Ralph Iwasawa theory for elliptic curves 1999 51 144

[16] Greenberg, Ralph, Vatsal, Vinayak On the Iwasawa invariants of elliptic curves Invent. Math. 2000 17 63

[17] Harris, Michael Eisenstein series on Shimura varieties Ann. of Math. (2) 1984 59 94

[18] Hida, Haruzo Elementary theory of 𝐿-functions and Eisenstein series 1993

[19] Hida, Haruzo Control theorems of 𝑝-nearly ordinary cohomology groups for 𝑆𝐿(𝑛) Bull. Soc. Math. France 1995 425 475

[20] Hida, Haruzo Automorphic induction and Leopoldt type conjectures for 𝐺𝐿(𝑛) Asian J. Math. 1998 667 710

[21] Hida, Haruzo Control theorems of coherent sheaves on Shimura varieties of PEL type J. Inst. Math. Jussieu 2002 1 76

[22] Hida, H. Non-vanishing modulo 𝑝 of Hecke 𝐿-values 2004

[23] Hida, H. 𝑝-adic automorphic forms on Shimura varieties 2004

[24] Hida, Haruzo Anticyclotomic main conjectures Doc. Math. 2006 465 532

[25] Hida, H. Non-vanishing modulo 𝑝 of Hecke 𝐿-values and application 2007

[26] Hida, Haruzo Irreducibility of the Igusa tower Acta Math. Sin. (Engl. Ser.) 2009 1 20

[27] Hida, H. Quadratic exercises in Iwasawa theory Int. Math. Res. Not. IMRN 2009

[28] Hida, Haruzo The Iwasawa 𝜇-invariant of 𝑝-adic Hecke 𝐿-functions Ann. of Math. (2) 2010 41 137

[29] Harris, Michael, Li, Jian-Shu, Skinner, Christopher M. 𝑝-adic 𝐿-functions for unitary Shimura varieties. I. Construction of the Eisenstein measure Doc. Math. 2006 393 464

[30] Hsieh, Ming-Lun The algebraic functional equation of Selmer groups for CM fields J. Number Theory 2010 1914 1924

[31] Hsieh, Ming-Lun Ordinary 𝑝-adic Eisenstein series and 𝑝-adic 𝐿-functions for unitary groups Ann. Inst. Fourier (Grenoble) 2011 987 1059

[32] Hsieh, Ming-Lun On the non-vanishing of Hecke 𝐿-values modulo 𝑝 Amer. J. Math. 2012 1503 1539

[33] Hsieh, M.-L. On the 𝜇-invariant of anticyclotomic 𝑝-adic 𝐿-functions for CM fields J. Reine Angew. Math. 2013

[34] Hida, H., Tilouine, J. Anti-cyclotomic Katz 𝑝-adic 𝐿-functions and congruence modules Ann. Sci. École Norm. Sup. (4) 1993 189 259

[35] Hida, H., Tilouine, J. On the anticyclotomic main conjecture for CM fields Invent. Math. 1994 89 147

[36] Jantzen, Jens Carsten Representations of algebraic groups 1987

[37] Katz, Nicholas M. 𝑝-adic 𝐿-functions for CM fields Invent. Math. 1978 199 297

[38] Katz, N. Serre-Tate local moduli 1981

[39] Kottwitz, Robert E. Points on some Shimura varieties over finite fields J. Amer. Math. Soc. 1992 373 444

[40] Langlands, Robert P. On the functional equations satisfied by Eisenstein series 1976

[41] Lan, K.-W. Arithmetic compactification of PEL-type Shimura varieties 2008

[42] Lan, Kai-Wen Comparison between analytic and algebraic constructions of toroidal compactifications of PEL-type Shimura varieties J. Reine Angew. Math. 2012 163 228

[43] Li, Jian-Shu Nonvanishing theorems for the cohomology of certain arithmetic quotients J. Reine Angew. Math. 1992 177 217

[44] Mainardi, F. Sur la conjecture principale de iwasawa pour les corps CM 2004

[45] Mainardi, Fabio On the main conjecture for CM fields Amer. J. Math. 2008 499 538

[46] Mauger, David Algèbres de Hecke quasi-ordinaires universelles Ann. Sci. École Norm. Sup. (4) 2004 171 222

[47] Murase, Atsushi, Sugano, Takashi Local theory of primitive theta functions Compositio Math. 2000 273 302

[48] Murase, Atsushi, Sugano, Takashi Fourier-Jacobi expansion of Eisenstein series on unitary groups of degree three J. Math. Sci. Univ. Tokyo 2002 347 404

[49] Murase, Atsushi, Sugano, Takashi On the Fourier-Jacobi expansion of the unitary Kudla lift Compos. Math. 2007

[50] Mumford, David Abelian varieties 1970

[51] Mazur, B., Wiles, A. Class fields of abelian extensions of 𝑄 Invent. Math. 1984 179 330

[52] Nekovã¡Å™, Jan Selmer complexes Astérisque 2006

[53] Perrin-Riou, B. Arithmétique des courbes elliptiques et théorie d’Iwasawa Mém. Soc. Math. France (N.S.) 1984 130

[54] Ribet, Kenneth A. A modular construction of unramified 𝑝-extensions of 𝑄(𝜇_{𝑝}) Invent. Math. 1976 151 162

[55] Rogawski, Jonathan D. Analytic expression for the number of points mod 𝑝 1992 65 109

[56] Rohrlich, David E. Root numbers of Hecke 𝐿-functions of CM fields Amer. J. Math. 1982 517 543

[57] Rubin, Karl The “main conjectures” of Iwasawa theory for imaginary quadratic fields Invent. Math. 1991 25 68

[58] Serre, Jean-Pierre Abelian 𝑙-adic representations and elliptic curves 1968

[59] Schémas en groupes. II: Groupes de type multiplicatif, et structure des schémas en groupes généraux, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3) 1962/1964

[60] Revêtements étales et groupe fondamental 1971

[61] Shimura, Goro The special values of the zeta functions associated with Hilbert modular forms Duke Math. J. 1978 637 679

[62] Shintani, T. On automorphic forms on unitary groups of oreder 3 1979

[63] Shimura, Goro On Eisenstein series Duke Math. J. 1983 417 476

[64] Shimura, G. Euler products and Eisenstein series 1997

[65] Shimura, Goro Abelian varieties with complex multiplication and modular functions 1998

[66] Skinner, C., Urban, E. The Iwasawa Main Conjectures for 𝐺𝐿₂ Inventiones Mathematicae 2013

[67] Tilouine, J., Urban, E. Several-variable 𝑝-adic families of Siegel-Hilbert cusp eigensystems and their Galois representations Ann. Sci. École Norm. Sup. (4) 1999 499 574

[68] Urban, E. On residually reducible representations on local rings J. Algebra 1999 738 742

[69] Urban, Eric Selmer groups and the Eisenstein-Klingen ideal Duke Math. J. 2001 485 525

[70] Urban, Eric Groupes de Selmer er fonctions L p-adiques pour les représentations modulaires adjointes 2006

[71] Wedhorn, Torsten Ordinariness in good reductions of Shimura varieties of PEL-type Ann. Sci. École Norm. Sup. (4) 1999 575 618

[72] Wiles, A. The Iwasawa conjecture for totally real fields Ann. of Math. (2) 1990 493 540

[73] Yager, Rodney I. On two variable 𝑝-adic 𝐿-functions Ann. of Math. (2) 1982 411 449

[74] Yang, Tonghai Theta liftings and Hecke 𝐿-functions J. Reine Angew. Math. 1997 25 53