Ordinary p-adic Eisenstein series and p-adic L-functions for unitary groups
Annales de l'Institut Fourier, Volume 61 (2011) no. 3, pp. 987-1059.

The purpose of this work is to carry out the first step in our four-step program towards the main conjecture for GL 2 ×𝒦 × by the method of Eisenstein congruence on GU(3,1), where 𝒦 is an imaginary quadratic field. We construct a p-adic family of ordinary Eisenstein series on the group of unitary similitudes GU(3,1) with the optimal constant term which is basically the product of the Kubota-Leopodlt p-adic L-function and a p-adic L-function for GL 2 ×𝒦 × . This construction also provides a different point of view of p-adic L-functions of GL 2 ×𝒦 × .

Le but de ce travail est d’accomplir le premier pas de notre programme vers la conjecture principale pour GL 2 ×𝒦 × , par la methode de congruences entre séries d’Eisenstein sur GU(3,1), où 𝒦 est d’un corps quadratique imaginaire. Nous construisons une famille p-adique de séries d’Eisenstein ordinaires sur le groupe de similitudes unitaires avec le terme constant optimal qui est essentiellement le produit de la fonction L p-adique de Kubota-Leopoldt et d’une fonction L p-adique pour GL 2 ×𝒦 × . Cette construction donne ainsi un nouveau point de vue sur la fonction L p-adique de GL 2 ×𝒦 × .

DOI: 10.5802/aif.2635
Classification: 11F33,  11F70,  11R23
Keywords: Eisenstein series on unitary groups, Iwasawa-Greenberg main conjectures
Hsieh, Ming-Lun 1

1 National Taiwan University Department of Mathematics Taipei (Taiwan)
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Hsieh, Ming-Lun. Ordinary $p$-adic Eisenstein series and $p$-adic $L$-functions for unitary groups. Annales de l'Institut Fourier, Volume 61 (2011) no. 3, pp. 987-1059. doi : 10.5802/aif.2635. http://archive.numdam.org/articles/10.5802/aif.2635/

[1] Bertolini, M.; Darmon, H. Iwasawa’s main conjecture for elliptic curves over anticyclotomic p -extensions, Ann. of Math. (2), Volume 162 (2005) no. 1, pp. 1-64 | DOI | MR | Zbl

[2] Chai, Ching-Li Compactification of Siegel moduli schemes, London Mathematical Society Lecture Note Series, 107, Cambridge University Press, Cambridge, 1985 | MR | Zbl

[3] Chai, Ching-Li Methods for p-adic monodromy, J. Inst. Math. Jussieu, Volume 7 (2008) no. 2, pp. 247-268 | DOI | MR | Zbl

[4] Coates, John Motivic p-adic L-functions, L -functions and arithmetic (Durham, 1989) (London Math. Soc. Lecture Note Ser.), Volume 153, Cambridge Univ. Press, Cambridge, 1991, pp. 141-172 | MR | Zbl

[5] Faltings, Gerd; Chai, Ching-Li Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 22, Springer-Verlag, Berlin, 1990 (with an appendix by David Mumford) | MR | Zbl

[6] Gelbart, Stephen; Piatetski-Shapiro, Ilya; Rallis, Stephen Explicit constructions of automorphic L -functions, Lecture Notes in Mathematics, 1254, Springer-Verlag, Berlin, 1987 | MR | Zbl

[7] Greenberg, Ralph Iwasawa theory and p-adic deformations of motives, Motives (Seattle, WA, 1991) (Proc. Sympos. Pure Math.), Volume 55, Amer. Math. Soc., Providence, RI, 1994, pp. 193-223 | MR | Zbl

[8] Grothendieck, A.; Demazure, M. 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). Dirigé par M. Demazure et A. Grothendieck, Lecture Notes in Mathematics, 152, Springer-Verlag, Berlin, 1962/1964 | MR

[9] Harris, Michael Eisenstein series on Shimura varieties, Ann. of Math. (2), Volume 119 (1984) no. 1, pp. 59-94 | DOI | MR | Zbl

[10] Harris, Michael; Li, Jian-Shu; Skinner, Christopher M. p-adic L-functions for unitary Shimura varieties. I. Construction of the Eisenstein measure, Doc. Math., Volume Extra Vol. (2006), p. 393-464 (electronic) | MR | Zbl

[11] Hida, Haruzo Elementary theory of L -functions and Eisenstein series, London Mathematical Society Student Texts, 26, Cambridge University Press, Cambridge, 1993 | MR | Zbl

[12] Hida, Haruzo Control theorems of coherent sheaves on Shimura varieties of PEL type, J. Inst. Math. Jussieu, Volume 1 (2002) no. 1, pp. 1-76 | DOI | MR | Zbl

[13] Hida, Haruzo p -adic automorphic forms on Shimura varieties, Springer Monographs in Mathematics, Springer-Verlag, New York, 2004 | MR | Zbl

[14] Hida, Haruzo; Tilouine, J. Katz p-adic L-functions, congruence modules and deformation of Galois representations, L -functions and arithmetic (Durham, 1989) (London Math. Soc. Lecture Note Ser.), Volume 153, Cambridge Univ. Press, Cambridge, 1991, pp. 271-293 | MR | Zbl

[15] Katz, Nicholas M. p-adic L-functions for CM fields, Invent. Math., Volume 49 (1978) no. 3, pp. 199-297 | DOI | MR | Zbl

[16] Kottwitz, R. Points on some Shimura varieties over finite fields, Journal of AMS, Volume 5 (1992) no. 2, pp. 373-443 | MR | Zbl

[17] Li, Jian-Shu Nonvanishing theorems for the cohomology of certain arithmetic quotients, J. Reine Angew. Math., Volume 428 (1992), pp. 177-217 | DOI | MR | Zbl

[18] Mazur, Barry; Wiles, A. Class fields of abelian extensions of Q, Invent. Math., Volume 76 (1984) no. 2, pp. 179-330 | DOI | MR | Zbl

[19] Mœglin, C.; Waldspurger, J.-L. Spectral decomposition and Eisenstein series, Cambridge Tracts in Mathematics, 113, Cambridge University Press, Cambridge, 1995 (Une paraphrase de l’Écriture [A paraphrase of Scripture]) | MR | Zbl

[20] Ribet, Kenneth A. A modular construction of unramified p-extensions of (μ p ), Invent. Math., Volume 34 (1976) no. 3, pp. 151-162 | DOI | MR | Zbl

[21] Shimura, Goro Confluent hypergeometric functions on tube domains, Math. Ann., Volume 260 (1982) no. 3, pp. 269-302 | DOI | MR | Zbl

[22] Shimura, Goro Euler products and Eisenstein series, CBMS Regional Conference Series in Mathematics, 93, published for the Conference Board of the Mathematical Sciences, Washington, DC, 1997 | MR

[23] Shimura, Goro; Taniyama, Yutaka Complex multiplication of abelian varieties and its applications to number theory, Publications of the Mathematical Society of Japan, 6, The Mathematical Society of Japan, Tokyo, 1961 | MR | Zbl

[24] Skinner, Christopher Towards Main Conjectures for Modular Forms, RIMS Kokyuroku, Volume 1468 (2006), pp. 149-157 | MR | Zbl

[25] Skinner, Christopher; Urban, Eric The Iwasawa main conjecture for GL 2 (Oct 2008) (preprint)

[26] Tilouine, J.; Urban, Eric Several-variable p-adic families of Siegel-Hilbert cusp eigensystems and their Galois representations, Ann. Sci. École Norm. Sup. (4), Volume 32 (1999) no. 4, pp. 499-574 | Numdam | MR | Zbl

[27] Urban, Eric Selmer groups and the Eisenstein-Klingen ideal, Duke Math. J., Volume 106 (2001) no. 3, pp. 485-525 | DOI | MR | Zbl

[28] Urban, Eric Groupes de Selmer et Fonctions L p -adiques pour les Représentations Modulaires Adjointes (2006) (preprint)

[29] Wiles, A. The Iwasawa conjecture for totally real fields, Ann. of Math. (2), Volume 131 (1990) no. 3, pp. 493-540 | DOI | MR | Zbl

[30] Zhang, B. Fourier-Jacobi Expansion of Eisenstein series on nonsplit unitary groups, Columbia University (2007) (Ph. D. Thesis)

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