Ordinary $p$-adic Eisenstein series and $p$-adic $L$-functions for unitary groups

Hsieh, Ming-Lun

doi:10.5802/aif.2635

Ordinary

p

-adic Eisenstein series and

p

-adic

L

-functions for unitary groups

Hsieh, Ming-Lun

Annales de l'Institut Fourier, Volume 61 (2011) no. 3, p. 987-1059

Abstract
Résumé

The purpose of this work is to carry out the first step in our four-step program towards the main conjecture for ${GL}_{2} \times 𝒦^{\times}$ by the method of Eisenstein congruence on $G U (3, 1)$ , where $𝒦$ is an imaginary quadratic field. We construct a $p$ -adic family of ordinary Eisenstein series on the group of unitary similitudes $G U (3, 1)$ 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 ${GL}_{2} \times 𝒦^{\times}$ . This construction also provides a different point of view of $p$ -adic $L$ -functions of ${GL}_{2} \times 𝒦^{\times}$ .

Le but de ce travail est d’accomplir le premier pas de notre programme vers la conjecture principale pour ${GL}_{2} \times 𝒦^{\times}$ , par la methode de congruences entre séries d’Eisenstein sur $G U (3, 1)$ , 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 ${GL}_{2} \times 𝒦^{\times}$ . Cette construction donne ainsi un nouveau point de vue sur la fonction $L$ $p$ -adique de ${GL}_{2} \times 𝒦^{\times}$ .

MR 2918724 | Zbl 1271.11051 | 1 citation in Numdam

DOI : https://doi.org/10.5802/aif.2635
Classification: 11F33, 11F70, 11R23
Keywords: Eisenstein series on unitary groups, Iwasawa-Greenberg main conjectures

@article{AIF_2011__61_3_987_0,
     author = {Hsieh, Ming-Lun},
     title = {Ordinary $p$-adic Eisenstein series and $p$-adic $L$-functions for unitary groups},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {3},
     year = {2011},
     pages = {987-1059},
     doi = {10.5802/aif.2635},
     mrnumber = {2918724},
     zbl = {1271.11051},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2011__61_3_987_0}
}

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://www.numdam.org/item/AIF_2011__61_3_987_0/

References

[1] Bertolini, M.; Darmon, H. Iwasawa’s main conjecture for elliptic curves over anticyclotomic $ℤ_{p}$ -extensions, Ann. of Math. (2), Tome 162 (2005) no. 1, pp. 1-64 | Article | MR 2178960 | Zbl 1093.11037

[2] Chai, Ching-Li Compactification of Siegel moduli schemes, Cambridge University Press, Cambridge, London Mathematical Society Lecture Note Series, Tome 107 (1985) | MR 853543 | Zbl 0578.14009

[3] Chai, Ching-Li Methods for $p$ -adic monodromy, J. Inst. Math. Jussieu, Tome 7 (2008) no. 2, pp. 247-268 | Article | MR 2400722 | Zbl 1140.14019

[4] Coates, John Motivic $p$ -adic $L$ -functions, $L$ -functions and arithmetic (Durham, 1989), Cambridge Univ. Press, Cambridge (London Math. Soc. Lecture Note Ser.) Tome 153 (1991), pp. 141-172 | MR 1110392 | Zbl 0725.11029

[5] Faltings, Gerd; Chai, Ching-Li Degeneration of abelian varieties, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Tome 22 (1990) (with an appendix by David Mumford) | MR 1083353 | Zbl 0744.14031

[6] Gelbart, Stephen; Piatetski-Shapiro, Ilya; Rallis, Stephen Explicit constructions of automorphic $L$ -functions, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1254 (1987) | MR 892097 | Zbl 0612.10022

[7] Greenberg, Ralph Iwasawa theory and $p$ -adic deformations of motives, Motives (Seattle, WA, 1991), Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) Tome 55 (1994), pp. 193-223 | MR 1265554 | Zbl 0819.11046

[8] Grothendieck, A.; Demazure, M. Schémas en groupes. II: Groupes de type multiplicatif, et structure des schémas en groupes généraux, Springer-Verlag, Berlin, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck, Lecture Notes in Mathematics, Tome 152 (1962/1964) | MR 274459

[9] Harris, Michael Eisenstein series on Shimura varieties, Ann. of Math. (2), Tome 119 (1984) no. 1, pp. 59-94 | Article | MR 736560 | Zbl 0589.10030

[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., Tome Extra Vol. (2006), p. 393-464 (electronic) | MR 2290594 | Zbl 1143.11019

[11] Hida, Haruzo Elementary theory of $L$ -functions and Eisenstein series, Cambridge University Press, Cambridge, London Mathematical Society Student Texts, Tome 26 (1993) | MR 1216135 | Zbl 0942.11024

[12] Hida, Haruzo Control theorems of coherent sheaves on Shimura varieties of PEL type, J. Inst. Math. Jussieu, Tome 1 (2002) no. 1, pp. 1-76 | Article | MR 1954939 | Zbl 1039.11041

[13] Hida, Haruzo $p$ -adic automorphic forms on Shimura varieties, Springer-Verlag, New York, Springer Monographs in Mathematics (2004) | MR 2055355 | Zbl 1055.11032

[14] Hida, Haruzo; Tilouine, J. Katz $p$ -adic $L$ -functions, congruence modules and deformation of Galois representations, $L$ -functions and arithmetic (Durham, 1989), Cambridge Univ. Press, Cambridge (London Math. Soc. Lecture Note Ser.) Tome 153 (1991), pp. 271-293 | MR 1110397 | Zbl 0739.11022

[15] Katz, Nicholas M. $p$ -adic $L$ -functions for CM fields, Invent. Math., Tome 49 (1978) no. 3, pp. 199-297 | Article | MR 513095 | Zbl 0417.12003

[16] Kottwitz, R. Points on some Shimura varieties over finite fields, Journal of AMS, Tome 5 (1992) no. 2, pp. 373-443 | MR 1124982 | Zbl 0796.14014

[17] Li, Jian-Shu Nonvanishing theorems for the cohomology of certain arithmetic quotients, J. Reine Angew. Math., Tome 428 (1992), pp. 177-217 | Article | MR 1166512 | Zbl 0749.11032

[18] Mazur, Barry; Wiles, A. Class fields of abelian extensions of $Q$ , Invent. Math., Tome 76 (1984) no. 2, pp. 179-330 | Article | MR 742853 | Zbl 0545.12005

[19] Mœglin, C.; Waldspurger, J.-L. Spectral decomposition and Eisenstein series, Cambridge University Press, Cambridge, Cambridge Tracts in Mathematics, Tome 113 (1995) (Une paraphrase de l’Écriture [A paraphrase of Scripture]) | MR 1361168 | Zbl 0846.11032

[20] Ribet, Kenneth A. A modular construction of unramified $p$ -extensions of $ℚ (μ_{p})$ , Invent. Math., Tome 34 (1976) no. 3, pp. 151-162 | Article | MR 419403 | Zbl 0338.12003

[21] Shimura, Goro Confluent hypergeometric functions on tube domains, Math. Ann., Tome 260 (1982) no. 3, pp. 269-302 | Article | MR 669297 | Zbl 0502.10013

[22] Shimura, Goro Euler products and Eisenstein series, published for the Conference Board of the Mathematical Sciences, Washington, DC, CBMS Regional Conference Series in Mathematics, Tome 93 (1997) | MR 1450866

[23] Shimura, Goro; Taniyama, Yutaka Complex multiplication of abelian varieties and its applications to number theory, The Mathematical Society of Japan, Tokyo, Publications of the Mathematical Society of Japan, Tome 6 (1961) | MR 125113 | Zbl 0112.03502

[24] Skinner, Christopher Towards Main Conjectures for Modular Forms, RIMS Kokyuroku, Tome 1468 (2006), pp. 149-157 | MR 2459294 | Zbl 1179.11037

[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), Tome 32 (1999) no. 4, pp. 499-574 | Numdam | MR 1693583 | Zbl 0991.11016

[27] Urban, Eric Selmer groups and the Eisenstein-Klingen ideal, Duke Math. J., Tome 106 (2001) no. 3, pp. 485-525 | Article | MR 1813234 | Zbl 1061.11027

[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), Tome 131 (1990) no. 3, pp. 493-540 | Article | MR 1053488 | Zbl 0719.11071

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