Non-abelian $p$-adic $L$-functions and Eisenstein series of unitary groups – The CM method

Bouganis, Thanasis

doi:10.5802/aif.2866

Non-abelian

p

-adic

L

-functions and Eisenstein series of unitary groups – The CM method
[

L

-fonctions

p

-adiques non-abéliennes et série d’Eisenstein pour les groupes unitaires – La méthode CM]

Bouganis, Thanasis ¹

¹ Department of Mathematical Sciences Durham University Science Laboratories, South Rd. Durham DH1 3LE (U.K.)

Annales de l'Institut Fourier, Tome 64 (2014) no. 2, pp. 793-891.

Résumé
Abstract

Dans cet article, nous démontrons divers cas particuliers de “congruences de torsion” entre les $L$ -fonctions $p$ -adiques abéliennes liées aux représentations automorphes de groupes unitaires définis. Ces congruences jouent un rôle central dans la théorie d’ Iwasawa non-commutative, ce qui a été mis en évidence par les résultats de Kakde, Ritter et Weiss sur la Conjecture Principale non-abélienne pour le motif de Tate. Nous nous attaquons à ces congruences pour un groupe unitaire défini général en $n$ variables, et obtenons des résultats plus explicites dans les cas $n = 1$ et $n = 2$ . Dans ces deux cas, nous expliquons aussi leur conséquences pour certains “motifs” particuliers, comme par exemple, les courbes elliptiques munies d’une multiplication complexe. Finalement, nous discutons d’un nouveau type de congruences que nous nommons “congruences de torsion modérées”.

In this work we prove various cases of the so-called “torsion congruences” between abelian $p$ -adic $L$ -functions that are related to automorphic representations of definite unitary groups. These congruences play a central role in the non-commutative Iwasawa theory as it became clear in the works of Kakde, Ritter and Weiss on the non-abelian Main Conjecture for the Tate motive. We tackle these congruences for a general definite unitary group of $n$ variables and we obtain more explicit results in the special cases of $n = 1$ and $n = 2$ . In both of these cases we also explain their implications for some particular “motives”, as for example elliptic curves with complex multiplication. Finally we also discuss a new kind of congruences, which we call “average torsion congruences”

MR Zbl

DOI : 10.5802/aif.2866

Classification : 11R23, 11F55, 11F67, 11M36
Keywords: ($p$-adic) $L$-functions, Eisenstein Series, Unitary Groups, Congruences
Mot clés : $L$-fonctions $p$-adiques, séries d’Eisenstein, Groupes unitaires, congruences

Affiliations des auteurs :

Bouganis, Thanasis ¹

¹ Department of Mathematical Sciences Durham University Science Laboratories, South Rd. Durham DH1 3LE (U.K.)

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     pages = {793--891},
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Bouganis, Thanasis. Non-abelian $p$-adic $L$-functions and Eisenstein series of unitary groups – The CM method. Annales de l'Institut Fourier, Tome 64 (2014) no. 2, pp. 793-891. doi : 10.5802/aif.2866. https://aif.centre-mersenne.org/articles/10.5802/aif.2866/

Bibliographie
Cité par

[1] Bouganis, Thanasis Non abelian $p$ -adic $L$ -functions and Eisenstein series of unitary groups II; the CM-method (in preparation)

[2] Bouganis, Thanasis Non abelian $p$ -adic $L$ -functions and Eisenstein series of unitary groups; the Constant Term Method (in preparation)

[3] Bouganis, Thanasis Special values of $L$ -functions and false Tate curve extensions, J. Lond. Math. Soc. (2), Volume 82 (2010) no. 3, pp. 596-620 (With an appendix by Vladimir Dokchitser) | DOI | MR | Zbl

[4] Bouganis, Thanasis Non-abelian congruences between special values of $L$ -functions of elliptic curves: the CM case, Int. J. Number Theory, Volume 7 (2011) no. 7, pp. 1883-1934 | DOI | MR | Zbl

[5] Bouganis, Thanasis; Nuccio, F. Kongruenzen zwischen abelschen pseudo-Maßen und die Shintani Zerlegung (preprint in German)

[6] Bouganis, Thanasis; Venjakob, Otmar On the non-commutative main conjecture for elliptic curves with complex multiplication, Asian J. Math., Volume 14 (2010) no. 3, pp. 385-416 | DOI | MR | Zbl

[7] 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

[8] Coates, John; Fukaya, Takako; Kato, Kazuya; Sujatha, Ramdorai; Venjakob, Otmar The ${GL}_{2}$ main conjecture for elliptic curves without complex multiplication, Publ. Math. Inst. Hautes Études Sci. (2005) no. 101, pp. 163-208 | DOI | EuDML | Numdam | MR | Zbl

[9] Delbourgo, Daniel; Ward, Thomas Non-abelian congruences between $L$ -values of elliptic curves, Ann. Inst. Fourier (Grenoble), Volume 58 (2008) no. 3, pp. 1023-1055 | DOI | EuDML | Numdam | MR | Zbl

[10] Delbourgo, Daniel; Ward, Thomas The growth of CM periods over false Tate extensions, Experiment. Math., Volume 19 (2010) no. 2, pp. 195-210 | DOI | MR | Zbl

[11] Dokchitser, T.; Dokchitser, V. Computations in non-commutative Iwasawa theory, Proc. Lond. Math. Soc. (3), Volume 94 (2007) no. 1, pp. 211-272 (With an appendix by J. Coates and R. Sujatha) | DOI | MR | Zbl

[12] Eischen, Ellen E. A $p$ -adic Eisenstein Measure for Unitary Groups, 2011 (Preprint arXiv:1106.3692v1, to appear in J. Reine Angew. Math.) | MR | Zbl

[13] Eischen, Ellen E. $p$ -adic differential operators on automorphic forms on unitary groups, Ann. Inst. Fourier (Grenoble), Volume 62 (2012) no. 1, pp. 177-243 | DOI | EuDML | Numdam | MR | Zbl

[14] Eischen, Ellen E.; Harris, M.; Li, J.-S.; Skinner, C. $p$ -adic $L$ -functions for unitary Shimura varieties, II (in preparation)

[15] Fukaya, Takako; Kato, Kazuya A formulation of conjectures on $p$ -adic zeta functions in noncommutative Iwasawa theory, Proceedings of the St. Petersburg Mathematical Society. Vol. XII (Amer. Math. Soc. Transl. Ser. 2), Volume 219 (2006), pp. 1-85 | MR | Zbl

[16] Garrett, Paul Values of Archimedean zeta integrals for unitary groups, Eisenstein series and applications (Progr. Math.), Volume 258, Birkhäuser Boston, Boston, MA, 2008, pp. 125-148 | MR | Zbl

[17] Gelbart, Stephen; Piatetski-Shapiro, Ilya; Rallis, Stephen Explicit constructions of automorphic $L$ -functions, Lecture Notes in Mathematics, 1254, Springer-Verlag, Berlin, 1987, pp. vi+152 | MR | Zbl

[18] Godement, Roger; Jacquet, Hervé Zeta functions of simple algebras, Lecture Notes in Mathematics, Vol. 260, Springer-Verlag, Berlin-New York, 1972, pp. ix+188 | MR | Zbl

[19] Hara, Takashi Iwasawa theory of totally real fields for certain non-commutative $p$ -extensions, J. Number Theory, Volume 130 (2010) no. 4, pp. 1068-1097 | DOI | MR | Zbl

[20] Harris, Michael Unitary groups and Base Change (notes available at http://www.math.jussieu.fr/~harris/)

[21] Harris, Michael $L$ -functions of $2 \times 2$ unitary groups and factorization of periods of Hilbert modular forms, J. Amer. Math. Soc., Volume 6 (1993) no. 3, pp. 637-719 | MR | Zbl

[22] Harris, Michael $L$ -functions and periods of polarized regular motives, J. Reine Angew. Math., Volume 483 (1997), pp. 75-161 | EuDML | MR | Zbl

[23] Harris, Michael A simple proof of rationality of Siegel-Weil Eisenstein series, Eisenstein series and applications (Progr. Math.), Volume 258, Birkhäuser Boston, Boston, MA, 2008, pp. 149-185 | MR | Zbl

[24] Harris, Michael; Li, Jian-Shu; Skinner, Christopher M. The Rallis inner product formula and $p$ -adic $L$ -functions, Automorphic representations, $L$ -functions and applications: progress and prospects (Ohio State Univ. Math. Res. Inst. Publ.), Volume 11, de Gruyter, Berlin, 2005, pp. 225-255 | MR | Zbl

[25] 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) no. Extra Vol., p. 393-464 (electronic) | EuDML | MR | Zbl

[26] Hida, H.; Tilouine, J. Anti-cyclotomic Katz $p$ -adic $L$ -functions and congruence modules, Ann. Sci. École Norm. Sup. (4), Volume 26 (1993) no. 2, pp. 189-259 | EuDML | Numdam | MR | Zbl

[27] Hida, Haruzo $p$ -adic automorphic forms on Shimura varieties, Springer Monographs in Mathematics, Springer-Verlag, New York, 2004, pp. xii+390 | MR | Zbl

[28] Hida, Haruzo Serre’s conjecture and base change for $GL (2)$ , Pure Appl. Math. Q., Volume 5 (2009) no. 1, pp. 81-125 | DOI | MR | Zbl

[29] Hsieh, Ming-Lun Ordinary $p$ -adic Eisenstein series and $p$ -adic $L$ -functions for unitary groups, Ann. Inst. Fourier (Grenoble), Volume 61 (2011) no. 3, pp. 987-1059 | DOI | EuDML | Numdam | MR | Zbl

[30] Hsieh, Ming-Lun Eisenstein congruence on unitary groups and Iwasawa main conjectures for CM fields, J. Amer. Math. Soc., Volume 27 (2014) no. 3, pp. 753-862 | DOI | MR | Zbl

[31] Kakde, Mahesh Proof of the main conjecture of noncommutative Iwasawa theory for totally real number fields in certain cases, J. Algebraic Geom., Volume 20 (2011) no. 4, pp. 631-683 | DOI | MR | Zbl

[32] Kakde, Mahesh From the classical to the noncommutative Iwasawa theory (for totally real number fields), Non-abelian fundamental groups and Iwasawa theory (London Math. Soc. Lecture Note Ser.), Volume 393, Cambridge Univ. Press, Cambridge, 2012, pp. 107-131 | MR | Zbl

[33] Kato, K. Iwasawa theory of totally real fields for Galois extensions of Heisenberg type (preprint)

[34] Katz, Nicholas M. $p$ -adic interpolation of real analytic Eisenstein series, Ann. of Math. (2), Volume 104 (1976) no. 3, pp. 459-571 | DOI | MR | Zbl

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

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

[37] Ritter, Jürgen; Weiss, Alfred Congruences between abelian pseudomeasures, Math. Res. Lett., Volume 15 (2008) no. 4, pp. 715-725 | DOI | MR | Zbl

[38] Ritter, Jürgen; Weiss, Alfred Congruences between abelian pseudomeasures, II, 2010 (Preprint arXiv:1001.2091v1)

[39] de Shalit, Ehud On monomial relations between $p$ -adic periods, J. Reine Angew. Math., Volume 374 (1987), pp. 193-207 | EuDML | MR | Zbl

[40] 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; by the American Mathematical Society, Providence, RI, 1997, pp. xx+259 | MR | Zbl

[41] Shimura, Goro Abelian varieties with complex multiplication and modular functions, Princeton Mathematical Series, 46, Princeton University Press, Princeton, NJ, 1998, pp. xvi+218 | MR | Zbl

[42] Shimura, Goro Arithmeticity in the theory of automorphic forms, Mathematical Surveys and Monographs, 82, American Mathematical Society, Providence, RI, 2000, pp. x+302 | MR | Zbl

[43] Skinner, Christopher; Urban, Eric The Iwasawa main conjectures for ${GL}_{2}$ , Invent. Math., Volume 195 (2014) no. 1, pp. 1-277 | DOI | MR | Zbl

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