Congruences among modular forms on U(2,2) and the Bloch-Kato conjecture
Annales de l'Institut Fourier, Volume 59 (2009) no. 1, p. 81-166

Let k be a positive integer divisible by 4, p>k a prime, f an elliptic cuspidal eigenform (ordinary at p) of weight k-1, level 4 and non-trivial character. In this paper we provide evidence for the Bloch-Kato conjecture for the motives ad 0 M(-1) and ad 0 M(2), where M is the motif attached to f. More precisely, we prove that under certain conditions the p-adic valuation of the algebraic part of the symmetric square L-function of f evaluated at k provides a lower bound for the p-adic valuation of the order of the Pontryagin dual of the Selmer group for the adjoint of the p-adic Galois representation attached to f restricted to the Gaussian field and twisted by the inverse of the cyclotomic character. Our method uses an idea of Ribet, in that we introduce an intermediate step and produce congruences between CAP and non-CAP modular forms on the unitary group U(2,2).

Soit k un entier strictement positif divisible par 4, soit p>k un nombre premier, et soit f une forme modulaire elliptique de poids k-1, de niveau 4, et de caractère non trivial qui est propre pour les opérateurs de Hecke et ordinaire en p. Dans cet article, on donne des résultats sur la conjecture de Bloch-Kato pour les motifs ad 0 M(-1) et ad 0 M(2), où M est le motif associé à f. Plus précisément, on démontre que, sous certaines hypothèses, la valuation p-adique de la part algébrique de la valeur de la fonction L du carré symétrique de f en k est bornée par la valuation p-adique de la valeur de l’ordre du groupe de Selmer de l’adjoint de la représentation p-adique galoisienne associée à f restreinte au corps gaussien et tordue par l’inverse du caractère cyclotomique. Notre méthode suit une idée de Ribet, dans le sens où nous introduisons une étape intermédiaire et produisons des congruences entre formes CAP et formes non CAP sur le groupe unitaire U(2,2).

DOI : https://doi.org/10.5802/aif.2427
Classification:  11F33,  11F55,  11F67,  11F80
Keywords: Automorphic forms on unitary groups, congruences, Selmer groups, Bloch-Kato conjecture
@article{AIF_2009__59_1_81_0,
     author = {Klosin, Krzysztof},
     title = {Congruences among modular forms on U(2,2) and the Bloch-Kato conjecture},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {59},
     number = {1},
     year = {2009},
     pages = {81-166},
     doi = {10.5802/aif.2427},
     mrnumber = {2514862},
     zbl = {1214.11055},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2009__59_1_81_0}
}
Congruences among modular forms on U(2,2) and the Bloch-Kato conjecture. Annales de l'Institut Fourier, Volume 59 (2009) no. 1, pp. 81-166. doi : 10.5802/aif.2427. http://www.numdam.org/item/AIF_2009__59_1_81_0/

[1] Bellaïche, J.; Chenevier, G. p -adic families of Galois representations and higher rank Selmer groups (Astérisque)

[2] Bellaïche, J.; Chenevier, G. Formes non tempérées pour U(3) et conjectures de Bloch-Kato, Ann. Sci. École Norm. Sup. (4), Tome 37 (2004) no. 4, pp. 611-662 | Numdam | MR 2097894 | Zbl pre02165094

[3] Berger, T. An Eisenstein ideal for imaginary quadratic fields and the Bloch-Kato conjecture for Hecke characters (2007) (arXiv:math.NT/0701177)

[4] Blasius, D.; Rogawski, J. D. Zeta functions of Shimura varieties, Motives (Seattle, WA, 1991), Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) Tome 55 (1994), pp. 525-571 | MR 1265563 | Zbl 0827.11033

[5] Bloch, S.; Kato, K. L-functions and Tamagawa numbers of motives, The Grothendieck Festschrift, Vol. I, Birkhäuser Boston, Boston, MA (Progr. Math.) Tome 86 (1990), pp. 333-400 | MR 1086888 | Zbl 0768.14001

[6] Braun, H. Hermitian modular functions, Ann. of Math. (2), Tome 50 (1949), pp. 827-855 | Article | MR 32699 | Zbl 0038.23803

[7] Braun, H. Hermitian modular functions. II. Genus invariants of Hermitian forms, Ann. of Math. (2), Tome 51 (1950), pp. 92-104 | Article | MR 32700 | Zbl 0038.23901

[8] Braun, H. Hermitian modular functions. III, Ann. of Math. (2), Tome 53 (1951), pp. 143-160 | Article | MR 39005 | Zbl 0041.41603

[9] Brown, J. Saito-Kurokawa lifts and applications to the Bloch-Kato conjecture, Compos. Math., Tome 143 (2007) no. 2, pp. 290-322 | MR 2309988 | Zbl pre05150511

[10] Bump, D. Automorphic forms and representations, Cambridge University Press, Cambridge, Cambridge Studies in Advanced Mathematics, Tome 55 (1997) | MR 1431508 | Zbl 0868.11022

[11] Darmon, H.; Diamond, F.; Taylor, R. Fermat’s last theorem, Elliptic curves, modular forms & Fermat’s last theorem (Hong Kong, 1993), Internat. Press, Cambridge, MA (1997), pp. 2-140 | MR 1605752 | Zbl 0877.11035

[12] Deligne, P. Valeurs de fonctions L et périodes d’intégrales, Automorphic forms, representations and L -functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Amer. Math. Soc., Providence, R.I. (Proc. Sympos. Pure Math., XXXIII) (1979), pp. 313-346 (With an appendix by N. Koblitz and A. Ogus) | Zbl 0449.10022

[13] Diamond, F.; Im, J. Modular forms and modular curves, Seminar on Fermat’s Last Theorem (Toronto, ON, 1993–1994), Amer. Math. Soc., Providence, RI (CMS Conf. Proc.) Tome 17 (1995), pp. 39-133 | MR 1357209 | Zbl 0853.11032

[14] Diamond, Fred; Flach, Matthias; Guo, Li The Tamagawa number conjecture of adjoint motives of modular forms, Ann. Sci. École Norm. Sup. (4), Tome 37 (2004) no. 5, pp. 663-727 | Numdam | MR 2103471 | Zbl 1121.11045

[15] Dummigan, N. Period ratios of modular forms, Math. Ann., Tome 318 (2000) no. 3, pp. 621-636 | Article | MR 1800772 | Zbl 1041.11037

[16] Dummigan, N. Symmetric square L-functions and Shafarevich-Tate groups, Experiment. Math., Tome 10 (2001) no. 3, pp. 383-400 | MR 1917426 | Zbl 1039.11029

[17] Dummigan, N. Symmetric square L-functions and Shafarevich-Tate groups, II (2008) (Preprint)

[18] Eisenbud, D. Commutative algebra, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 150 (1995) (With a view toward algebraic geometry) | MR 1322960 | Zbl 0819.13001

[19] Fontaine, J.-M. Sur certains types de représentations p-adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti-Tate, Ann. of Math. (2), Tome 115 (1982) no. 3, pp. 529-577 | Article | MR 657238 | Zbl 0544.14016

[20] Fontaine, J.-M.; Perrin-Riou, B. Autour des conjectures de Bloch et Kato: cohomologie galoisienne et valeurs de fonctions L, Motives (Seattle, WA, 1991), Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) Tome 55 (1994), pp. 599-706 | MR 1265546

[21] Freitag, E. Siegelsche Modulfunktionen, Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 254 (1983) | MR 871067 | Zbl 0498.10016

[22] Gelbart, S. Automorphic forms on adèle groups, Princeton University Press, Princeton, N.J. (1975) (Annals of Mathematics Studies, No. 83) | MR 379375 | Zbl 0329.10018

[23] Gritsenko, V. A. The Maass space for SU (2,2). The Hecke ring, and zeta functions, Trudy Mat. Inst. Steklov., Tome 183 (1990), p. 68-78, 223–225 (Translated in Proc. Steklov Inst. Math. 1991, no. 4, 75–86, Galois theory, rings, algebraic groups and their applications (Russian)) | MR 1092016 | Zbl 0764.11026

[24] Gritsenko, V. A. Parabolic extensions of the Hecke ring of the general linear group. II, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), Tome 183 (1990) no. Modul. Funktsii i Kvadrat. Formy. 1, p. 56-76, 165, 167 | MR 1075006 | Zbl 0748.11026

[25] Hida, H. Galois representations and the theory of p-adic Hecke algebras, Sūgaku, Tome 39 (1987) no. 2, pp. 124-139 (Sugaku Expositions 2 (1989), no. 1, 75–102) | MR 904860 | Zbl 0641.10025

[26] Hida, H. 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

[27] Hida, H. Modular forms and Galois cohomology, Cambridge University Press, Cambridge, Cambridge Studies in Advanced Mathematics, Tome 69 (2000) | MR 1779182 | Zbl 0952.11014

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

[29] Hina, T.; Sugano, T. On the local Hecke series of some classical groups over 𝔭-adic fields, J. Math. Soc. Japan, Tome 35 (1983) no. 1, pp. 133-152 | Article | MR 679080 | Zbl 0496.14014

[30] Ikeda, T. On the lifting of hermitian modular forms (2007) (Preprint) | MR 2457521 | Zbl pre05359275

[31] Iwaniec, H. Topics in classical automorphic forms, American Mathematical Society, Providence, RI, Graduate Studies in Mathematics, Tome 17 (1997) | MR 1474964 | Zbl 0905.11023

[32] Kim, H. Automorphic L-functions, Lectures on automorphic L -functions, Amer. Math. Soc., Providence, RI (Fields Inst. Monogr.) Tome 20 (2004), pp. 97-201 | MR 2071507

[33] Kings, G. The Bloch-Kato conjecture on special values of L-functions. A survey of known results, J. Théor. Nombres Bordeaux, Tome 15 (2003) no. 1, pp. 179-198 (Les XXIIèmes Journées Arithmetiques (Lille, 2001)) | Article | Numdam | MR 2019010 | Zbl 1050.11063

[34] Klosin, K. Congruences among automorphic forms on the unitary group U ( 2 , 2 ) , University of Michigan, Ann Arbor, Thesis (2006)

[35] Klosin, K. Adelic Maass spaces on U ( 2 , 2 ) (2007) (arXiv:math.NT/0706.2828)

[36] Kojima, H. An arithmetic of Hermitian modular forms of degree two, Invent. Math., Tome 69 (1982) no. 2, pp. 217-227 | Article | MR 674402 | Zbl 0502.10011

[37] Krieg, A. The Maaß spaces on the Hermitian half-space of degree 2, Math. Ann., Tome 289 (1991) no. 4, pp. 663-681 | Article | MR 1103042 | Zbl 0713.11033

[38] Langlands, R. P. On the functional equations satisfied by Eisenstein series, Springer-Verlag, Berlin (1976) (Lecture Notes in Mathematics, Vol. 544) | MR 579181 | Zbl 0332.10018

[39] Mazur, B. Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. (1977) no. 47, p. 33-186 (1978) | Article | Numdam | MR 488287 | Zbl 0394.14008

[40] Mazur, B. An introduction to the deformation theory of Galois representations, Modular forms and Fermat’s last theorem (Boston, MA, 1995), Springer, New York (1997), pp. 243-311 | MR 1638481 | Zbl 0901.11015

[41] Mazur, B.; 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

[42] Miyake, T. Modular forms, Springer-Verlag, Berlin (1989) (Translated from the Japanese by Yoshitaka Maeda) | MR 1021004 | Zbl 0701.11014

[43] Mœglin, C.; Waldspurger, J.-L. Le spectre résiduel de GL (n), Ann. Sci. École Norm. Sup. (4), Tome 22 (1989) no. 4, pp. 605-674 | Numdam | MR 1026752 | Zbl 0696.10023

[44] Raghavan, S.; Sengupta, J. A Dirichlet series for Hermitian modular forms of degree 2, Acta Arith., Tome 58 (1991) no. 2, pp. 181-201 | MR 1121080 | Zbl 0685.10022

[45] Ribet, K. 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

[46] Rubin, K. Euler systems, Princeton University Press, Princeton, NJ, Annals of Mathematics Studies, Tome 147 (2000) (Hermann Weyl Lectures. The Institute for Advanced Study) | MR 1749177 | Zbl 0977.11001

[47] Schmidt, C.-G. p-adic measures attached to automorphic representations of GL (3), Invent. Math., Tome 92 (1988) no. 3, pp. 597-631 | Article | MR 939477 | Zbl 0656.10023

[48] Serre, J.-P. Groupes de Lie l-adiques attachés aux courbes elliptiques, Les Tendances Géom. en Algébre et Théorie des Nombres, Éditions du Centre National de la Recherche Scientifique, Paris (1966), pp. 239-256 | MR 218366 | Zbl 0148.41502

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

[50] Shimura, G. 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

[51] Shimura, G. Arithmeticity in the theory of automorphic forms, American Mathematical Society, Providence, RI, Mathematical Surveys and Monographs, Tome 82 (2000) | MR 1780262 | Zbl 0967.11001

[52] Skinner, C. M. Selmer groups (2004) (Preprint)

[53] Sturm, J. Special values of zeta functions, and Eisenstein series of half integral weight, Amer. J. Math., Tome 102 (1980) no. 2, pp. 219-240 | Article | MR 564472 | Zbl 0433.10015

[54] Taylor, R.; Wiles, A. Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2), Tome 141 (1995) no. 3, pp. 553-572 | Article | MR 1333036 | Zbl 0823.11030

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

[56] Vatsal, V. Canonical periods and congruence formulae, Duke Math. J., Tome 98 (1999) no. 2, pp. 397-419 | Article | MR 1695203 | Zbl 0979.11027

[57] Vatsal, V. Special values of anticyclotomic L-functions, Duke Math. J., Tome 116 (2003) no. 2, pp. 219-261 | Article | MR 1953292 | Zbl 1065.11048

[58] Washington, L. C. Galois cohomology, Modular forms and Fermat’s last theorem (Boston, MA, 1995), Springer, New York (1997), pp. 101-120 | MR 1638477 | Zbl 0928.12003

[59] Washington, L. C. Introduction to cyclotomic fields, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 83 (1997) | MR 1421575 | Zbl 0966.11047

[60] 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

[61] Wiles, A. Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2), Tome 141 (1995) no. 3, pp. 443-551 | Article | MR 1333035 | Zbl 0823.11029