The Tamagawa number conjecture of adjoint motives of modular forms
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 37 (2004) no. 5, pp. 663-727.
@article{ASENS_2004_4_37_5_663_0,
     author = {Diamond, Fred and Flach, Matthias and Guo, Li},
     title = {The {Tamagawa} number conjecture of adjoint motives of modular forms},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {663--727},
     publisher = {Elsevier},
     volume = {Ser. 4, 37},
     number = {5},
     year = {2004},
     doi = {10.1016/j.ansens.2004.09.001},
     mrnumber = {2103471},
     zbl = {02136287},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.ansens.2004.09.001/}
}
TY  - JOUR
AU  - Diamond, Fred
AU  - Flach, Matthias
AU  - Guo, Li
TI  - The Tamagawa number conjecture of adjoint motives of modular forms
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2004
SP  - 663
EP  - 727
VL  - 37
IS  - 5
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.ansens.2004.09.001/
DO  - 10.1016/j.ansens.2004.09.001
LA  - en
ID  - ASENS_2004_4_37_5_663_0
ER  - 
%0 Journal Article
%A Diamond, Fred
%A Flach, Matthias
%A Guo, Li
%T The Tamagawa number conjecture of adjoint motives of modular forms
%J Annales scientifiques de l'École Normale Supérieure
%D 2004
%P 663-727
%V 37
%N 5
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.ansens.2004.09.001/
%R 10.1016/j.ansens.2004.09.001
%G en
%F ASENS_2004_4_37_5_663_0
Diamond, Fred; Flach, Matthias; Guo, Li. The Tamagawa number conjecture of adjoint motives of modular forms. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 37 (2004) no. 5, pp. 663-727. doi : 10.1016/j.ansens.2004.09.001. http://www.numdam.org/articles/10.1016/j.ansens.2004.09.001/

[1] Grothendieck A. et al. , Séminaire de Géométrie Algébrique 1, Lecture Notes in Math., vol. 224, Springer-Verlag, 1971.

[2] Artin M. et al. , Séminaire de Géométrie Algébrique 4, vol. 3, Lecture Notes in Math., vol. 305, Springer-Verlag, 1973. | MR

[3] Berger L., Représentations p-adiques et équations différentielles, Invent. Math. 148 (2) (2002) 219-284. | MR | Zbl

[4] Bloch S., Kato K., L-functions and Tamagawa numbers of motives, in: The Grothendieck Festschrift, vol. 1, Birkhäuser, 1990, pp. 333-400. | MR | Zbl

[5] Breuil C., Conrad B., Diamond F., Taylor R., On the modularity of elliptic curves over Q: Wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001) 843-939. | MR | Zbl

[6] Brown K., Cohomology of Groups, Springer-Verlag, 1982. | MR | Zbl

[7] Burns D., Flach M., Tamagawa numbers for motives with (noncommutative) coefficients, Documenta Mathematica 6 (2001) 501-570. | MR | Zbl

[8] Burns D., Greither C., On the equivariant Tamagawa number conjecture for Tate motives, Invent. Math. 153 (2003) 303-359. | MR | Zbl

[9] Carayol H., Sur les représentations -adiques attachées aux formes modulaires de Hilbert, C. R. Acad. Sci., Paris 296 (1983) 629-632. | MR | Zbl

[10] Coates J., Schmidt C.-G., Iwasawa theory for the symmetric square of an elliptic curve, J. Reine Angew. Math. 375/376 (1987) 104-156. | MR | Zbl

[11] Coates J., Wiles A., On the conjecture of Birch and Swinnerton-Dyer, Invent. Math. 39 (1977) 223-251. | MR | Zbl

[12] Colmez P., Théorie d'Iwasawa des représentations de de Rham d'un corps local, Ann. of Math. (2) 148 (2) (1998) 485-571. | MR | Zbl

[13] Conrad B., Diamond F., Taylor R., Modularity of certain potentially Barsotti-Tate Galois representations, J. AMS 12 (1999) 521-567. | MR | Zbl

[14] Darmon H., Diamond F., Taylor R., Fermat's Last Theorem, in: Current Development in Mathematics, International Press, 1995, pp. 1-154. | MR | Zbl

[15] Deligne P., Formes modulaires et représentations -adiques, in: Séminaire Bourbaki 1968/1969, exposé 255, Lecture Notes in Math., vol. 179, 1969, pp. 139-172. | Numdam | Zbl

[16] Deligne P., Les constantes de l'équation fonctionnelle des fonctions L, in: Modular Functions of One Variable. II, Lecture Notes in Math., vol. 349, Springer-Verlag, Berlin, 1997, pp. 501-595. | MR | Zbl

[17] Deligne P., Valeurs de fonctions L et périodes d'intégrales, in: Automorphic Forms, Representations and L-functions, Proc. Symp. Pure Math., American Math. Soc., vol. 33, 1979, pp. 313-346. | MR | Zbl

[18] Deligne P., Rapoport M., Les schémas de modules de courbes elliptiques, in: Lecture Notes in Math., vol. 349, 1973, pp. 143-316. | MR | Zbl

[19] De Smit B., Lenstra H., Explicit construction of universal deformation rings, in: Elliptic Curves, Modular Forms and Fermat's Last Theorem, International Press, Cambridge, 1995, pp. 313-326. | MR | Zbl

[20] Diamond F., Congruence primes for cusp forms of weight k2, Astérisque 196-197 (1991) 205-213. | MR | Zbl

[21] Diamond F., The refined conjecture of Serre, in: Elliptic Curves, Modular Forms and Fermat's Last Theorem, International Press, Cambridge, 1995, pp. 22-37. | MR | Zbl

[22] Diamond F., On deformation rings and Hecke rings, Annals of Math. 144 (1996) 137-166. | MR | Zbl

[23] Diamond F., An extension of Wiles' results, in: Modular Forms and Fermat's Last Theorem, Springer-Verlag, 1997, pp. 475-489. | MR | Zbl

[24] Diamond F., The Taylor-Wiles construction and multiplicity one, Invent. Math. 128 (1997) 379-391. | MR | Zbl

[25] Diamond, F., Flach, M., Guo, L., Adjoint motives of modular forms and the Tamagawa number conjecture, preprint.

[26] Diamond F., Taylor R., Non-optimal levels of mod modular representations, Invent. Math. 115 (1994) 435-462. | MR | Zbl

[27] Diamond F., Taylor R., Lifting modular mod representations, Duke Math. J. 74 (1994) 253-269. | MR | Zbl

[28] Dickinson M., On the modularity of certain 2-adic Galois representations, Duke Math. J. 109 (2001) 319-382. | MR | Zbl

[29] Dimitrov, M., Galois representations modulo p and cohomology of Hilbert modular varieties, Prépub. Math. de l'Univ. Paris 13, 2004-02. | MR

[30] Edixhoven B., Serre's conjecture, in: Modular Forms and Fermat's Last Theorem, Springer-Verlag, 1997, pp. 209-242. | MR | Zbl

[31] Faltings G., Crystalline cohomology and p-adic étale cohomology, in: Algebraic Analysis, Geometry and Number Theory, The John Hopkins University Press, 1989, pp. 25-80. | MR | Zbl

[32] Flach, M., Selmer groups for the symmetric square of an elliptic curve, thesis, Cambridge University, 1990.

[33] Flach M., A generalization of the Cassels-Tate pairing, J. Reine Angew. Math. 412 (1990) 113-127. | MR | Zbl

[34] Flach M., A finiteness theorem for the symmetric square of an elliptic curve, Invent. Math. 109 (1992) 307-327. | MR | Zbl

[35] Flach M., The equivariant Tamagawa number conjecture: a survey, in: Sands J., (Eds.), Proceedings of a Conference on Stark's Conjecture, Baltimore, 2002, Contemp. Math. Ser., AMS, 2004. | MR | Zbl

[36] Fontaine J.-M., Modules galoisiens, modules filtrés et anneaux de Barsotti-Tate, in: Journées de Géométrie Algébrique de Rennes (III), Astérisque, vol. 65, Soc. Math. de France, 1979, pp. 3-80. | Numdam | MR | Zbl

[37] 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. Math. 115 (1982) 529-577. | MR | Zbl

[38] Fontaine J.-M., Valeurs spéciales des fonctions L des motifs, in: Séminaire Bourbaki, exposé 751, février 1992, Astérisque, vol. 206, 1992, pp. 205-249. | Numdam | MR | Zbl

[39] Fontaine J.-M., Laffaille G., Construction de représentations p-adiques, Ann. Sci. Éc. Norm. Sup. 15 (1982) 547-608. | Numdam | MR | Zbl

[40] Fontaine J.-M., Mazur B., Geometric Galois representations, in: Elliptic Curves, Modular Forms and Fermat's Last Theorem, International Press, 1995, pp. 41-78. | MR | Zbl

[41] Fontaine J.-M., Perrin-Riou B., Autour des conjectures de Bloch et Kato: cohomologie galoisienne et valeurs de fonction L, in: Motives, Proc. Symp. in Pure Math., vol. 55, 1994, pp. 599-706, Part 1. | MR | Zbl

[42] Fujiwara, K., Deformation rings and Hecke algebras in the totally real case, preprint. | MR

[43] Gelbart S., Jacquet H., A relation between automorphic representations of GL2 and GL3, Ann. Sci. Éc. Norm. Sup., IV. Ser. 11 (1978) 471-542. | Numdam | MR | Zbl

[44] Gérardin P., Facteurs locaux des algèbres simples de rang 4. I, in: Groupes Réductifs et Formes Automorphes, I (Paris, 1976-77), Univ. Paris VII, 1978, pp. 37-77. | MR

[45] Gross B., Zagier D., Heegner points and derivatives of L-series, Invent. Math. 84 (1986) 225-320. | MR | Zbl

[46] Grothendieck A., Murre J.P., The Tame Fundamental Group of a Formal Neighbourhood of a Divisor with Normal Crossings on a Scheme, Springer-Verlag, 1971. | MR | Zbl

[47] Guo L., General Selmer groups and critical values of Hecke L-functions, Math. Ann. 297 (1993) 221-233. | MR | Zbl

[48] Guo L., On the Bloch-Kato conjecture for Hecke L-functions, J. Number Theory 57 (1996) 340-365. | MR | Zbl

[49] Harrison, M., On the conjecture of Bloch-Kato for Grössencharacters over Qi, Ph.D. thesis, Cambridge University, 1993.

[50] Hartshorne R., Algebraic Geometry, Springer-Verlag, Berlin, 1977. | MR | Zbl

[51] Hida H., Congruences of cusp forms and special values of their zeta functions, Invent. Math. 63 (1981) 225-261. | MR | Zbl

[52] Huber A., Kings G., Bloch-Kato conjecture and main conjecture of Iwasawa theory for Dirichlet characters, Duke Math. J. 119 (2003) 393-464. | MR | Zbl

[53] Jannsen U., Mixed Motives and Algebraic K-theory, Lect. Notes in Math., vol. 1400, Springer, 1990. | MR | Zbl

[54] Kato K., Logarithmic structures of Fontaine-Illusie, in: Algebraic Analysis, Geometry and Number Theory, The John Hopkins University Press, 1989, pp. 191-224. | MR | Zbl

[55] Kato K., Iwasawa theory and p-adic Hodge theory, Kodai Math. J. 16 (1993) 1-31. | MR | Zbl

[56] Kato K., Euler systems, Iwasawa theory and Selmer groups, Kodai Math. J. 22 (1999) 313-372. | MR | Zbl

[57] Kings G., The Tamagawa number conjecture for CM elliptic curves, Invent. Math. 143 (2001) 571-627. | MR | Zbl

[58] Kolyvagin V.A., The Mordell-Weil and Shafarevich-Tate groups for Weil elliptic curves, Izv. Akad. Nauk SSSR Ser. Mat. 52 (6) (1988) 1154-1180, 1327. Translation in, Math. USSR-Izv. 33 (3) (1989) 473-499. | MR | Zbl

[59] Kolyvagin V.A., Logachev Yu.D., Finiteness of Ш over totally real fields, Izv. Akad. Nauk SSSR Ser. Mat. 55 (4) (1991) 851-876, Translation in, Math. USSR-Izv. 39 (1) (1992) 829-853. | Zbl

[60] Lenstra H., Complete intersections and Gorenstein rings, in: Elliptic Curves, Modular Forms and Fermat's Last Theorem, International Press, Cambridge, 1995. | MR | Zbl

[61] Mazur B., An introduction to the deformation theory of Galois representations, in: Modular Forms and Fermat's Last Theorem, Springer-Verlag, 1997, pp. 243-311. | MR | Zbl

[62] Mazur B., Wiles A., Class fields of abelian extensions of Q, Invent. Math. 76 (1984) 179-330. | MR | Zbl

[63] Miyake T., Modular Forms, Springer-Verlag, 1989. | MR | Zbl

[64] Nekovar J., On the p-adic height of Heegner cycles, Math. Ann. 302 (4) (1995) 609-686. | MR | Zbl

[65] Ogg , On a convolution of L-series, Invent. Math. 7 (1969) 297-312. | MR | Zbl

[66] Perrin-Riou B., Théorie d'Iwasawa des représentations p-adiques sur un corps local, Invent. Math 115 (1994) 81-149. | MR | Zbl

[67] Perrin-Riou B., Fonctions L p-adiques des représentations p-adiques, Astérisque 229 (1995). | Numdam | MR | Zbl

[68] Ribet K., Congruence relations between modular forms, Proc. ICM 17 (1983) 503-514. | MR | Zbl

[69] Rubin K., Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication, Invent. Math. 89 (1987) 527-560. | MR | Zbl

[70] Rubin K., The “main conjecture” of Iwasawa theory for imaginary quadratic fields, Invent. Math. 103 (1991) 25-68. | Zbl

[71] Saito T., Modular forms and p-adic Hodge theory, Invent. Math. 129 (1997) 607-620. | MR | Zbl

[72] Savitt, D., Modularity of some potentially Barsotti-Tate Galois representations, Thesis, Harvard University, 2001.

[73] Schmidt C.-G., p-adic measures attached to automorphic representations of GL3, Invent. Math. 92 (1988) 597-631. | MR | Zbl

[74] Scholl A.J., Modular forms and de Rham cohomology; Atkin-Swinnerton-Dyer congruences, Invent. Math. 79 (1985) 49-77. | MR | Zbl

[75] Scholl A.J., Motives for modular forms, Invent. Math. 100 (1990) 419-430. | MR | Zbl

[76] Serre J.-P., Géométrie algébrigue et géométrie analytique, Ann. Inst. Fourier 6 (1956) 1-42. | Numdam | MR | Zbl

[77] Serre J.-P., Local Fields, Springer-Verlag, 1979. | MR | Zbl

[78] Serre J.-P., Sur les représentations modulaires de degré 2 de Gal (Q ¯/Q), Duke Math. J. 54 (1987) 179-230. | MR | Zbl

[79] Shimura G., Introduction to the Arithmetic Theory of Automorphic Functions, Iwanami Shoten and Princeton University Press, 1971. | MR | Zbl

[80] Shimura G., On the holomorphy of certain Dirichlet series, Proc. London Math. Soc. 31 (1975) 79-98. | MR | Zbl

[81] Shimura G., On the periods of modular forms, Math. Ann. 229 (1977) 211-221. | MR | Zbl

[82] Skinner C.M., Wiles A., Ordinary representations and modular forms, Proc. Nat. Acad. Sci. USA 94 (20) (1997) 10520-10527. | MR | Zbl

[83] Skinner C.M., Wiles A., Residually reducible representations and modular forms, IHÉS Publ. 89 (1999) 5-126, (2000). | Numdam | MR | Zbl

[84] Sturm J., Special values of zeta functions and Eisenstein series of half integral weight, Amer. J. Math. 102 (1980) 219-240. | MR | Zbl

[85] Sturm J., Evaluation of the symmetric square at the near center point, Amer. J. Math. 111 (1989) 585-598. | MR | Zbl

[86] Taylor R., Wiles A., Ring theoretic properties of certain Hecke algebras, Annals of Math. 141 (1995) 553-572. | MR | Zbl

[87] Tsuji T., p-adic étale cohomology and crystalline cohomology in the semi-stable reduction case, Invent. Math. 137 (2000) 233-411. | MR | Zbl

[88] Wiles A., Modular elliptic curves and Fermat's Last Theorem, Annals of Math. 141 (1995) 443-551. | MR | Zbl

[89] Zhang S., Heights of Heegner cycles and derivatives of L-series, Invent. Math. 130 (1997) 99-152. | MR | Zbl

Cited by Sources: