$p$-adic Hodge theory and values of zeta functions of modular forms

p

-adic Hodge theory and values of zeta functions of modular forms

Kato, Kazuya

Cohomologie

p

-adiques et applications arithmétiques (III), Astérisque, no. 295 (2004), pp. 117-290.

MR Zbl | 20 citations dans Numdam

@incollection{AST_2004__295__117_0,
     author = {Kato, Kazuya},
     title = {$p$-adic {Hodge} theory and values of zeta functions of modular forms},
     booktitle = {Cohomologie $p$-adiques et applications arithm\'etiques (III)},
     editor = {Berthelot Pierre and Fontaine Jean-Marc and Illusie Luc and Kato Kazuya and Rapoport Michael},
     series = {Ast\'erisque},
     pages = {117--290},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {295},
     year = {2004},
     mrnumber = {2104361},
     zbl = {1142.11336},
     language = {en},
     url = {http://www.numdam.org/item/AST_2004__295__117_0/}
}

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Kato, Kazuya. $p$-adic Hodge theory and values of zeta functions of modular forms, dans Cohomologie $p$-adiques et applications arithmétiques (III), Astérisque, no. 295 (2004), pp. 117-290. http://www.numdam.org/item/AST_2004__295__117_0/

Bibliographie
Cité par

[AV] Y. Amice & J. Vélu - Distributions $p$ -adiques associées aux séries de Hecke, in Journées arithmétiques de Bordeaux, Astérisque, vol. 24-25, Société Mathématique de France, 1975, p. 119-131. | Numdam | MR | Zbl

[AS] A. Ash & G. Stevens - Modular forms in characteristic $ℓ$ and special values of their $L$ -functions, Duke Math. J. 53 (1986), p. 849-868. | DOI | MR | Zbl

[AL] A. O. L. Atkin & J. Lehner - Hecke operators on $Γ_{0} (m)$ , Math. Ann. 185 (1970), p. 134-160. | DOI | EuDML | MR | Zbl

[Be] A. Beilinson - Higher regulators and values of $L$ -functions, J. Soviet Math. 30 (1985), p. 2036-2070. | DOI | Zbl

[BBM] P. Berthelot, L. Breen & W. Messing - Théorie de Dieudonné cristalline II, Lect. Notes in Math., vol. 930, Springer, 1986, p. 849-868. | MR | Zbl

[BD] M. Bertolini & H. Darmon - A rigid analytic Gross-Zagier formula and arithmetic applications, with Appendix by B. Edixhoven, Ann. of Math. 146 (1997), p. 117-147. | DOI | MR | Zbl

[BK1] S. Bloch & K. Kato - $p$ -adic etale cohomology, Publ. Math. Inst. Hautes Études Sci. 63 (1986), p. 107-152. | DOI | EuDML | Numdam | MR | Zbl

[BK2] S. Bloch & K. Kato - Tamagawa numbers of motives and $L$ -functions, in The Grothendieck Festschrift, 1, Progress in Math., vol. 86, Birkäuser, 1990, p. 333-400. | MR | Zbl

[BCDT] C. Breuil, B. Conrad, F. Diamond & R. Taylor - On the modularity of elliptic curves over $ℚ$ , or $3$ -adic exercises, J. Amer. Math. Soc. 14 (2001), p. 849-939. | DOI | MR | Zbl

[Ca] H. Carayol - Sur les représentations $l$ -adiques associées aux formes modulaires de Hilbert, Ann. scient. Éc. Norm. Sup. 4e série 19 (1986), p. 409-468. | DOI | EuDML | Numdam | MR | Zbl

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

[CR1] R. Coleman - Division values in local fields, Invent. Math. 53 (1979), p. 91-116. | DOI | EuDML | MR | Zbl

[CR2] R. Coleman - Local units modulo cyclotomic units, Proc. Amer. Math. Soc. 89 (1983), p. 1-7. | DOI | Zbl

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

[DJ] J. Dee - Selmer groups of Hecke characters and Chow groups of self products of CM elliptic curves, preprint, http://www.math.uiuc.edu/Algebraic-Number-Theory/0162/.

[Del] P. Deligne - Formes modulaires et représentations $ℓ$ -adiques, Lect. Notes in Math., vol. 179, Springer, 1971, p. 139-172. | EuDML | Numdam | MR | Zbl

[De2] P. Deligne - Formes modulaires et représentations de $G L (2)$ , Lect. Notes in Math., vol. 349, Springer, 1973, p. 55-105. | MR | Zbl

[DR] P. Deligne & M. Rapoport - Les schémas de modules des courbes elliptiques, Lect. Notes in Math., vol. 349, Springer, 1973, p. 143-316. | MR | Zbl

[Fa1] G. Faltings - Hodge-Tate decompositions and modular forms, Math. Ann. 278 (1987), p. 133-149. | DOI | EuDML | MR | Zbl

[Fa2] G. Faltings - Crystalline cohomology and $p$ -adic Galois representations, in Algebraic Analysis, Geometry, and Number Theory, Johns Hopkins Univ. Press, 1989, p. 25-80. | MR | Zbl

[Fa3] G. Faltings - Crystalline cohomology of semi-stable curves, the $ℚ_{p}$ -theory, J. Alg. Geom. 6 (1997), p. 1-18. | MR | Zbl

[Fa4] G. Faltings - Almost étale extensions, in Cohomologie $p$ -adiques et applications arithmétiques. II, Astérisque, vol. 279, Société Mathématique de France, 2002, p. 185-270. | Numdam | MR | Zbl

[Fla] M. Flach - A generalisation of the Cassels-Tate pairing, J. reine angew. Math. 412 (1990), p. 113-127. | EuDML | MR | Zbl

[Fo1] J.-M. Fontaine - Modules galoisiens, modules filtrés et anneau de Barsotti-Tate, in Journées de Géométrie algébrique de Rennes, Astérisque, vol. 63, Société Mathématique de France, 1979, p. 3-80. | Numdam | MR | Zbl

[Fo2] J.-M. Fontaine - Sur certains types de représentations $p$ -adiques d'un corps local : construction d'un anneau de Barsotti-Tate, Ann. of Math. 115 (1982), p. 547-608. | MR | Zbl

[Fo3] J.-M. Fontaine - Représentations $p$ -adiques semi-stables, in Périodes $p$ -adiques (séminaire de Bures, 1988), Astérisque, vol. 223, Société Mathématique de France, 1994, p. 113-184. | Numdam | MR | Zbl

[FL] J.-M. Fontaine & G. Lafaille - Constructions de représentations $p$ -adiques, Ann. scient. Éc. Norm. Sup. 4e série 15 (1982), p. 547-608. | DOI | EuDML | Numdam | MR | Zbl

[FM] J.-M. Fontaine & W. Messing - $p$ -adic periods and $p$ -adic étale cohomology, Contemp. Math. 67 (1987), p. 179-207. | DOI | MR | Zbl

[FP] J.-M. Fontaine & B. Perrin-Riou - Autour des conjectures de Bloch et Kato, cohomologie galoisienne et valeurs de fonctions $L$ , Proc. Symp. Pure Math., vol. 55, American Mathematical Society, 1994, p. 599-706. | DOI | MR | Zbl

[Gr1] R. Greenberg - Iwasawa theory for $p$ -adic representations, Adv. Stud. in Pure Math., vol. 17, 1989, p. 97-137. | DOI | MR | Zbl

[Gr2] R. Greenberg - Iwasawa theory for motives, in $L$ -functions and arithmetic (Durham, 1989), London Math. Soc. Lecture Note Ser., vol. 153, Cambridge University Press, 1991, p. 211-233. | MR | Zbl

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

[Ha] B. Han - On Bloch-Kato's Tamagawa number conjecture for Hecke characters of imaginary quadratic fields, preprint.

[HK] O. Hyodo & K. Kato - Semi-stable reduction and crystalline cohomology with log poles, in Périodes $p$ -adiques (séminaire de Bures, 1988), Astérisque, vol. 223, Société Mathématique de France, 1994, p. 221-268. | Numdam | MR | Zbl

[Iw] K. Iwasawa - On $ℤ_{ℓ}$ -extensions of algebraic number fields, Ann. of Math. 98 (1973), p. 246-326. | DOI | MR | Zbl

[JS] H. Jaquet & J. Shalika - A nonvanishing theorem for zeta functions of $G L_{n}$ , Invent. Math. 38 (1976), p. 1-16. | DOI | EuDML | MR | Zbl

[KK1] K. Kato - Iwasawa theory and $p$ -adic Hodge theory, Kodai Math. J. 16 (1993), p. 1-31. | DOI | MR | Zbl

[KK2] K. Kato - Lectures on the approach to Iwasawa theory for Hasse-Weil $L$ -functions via $B_{d R}$ . I., in Arithmetic algebraic geometry (Trento, 1991), Lect. Notes in Math., vol. 1553, Springer, 1993, p. 50-163. | DOI | MR | Zbl

[KK3] K. Kato - Generalized explicit reciprocity laws, in Algebraic number theory (Hapcheon/Saga, 1996), Adv. Stud in Contemp. Math., vol. 1, 1999, p. 57-126. | MR | Zbl

[KK4] K. Kato - Euler systems, Iwasawa theory and Selmer groups, Kodai Math. J. 22 (1999), p. 313-372. | DOI | MR | Zbl

[KKT] K. Kato, M. Kurihara & T. Tsuji - Local Iwasawa theory of Perrin-Riou and syntomic complexes, I (preprint)

[KKT] K. Kato, M. Kurihara & T. Tsuji - Local Iwasawa theory of Perrin-Riou and syntomic complexes II (in preparation).

[KN] N. Katz - $p$ -adic interpolations of real analytic Eisenstein series, Ann. of Math. 104 (1976), p. 459-571. | DOI | MR | Zbl

[KM] N. Katz & B. Mazur - Arithmetic moduli of elliptic curves, Ann. of Math. Studies, vol. 108, Princeton Univ. Press, 1985. | MR | Zbl

[Ko] V. A. Kolyvagin - Euler Systems, in The Grothendieck Festschrift, 2, Progress in Math., vol. 88, Birkäuser, 1990, p. 435-483. | MR | Zbl

[KoL] V. A. Kolyvagin & D.Yu. Logachëv - Finiteness of the Shafarevich-Tate group and the group of rational points for some modular abelian varieties, Algebra i Analiz 1 (1989), no. 5, p. 171-196, translation in Leningrad Math. J. 1 (1990), no. 5, p. 1229-1253. | MR | Zbl

[KL] D. Kubert & S. Lang - Units in the modular function fields II, Math. Ann. 218 (1975), p. 175-189. | DOI | EuDML | MR | Zbl

[LG] G. Lafaille - Groupes $p$ -divisibles et modules filtrés, le cas peu ramifié, Bull. Soc. math. France 108 (1980), p. 187-206. | DOI | EuDML | Numdam | MR | Zbl

[Ma1] B. Mazur - Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18 (1972), p. 183-266. | DOI | EuDML | MR | Zbl

[Ma2] B. Mazur - Notes on étale cohomology of number fields, Ann. scient. Éc. Norm. Sup. 4e série 6 (1973), p. 521-556. | DOI | EuDML | Numdam | MR | Zbl

[MTT] B. Mazur, J. Tate & J. Teitelbaum - On $p$ -adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math. 84 (1986), p. 1-48. | DOI | EuDML | MR | Zbl

[MW] B. Mazur & A. Wiles - Class fields of abelian extensions of $ℚ$ , Invent. Math. 76 (1984), p. 179-330. | DOI | EuDML | MR | Zbl

[Ne] J. Nekovář - Kolyvagin's method for Chow groups of Kuga-Sato varieties, Invent. Math. 107 (1992), p. 99-125. | DOI | EuDML | MR | Zbl

[Pe0] B. Perrin-Riou - Théorie d'Iwasawa et hauteurs $p$ -adiques, Invent. Math. 109 (1992), p. 137-185. | DOI | EuDML | MR | Zbl

[Pe1] B. Perrin-Riou - Fonctions $L p$ -adiques d'une courbe elliptique et points rationnels, Ann. Inst. Fourier (Grenoble) 43 (1993), p. 945-995. | DOI | EuDML | Numdam | MR | Zbl

[Pe2] B. Perrin-Riou - Théorie d'Iwasawa des représentations $p$ -adiques sur un corps local, Invent. Math. 115 (1994), p. 81-149. | DOI | EuDML | MR | Zbl

[Pe3] B. Perrin-Riou - Fonctions $L p$ -adiques des représentations p-adiques, Astérisque, vol. 229, Société Mathématique de France, 1995. | Numdam | MR | Zbl

[Pe4] B. Perrin-Riou - Systèmes d'Euler $p$ -adiques et théorie d'Iwasawa, Ann. Inst. Fourier (Grenoble) 48 (1998), p. 1231-1307. | DOI | EuDML | Numdam | MR | Zbl

[Pe5] B. Perrin-Riou - Théorie d'Iwasawa des représentations $p$ -adiques semi-stables, Mém. Soc. math. France (N.S.), vol. 84, Société Mathématique de France, 2001. | Numdam | MR | Zbl

[Ri1] K. Ribet - On $ℓ$ -adic representations attached to modular forms, Invent. Math. 28 (1975), p. 245-275. | DOI | EuDML | MR | Zbl

[Ri2] K. Ribet - Galois representations attached to eigenforms with nebentypes, Lect. Notes in Math., vol. 601, Springer, 1977, p. 17-52. | MR | Zbl

[Ri3] K. Ribet - On $ℓ$ -adic representations attached to modular forms II, Glasgow Math. J. 27 (1985), p. 185-194. | DOI | MR | Zbl

[Ro1] D. Rohrlich - On $L$ -functions of elliptic curves and cyclotomic towers, Invent. Math. 75 (1984), p. 409-423. | DOI | EuDML | MR | Zbl

[Ro2] D. Rohrlich - $L$ -functions and division towers, Math. Ann. 281 (1988), p. 611-632. | DOI | EuDML | MR | Zbl

[Ru1] K. Rubin - Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer, Invent. Math. 64 (1981), p. 455-470. | DOI | EuDML | MR | Zbl

[Ru2] K. Rubin - The "main conjectures" of Iwasawa theory for imaginary quadratic fields, Invent. Math. 103 (1991), p. 25-68. | DOI | EuDML | MR | Zbl

[Ru3] K. Rubin - Euler systems and modular elliptic curves, in Galois representations in arithmetic algebraic geometry, London Math. Soc. Lecture Notes, vol. 254, Cambridge Univ. Press, 1998, p. 351-367. | DOI | MR | Zbl

[Ru4] K. Rubin - Euler systems, Hermann Weyl lectures, Ann. of Math. Studies, vol. 147, Princeton Univ. Press, 2000. | MR | Zbl

[Sa1] T. Saito - Modular forms and $p$ -adic Hodge theory, Invent. Math. 129 (1997), p. 607-620. | DOI | MR | Zbl

[Sa2] T. Saito - Weight-Monodromy conjecture for $ℓ$ -adic representations associated to modular forms, in The Arithmetic and Geometry of Algebraic cycles, Kluwer academic Publishers, 2000, p. 427-431. | DOI | MR | Zbl

[Scp] P. Schneider - Motivic Iwasawa theory, Adv. Stud. Pure Math., vol. 17, 1989, p. 421-456. | DOI | MR | Zbl

[Sc1] A. Scholl - Motives for modular forms, Invent. Math. 100 (1990), p. 419-430. | DOI | EuDML | MR | Zbl

[Sc2] A. Scholl - An introduction to Kato's Euler systems, in Galois representations in arithmetic algebraic geometry, London Math. Soc. Lecture Notes, vol. 254, Cambridge Univ. Press, 1998, p. 379-460. | DOI | MR | Zbl

[Se1] J.-P. Serre - Cohomologie Galoisienne, Lect. Notes in Math., vol. 5, Springer, 1965. | MR | Zbl

[Se2] J.-P. Serre - Abelian $ℓ$ -adic representations and elliptic curves, Benjamin, 1968. | MR | Zbl

[Se3] J.-P. Serre - Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent Math. 15 (1972), p. 259-331. | DOI | EuDML | MR | Zbl

[ST] J.-P. Serre & J. Tate - Good reduction of abelian varieties, Ann. of Math. 88 (1968), p. 492-517. | DOI | MR | Zbl

[Sh] G. Shimura - The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math. 29 1976, p. 783-804. | DOI | MR | Zbl

[So] C. Soulé - On higher $p$ -adic regulators, Lect. Notes in Math., vol. 854, Springer, 1981, p. 371-401. | MR | Zbl

[Ta1] J. Tate - Duality theorems in Galois cohomology over number fields, in Proc. Intern. Cong. Math. (Stockholm), 1962, p. 234-241. | Zbl

[Ta2] J. Tate - On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, in Dix exposés sur la cohomologie des schémas, North Holland, 1968, Sém. Bourbaki exposé n° 306, p. 189-214. | Numdam | MR | Zbl

[Th] F. Thaine - On the ideal class groups of real abelian number fields, Ann. of Math. (2) 128 (1998), p. 1-18. | MR | Zbl

[TT] T. Tsuji - $p$ -adic étale cohomology and crystalline cohomology in the semi-stable reduction case, Invent. Math. 137 (1999), p. 233-411. | DOI | MR | Zbl

[TN] N. Tsuzuki - Variations of $p$ -adic de Rham structures, preprint.

[Vi] M. M. Vishik - Non-archimedian measures connected with Dirichlet series, Math. USSR Sbornik 28 (1976), p. 216-228. | DOI | Zbl

[Vo] S. V. Vostokov - Explicit construction of class field theory for multidimensional local field, Math. USSR Izv. 26 (1986), p. 263-287. | DOI | Zbl

[We] A. Weil - Elliptic functions according to Eisenstein and Kronecker, Ergebnisse der Math., vol. 88, Springer, 1976. | MR | Zbl

[Wi] A. Wiles - Modular elliptic curves and Fermat's last theorem, Ann. of Math. 141 (1995), p. 443-551. | DOI | MR | Zbl