The p -adic Birch and Swinnerton-Dyer’s conjecture

Pierre Colmez

Séminaire Bourbaki (2002-2003)

  • Volume: 45, page 251-320
  • ISSN: 0303-1179

Abstract

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The classical Birch and Swinnerton-Dyer’s conjecture asserts that the order r of the zero at s = 1 of the L -function of an elliptic curve  E defined over  𝐐 is equal to the rank r of its group of rational points. This is a theorem if r = 0 or 1 , but there is no result relating r and r if r 2 . We will explain how Kato proves that the p -adic L function attached to E has, at s = 1 , a zero of order at least r .

How to cite

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Colmez, Pierre. "La conjecture de Birch et Swinnerton-Dyer $\mathbf {p}$-adique." Séminaire Bourbaki 45 (2002-2003): 251-320. <http://eudml.org/doc/252144>.

@article{Colmez2002-2003,
abstract = {La conjecture de Birch et Swinnerton-Dyer prédit que l’ordre $r_\infty $ du zéro en $s=1$ de la fonction $L$ d’une courbe elliptique $E$ définie sur $\{\mathbf \{Q\}\}$ est égal au rang $r$ du groupe de ses points rationnels. On sait démontrer cette conjecture si $r_\infty =0$ ou $1$, mais on n’a aucun résultat reliant $r_\infty $ et $r$ si $r_\infty \ge 2$. Nous expliquerons comment Kato démontre que la fonction $L$$p$-adique attachée à $E$ a, en $s=1$, un zéro d’ordre supérieur ou égal à $r$.},
author = {Colmez, Pierre},
journal = {Séminaire Bourbaki},
keywords = {elliptic curve; $p$-adic $L$ function},
language = {fre},
pages = {251-320},
publisher = {Association des amis de Nicolas Bourbaki, Société mathématique de France},
title = {La conjecture de Birch et Swinnerton-Dyer $\mathbf \{p\}$-adique},
url = {http://eudml.org/doc/252144},
volume = {45},
year = {2002-2003},
}

TY - JOUR
AU - Colmez, Pierre
TI - La conjecture de Birch et Swinnerton-Dyer $\mathbf {p}$-adique
JO - Séminaire Bourbaki
PY - 2002-2003
PB - Association des amis de Nicolas Bourbaki, Société mathématique de France
VL - 45
SP - 251
EP - 320
AB - La conjecture de Birch et Swinnerton-Dyer prédit que l’ordre $r_\infty $ du zéro en $s=1$ de la fonction $L$ d’une courbe elliptique $E$ définie sur ${\mathbf {Q}}$ est égal au rang $r$ du groupe de ses points rationnels. On sait démontrer cette conjecture si $r_\infty =0$ ou $1$, mais on n’a aucun résultat reliant $r_\infty $ et $r$ si $r_\infty \ge 2$. Nous expliquerons comment Kato démontre que la fonction $L$$p$-adique attachée à $E$ a, en $s=1$, un zéro d’ordre supérieur ou égal à $r$.
LA - fre
KW - elliptic curve; $p$-adic $L$ function
UR - http://eudml.org/doc/252144
ER -

References

top
  1. [1] A. Agboola & B. Howard – “Anticyclotomic Iwasawa theory of CM elliptic curves”, preprint, 2003. Zbl1130.11058MR2266884
  2. [2] Y. Amice – “Interpolation p -adique”, Bull. Soc. math. France 92 (1964), p. 117–180. Zbl0158.30201MR188199
  3. [3] —, “Duals”, in Proc. of a conf. on p -adic analysis (Nijmegen, 1978), Nijmegen, Math. Institut Katholische Univ., 1978, p. 1–15. 
  4. [4] 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. Zbl0332.14010MR447195
  5. [5] K. Barré-Sirieix, G. Diaz, F. Gramain & G. Philibert – “Une preuve de la conjecture de Mahler-Manin”, Invent. Math.124 (1996), p. 1–9. Zbl0853.11059MR1369409
  6. [6] D. Barsky – “Fonctions zêta p -adiques d’une classe de rayon des corps totalement réels”, Groupe d’études d’analyse ultramétrique, 1977-1978 ; errata 1978-1979. Zbl0406.12008
  7. [7] A. Beilinson – “Higher regulators and values of L -functions”, J. Soviet Math.30 (1985), p. 2036–2070. Zbl0588.14013MR760999
  8. [8] —, “Higher regulators of modular curves”, Contemp. Math.55 (1986), p. 1–34. MR862627
  9. [9] J. Bellaïche – “Congruences endoscopiques et représentations galoisiennes”, Thèse, Université Paris 11, 2002. 
  10. [10] J. Bellaïche & G. Chenevier – “Formes non tempérées pour U ( 3 ) et conjectures de Bloch-Kato”, Ann. scient. Éc. Norm. Sup. 4 e série (à paraître). Zbl1201.11051
  11. [11] D. Benois – “On Iwasawa theory of crystalline representations”, Duke Math. J.104 (2000), p. 211–267. Zbl0996.11072MR1773559
  12. [12] L. Berger – “Représentations p -adiques et équations différentielles”, Invent. Math.148 (2002), p. 219–284. Zbl1113.14016MR1906150
  13. [13] —, “Représentations de de Rham et normes universelles”, Bull. Soc. math. France (à paraître). Zbl1122.11036
  14. [14] M. Bertolini & H. Darmon – “Heegner points on Mumford-Tate curves”, Invent. Math.126 (1996), p. 413–456. Zbl0882.11034MR1419003
  15. [15] —, “A rigid analytic Gross-Zagier formula and arithmetic applications, with Appendix by B. Edixhoven”, Ann. of Math.146 (1997), p. 117–147. Zbl1029.11027MR1469318
  16. [16] —, “Heegner points, p -adic L -functions, and the Cerednik-Drinfeld uniformisation”, Invent. Math.131 (1998), p. 453–491. Zbl0899.11029MR1614543
  17. [17] —, “ p -adic periods, p -adic L -functions and the p -adic uniformisation of Shimura curves”, Duke Math. J.98 (1999), p. 305–334. Zbl1037.11045MR1695201
  18. [18] —, “The p -adic L -functions of modular elliptic curves”, in 2001 and Beyond, Springer-Verlag, 2001. Zbl1048.11051MR1852156
  19. [19] —, “Iwasawa’s main conjecture for elliptic curves in the anticyclotomic setting”, preprint. 
  20. [20] M. Bertolini, H. Darmon, A. Iovita & M. Spiess – “Teitelbaum’s conjecture in the anticyclotomic setting”, Amer. J. Math.124 (2002), p. 411–449. Zbl1079.11036MR1890998
  21. [21] D. Bertrand – “Relations d’orthogonalité sur les groupes de Mordell-Weil”, in Séminaire de théorie des nombres, Paris 1984-85, Progress in Math., vol. 63, Birkhäuser, 1986, p. 33–39. Zbl0607.14014MR897339
  22. [22] A. Besser – “Syntomic regulators and p -adic integration I : rigid syntomic regulators”, Israel J. Math.120 (2000), p. 291–334. Zbl1001.19003MR1809626
  23. [23] —, “Syntomic regulators and p -adic integration II : K 2 of curves”, Israel J. Math.120 (2000), p. 335–359. Zbl1001.19004MR1809627
  24. [24] —, “The p -adic height pairings of Coleman-Gross and Nekovář”, in Proceedings of CNTA7, Montréal, à paraître. Zbl1153.11316
  25. [25] B. Birch & H. Swinnerton-Dyer – “Notes on elliptic curves. I”, J. reine angew. Math. 212 (1963), p. 7–25. Zbl0118.27601MR146143
  26. [26] —, “Notes on elliptic curves. II”, J. reine angew. Math. 218 (1965), p. 79–108. Zbl0147.02506MR179168
  27. [27] S. Bloch & K. Kato – “Tamagawa numbers of motives and L -functions”, in The Grothendieck Festschrift, vol. 1, Progress in Math., vol. 86, Birkäuser, 1990, p. 333–400. Zbl0768.14001MR1086888
  28. [28] J.-B. Bost – “Algebraic leaves of algebraic foliations over number fields”, Publ. Math. Inst. Hautes Études Sci.93 (2001), p. 161–221. Zbl1034.14010MR1863738
  29. [29] C. Breuil, B. Conrad, F. Diamond & R. Taylor – “On the modularity of elliptic curves over 𝐐 : wild 3-adic exercises”, J. Amer. Math. Soc.14 (2001), p. 843–939. Zbl0982.11033MR1839918
  30. [30] A. Brumer – “On the units of algebraic number fields”, Mathematika14 (1967), p. 121–124. Zbl0171.01105MR220694
  31. [31] H. Carayol – “Sur les représentations l -adiques associées aux formes modulaires de Hilbert”, Ann. scient. Éc. Norm. Sup. 4 e série 19 (1986), p. 409–468. Zbl0616.10025MR870690
  32. [32] J. Cassels – “Arithmetic on an elliptic curve”, in Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, 1963, p. 234–246. Zbl0118.27701MR175891
  33. [33] —, “Diophantine equations with special reference to elliptic curves”, J. London Math. Soc. (2) 41 (1966), p. 193–291. Zbl0138.27002MR199150
  34. [34] P. Cassou-Noguès – “Valeurs aux entiers négatifs des fonctions zêta et fonctions zêta p -adiques”, Invent. Math.51 (1979), p. 29–59. Zbl0408.12015MR524276
  35. [35] A. Chambert-Loir – “Théorèmes d’algébricité en géométrie diophantienne (d’après J.-B. Bost, Y. André, D. et G. Chudnovsky)”, in Sém. Bourbaki 2000/01, Astérisque, vol. 282, Société Mathématique de France, 2002, exp. no 886, p. 175–209. Zbl1044.11055MR1975179
  36. [36] F. Cherbonnier & P. Colmez – “Représentations p -adiques surconvergentes”, Invent. Math.133 (1998), p. 581–611. Zbl0928.11051MR1645070
  37. [37] —, “Théorie d’Iwasawa des représentations p -adiques d’un corps local”, J. Amer. Math. Soc.12 (1999), p. 241–268. Zbl0933.11056MR1626273
  38. [38] D. Chudnovsky & G. Chudnovsky – “Padé approximations and Diophantine geometry”, Proc. Nat. Acad. Sci. U.S.A.82 (1985), p. 2212–2216. Zbl0577.14034MR788857
  39. [39] J. Coates – “The work of Mazur and Wiles on cyclotomic fields”, in Sém. Bourbaki 1980/81, Lect. Notes in Math., vol. 901, Springer, 1981, exp. no 575, p. 220–242. Zbl0506.12001MR647499
  40. [40] —, “The work of Gross and Zagier on Heegner points and the derivatives of L -series”, in Sém. Bourbaki 1984/85, Astérisque, vol. 133-134, Société Mathématique de France, 1986, exp. no 635, p. 57–72. Zbl0641.14013MR837214
  41. [41] J. Coates & A. Wiles – “On the conjecture of Birch and Swinnerton-Dyer”, Invent. Math.39 (1977), p. 223–251. Zbl0359.14009MR463176
  42. [42] —, “On p -adic L -functions and elliptic units”, J. Austral. Math. Soc. Ser. A26 (1978), p. 1–25. Zbl0442.12007MR510581
  43. [43] R. Coleman – “Division values in local fields”, Invent. Math.53 (1979), p. 91–116. Zbl0429.12010MR560409
  44. [44] —, “The dilogarithm and the norm residue symbol”, Bull. Soc. math. France 109 (1981), p. 373–402. Zbl0493.12019MR660143
  45. [45] —, “A p -adic Shimura isomorphism and p -adic periods of modular forms”, Contemp. Math.165 (1994), p. 21–51. Zbl0838.11033MR1279600
  46. [46] R. Coleman & B. Gross – “ p -adic heights on curves”, Adv. in Math.17 (1989), p. 73–81. Zbl0758.14009MR1097610
  47. [47] R. Coleman & A. Iovita – “Hidden structures on semi-stable curves”, preprint, 2003. Zbl1251.11047
  48. [48] P. Colmez – “Résidu en s = 1 des fonctions zêta p -adiques”, Invent. Math.91 (1988), p. 371–389. Zbl0651.12010MR922806
  49. [49] —, Intégration sur les variétés p -adiques, Astérisque, vol. 248, Société Mathématique de France, 1998. Zbl0930.14013MR1645429
  50. [50] —, “Représentations p -adiques d’un corps local”, in Proceedings of the International Congress of Mathematicians II (Berlin 1998), Doc. Mat. Extra, vol. II, Deutsche Math. Verein., 1998, p. 153–162. 
  51. [51] —, “Théorie d’Iwasawa des représentations de de Rham d’un corps local”, Ann. of Math.148 (1998), p. 485–571. Zbl0928.11045MR1668555
  52. [52] —, “Fonctions L p -adiques”, in Sém. Bourbaki 1998/99, Astérisque, vol. 266, Société Mathématique de France, 2000, exp. no 851, p. 21–58. MR1772669
  53. [53] —, “Arithmétique de la fonction zêta”, in La fonction zêta, Journées X-UPS, Éditions de l’École polytechnique, Palaiseau, 2002, p. 37–164. 
  54. [54] —, “Espaces de Banach de dimension finie”, J. Inst. Math. Jussieu1 (2002), p. 331–439. Zbl1044.11102MR1956055
  55. [55] —, “Invariants et dérivées de valeurs propres de frobenius”, preprint, 2003. 
  56. [56] —, “Les conjectures de monodromie p -adiques”, in Sém. Bourbaki 2001/02, Astérisque, vol. 290, Société Mathématique de France, 2003, exp. no 897, p. 53–101. Zbl1127.12301MR2074051
  57. [57] P. Colmez & J.-M. Fontaine – “Construction des représentations p -adiques semi-stables”, Invent. Math.140 (2000), p. 1–43. Zbl1010.14004MR1779803
  58. [58] C. Cornut – “Mazur’s conjecture on higher Heegner points”, Invent. Math.148 (2002), p. 495–523. Zbl1111.11029MR1908058
  59. [59] H. Darmon – “Integration on p × and arithmetic applications”, Ann. of Math.154 (2001), p. 589–639. Zbl1035.11027MR1884617
  60. [60] H. Darmon & A. Iovita – “The anticyclotomic main conjecture for supersingular elliptic curves”, preprint, 2003. Zbl1146.11057
  61. [61] D. Delbourgo – “On the p -adic Birch, Swinnerton-Dyer conjecture for non-semistable reduction”, J. Number Theory95 (2002), p. 38–71. Zbl1013.11029MR1916079
  62. [62] P. Deligne – “Formes modulaires et représentations -adiques”, in Sém. Bourbaki 1968/69, Lect. Notes in Math., vol. 179, Springer, 1971, exp. no 343, p. 139–172. Zbl0206.49901
  63. [63] —, “Valeurs de fonctions L et périodes d’intégrales”, in Automorphic forms, representations and L -functions, Proc. Symp. Pure Math., vol. 33, American Mathematical Society, 1979, p. 313–346. Zbl0449.10022
  64. [64] —, “Preuve des conjectures de Tate et de Shafarevitch (d’après G. Faltings)”, in Sém. Bourbaki 1983/84, Astérisque, vol. 121-122, Société Mathématique de France, 1985, exp. no 616, p. 25–41. Zbl0591.14026
  65. [65] P. Deligne & K. Ribet – “Values of Abelian L -functions at Negative Integers Over Totally Real Fields”, Invent. Math.59 (1980), p. 227–286. Zbl0434.12009MR579702
  66. [66] C. Deninger & A. Scholl – “The Beilinson conjectures”, in L -functions and arithmetic (Durham, 1989), London Math. Soc. Lecture Note Ser., vol. 153, Cambridge Univ. Press, 1991, p. 173–209. Zbl0729.14002MR1110393
  67. [67] M. Deuring – “Die Zetafunktion einer algebraischen Kurve vom Geschlecte Eins, I–IV”, Gött. Nac. (1953-1957). Zbl0064.27401MR61133
  68. [68] B. Edixhoven – “Rational elliptic curves are modular (after Breuil, Conrad, Diamond and Taylor)”, in Sém. Bourbaki 1999/2000, Astérisque, vol. 276, Société Mathématique de France, 2002, exp. no 871, p. 161–188. Zbl0998.11030MR1886760
  69. [69] M. Eichler – “Quaternäre quadratische Formen und die Riemannsche Vermutung für die Konguenzzetafunktion”, Archiv der Mat.5 (1954), p. 355–366. Zbl0059.03804MR63406
  70. [70] G. Faltings – “Endlichkeitssätze für abelsche Varietäten über Zahlkörpern”, Invent. Math.73 (1983), p. 349–366. Zbl0588.14026MR718935
  71. [71] —, “Almost étale extensions”, in Cohomologies p -adiques et applications arithmétiques (II) (P. Berthelot, J.-M. Fontaine, L. Illusie, K. Kato M. Rapoport, éds.), Astérisque, vol. 279, Société Mathématique de France, 2002, p. 185–270. Zbl1052.00008
  72. [72] J.-M. Fontaine – “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.115 (1982), p. 529–577. Zbl0544.14016MR657238
  73. [73] —, “Représentations p -adiques des corps locaux”, in The Grothendieck Festschrift, vol. 2, Progress in Math., vol. 87, Birkäuser, 1991, p. 249–309. Zbl0743.11066
  74. [74] —, “Valeurs spéciales de fonctions L des motifs”, in Sém. Bourbaki 1991/92, Astérisque, vol. 206, Société Mathématique de France, 1992, exp. no 751, p. 205–249. Zbl0799.14006
  75. [75] —, “Le corps des périodes p -adiques”, in Périodes p -adiques, Astérisque, vol. 223, Société Mathématique de France, 1994, exposé II, p. 59–102. Zbl0802.00019
  76. [76] J.-M. Fontaine & B. Perrin-Riou – “Autour des conjectures de Bloch et Kato : cohomologie galoisienne et valeurs de fonctions L ”, in Motives (Seattle), part 1, Proc. Symp. Pure Math., vol. 55, 1994, p. 599–706. Zbl0821.14013MR1265546
  77. [77] T. Fukaya – “The theory of Coleman power series for K 2 ”, J. Algebraic Geom.12 (2003), p. 1–80. Zbl1053.11088MR1948685
  78. [78] —, “Coleman power series for K -groups and explicit reciprocity laws”, preprint, 2003. MR2000771
  79. [79] M. Gealy – “Special values of p -adic L -functions associated to modular forms”, preprint, 2003. 
  80. [80] E. Ghate & V. Vatsal – “On the local behaviour of Λ -adic representations”, preprint, 2003. 
  81. [81] D. Goldfeld – “The class number of quadratic fields and the conjectures of Birch and Swinnerton-Dyer”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), p. 624–663. Zbl0345.12007MR450233
  82. [82] R. Greenberg – “On the Birch and Swinnerton-Dyer conjecture”, Invent. Math.72 (1983), p. 241–265. Zbl0546.14015MR700770
  83. [83] R. Greenberg & G. Stevens – “ p -adic L -functions and p -adic periods of modular forms”, Invent. Math.111 (1993), p. 407–447. Zbl0778.11034MR1198816
  84. [84] B. Gross & D. Zagier – “Heegner points and derivatives of L -series”, Invent. Math.84 (1986), p. 225–320. Zbl0608.14019MR833192
  85. [85] L. Guo – “General Selmer groups and critical values of Hecke L -functions”, Math. Ann.297 (1993), p. 221–233. Zbl0789.14018MR1241803
  86. [86] L. Herr – “Sur la cohomologie galoisienne des corps p -adiques”, Bull. Soc. math. France 126 (1998), p. 563–600. Zbl0967.11050MR1693457
  87. [87] H. Hida – “Anticyclotomic Main Conjectures”, preprint, 2003. Zbl1200.11082MR2290595
  88. [88] H. Hida & J. Tilouine – “Anti-cyclotomic Katz p -adic L -functions and congruence modules”, Ann. scient. Éc. Norm. Sup. 4 e série 26 (1993), p. 189–259. Zbl0778.11061MR1209708
  89. [89] —, “On the anticyclotomic main conjecture for CM fields”, Invent. Math.117 (1994), p. 89–147. Zbl0819.11047MR1269427
  90. [90] O. Hyodo – “On the Hodge-Tate decomposition in the imperfect residue field case”, J. reine angew. Math. 365 (1986), p. 97–113. Zbl0571.14004MR826154
  91. [91] A. Iovita & R. Pollack – “Iwasawa theory of Elliptic Curves at Supersingular Primes over 𝐙 p -extensions of number fields”, in MSRI Proceedings of a Conference on “Rankin’s method in arithmetic”, à paraître. Zbl1114.11053
  92. [92] A. Iovita & M. Spiess – “Derivatives of p -adic L -functions, Heegner cycles and monodromy modules attached to modular forms”, Invent. Math.154 (2003), p. 333–384. Zbl1099.11032MR2013784
  93. [93] A. Iovita & A. Werner – “ p -adic height pairings on abelian varieties with semistable ordinary reduction”, J. reine angew. Math. 564 (2003), p. 181–203. Zbl1168.11314MR2021039
  94. [94] K. Iwasawa – “On explicit formulas for the norm residue symbol”, J. Math. Soc. Japan20 (1968), p. 151–164. Zbl0256.12013MR229609
  95. [95] H. Jacquet & J. Shalika – “A non vanishing theorem for zeta functions of 𝐆𝐋 n ”, Invent. Math.38 (1976), p. 1–16. Zbl0349.12006MR432596
  96. [96] K. Kato – “The explicit reciprocity law and the cohomology of Fontaine-Messing”, Bull. Soc. math. France 119 (1991), p. 397–441. Zbl0752.14015MR1136845
  97. [97] —, “Lectures on the approach to Iwasawa theory for Hasse-Weil L -functions via 𝐁 dR ”, in Arithmetic Algebraic Geometry, Lect. Notes in Math., vol. 1553, Springer, 1993. Zbl0815.11051MR1338860
  98. [98] —, “Euler systems, Iwasawa theory and Selmer groups”, Kodai Math. J.22 (1999), p. 313–372. Zbl0993.11033MR1727298
  99. [99] —, “Generalized explicit reciprocity laws”, in Algebraic number theory (Hapcheon/Saga, 1996), Adv. Stud. Contemp. Math. (Pusan), vol. 1, 1999, p. 57–126. Zbl0411.12005MR1701912
  100. [100] —, “Hodge theory and values of zeta functions of modular forms”, in Cohomologies p -adiques et applications arithmétiques (III) (P. Berthelot, J.-M. Fontaine, L. Illusie, K. Kato M. Rapoport, éds.), Astérisque, Société Mathématique de France, à paraître. Zbl1142.11336
  101. [101] K. Kato, M. Kurihara & T. Tsuji – “Local Iwasawa theory of Perrin-Riou and syntomic complexes”, preprint, 1996. 
  102. [102] —, Cours au centre Émile Borel, premier semestre 1997. 
  103. [103] M. Kisin – “Overconvergent modular forms and the Fontaine-Mazur conjecture”, Invent. Math.153 (2003), p. 373–454. Zbl1045.11029MR1992017
  104. [104] S. Kobayashi – “Iwasawa theory for elliptic curves at supersingular primes”, Invent. Math.152 (2003), p. 1–36. Zbl1047.11105MR1965358
  105. [105] V. Kolyvagin – “Euler systems”, in The Grothendieck Festschrift, vol. 2, Progress in Math., vol. 87, Birkhäuser, 1990, p. 436–483. Zbl0742.14017MR1106906
  106. [106] D. Kubert & S. Lang – “Units in the modular function fields II”, Math. Ann.218 (1975), p. 175–189. Zbl0311.14005MR437497
  107. [107] M. Kurihara – “On the Tate Shafarevich groups over cyclotomic fields of an elliptic curve with supersingular reduction. I”, Invent. Math.149 (2002), p. 195–224. Zbl1033.11028MR1914621
  108. [108] S. Lang – “Sur la conjecture de Birch-Swinnerton-Dyer (d’après J. Coates et A. Wiles)”, in Sém. Bourbaki 1976/77, Lect. Notes in Math., vol. 677, Springer, 1978, exp. no 503, p. 189–200. Zbl0422.14013MR521769
  109. [109] —, “Les formes bilinéaires de Néron et Tate”, in Sém. Bourbaki 1963/64, Société Mathématique de France, 1995, exp. no 274, rééd. Sém Bourbaki 1948-1968, vol. 8. Zbl0138.42101
  110. [110] Y. Manin – “Periods of cusp forms, and p -adic Hecke series”, Math. USSR-Sb. 92 (1973), p. 371–393. Zbl0293.14008MR345909
  111. [111] B. Mazur – “Rational points of abelian varieties in towers of number fields”, Invent. Math.18 (1972), p. 183–266. Zbl0245.14015MR444670
  112. [112] —, “Modular curves and arithmetic”, in Proceedings of the International Congress of Mathematicians (Warsaw, 1983), PWN, Warsaw, 1984, p. 185–211. MR804682
  113. [113] —, “On monodromy invariants occuring in global arithmetic, and Fontaine’s theory”, Contemp. Math.165 (1994), p. 1–20. Zbl0846.11039MR1279599
  114. [114] B. Mazur & K. Rubin – “Elliptic curves and class field theory”, in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, 2002, p. 185–195. Zbl1036.11023MR1957032
  115. [115] —, Kolyvagin systems, Mem. Amer. Math. Soc., vol. 168, American Mathematical Society, 2004. MR2031496
  116. [116] B. Mazur & P. Swinnerton-Dyer – “Arithmetic of Weil curves”, Invent. Math.25 (1974), p. 1–61. Zbl0281.14016MR354674
  117. [117] B. Mazur & J. Tate – “Canonical height pairings via biextensions”, in Arithmetic and Geometry : Papers dedicated to I.R. Shafarevich, Progress in Math., vol. 35, Birkhäuser, 1983, p. 195–238. Zbl0574.14036MR717595
  118. [118] —, “The p -adic sigma function”, Duke Math. J.62 (1991), p. 663–688. Zbl0735.14020MR1104813
  119. [119] 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. Zbl0699.14028MR830037
  120. [120] B. Mazur & J. Tilouine – “Représentations galoisiennes, différentielles de Kähler et “conjectures principales””, Publ. Math. Inst. Hautes Études Sci. 71 (1990), p. 65–103. Zbl0744.11053MR1079644
  121. [121] B. Mazur & A. Wiles – “Class fields of abelian extensions of 𝐐 ”, Invent. Math.76 (1984), p. 179–330. Zbl0545.12005MR742853
  122. [122] J.-F. Mestre – “Formules explicites et minorations de conducteurs de variétés algébriques”, Comp. Math.58 (1986), p. 209–232. Zbl0607.14012MR844410
  123. [123] J. Nekovář – “Kolyvagin’s method for Chow groups of Kuga-Sato varieties”, Invent. Math.107 (1992), p. 99–125. Zbl0729.14004MR1135466
  124. [124] —, “On p -adic height pairings”, in Séminaire de théorie des nombres 1990-1991, Progress in Math., vol. 108, Birkhäuser, 1993, p. 127–202. Zbl0859.11038MR1263527
  125. [125] —, “On the p -adic heights of Heegner cycles”, Math. Ann.302 (1995), p. 609–686. Zbl0841.11025MR1343644
  126. [126] —, “On the parity of ranks of Selmer groups II”, C. R. Acad. Sci. Paris Sér. I Math.332 (2001), p. 99–104. Zbl1090.11037MR1813764
  127. [127] A. Néron – “Quasi-fonctions et hauteurs sur les variétés abéliennes”, Ann. of Math.82 (1965), p. 249–331. Zbl0163.15205MR179173
  128. [128] —, “Fonctions thêta p -adiques et hauteurs p -adiques”, in Séminaire de Théorie des Nombres, Paris 1980-1981, Progress in Math., vol. 22, Birkhäuser, 1982. Zbl0492.14035
  129. [129] J. Oesterlé – “Nombres de classes des corps quadratiques imaginaires”, in Sém. Bourbaki 1983/84, Astérisque, vol. 121-122, Société Mathématique de France, 1985, exp. no 631, p. 309–323. Zbl0551.12003MR768967
  130. [130] —, “Travaux de Wiles (et Taylor,...). II”, in Sém. Bourbaki 1994/95, Astérisque, vol. 237, Société Mathématique de France, 1996, exp. no 804, p. 333–355. MR1423631
  131. [131] A. Panchishkin – “A new method of constructing p -adic L -functions associated with modular forms”, Mosc. Math. J.2 (2002), p. 313–328. Zbl1011.11026MR1944509
  132. [132] —, “Two variable p -adic L functions attached to eigenfamilies of positive slope”, Invent. Math.154 (2003), p. 551–615. Zbl1065.11025MR2018785
  133. [133] B. Perrin-Riou – “Hauteurs p -adiques”, in Séminaire de Théorie des Nombres, Paris 1982-1983, Progress in Math., vol. 51, Birkhäuser, 1984. Zbl0585.14017MR791597
  134. [134] —, “Points de Heegner et dérivées de fonctions L p -adiques”, Invent. Math.89 (1987), p. 455–510. Zbl0645.14010MR903381
  135. [135] —, “Travaux de Kolyvagin et Rubin”, in Sém. Bourbaki 1989/90, Astérisque, vol. 189-190, Société Mathématique de France, 1990, exp. no 717, p. 69–106. MR1099872
  136. [136] —, “Théorie d’Iwasawa et hauteurs p -adiques”, Invent. Math.109 (1992), p. 137–185. Zbl0781.14013MR1168369
  137. [137] —, “Fonctions L p -adiques d’une courbe elliptique et points rationnels”, Ann. Inst. Fourier (Grenoble) 43 (1993), p. 945–995. Zbl0840.11024MR1252935
  138. [138] —, “Théorie d’Iwasawa des représentations p -adiques sur un corps local”, Invent. Math.115 (1994), p. 81–149. Zbl0838.11071MR1248080
  139. [139] —, Fonctions L p -adiques des représentations p -adiques, Astérisque, vol. 229, Société Mathématique de France, 1995. Zbl0845.11040MR1327803
  140. [140] —, “Systèmes d’Euler et représentations p -adiques”, Ann. Inst. Fourier (Grenoble) 48 (1998), p. 1231–1307. Zbl0930.11078MR1662231
  141. [141] —, “Représentations p -adiques et normes universelles I, le cas cristallin”, J. Amer. Math. Soc.13 (2000), p. 533–551. Zbl1024.11069MR1758753
  142. [142] —, “Arithmétique des courbes elliptiques à réduction supersingulière en p ”, preprint, 2001. Zbl1061.11031
  143. [143] —, Théorie d’Iwasawa des représentations p -adiques semi-stables, Mém. Soc. math. France (N.S.), vol. 84, Société Mathématique de France, 2001. Zbl1031.11064
  144. [144] —, “Quelques remarques sur la théorie d’Iwasawa des courbes elliptiques”, in Number theory for the millennium, III (Urbana, IL, 2000), 2002, p. 119–147. Zbl1047.11056
  145. [145] R. Pollack – “On the p -adic L -function of a modular form at a supersingular prime”, Duke Math. J.118 (2003), p. 523–558. Zbl1074.11061MR1983040
  146. [146] R. Pollack & K. Rubin – “The main conjecture for CM elliptic curves at supersingular primes”, Ann. of Math.159 (2004), p. 447–464. Zbl1082.11035MR2052361
  147. [147] R. Pollack & G. Stevens – “The “missing” p -adic L -function”, preprint, 2003. MR1988578
  148. [148] K. Ribet – “Galois representations attached to modular forms”, Invent. Math.28 (1975), p. 245–275. Zbl0302.10027MR419358
  149. [149] —, “A modular construction of unramified extensions of 𝐐 ( ζ p ) ”, Invent. Math.34 (1976), p. 151–162. Zbl0338.12003MR419403
  150. [150] —, “Galois representations attached to modular forms II”, Glasgow Math. J.27 (1985), p. 185–194. Zbl0596.10027MR819838
  151. [151] D. Rohrlich – “On L -functions of elliptic curves and cyclotomic towers”, Invent. Math.75 (1984), p. 409–423. Zbl0565.14006MR735333
  152. [152] K. Rubin – “Elliptic curves and 𝐙 p -extensions”, Comp. Math.56 (1985), p. 237–250. Zbl0599.14028MR809869
  153. [153] —, “Local units, elliptic units, Heegner points, and elliptic curves”, Invent. Math.88 (1987), p. 405–422. Zbl0623.14006MR880958
  154. [154] —, “Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication”, Invent. Math.89 (1987), p. 527–560. Zbl0628.14018MR903383
  155. [155] —, “The “main conjectures” of Iwasawa theory for imaginary quadratic fields”, Invent. Math. 103 (1991), p. 25–68. Zbl0737.11030MR1079839
  156. [156] —, Euler systems, Ann. of Math. Studies, vol. 147, Princeton Univ. Press, 2000. 
  157. [157] P. Schneider – “ p -adic height pairings, II”, Invent. Math.79 (1985), p. 329–374. Zbl0571.14021MR778132
  158. [158] P. Schneider & J. Teitelbaum – “ p -adic Fourier theory”, Doc. Math.6 (2001), p. 447–481. Zbl1028.11069MR1871671
  159. [159] A. Scholl – “Motives for modular forms”, Invent. Math.100 (1990), p. 419–430. Zbl0760.14002MR1047142
  160. [160] —, “Remarks on special values of L -functions”, in L -functions and Arithmetic, Proc. of the Durham Symp., London Math. Soc. L.N.S., vol. 153, Cambridge University Press, 1991, p. 373–392. Zbl0817.14007MR1110402
  161. [161] —, “An introduction to Kato’s Euler systems”, in Galois representations in arithmetic algebraic geometry, Cambridge University Press, 1998, p. 379–460. Zbl0952.11015MR1696501
  162. [162] —, “Higher regulators and special values of L -functions of modular forms”, en préparation. 
  163. [163] —, “Zeta elements for higher weight modular forms”, en préparation. 
  164. [164] J.-P. Serre – Abelian -adic representations and elliptic curves, W. A. Benjamin, 1968. Zbl0186.25701MR263823
  165. [165] —, “Formes modulaires et fonctions zêta p -adiques”, in Modular functions of one variable III, Lect. Notes in Math., vol. 350, Springer, 1972, p. 191–268. Zbl0277.12014
  166. [166] —, “Propriétés galoisiennes des points d’ordre fini des courbes elliptiques”, Invent. Math.15 (1972), p. 259–331. Zbl0235.14012MR387283
  167. [167] —, “Travaux de Wiles (et Taylor,...). I”, in Sém. Bourbaki 1994/95, Astérisque, vol. 237, Société Mathématique de France, 1996, exp. no 803, p. 319–332 MR1423630
  168. [168] —, lettre du 13/11/59, in Correspondance Grothendieck-Serre, Documents Mathématiques, vol. 2 Société Mathématique de France, 2001. Zbl0986.01019
  169. [169] G. Shimura – “Correspondances modulaires et les fonctions ζ de courbes algébriques”, J. Math. Soc. Japan10 (1958), p. 1–28. Zbl0081.07603MR95173
  170. [170] —, “Sur les intégrales attachées aux formes automorphes”, J. Math. Soc. Japan11 (1959), p. 291–311. Zbl0090.05503MR120372
  171. [171] —, Introduction to the arithmetic theory of automorphic functions, Kan Memorial Lectures 1, vol. 11, Math. Soc. of Japan, 1971. Zbl0872.11023
  172. [172] —, “On elliptic curves with complex multiplication as factors of the Jacobians of modular function fields”, Nagoya Math. J.43 (1971), p. 199–208. Zbl0225.14015MR296050
  173. [173] —, “On the factors of the jacobian variety of a modular function field”, J. Math. Soc. Japan25 (1973), p. 523–544. Zbl0266.14017MR318162
  174. [174] —, “The special values of the zeta functions associated with cusp forms”, Comm. Pure Appl. Math.29 (1976), p. 783–804. Zbl0348.10015MR434962
  175. [175] C. Skinner & E. Urban – “Sur les déformations p -adiques des formes de Saito-Kurokawa”, C. R. Acad. Sci. Paris Sér. I Math.335 (2002), p. 581–586. Zbl1024.11030MR1941298
  176. [176] —, en préparation. 
  177. [177] C. Soulé – “Régulateurs”, in Sém. Bourbaki 1984/85, Astérisque, vol. 133-134, Société Mathématique de France, 1986, exp. no 644, p. 237–253. Zbl0617.14008MR837223
  178. [178] —, “Éléments cyclotomiques en K -théorie”, in Journées Arithmétiques, (Besançon, 1985), Astérisque, vol. 147-148, Société Mathématique de France, 1987, p. 225–257. Zbl0632.12014
  179. [179] G. Stevens – “Coleman’s -invariant and families of modular forms”, preprint, 1996. MR2667884
  180. [180] —, Cours au centre Émile Borel, premier semestre 2000. 
  181. [181] J. Tate – “ p -divisible groups”, in Proc. of a conference on local fields, Nuffic Summer School at Driebergen, Springer, Berlin, 1967, p. 158–183. Zbl0157.27601MR231827
  182. [182] —, “A review of non-Archimedean elliptic functions”, in Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993), Ser. Number Theory, I, Internat. Press, Cambridge, MA, 1995, p. 162–184. Zbl1071.11508MR1363501
  183. [183] —, “On the conjectures of Birch and Swinnerton-Dyer and a geometric analog”, in Sém. Bourbaki 1965/66, Société Mathématique de France, 1995, exp. no 306, rééd. Sém. Bourbaki 1948-1968, vol. 9. Zbl0199.55604MR1610977
  184. [184] F. Thaine – “On the ideal class group of real abelian number fields”, Ann. of Math.128 (1988), p. 1–18. Zbl0665.12003MR951505
  185. [185] J. Tilouine – “Sur la conjecture principale anticyclotomique”, Duke Math. J.59 (1989), p. 629–673. Zbl0707.11079MR1046742
  186. [186] T. Tsuji – “ p -adic étale cohomology and crystalline cohomology in the semi-stable reduction case”, Invent. Math.137 (1999), p. 233–411. Zbl0945.14008MR1705837
  187. [187] —, “Semi-stable conjecture of Fontaine-Jannsen : a survey”, in Cohomologies p -adiques et applications arithmétiques (II) (P. Berthelot, J.-M. Fontaine, L. Illusie, K. Kato M. Rapoport, éds.), Astérisque, vol. 279, Société Mathématique de France, 2002, p. 323–370. Zbl1041.14003MR1922833
  188. [188] V. Vatsal – “Uniform distribution of Heegner points”, Invent. Math.148 (2002), p. 1–46. Zbl1119.11035MR1892842
  189. [189] M. Vishik – “Non-archimedian measures connected with Dirichlet series”, Math. USSR-Sb. 28 (1976), p. 216–228. Zbl0369.14010
  190. [190] M. Waldschmidt – “Sur la nature arithmétique des valeurs de fonctions modulaires”, in Sém. Bourbaki 1996/97, Astérisque, vol. 245, Société Mathématique de France, 1997, exp. no 824, p. 105–140. Zbl0908.11029MR1627109
  191. [191] A. Weil – Elliptic functions according to Eisenstein and Kronecker, Erg. der Math., vol. 88, Springer-Verlag, 1976. Zbl0318.33004MR562289
  192. [192] A. Wiles – “Higher explicit reciprocity laws”, Ann. of Math.107 (1978), p. 235–254. Zbl0378.12006MR480442
  193. [193] —, “Modular elliptic curves and Fermat’s last theorem”, Ann. of Math.141 (1995), p. 443–551. Zbl0823.11029MR1333035
  194. [194] Y. Zarhin – “ p -adic heights on abelian varieties”, in Séminaire de Théorie des Nombres, Paris 1987-1988, Progress in Math., vol. 81, Birkhäuser, 1989. Zbl0707.14040MR1042777

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