PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Tytuł artykułu

Modularność krzywych eliptycznych i Wielkie Twierdzenie Fermata

Treść / Zawartość
Identyfikatory
Warianty tytułu
Języki publikacji
PL
Abstrakty
Słowa kluczowe
Rocznik
Strony
3--47
Opis fizyczny
Bibliogr. 132 poz.
Twórcy
  • Instytut Matematyki Uniwersytetu Szczecińskiego ul- Wielkopolska 15 70-451 Szczecin
Bibliografia
  • [1] M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley 1969.
  • [2] A. O. L. Atkin, J. Lebner, Hecke operators on Γ_0(N), Math. Ann. 185 (1970), 134-160.
  • [3] S. Balcerzyk, Wstęp do Algebry Homologicznej, Biblioteka Matematyczna 34, PWN Warszawa 1972.
  • [4] S. Вalcerzуk, T. Józefiak, Pierścienie Przemienne, Biblioteka Matematyczna 58, PWN Warszawa 1985.
  • [5] A. Beilinson, V. Drinfe1d, Quantization of Hitchin’s integrable system and Hecke eigensheaves, preprint 2004.
  • [6] G. V. Belyi, On Galois extensions of a maximal cyclotomie field, Math. USSR Izv. 14 (1980), 247-256; Izv. AN SSSR, ser. Mat. 43 (1979), 267-276.
  • [7] M. A. Bennett, C. Skinner, Ternary diophantine equations via Galois representations and modular forms, Canad. J. Math. 56 (2004), 23-54.
  • [8] Z. I. Воrevich, I. R. Shafarevich, Number Theory, Moscow 1985 (ros.).
  • [9] J.-F. Boutot, H. Carayol, Uniformisation p-adique des courbes de Shimura: les théorème de Cerednik et Drinfeld, Astérisque 196-197 (2001), 45-158.
  • [10] С. Вreuil, В. Conrad, F. Diamond, R. Taylor, On the modularity of elliptic curves over Q: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), 843-939.
  • [11] J. Вrowkin, Teoria Ciał. Biblioteka Matematyczna 49, PWN Warszawa 1977.
  • [12] J. Вrowkin Siódmy problem milenijny: Hipoteza Bircha i Swinnertona-Dyera, Wiadomości Matematyczne 39 (2003), 1-25.
  • [13] Y. Bugeaud, M. Mignotte, S. Siksek, Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers, Ann. Math. 163 (2006), 969-1018.
  • [14] Y. Bugeaud, M. Mignotte, S. Siksek, Classical and modular approaches to exponential Diophantine equations II. The Lebesgue-Nagell equation, Compos. Math. 142 (2006), 31-32.
  • [15] Algebraic Number Theory (J. W. S. Cassels, A. Fröhlich, eds). Academie Press 1967.
  • [16] Elliptic Curves, Modular Forms, and Fermat’s Last Theorem (J. Coates and al., eds.), Intern. Press, Cambridge 1995.
  • [17] I. Connell, Computing root numbers of elliptic curves over Q, Manusc. Math. 82 (1994), 93-104.
  • [18] B. Conrad, Ramified deformation problems, Duke Math. J. 97 (1999), 439-514.
  • [19] B. Conrad, F. Diamond, R. Taylor, Modularity of certain potentially Barsotti-Tate Galois representations, J. Amer. Math. Soc. 12 (1999), 521-567.
  • [20] Modular Forms and Fermat’s Last Theorem (G. Cornell, J. H. Silverman and G. Stevens, eds.), Springer 1997.
  • [21] J. Cremona, Algorithms for Modular Elliptic Curves. Cambridge Univ. Press, Cambridge 1997.
  • [22] A. Dąbrowski, On the integers represented by x^4 - у^4, Bull. Austral. Math. Soc. 76 (2007), 133-136.
  • [23] H. Darmon, Rigid local systems, Hilbert modular forms, and Fermat’s last theorem, Duke Math. J. 102 (2000), 413-449.
  • [24] H. Darmon, The Shimura-Taniyama conjecture (d’après Wiles), Russian Math. Surveys 50 (1995), 503-548.
  • [25] H. Darmon, A. Granville, On the equations z^m = F(x, y) and Ax^p + By^q = Cz^r, Bull. London Math. Soc. 27 (1995), 513-543.
  • [26] H. Darmon, L. Merel, Winding quotients and some variants of Fermat’s Last Theorem, J. reine angew. Math. 490 (1997), 81-100.
  • [27] H. Darmon, F. Diamond, R. Taylor, Fermat’s Last Theorem. In: Current Developments in Mathematics, International Press 1995.
  • [28] P. Deligne, Formes modulaires et représentations l-adiques, Sém. Bourbaki No. 355, Lecture Notes in Math. 179, Springer-Verlag 1971, 139-172.
  • [29] P. Deligne, La conjecture de Weil II, Publ. Math. IHES 52 (1980), 137-252.
  • [30] P. Deligne, M. Rapoport, Les schémas de modules de courbes elliptiques. In: Modular Functions of One Variable II, Lecture Notes in Math. 349, Springer-Verlag 1973, 143-316.
  • [31] P. Deligne, J.-P. Serre, Formes modulaires de poids 1, Ann. Sci. Ec. Norm. Sup. 7 (1974), 507-530.
  • [32] M. Deuring, Die Zetafunktion einer algebraischen Kurve vom Geschlecht Eins, I, II, III, IV, Nachr. Akad. Wiss. Göttingen (1953), 85-94; (1955), 13-42; (1956), 37-76; (1957), 55-80.
  • [33] F. Diamond, On deformation rings and Hecke rings, Ann. Math. 144 (1996), 137-166.
  • [34] F. Diamond, An extension of Wiles’ results. In: Modular Forms and Fermat’s Last Theorem (eds. G. Cornell et al.), Springer-Verlag 1997, 475-489.
  • [35] F. Diamond, J. Shurman, A First Course in Modular Forms. Graduate Texts in Math. 228, Springer 2005.
  • [36] V. G. Drinfeld, Moduli varieties of F-sheaves, Funct. Anal. Appl. 21 (1987), 107-122.
  • [37] B. Edixhoven, Rational elliptic curves are modular [after Breuil, Conrad, Diamond and Taylor], Sém. Bourbaki, exp. 871 (1999-2000).
  • [38] H. Edwards, Fermat’s Last Theorem, a Genetic Introduction to Algebraic Number Theory. Graduate Texts in Math. 64, Springer 1977.
  • [39] J. Ellenberg, Galois representations attached to Q-curves and the generalized Fermat equation A^4 + B^2 = C^p, Amer. J. Math. 126 (2004), 763-787.
  • [40] J. Ellenberg, Serve’s conjecture over F_9, Ann. Math. 161 (2005), 1111-1142.
  • [41] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpem, Invent. Math. 73 (1983), 349-366; Erratum: 75 (1984), 381.
  • [42] G. Faltings, The proof of Fermat’s Last Theorem by R. Taylor and A. Wiles, Notices of the Amer. Math. Soc. 42 (1995), 743-746.
  • [43] M. Flach, A finiteness theorem for the symmetric square of an elliptic curve, Invent. Math. 109 (1992), 307-327.
  • [44] J.-M. Fontaine, B. Mazur, Geometric Galois reprezentations. In: Elliptic Curves, Modular Forms, and Fermat’s Last Theorem (eds. J. Coates et al.), Intern. Press, Cambridge 1995, 41-78.
  • [45] E. Frenkel, D. Gaitsgоrу, K. Vilоnеn, On the geometric Langlands conjecture, J. Amer. Math. Soc. 15 (2002), 367-417.
  • [46] G. Frey, Links between stable elliptic curves and certain diophantine equations, Ann. Univ. Saraviensis, Ser. Math. 1 (1986), 1-40.
  • [47] K. Fujiwara, Deformation rings and Hecke algebras in totally real case, preprint 1999.
  • [48] F. Gouvea, A marvelous proof, Amer. Math. Monthly 101 (1994), 203-222.
  • [49] B. H. Gross, D. B. Zagier, Heegner points and derivatives of L-series, Invent. Math. 84 (1986), 225-320.
  • [50] E. Halberstadt, Signes locaux des courbes elliptiques en 2 et 3, C. R. Acad. Sci. Paris 326 (1998), 1047-1052.
  • [51] E. Halberstadt, A. Kraus, Courbes de Fermat: résultats et problèmes, J. reine ang. Math. 548 (2002), 167-234.
  • [52] M. Harris, R. Taylor, The geometry and cohomology of some simple Shimura varieties. Ann. Math. Studies 151, Princeton Univ. Press 2001.
  • [53] R. Hartshorne, Algebraic Geometry. Graduate Texts in Math. 52, Springer 1977.
  • [54] E. Hecke, Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung, Math. Ann. 112 (1936), 664-699.
  • [55] Y. Hellegouarch, Invitation aux Mathématiques de Fermat-Wiles. Masson, Paris 1997.
  • [56] G. Henniart, Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique, Invent. Math. 139 (2000), 439-455.
  • [57] H. Hida, Modular Forms and Galois Cohomology. Cambridge Studies in Adv. Math., Cambridge Univ. Press 2000.
  • [58] H. Hida, Hilbert Modular Forms and Iwasawa Theory. Oxford Univ. Press 2006.
  • [59] T. Honda, I. Miyawaki, Zeta-functions of elliptic curves of 2-power conductors, J. Math. Soc. Japan 26 (1974), 362-373.
  • [60] K. Hulek, R. Klооsterman, M. Schütt, Modularity of Calabi-Yau varieties, arXiv:math.AG/0601238 v2, 31 Jan 2006
  • [61] J. Igusa, Kroneckerian model of fields of elliptic modular functions, Amer. J. Math. 81 (1959), 561-577.
  • [62] K. Ireland, M. Rosen, A Classical Introduction to Modem Number Theory. Graduate Texts in Math. 84, Springer 1982.
  • [63] N. Katz, B. Mazur, Arithmetic Moduli of Elliptic Curves. Annals of Math. Studies 108, Princeton Univ. Press, Princeton 1985.
  • [64] С. Khare, On isomorphisms between deformation rings and Hecke rings, Invent. Math. 154 (2003), 199-222.
  • [65] С. Khare, Modularity of p-adic Galois representations via p-adic approximations, J. Théorie des Nombres de Bordeaux 16 (2004), 179-185.
  • [66] M. Kisin, Overconvergent modular forms and the Fontaine-Mazur conjecture, Invent. Math. 153 (2003), 373-454.
  • [67] N. Koblitz, Introduction to Elliptic Curves and Modular Forms. Graduate Texts in Math. 97, Springer-Verlag 1984.
  • [68] V. А. Kоlуvagin, Finiteness of E(Q) and Ш(E/Q) for a subclass of Weil curves (Russian) Izv. Akad. Nauk Ser. Mat. 52 (1988), 1154-1180.
  • [69] V. A. Kоlуvagin, Euler systems. In: The Grothendieck Festschrift, vol. II (P. Cartier et al., eds.), Birkhäuser, Boston 1990, 435-483.
  • [70] A. Kraus, Majorations effectives pour l’equation de Fermat généralisée, Can. J. Math. 49 (1997), 1139-1161.
  • [71] E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry. Birkhäuser 1985.
  • [72] L. Lafforgue, Chtoucas de Drinfeld et correspondance de Langlands, Invent. Math. 147 (2002), 1-241.
  • [73] S. Lang, Some history of the Shimura-Taniyama conjecture, Notices Amer. Math. Soc. 42 (1995), 1301-1307.
  • [74] S. Lang, Algebra. PWN, Warszawa 1983 (przekład z j. ang.).
  • [75] A. Langer, Program Langlandsa według Lafforgue’a, Wiadomości Matematyczne 39 (2003), 39-46.
  • [76] R. P. Langlands, Base Change for GL(2). Annals of Math. Studies 96, Princeton Univ. Press, Princeton 1980.
  • [77] R. P. Langlands, Problems in the theory of automorphic forms. In: Lecture Notes in Math. 170 (1970), 18-61.
  • [78] G. Laumon, M. Rapoport, U. Stuhler, D-elliptic sheaves and the Langlands correspondence, Invent. Math. 113 (1993), 217-338.
  • [79] H.W. Lenstra, Complete intersections and Gorenstein rings. In: Elliptic Curves, Modular Forms, and Fermat’s Last Theorem (eds. J. Coates et al.), Intern. Press, Cambridge 1995, 99-109.
  • [80] M. Ligozat, Courbes modulaires de genre 1, Bull. Soc. Math. France, Suppl., Mém. 43 (1975), 80 stron.
  • [81] J. Manoharmayum, On the modularity of certain GL_2( F_7) Galois representations, Math. Res. Letters 8 (2001), 703-712.
  • [82] Y. Martin, K. Оno, Eta-quotients and elliptic curves, Proc. AMS 125 (1997), 3169-3176.
  • [83] B. Mazur, An introduction to the deformation theory of Galois representations. In: Modular Forms and Fermat’s Last Theorem (eds. G. Cornell et al.), Springer- Verlag 1997, 243-311.
  • [84] В. Mazur, Modular curves and the Eisenstein ideal, Publ. Math. IHES 47 (1977), 33-186.
  • [85] B. Mazur, Perturbations, deformations, and variations (and „nearl-misses”) in geometry, physics, and number theory, Bull. Amer. Math. Soc. 41 (2004), 307-336.
  • [86] L. Merel, Arithmetic of elliptic curves and diophantine equations, J. Théorie des Nombres de Bordeaux 11 (1999), 173-200.
  • [87] P. Mihăilescu, Primary cyclotomie units and a proof of Catalan’s conjecture, J. reine angew. Math. 572 (2004), 167-195.
  • [88] W. Narkiewicz, Wielkie Twierdzenie Fermata, Wiad. Mat. 30 (1993), 1-16.
  • [89] W. Narkiewicz, Teoria Liczb. Biblioteka Matematyczna 50, PWN Warszawa 1977.
  • [90] A. Nitaj, The abc conjecture homepage, http://www.math.unicaen.fr/~nitaj/abc.html.
  • [91] J. Oesterlé, Nouvelles approaches du théorème de Fermat, Sém. Bourbaki, exp. 694, Astérisque 161-162 (1988), 165-186.
  • [92] J. Oesterlé, Travaux de Wiles (et Taylor,...), Partie II. Sém. Bourbaki, exp. 804, Astérisque 237 (1996), 333-355.
  • [93] A. P. Ogg, Abelian curves of 2-power conductor, Proc. Camb. Phil. Soc. 62 (1966), 143-148.
  • [94] R. Ramakrishna, On a variation of Mazur’s deformation functor, Compos. Math. 87 (1993), 269-286.
  • [95] M. Reid, Undergraduate Algebraic Geometry. London Math. Student Texts 12, Cambridge Univ. Press 1988.
  • [96] P. Ribenboim, Wielkie Twierdzenie Fermata dla Laików, WNT, Warszawa 2001 (przekład z j. ang.).
  • [97] K. Ribet, On modular representations of Gal(\bar{Q}/Q) arising from modular forms, Invent. Math. 100 (1990), 431-476.
  • [98] K. Ribet, Abelian varieties over Q and modular forms. In: Modular curves and abelian varieties, Progress in Math. 224, Birkhäuser 2004, 241-261.
  • [99] K. Rubin, Modularity of mod 5 representations. In: Modular Forms and Fermat’s Last Theorem (eds. G. Cornell et al.), Springer-Verlag 1997, 463-474.
  • [100] K. Rubin, A. Silverberg, Families of elliptic curves with constant mod p representations. In: Elliptic Curves, Modular Forms, and Fermat’s Last Theorem (eds. J. Coates et al.), Intern. Press, Cambridge 1995, 148-161.
  • [101] K. Rubin, A. Silverberg, A report on Wiles’ Cambridge lectures, Bull. Amer. Math. Soc. 31 (1994), 15-38.
  • [102] D. Savitt, Modularity of some potentially Barsotti-Tate Galois representations, Compos. Math. 140 (2004), 31-63.
  • [103] J.-P. Serre, Sur les représentations modulaires de degré 2 de Gal(\bar{Q}/Q), Duke Math. J. 54 (1987), 179-230.
  • [104] J.-P. Serre, Propriétes galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259-331.
  • [105] J.-P. Serre, Abelian l-adic Representations and Elliptic Curves. New York-Amsterdam 1968.
  • [106] J.-P. Serre, Travaux de Wiles (et Taylor,...), Partie I. Sém. Bourbaki, exp. 803, Astérisque 237 (1996), 319-332.
  • [107] J.-P. Serre, Lectures on the Mordell-Weil Theorem. Aspects of Math. 15, Braunschweig 1989.
  • [108] J.-P. Serre, J. Tate, Good reduction of abelian varieties, Ann. Math. 88 (1968), 492-517.
  • [109] I. R. Shafarevich, Basic Algebraic Geometry, Moscow 1988 (ros.).
  • [11О] I. R. Shafarevich, Algebraic number fields. In: Proc. Intern. Congr. Math., Stockholm 1962, 163-176.
  • [111] N. I. Shepherd-Barron, R. Taylor, Mod 2 and mod 5 icosahedral representations, J. Arner. Math. Soc. 10 (1997), 283-298.
  • [112] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions. Iwanami Shoten and Princeton Univ. Press, Princeton 1971.
  • [113] G. Shimura, On elliptic curves with complex multiplication as factors of the Jacobians of modular function fields, Nagoya Math. J. 43 (1971), 199-208.
  • [114] G. Shimura, Yutaka Taniyama and his time. Very personal recollections, Bull. London Math. Soc. 21 (1989), 186-196.
  • [115] A. Silverberg, Explicit families of elliptic curves with prescribed mod N representations. In: Modular Forms and Fermat’s Last Theorem (eds. G. Cornell et al.), Springer-Verlag 1997, 447-461.
  • [116] J. Silverman, The Arithmetic of Elliptic Curves. Graduate Texts in Math. 106, Springer-Verlag 1985.
  • [117] J. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Graduate Texts in Math. 151, Springer-Verlag 1994.
  • [118] S. Singh, Tajemnica Fermata, Prószyński i S-ka 1999 (przekład z j. ang.).
  • [119] C. Skinner, A. Wiles, Residually reducible representations and modular forms, Publ. Math. IHES 89 (1999), 5-126.
  • [120] C. Skinner, A. Wiles, Nearly ordinary deformations of irreducible residual representations, Ann. Fac. Sci. Toulouse Math. 10 (2001), 185-215.
  • [121] R. Taylor, Automorphy for some l-adic lifts of automorphic mod l representations II, preprint 2006.
  • [122] R. Taylor, Remarks on a conjecture of Fomtaine and Mazur, J. Inst. Math. Jussieu 1 (2002), 1-19.
  • [123] R. Taylor, A. Wiles, Ring theoretic properties of certain Hecke algebras, Ann. Math. 141 (1995), 553-572.
  • [124] T. Tsuji, p-adic étale cohomology and cristalline cohomology in the semi-stable reduction case, Invent. Math. 137 (1999), 233-411.
  • [125] J. Tunnell, Artin’s conjecture for representations of octahedral type, Bull. A.M.S. 5 (1981), 173-175.
  • [126] A. J. van der Poorten, Notes on Fermat’s Last Theorem. Canadian Math. Soc. Series of Monographs and Adv. Texts, A Wiley-Interscience Publication 1996.
  • [127] P. Vоjta, Diophantine Approximation and Value Distribution Theory. Lect. Notes in Math. 476, Springer 1987.
  • [128] L. C. Washington, Introduction to Cyclotomic Fields, Graduate Texts in Math. 83, Springer 1997.
  • [129] A. Weil, Über die Bestimmung Dirichletseher Reichen durch Funktionalgleichungen, Math. Ann. 168 (1967), 165-172.
  • [130] W. Więsław, Matematyka i Jej Historia. Opole 1997.
  • [131] A. Wiles, Modular elliptic curves and Fermat’s Last Theorem, Ann. Math. 141 (1995), 443-551.
  • [132] J.-P. Wintenberger, La conjecture de modularité de Serre: le cas de conducteur 1 [d’après C. Khare], Sém. Bourbaki, exp. 956 (2005-2006).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-db791f91-60a5-4c0e-9b96-ecabf408ce11
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.