Arithmetic of elliptic curves and diophantine equations

Loïc Merel

Journal de théorie des nombres de Bordeaux (1999)

  • Volume: 11, Issue: 1, page 173-200
  • ISSN: 1246-7405

Abstract

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We give a survey of methods used to connect the study of ternary diophantine equations to modern techniques coming from the theory of modular forms.

How to cite

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Merel, Loïc. "Arithmetic of elliptic curves and diophantine equations." Journal de théorie des nombres de Bordeaux 11.1 (1999): 173-200. <http://eudml.org/doc/248334>.

@article{Merel1999,
abstract = {We give a survey of methods used to connect the study of ternary diophantine equations to modern techniques coming from the theory of modular forms.},
author = {Merel, Loïc},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {Fermat equation; Dénes’ equations; conjecture; curves of Frey; degree conjecture; survey; modular forms; Galois representation; modular elliptic curve},
language = {eng},
number = {1},
pages = {173-200},
publisher = {Université Bordeaux I},
title = {Arithmetic of elliptic curves and diophantine equations},
url = {http://eudml.org/doc/248334},
volume = {11},
year = {1999},
}

TY - JOUR
AU - Merel, Loïc
TI - Arithmetic of elliptic curves and diophantine equations
JO - Journal de théorie des nombres de Bordeaux
PY - 1999
PB - Université Bordeaux I
VL - 11
IS - 1
SP - 173
EP - 200
AB - We give a survey of methods used to connect the study of ternary diophantine equations to modern techniques coming from the theory of modular forms.
LA - eng
KW - Fermat equation; Dénes’ equations; conjecture; curves of Frey; degree conjecture; survey; modular forms; Galois representation; modular elliptic curve
UR - http://eudml.org/doc/248334
ER -

References

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  1. [1] D. Abramovich, Formal finiteness and the torsion conjecture on elliptic curves. A footnote to a paper: "Rational torsion of prime order in elliptic curves over number fields" by S. Kamienny and B. Mazur. Astérisque 228 (1995), Columbia University Number Theory Seminar (New- York, 1992), 5-17. Zbl0846.14013MR1330925
  2. [2] A. Ash & G. StevensModular forms in characteristic l and special values of their L-functions. Duke Math. J.53 (1986), no.3, 849-868. Zbl0618.10026MR860675
  3. [3] F. Beukers, The diophantine equation AxP + Byq = Czr. preprint 1995. MR1487980
  4. [4] D. Bump, S. Friedberg & J. Hoffstein, Nonvanishing theorem, for L-functions of modular forms and their derivatives. Invent. Math.102 (1990), 543-618. Zbl0721.11023MR1074487
  5. [5] H. Carayol,Sur les représentations λ-adiques associées aux formes modulaires de Hilbert. Ann. Sci. de l'ENS19 (1986), 409-468. Zbl0616.10025
  6. [6] I. Chen, The Jacobian of non-split Cartan modular curves. To appear in the Proceedings of the London Mathematical Society. Zbl0903.11019MR1625491
  7. [7] J. Cremona, Computing the degree of a modular parametrization, in Algorithmic number theory (Ithaca, NY, 1994), 134-142, Lecture Notes in Comput. Sci. 877, Springer, Berlin, 1994. Zbl0840.14018MR1322718
  8. [8] H. Darmon, The equations xn + yn = z2 and xn + yn = z3. Internat. Math. Res. Notices10 (1993), 263-274. Zbl0805.11028MR1242931
  9. [9] H. Darmon, Serre's conjecture, in Seminar on Fermat's last Theorem. CMS Conference Proceedings 17, American Mathematical Society, Providence, 135-155. Zbl0848.11019MR1357210
  10. [10] H. Darmon, Faltings plus epsilon, Wiles plus epsilon and the Generalized Fermat Equation. preprint 1997. Zbl0932.11022MR1479291
  11. [11] H. Darmon, Faltings plus epsilon, Wiles plus epsilon, and the Generalized Fermat Equation. preprint1997. Zbl0932.11022MR1479291
  12. [12] H. Darmon, A. Granville, On the equations xP + yq = zr and zm = f(x, y). Bulletin of the London Math. Society, no 129, 27 part 6, November (1995), 513-544. Zbl0838.11023MR1348707
  13. [13] H. Darmon & L. Merel, Winding quotients and some variants of Fermat's last theorem. To appear in Crelle. Zbl0976.11017
  14. [14] P. Deligne & M. Rappoport, Les schémas de module des courbes elliptiques, in Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), 143-316, Lecture Notes in mathematics 349, Springer, Berlin, 1975. Zbl0281.14010MR337993
  15. [15] P. Dénes, Über die Diophantische Gleichung x + y = cz. Acta Math.88 (1952), 241-251. Zbl0048.27503MR68560
  16. [16] F. Diamond, On deformation rings and Hecke rings. Ann. of Math. (2) 144 (1996), no. 1, 137-166. Zbl0867.11032MR1405946
  17. [17] F. Diamond, K. Kramer, Modularity of a family of elliptic curves. Math. Res. Letters2 (1995), 299-304. Zbl0867.11041MR1338788
  18. [18] L.E. Dickson, History of the theory of numbers. Chelsea, New York, 1971. JFM60.0817.03
  19. [19] V. Drinfeld, Two theorems on modulars curves. Funct. anal. appl.2 (1973), 155-156. Zbl0285.14006MR318157
  20. [20] B. Edixhoven, On a result of Imin Chen. preprint 1995. To appear in: Séminaire de théorie des nombres de Paris, 1995-96, Cambridge University Press. 
  21. [21] N. Elkies, Wiles minus epsilon implies Fermat, in Elliptic curves, modular forms and Fermat's Last Theorem (Hong-Kong1993). J. Coates, S-T. Yau, eds., Internat. Press, Cambridge, MA, 1995, 38-40. Zbl0836.11013MR1363494
  22. [22] G. Frey, Links between stable elliptic curves and certain diophantine equations. Ann. Univ. Saraviensis, Ser. Math1 (1986), 1-40. Zbl0586.10010MR853387
  23. [23] G. Frey, Links between solutions of A - B = C and elliptic curves. Lect. Notes in Math. 1380 (1989), 31-62. Zbl0688.14018MR1009792
  24. [24] G. Frey, On elliptic curves with isomorphic torsion structures and corresponding curves of genus 2, in Elliptic curves, modular forms and Fermat's Last Theorem (Hong-Kong,1993). J. Coates, S-T. Yau, eds., Internat. Press, Cambridge, MA, 1995, 79-98. Zbl0856.11026MR1363496
  25. [25] G. Frey, On ternary relations of Fermat type and relations with elliptic curves. preprint 1996. 
  26. [26] A. Granville, On the number of solutions of the generalized Fermat equation, in Number Theory (Halifax, NS, 1994), 197-207, CMS Conf. Proc., 15, Amer. Math. Soc., Providence, RI, (1994) . Zbl0836.11014MR1353932
  27. [27] B. Gross & G. Lubin, The Eisenstein descent on J0(N). Invent. Math.83 (1986), 303-319. Zbl0594.14027MR818355
  28. [28] B. Gross & D. Zagier, Heegner points and derivatives of L-series. Invent. Math.84 (1986), 225-320. Zbl0608.14019MR833192
  29. [29] A. Grothendieck, Esquisse d'un programme. 1984. 
  30. [30] Y. Hellegouarch, Points d'ordre 2ph sur les courbes elliptiques. Acta Arith.26 (1974/75), no. 3, 253-263. Zbl0264.14007MR379507
  31. [31] Y. Hellegouarch, Thèse. Université de Besançon, 1972. 
  32. [32] M. Kenku & F. Momose, Torsion points on elliptic curves defined over quadratic fields. Nagoya Mathematical Journal109 (1988), 125-149. Zbl0647.14020MR931956
  33. [33] S. Kamienny, Points on Shimura curves over fields of even degree. Math. Ann.286 (1990), 731-734. Zbl0693.14010MR1045399
  34. [34] S. Kamienny, Torsion points of elliptic curves over fields of higher degree. International Mathematics Research Notices6 (1992), 129-133. Zbl0807.14022MR1167117
  35. [35] S. Kamienny, Torsion points on elliptic curves and q-coefficients of modular forms. Invent. Math.109 (1992), 221-229. Zbl0773.14016MR1172689
  36. [36] K. Kato, p-adic Hodge theory and special values of zeta functions of elliptic cusp forms. to appear. 
  37. [37] K. Kato, Euler systems, Iwasawa theory, and Selmer groups. preprint. Zbl0993.11033MR1727298
  38. [38] K. Kato, Generalized explicit reciprocity laws. preprint. Zbl1024.11029MR1701912
  39. [39] V.A. Kolyvagin & D. Logachev, Finiteness of the Shafarevich-Tate group and the group of rational points for some modular abelian varieties, Leningrad Math. J., vol. 1 no. 5 (1990), 1229-1253. Zbl0728.14026MR1036843
  40. [40] A. Kraus.Une remarque sur les points de torsion des courbes elliptiques. C. R. Acad. Sci. Paris321, Série I (1995), 1143-1146. Zbl0862.11037MR1360773
  41. [41] A. Kraus, Sur certaines variantes de l'équation de Fermat. preprint 1997. 
  42. [42] G. Ligozat, Courbes modulaires de niveau 11, in Modular functions of one variable V. Lecture Notes in Math. 601 (1977), 115-152. Zbl0357.14006MR463118
  43. [43] S. Ling & J. Oesterlé, The Shimura subgroup of Jo(N), in Courbes modulaires et courbes de Shimura. Astérisque196-197, (1991), 171-203. Zbl0781.14015MR1141458
  44. [44] L. Mai, R. Murty, The Phragmen-Lindelof theorem and modular elliptic curves, in The Rademacher legacy to mathematics University Park, PA, 1992, 335-340, Contemp. Math., 166, Amer. Math. Soc., Providence, RI, 1994. Zbl0822.11047MR1284072
  45. [45] Y. Manin, Parabolic points and zeta functions on modular curves. Math. USSR Izvestija6, no. 1 (1972), 19-64. Zbl0248.14010MR314846
  46. [46] Y. Manin, Modular forms and number theory. In the proceedings of the international congress of mathematicians 1978, (1980), 177-186. Zbl0421.10016MR562606
  47. [47] D. Masser, G. Wüstholz, Galois properties of division fields of elliptic curves. Bull. London Math. Soc.25 (1993), no. 3, 247-254. Zbl0809.14026MR1209248
  48. [48] B. Mazur, H.P.F. Swinnerton-Dyer, The arithmetic of Weil curves. Invent. Math.25 (1974), 1-61. Zbl0281.14016MR354674
  49. [49] B. Mazur, Modular curves and the Eisenstein ideal. Publ. Math. IHES47 (1977), 33-186. Zbl0394.14008MR488287
  50. [50] B. Mazur, Rational isogenies of prime degree. Invent. Math.44 (1978), 129-162. Zbl0386.14009MR482230
  51. [51] B. Mazur, Questions about number, in New Directions in Mathematics. to appear. 
  52. [52] B. Mazur, Courbes elliptiques et symboles modulaires. Séminaire Bourbaki414, Lecture Notes in mathematics317(1973), 277-294. Zbl0276.14012MR429921
  53. [53] B. Mazur, Letter to J. Ellenberg. 
  54. [54] L. Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent. Math.124 (1996), no. 1-3, 437-449. Zbl0936.11037MR1369424
  55. [55] L. Merel, Homologie des courbes modulaires affines et paramétrisations modulaires, in Elliptic curves, modular forms, and Fermat's last theorem (Hong-Kong1993). J. Coates, S.-T. Yau, eds, Internat. Press, Cambridge, MA, 1995, 110-130. Zbl0845.11023MR1363498
  56. [56] F. Momose, Rational points on the modular curves Xsplit (p). Compositio Math.52 (1984), 115-137. Zbl0574.14023MR742701
  57. [57] K. Murty, R. Murty, Mean values of derivatives of L-series. Ann. Math.133 (1991), 447-475. Zbl0745.11032MR1109350
  58. [58] A. Nitaj, La conjecture abc. Enseign. Math.42 (1996), no. 1-2, 3-24. Zbl0856.11014MR1395039
  59. [59] J. Oesterlé, Nouvelles approches du "théorème" de Fermat. Sém. Bourbaki 694, Astérisque 161-162, S.M.F. (1988), 165-186. Zbl0668.10024MR992208
  60. [60] I. Papadopoulos, Sur la classification de Néron des courbes elliptiques. J. Number Theory44 (1993), no.2, 119-152. Zbl0786.14020MR1225948
  61. [61] P. Parent, Bornes effectives pour la torsion des courbes elliptiques sur les corps de nombres. prépublication 95-33, Institut de recherches mathématiques de Rennes (1995). 
  62. [62] B. Poonen, Some diophantine equations of the form xn + yn = zm. to appear. MR1655978
  63. [63] K. Ribet, On modular representations of Gal(/Q) arising from modular forms. Invent. Math.100 (1990), 431-476. Zbl0773.11039MR1047143
  64. [64] K. Ribet, On the equation aP + 2αbp + cP = 0. Acta Arith.79, no. 1, (1997), 7-16. Zbl0877.11015
  65. [65] K. Rubin et A. Silverberg, A report on Wiles' Cambridge lecture. Bull. Amer. Math. Soc. (N.S.) 31 (1994), no.1, 15-38. Zbl0924.11046MR1256978
  66. [66] Serre J.-P., Propriétés galoisiennes des points d'ordre fini des courbes elliptiques. Invent. Math.15 (1972), 259-331. Zbl0235.14012MR387283
  67. [67] J.-P. Serre, Sur les représentations modulaires de degré 2 de Gal(/Q). Duke Math. J.54. no. 1 (1987), 179-230. Zbl0641.10026MR885783
  68. [68] J.-P. Serre, Propriétés conjecturales des groupes de Galois motiviques et des représentations l-adiques. Proceedings of Symposia in Pure Mathematics, 55 (1994), Part 1, 377-400. Zbl0812.14002MR1265537
  69. [69] J.-P. Serre, Travaux de Wiles (et Taylor,...), Partie I. Séminaire Bourbaki, 803, Juin 1995, Astérisque237 (1996), 5, 319-332. Zbl0957.11027MR1423630
  70. [70] J.-P. Serre, Quelques applications du théorème de densité de Chebotarev. Pub. Math. I.H.E.S54 (1981), 123-201. Zbl0496.12011MR644559
  71. [71] J.-P. Serre, Œuvres, vol. III, Springer-Verlag. 
  72. [72] J.-P. Serre, Représentations linéaires des groupes finis. Hermann, Paris, 1978. Zbl0407.20003MR543841
  73. [73] J. Silverman, Heights and elliptic curves. in Arithmetic geometry (Storrs, Conn., 1984). Springer, New-York, 1986, 151-166. Zbl0603.14020MR861979
  74. [74] L. Szpiro, Discriminants et conducteurs de courbes elliptiques, in Séminaire sur les pinceaux de courbes elliptiques (Paris, 1988). Astérisque183 (1990), 7-18. Zbl0742.14026MR1065151
  75. [75] R. Taylor, A. Wiles, Ring-theoretic properties of certain Hecke algebras. Ann. of Math.141 (1995), 553-572. Zbl0823.11030MR1333036
  76. [76] P. Vojta, Diophantine approximation and value distribution theory. Lecture Notes in Mathematics, 1239, Springer-Verlag, Berlin, 1987. Zbl0609.14011MR883451
  77. [77] A. Wiles, Modular elliptic curves and Fermat's Last Theorem. Ann. of Math.141 (1995), 443-551. Zbl0823.11029MR1333035
  78. [78] D. Zagier, Modular parametrizations of elliptic curves. Can. Math. Bull.28 (1985), 372-384. Zbl0579.14027MR790959

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