Nous décrivons un panorama des méthodes reliant l'étude des équations diophantiennes ternaires aux techniques modernes issues de la théorie des formes modulaires.
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.
@article{JTNB_1999__11_1_173_0, author = {Merel, Lo{\"\i}c}, title = {Arithmetic of elliptic curves and diophantine equations}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {173--200}, publisher = {Universit\'e Bordeaux I}, volume = {11}, number = {1}, year = {1999}, mrnumber = {1730439}, zbl = {0964.11028}, language = {en}, url = {http://archive.numdam.org/item/JTNB_1999__11_1_173_0/} }
TY - JOUR AU - Merel, Loïc TI - Arithmetic of elliptic curves and diophantine equations JO - Journal de théorie des nombres de Bordeaux PY - 1999 SP - 173 EP - 200 VL - 11 IS - 1 PB - Université Bordeaux I UR - http://archive.numdam.org/item/JTNB_1999__11_1_173_0/ LA - en ID - JTNB_1999__11_1_173_0 ER -
Merel, Loïc. Arithmetic of elliptic curves and diophantine equations. Journal de théorie des nombres de Bordeaux, Tome 11 (1999) no. 1, pp. 173-200. http://archive.numdam.org/item/JTNB_1999__11_1_173_0/
[1] 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. | MR | Zbl
,[2] Modular forms in characteristic l and special values of their L-functions. Duke Math. J. 53 (1986), no.3, 849-868. | MR | Zbl
&[3] The diophantine equation AxP + Byq = Czr. preprint 1995. | MR
,[4] Nonvanishing theorem, for L-functions of modular forms and their derivatives. Invent. Math. 102 (1990), 543-618. | MR | Zbl
, & ,[5] Sur les représentations λ-adiques associées aux formes modulaires de Hilbert. Ann. Sci. de l'ENS 19 (1986), 409-468. | Numdam | Zbl
,[6] The Jacobian of non-split Cartan modular curves. To appear in the Proceedings of the London Mathematical Society. | MR | Zbl
,[7] 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. | MR | Zbl
,[8] The equations xn + yn = z2 and xn + yn = z3. Internat. Math. Res. Notices 10 (1993), 263-274. | MR | Zbl
,[9] Serre's conjecture, in Seminar on Fermat's last Theorem. CMS Conference Proceedings 17, American Mathematical Society, Providence, 135-155. | MR | Zbl
,[10] Faltings plus epsilon, Wiles plus epsilon and the Generalized Fermat Equation. preprint 1997. | MR
,[11] Faltings plus epsilon, Wiles plus epsilon, and the Generalized Fermat Equation. preprint 1997. | MR
,[12] 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. | MR | Zbl
, ,[13] Winding quotients and some variants of Fermat's last theorem. To appear in Crelle. | Zbl
& ,[14] 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. | MR | Zbl
& ,[15] Über die Diophantische Gleichung x + y = cz. Acta Math. 88 (1952), 241-251. | MR | Zbl
,[16] On deformation rings and Hecke rings. Ann. of Math. (2) 144 (1996), no. 1, 137-166. | MR | Zbl
,[17] Modularity of a family of elliptic curves. Math. Res. Letters 2 (1995), 299-304. | MR | Zbl
, ,[18] History of the theory of numbers. Chelsea, New York, 1971. | JFM
,[19] Two theorems on modulars curves. Funct. anal. appl. 2 (1973), 155-156. | MR | Zbl
,[20] On a result of Imin Chen. preprint 1995. To appear in: Séminaire de théorie des nombres de Paris, 1995-96, Cambridge University Press.
,[21] Wiles minus epsilon implies Fermat, in Elliptic curves, modular forms and Fermat's Last Theorem (Hong-Kong 1993). J. Coates, S-T. Yau, eds., Internat. Press, Cambridge, MA, 1995, 38-40. | MR | Zbl
,[22] Links between stable elliptic curves and certain diophantine equations. Ann. Univ. Saraviensis, Ser. Math 1 (1986), 1-40. | MR | Zbl
,[23] Links between solutions of A - B = C and elliptic curves. Lect. Notes in Math. 1380 (1989), 31-62. | MR | Zbl
,[24] 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. | MR | Zbl
,[25] On ternary relations of Fermat type and relations with elliptic curves. preprint 1996.
,[26] 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) . | MR | Zbl
,[27] The Eisenstein descent on J0(N). Invent. Math. 83 (1986), 303-319. | MR | Zbl
& ,[28] Heegner points and derivatives of L-series. Invent. Math. 84 (1986), 225-320. | MR | Zbl
& ,[29] Esquisse d'un programme. 1984.
,[30] Points d'ordre 2ph sur les courbes elliptiques. Acta Arith. 26 (1974/75), no. 3, 253-263. | MR | Zbl
,[31] Thèse. Université de Besançon, 1972.
,[32] Torsion points on elliptic curves defined over quadratic fields. Nagoya Mathematical Journal 109 (1988), 125-149. | MR | Zbl
& ,[33] Points on Shimura curves over fields of even degree. Math. Ann. 286 (1990), 731-734. | MR | Zbl
,[34] Torsion points of elliptic curves over fields of higher degree. International Mathematics Research Notices 6 (1992), 129-133. | MR | Zbl
,[35] Torsion points on elliptic curves and q-coefficients of modular forms. Invent. Math. 109 (1992), 221-229. | MR | Zbl
,[36] p-adic Hodge theory and special values of zeta functions of elliptic cusp forms. to appear.
,[37] Euler systems, Iwasawa theory, and Selmer groups. preprint. | MR
,[38] Generalized explicit reciprocity laws. preprint. | MR
,[39] 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. | MR | Zbl
& ,[40] Une remarque sur les points de torsion des courbes elliptiques. C. R. Acad. Sci. Paris 321, Série I (1995), 1143-1146. | MR | Zbl
[41] Sur certaines variantes de l'équation de Fermat. preprint 1997.
,[42] Courbes modulaires de niveau 11, in Modular functions of one variable V. Lecture Notes in Math. 601 (1977), 115-152. | MR | Zbl
,[43] The Shimura subgroup of Jo(N), in Courbes modulaires et courbes de Shimura. Astérisque 196-197, (1991), 171-203. | MR | Zbl
& ,[44] 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. | MR | Zbl
, ,[45] Parabolic points and zeta functions on modular curves. Math. USSR Izvestija 6, no. 1 (1972), 19-64. | MR | Zbl
,[46] Modular forms and number theory. In the proceedings of the international congress of mathematicians 1978, (1980), 177-186. | MR | Zbl
,[47] Galois properties of division fields of elliptic curves. Bull. London Math. Soc. 25 (1993), no. 3, 247-254. | MR | Zbl
, ,[48] The arithmetic of Weil curves. Invent. Math. 25 (1974), 1-61. | MR | Zbl
, ,[49] Modular curves and the Eisenstein ideal. Publ. Math. IHES 47 (1977), 33-186. | Numdam | MR | Zbl
,[50] Rational isogenies of prime degree. Invent. Math. 44 (1978), 129-162. | MR | Zbl
,[51] Questions about number, in New Directions in Mathematics. to appear.
,[52] Courbes elliptiques et symboles modulaires. Séminaire Bourbaki 414, Lecture Notes in mathematics 317(1973), 277-294. | Numdam | MR | Zbl
,[53] Letter to J. Ellenberg.
,[54] Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent. Math. 124 (1996), no. 1-3, 437-449. | MR | Zbl
,[55] Homologie des courbes modulaires affines et paramétrisations modulaires, in Elliptic curves, modular forms, and Fermat's last theorem (Hong-Kong 1993). J. Coates, S.-T. Yau, eds, Internat. Press, Cambridge, MA, 1995, 110-130. | MR | Zbl
,[56] Rational points on the modular curves Xsplit (p). Compositio Math. 52 (1984), 115-137. | Numdam | MR | Zbl
,[57] Mean values of derivatives of L-series. Ann. Math. 133 (1991), 447-475. | MR | Zbl
, ,[58] La conjecture abc. Enseign. Math. 42 (1996), no. 1-2, 3-24. | MR | Zbl
,[59] Nouvelles approches du "théorème" de Fermat. Sém. Bourbaki 694, Astérisque 161-162, S.M.F. (1988), 165-186. | Numdam | MR | Zbl
,[60] Sur la classification de Néron des courbes elliptiques. J. Number Theory 44 (1993), no.2, 119-152. | MR | Zbl
,[61] Bornes effectives pour la torsion des courbes elliptiques sur les corps de nombres. prépublication 95-33, Institut de recherches mathématiques de Rennes (1995).
,[62] Some diophantine equations of the form xn + yn = zm. to appear. | MR
,[63] On modular representations of Gal(/Q) arising from modular forms. Invent. Math. 100 (1990), 431-476. | MR | Zbl
,[64] On the equation aP + 2αbp + cP = 0. Acta Arith. 79, no. 1, (1997), 7-16. | Zbl
,[65] A report on Wiles' Cambridge lecture. Bull. Amer. Math. Soc. (N.S.) 31 (1994), no.1, 15-38. | MR | Zbl
et ,[66] Propriétés galoisiennes des points d'ordre fini des courbes elliptiques. Invent. Math. 15 (1972), 259-331. | MR | Zbl
,[67] Sur les représentations modulaires de degré 2 de Gal(/Q). Duke Math. J. 54. no. 1 (1987), 179-230. | MR | Zbl
,[68] Propriétés conjecturales des groupes de Galois motiviques et des représentations l-adiques. Proceedings of Symposia in Pure Mathematics, 55 (1994), Part 1, 377-400. | MR | Zbl
,[69] Travaux de Wiles (et Taylor,...), Partie I. Séminaire Bourbaki, 803, Juin 1995, Astérisque 237 (1996), 5, 319-332. | Numdam | MR | Zbl
,[70] Quelques applications du théorème de densité de Chebotarev. Pub. Math. I.H.E.S 54 (1981), 123-201. | Numdam | MR | Zbl
,[71] Œuvres, vol. III, Springer-Verlag.
,[72] Représentations linéaires des groupes finis. Hermann, Paris, 1978. | MR | Zbl
,[73] Heights and elliptic curves. in Arithmetic geometry (Storrs, Conn., 1984). Springer, New-York, 1986, 151-166. | MR | Zbl
,[74] Discriminants et conducteurs de courbes elliptiques, in Séminaire sur les pinceaux de courbes elliptiques (Paris, 1988). Astérisque 183 (1990), 7-18. | Numdam | MR | Zbl
,[75] Ring-theoretic properties of certain Hecke algebras. Ann. of Math. 141 (1995), 553-572. | MR | Zbl
, ,[76] Diophantine approximation and value distribution theory. Lecture Notes in Mathematics, 1239, Springer-Verlag, Berlin, 1987. | MR | Zbl
,[77] Modular elliptic curves and Fermat's Last Theorem. Ann. of Math. 141 (1995), 443-551. | MR | Zbl
,[78] Modular parametrizations of elliptic curves. Can. Math. Bull. 28 (1985), 372-384. | MR | Zbl
,