Arithmetic of elliptic curves and diophantine equations
Journal de théorie des nombres de Bordeaux, Volume 11 (1999) no. 1, pp. 173-200.

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.

Nous décrivons un panorama des méthodes reliant l'étude des équations diophantiennes ternaires aux techniques modernes issues de la théorie des formes modulaires.

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Merel, Loïc. Arithmetic of elliptic curves and diophantine equations. Journal de théorie des nombres de Bordeaux, Volume 11 (1999) no. 1, pp. 173-200. http://www.numdam.org/item/JTNB_1999__11_1_173_0/

[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. | MR | Zbl

[2] A. Ash & G. Stevens Modular forms in characteristic l and special values of their L-functions. Duke Math. J. 53 (1986), no.3, 849-868. | MR | Zbl

[3] F. Beukers, The diophantine equation AxP + Byq = Czr. preprint 1995. | MR

[4] D. Bump, S. Friedberg & J. Hoffstein, Nonvanishing theorem, for L-functions of modular forms and their derivatives. Invent. Math. 102 (1990), 543-618. | MR | Zbl

[5] H. Carayol,Sur les représentations λ-adiques associées aux formes modulaires de Hilbert. Ann. Sci. de l'ENS 19 (1986), 409-468. | Numdam | Zbl

[6] I. Chen, The Jacobian of non-split Cartan modular curves. To appear in the Proceedings of the London Mathematical Society. | MR | Zbl

[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. | MR | Zbl

[8] H. Darmon, The equations xn + yn = z2 and xn + yn = z3. Internat. Math. Res. Notices 10 (1993), 263-274. | MR | Zbl

[9] H. Darmon, Serre's conjecture, in Seminar on Fermat's last Theorem. CMS Conference Proceedings 17, American Mathematical Society, Providence, 135-155. | MR | Zbl

[10] H. Darmon, Faltings plus epsilon, Wiles plus epsilon and the Generalized Fermat Equation. preprint 1997. | MR

[11] H. Darmon, Faltings plus epsilon, Wiles plus epsilon, and the Generalized Fermat Equation. preprint 1997. | MR

[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. | MR | Zbl

[13] H. Darmon & L. Merel, Winding quotients and some variants of Fermat's last theorem. To appear in Crelle. | Zbl

[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. | MR | Zbl

[15] P. Dénes, Über die Diophantische Gleichung x + y = cz. Acta Math. 88 (1952), 241-251. | MR | Zbl

[16] F. Diamond, On deformation rings and Hecke rings. Ann. of Math. (2) 144 (1996), no. 1, 137-166. | MR | Zbl

[17] F. Diamond, K. Kramer, Modularity of a family of elliptic curves. Math. Res. Letters 2 (1995), 299-304. | MR | Zbl

[18] L.E. Dickson, History of the theory of numbers. Chelsea, New York, 1971. | JFM

[19] V. Drinfeld, Two theorems on modulars curves. Funct. anal. appl. 2 (1973), 155-156. | MR | Zbl

[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] N. Elkies, 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] G. Frey, Links between stable elliptic curves and certain diophantine equations. Ann. Univ. Saraviensis, Ser. Math 1 (1986), 1-40. | MR | Zbl

[23] G. Frey, Links between solutions of A - B = C and elliptic curves. Lect. Notes in Math. 1380 (1989), 31-62. | MR | Zbl

[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. | MR | Zbl

[25] G. Frey, On ternary relations of Fermat type and relations with elliptic curves. preprint 1996.

[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) . | MR | Zbl

[27] B. Gross & G. Lubin, The Eisenstein descent on J0(N). Invent. Math. 83 (1986), 303-319. | MR | Zbl

[28] B. Gross & D. Zagier, Heegner points and derivatives of L-series. Invent. Math. 84 (1986), 225-320. | MR | Zbl

[29] A. Grothendieck, Esquisse d'un programme. 1984.

[30] Y. Hellegouarch, Points d'ordre 2ph sur les courbes elliptiques. Acta Arith. 26 (1974/75), no. 3, 253-263. | MR | Zbl

[31] Y. Hellegouarch, Thèse. Université de Besançon, 1972.

[32] M. Kenku & F. Momose, Torsion points on elliptic curves defined over quadratic fields. Nagoya Mathematical Journal 109 (1988), 125-149. | MR | Zbl

[33] S. Kamienny, Points on Shimura curves over fields of even degree. Math. Ann. 286 (1990), 731-734. | MR | Zbl

[34] S. Kamienny, Torsion points of elliptic curves over fields of higher degree. International Mathematics Research Notices 6 (1992), 129-133. | MR | Zbl

[35] S. Kamienny, Torsion points on elliptic curves and q-coefficients of modular forms. Invent. Math. 109 (1992), 221-229. | MR | Zbl

[36] K. Kato, p-adic Hodge theory and special values of zeta functions of elliptic cusp forms. to appear.

[37] K. Kato, Euler systems, Iwasawa theory, and Selmer groups. preprint. | MR

[38] K. Kato, Generalized explicit reciprocity laws. preprint. | MR

[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. | MR | Zbl

[40] A. Kraus. Une remarque sur les points de torsion des courbes elliptiques. C. R. Acad. Sci. Paris 321, Série I (1995), 1143-1146. | MR | Zbl

[41] A. Kraus, Sur certaines variantes de l'équation de Fermat. preprint 1997.

[42] G. Ligozat, Courbes modulaires de niveau 11, in Modular functions of one variable V. Lecture Notes in Math. 601 (1977), 115-152. | MR | Zbl

[43] S. Ling & J. Oesterlé, The Shimura subgroup of Jo(N), in Courbes modulaires et courbes de Shimura. Astérisque 196-197, (1991), 171-203. | MR | Zbl

[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. | MR | Zbl

[45] Y. Manin, Parabolic points and zeta functions on modular curves. Math. USSR Izvestija 6, no. 1 (1972), 19-64. | MR | Zbl

[46] Y. Manin, Modular forms and number theory. In the proceedings of the international congress of mathematicians 1978, (1980), 177-186. | MR | Zbl

[47] D. Masser, G. Wüstholz, Galois properties of division fields of elliptic curves. Bull. London Math. Soc. 25 (1993), no. 3, 247-254. | MR | Zbl

[48] B. Mazur, H.P.F. Swinnerton-Dyer, The arithmetic of Weil curves. Invent. Math. 25 (1974), 1-61. | MR | Zbl

[49] B. Mazur, Modular curves and the Eisenstein ideal. Publ. Math. IHES 47 (1977), 33-186. | Numdam | MR | Zbl

[50] B. Mazur, Rational isogenies of prime degree. Invent. Math. 44 (1978), 129-162. | MR | Zbl

[51] B. Mazur, Questions about number, in New Directions in Mathematics. to appear.

[52] B. Mazur, Courbes elliptiques et symboles modulaires. Séminaire Bourbaki 414, Lecture Notes in mathematics 317(1973), 277-294. | Numdam | MR | Zbl

[53] B. Mazur, Letter to J. Ellenberg.

[54] L. Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent. Math. 124 (1996), no. 1-3, 437-449. | MR | Zbl

[55] L. Merel, 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] F. Momose, Rational points on the modular curves Xsplit (p). Compositio Math. 52 (1984), 115-137. | Numdam | MR | Zbl

[57] K. Murty, R. Murty, Mean values of derivatives of L-series. Ann. Math. 133 (1991), 447-475. | MR | Zbl

[58] A. Nitaj, La conjecture abc. Enseign. Math. 42 (1996), no. 1-2, 3-24. | MR | Zbl

[59] J. Oesterlé, Nouvelles approches du "théorème" de Fermat. Sém. Bourbaki 694, Astérisque 161-162, S.M.F. (1988), 165-186. | Numdam | MR | Zbl

[60] I. Papadopoulos, Sur la classification de Néron des courbes elliptiques. J. Number Theory 44 (1993), no.2, 119-152. | MR | Zbl

[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] B. Poonen, Some diophantine equations of the form xn + yn = zm. to appear. | MR

[63] K. Ribet, On modular representations of Gal(/Q) arising from modular forms. Invent. Math. 100 (1990), 431-476. | MR | Zbl

[64] K. Ribet, On the equation aP + 2αbp + cP = 0. Acta Arith. 79, no. 1, (1997), 7-16. | Zbl

[65] K. Rubin et A. Silverberg, A report on Wiles' Cambridge lecture. Bull. Amer. Math. Soc. (N.S.) 31 (1994), no.1, 15-38. | MR | Zbl

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

[67] J.-P. Serre, Sur les représentations modulaires de degré 2 de Gal(/Q). Duke Math. J. 54. no. 1 (1987), 179-230. | MR | Zbl

[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. | MR | Zbl

[69] J.-P. Serre, Travaux de Wiles (et Taylor,...), Partie I. Séminaire Bourbaki, 803, Juin 1995, Astérisque 237 (1996), 5, 319-332. | Numdam | MR | Zbl

[70] J.-P. Serre, Quelques applications du théorème de densité de Chebotarev. Pub. Math. I.H.E.S 54 (1981), 123-201. | Numdam | MR | Zbl

[71] J.-P. Serre, Œuvres, vol. III, Springer-Verlag.

[72] J.-P. Serre, Représentations linéaires des groupes finis. Hermann, Paris, 1978. | MR | Zbl

[73] J. Silverman, Heights and elliptic curves. in Arithmetic geometry (Storrs, Conn., 1984). Springer, New-York, 1986, 151-166. | MR | Zbl

[74] L. Szpiro, 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] R. Taylor, A. Wiles, Ring-theoretic properties of certain Hecke algebras. Ann. of Math. 141 (1995), 553-572. | MR | Zbl

[76] P. Vojta, Diophantine approximation and value distribution theory. Lecture Notes in Mathematics, 1239, Springer-Verlag, Berlin, 1987. | MR | Zbl

[77] A. Wiles, Modular elliptic curves and Fermat's Last Theorem. Ann. of Math. 141 (1995), 443-551. | MR | Zbl

[78] D. Zagier, Modular parametrizations of elliptic curves. Can. Math. Bull. 28 (1985), 372-384. | MR | Zbl