Bounds for the order of the Tate-Shafarevich group

Dorian Goldfeld; Lucien Szpiro

Compositio Mathematica (1995)

  • Volume: 97, Issue: 1-2, page 71-87
  • ISSN: 0010-437X

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Goldfeld, Dorian, and Szpiro, Lucien. "Bounds for the order of the Tate-Shafarevich group." Compositio Mathematica 97.1-2 (1995): 71-87. <http://eudml.org/doc/90384>.

@article{Goldfeld1995,
author = {Goldfeld, Dorian, Szpiro, Lucien},
journal = {Compositio Mathematica},
keywords = {order of the Tate-Shafarevich group; conductor; discriminant; modular elliptic curves; Birch-Swinnerton-Dyer conjecture},
language = {eng},
number = {1-2},
pages = {71-87},
publisher = {Kluwer Academic Publishers},
title = {Bounds for the order of the Tate-Shafarevich group},
url = {http://eudml.org/doc/90384},
volume = {97},
year = {1995},
}

TY - JOUR
AU - Goldfeld, Dorian
AU - Szpiro, Lucien
TI - Bounds for the order of the Tate-Shafarevich group
JO - Compositio Mathematica
PY - 1995
PB - Kluwer Academic Publishers
VL - 97
IS - 1-2
SP - 71
EP - 87
LA - eng
KW - order of the Tate-Shafarevich group; conductor; discriminant; modular elliptic curves; Birch-Swinnerton-Dyer conjecture
UR - http://eudml.org/doc/90384
ER -

References

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  5. 5 Goldfeld, D.: Modular elliptic curves and diophantine problems, in Number Theory, Proc. Conf. of the Canad. Number Theory Assoc., Banff, C Alberta, Canada, 1988, pp. 157-176. Zbl0715.14014MR1106659
  6. 6 Gross, B.H.: Kolyvagin's work on elliptic curves, in L-functions and Arithmetic, Proc. of the Durham Symp., 1989, pp. 235-256. Zbl0743.14021MR1110395
  7. 7 Hindry, M. and Silverman, J.H.: The canonical height and integral points on elliptic curves, Invent. Math.93 (1988), 419-450. Zbl0657.14018MR948108
  8. 8 Kolyvagin, V.A.: Finiteness of E(Q) and III(E/Q) for a class of Weil curves, Izv. Akad. Nauk SSSR52 (1988). Zbl0662.14017
  9. 9 Kohnen, W. and Zagier, D.B.: Values of L-series of modular forms at the centre of the critical strip, Invent. Math.64 (1981), 175-198. Zbl0468.10015MR629468
  10. 10 Lang, S.: Conjectured diophantine estimates on elliptic curves, in Arithmetic and Geometry, Papers dedicated to I.R. Shafarevich on the occasion of his sixtieth birthday, Vol. I, Arithmetic, Birkhäuser, 1983, pp. 155-172. Zbl0529.14017MR717593
  11. 11 Mazur, B.: Modular curves and the Eisenstein ideal, IHES Publ. Math.47 (1977), 33-186. Zbl0394.14008MR488287
  12. 12 Milne, J.S.: On a conjecture of Artin and Tate, Annals of Math.102 (1975), 517-533. Zbl0343.14005MR414558
  13. 13 Pesenti, J. and Szpiro, L.: Discriminant et conducteur des courbes elliptiques non semi-stable, à paraitre. Zbl0742.14026
  14. 14 Rubin, K.: Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer, Invent. Math.64 (1981), 455-470. Zbl0506.14039MR632985
  15. 15 Silverman, J.: The Arithmetic of Elliptic Curves, Graduate Texts in Math.106, Springer-Verlag, 1986. Zbl0585.14026MR817210
  16. 16 Szpiro, L.: Propriétés numériques du faisceau dualisant relatif, in Pinceaux de Courbes de Genre au Moins Deux, Asterisque86 (1981), 44-78. Zbl0517.14006
  17. 17 Szpiro, L.: Discriminant et conducteur, in Seminaire sur les Pinceaux de Courbes Elliptiques, Asterisque183 (1990), 7-17. Zbl0742.14026MR1065151
  18. 18 Tate, J.: An algorithm for determining the type of a singular fiber in an elliptic pencil, in Modular Functions of One Variable IV, Lecture Notes in Math.476, Springer-Verlag, 1975, pp. 33-52. Zbl1214.14020MR393039
  19. 19 Tate, J.: On a conjecture of Birch and Swinnerton-Dyer and a geometric analogue, Seminaire N. Bourbaki, Exposé 306, 1966. Zbl0199.55604
  20. 20 Taylor, R. and Wiles, A.: Ring theoretic properties of certain Hecke algebras, to appear. Zbl0823.11030
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  24. 24 Wiles, A.: Modular elliptic curves and Fermat's last theorem, to appear. Zbl0823.11029

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