Rank zero quadratic twists of modular elliptic curves

Ken Ono

Compositio Mathematica (1996)

  • Volume: 104, Issue: 3, page 293-304
  • ISSN: 0010-437X

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Ono, Ken. "Rank zero quadratic twists of modular elliptic curves." Compositio Mathematica 104.3 (1996): 293-304. <http://eudml.org/doc/90491>.

@article{Ono1996,
author = {Ono, Ken},
journal = {Compositio Mathematica},
keywords = {modular form; non-congruent numbers; modular elliptic curve; quadratic twists; Mordell-Weil rank zero; arithmetic progressions; Shimura correspondence},
language = {eng},
number = {3},
pages = {293-304},
publisher = {Kluwer Academic Publishers},
title = {Rank zero quadratic twists of modular elliptic curves},
url = {http://eudml.org/doc/90491},
volume = {104},
year = {1996},
}

TY - JOUR
AU - Ono, Ken
TI - Rank zero quadratic twists of modular elliptic curves
JO - Compositio Mathematica
PY - 1996
PB - Kluwer Academic Publishers
VL - 104
IS - 3
SP - 293
EP - 304
LA - eng
KW - modular form; non-congruent numbers; modular elliptic curve; quadratic twists; Mordell-Weil rank zero; arithmetic progressions; Shimura correspondence
UR - http://eudml.org/doc/90491
ER -

References

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  2. 2 Deligne, P.: La conjecture de Weil I, Publ. Math. I.H.E.S.43 (1974), 273-307. Zbl0287.14001MR340258
  3. 3 Diamond, F. and Kramer, K.: Modularity of a family of elliptic curves, Math. Research Letters (2)(3) (1995), 299-304. Zbl0867.11041MR1338788
  4. 4 Gouvêa, F. and Mazur, B.: The square-free sieve and the rank of elliptic curves, J. Amer. Math. Soc.4 (1991), 1-23. Zbl0725.11027MR1080648
  5. 5 Husemöller, D.: Elliptic curves, Springer-Verlag, New York, 1987. Zbl0605.14032MR868861
  6. 6 Knapp, A.: Elliptic curves, Princeton Univ. Press, 1992. Zbl0804.14013MR1193029
  7. 7 Koblitz, N.: Introduction to elliptic curves and modular forms, Springer-Verlag, 1984. Zbl0553.10019MR766911
  8. 8 Kohnen, W.: A remark on the Shimura correspondence, Glasgow Math. J.30 (1988), 285-291. Zbl0659.10024MR964575
  9. 9 Kolyvagin, V.A.: Finiteness of E(Q) and the Tate-Shafarevich group of E(Q) for a subclass of Weil curves (Russian), Izv. Akad. Nauk, USSR, ser. Matem. 52 (1988). Zbl0662.14017
  10. 10 Lieman, D.: Nonvanishing of L-series associated to cubic twists of elliptic curves, Jour Annals of Math.40 (1994), 81-108. Zbl0817.11029MR1289492
  11. 11 Mai, L. and Murty, M.R.: A note on quadratric twists of an elliptic curve, Elliptic curves and related topics, Ed. H. Kisilevsky and M. R. Murty, Amer. Math. Soc. [CRM Proceedings and Lecture Notes] (1994), 121-124. Zbl0806.14025MR1260959
  12. 12 Murty, M.R. and Murty, V.K.: Mean values of derivatives of modular L-series, Annals of Math.133 (1991) 447-475. Zbl0745.11032MR1109350
  13. 13 Ono, K.: Euler's concordant forms, Acta. Arith., to appear. Zbl0863.11038
  14. 14 Silverman, J.: The arithmetic of elliptic curves, Springer-Verlag, New York, 1986. Zbl0585.14026MR817210
  15. 15 Shimura, G.: On modular forms of half-integral weight, Annals of Math.97 (1973), 440-481. Zbl0266.10022MR332663
  16. 16 Stewart, C.L. and Top, J.: On ranks of twists of elliptic curves and power-free values of binary forms, preprint. Zbl0857.11026
  17. 17 Tunnell, J.: A classical Diophantine problem and modular forms of weight 3/2, Inventiones Math.72 (1983), 323-334. Zbl0515.10013MR700775
  18. 18 Vignéras, M.-F.: Facteurs gamma et equations fonctionelles, Springer Lect. in Math.627 (1977), 79-103. Zbl0373.10027MR485739
  19. 19 Waldspurger, J.L.: Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures et Appl.60 (1981), 375-484. Zbl0431.10015MR646366
  20. 20 Wiles, A.: Modular elliptic curves and Fermat's Last Theorem, Ann. Math.141(3) (1995), 443-551. Zbl0823.11029MR1333035

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