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On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank $ 3$


Authors: Joe P. Buhler, Benedict H. Gross and Don B. Zagier
Journal: Math. Comp. 44 (1985), 473-481
MSC: Primary 11G40; Secondary 14G25
DOI: https://doi.org/10.1090/S0025-5718-1985-0777279-X
MathSciNet review: 777279
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Abstract: The elliptic curve $ {y^2} = 4{x^3} - 28x + 25$ has rank 3 over Q. Assuming the Weil-Taniyama conjecture for this curve, we show that its L-series $ L(s)$ has a triple zero at $ s = 1$ and compute $ {\lim _{s \to 1}}L(s)/{(s - 1)^3}$ to 28 decimal places; its value agrees with the product of the regulator and real period, in accordance with the Birch-Swinnerton-Dyer conjecture if III is trivial.


References [Enhancements On Off] (What's this?)

  • [1] A. Brumer & K. KRAMER, "The rank of elliptic curves," Duke Math. J., v. 44, 1977, pp. 715-743. MR 0457453 (56:15658)
  • [2] B. Gross & D. Zagier, "Points de Heegner et dérivées de fonctions L," C. R. Acad. Sci. Paris, v. 297, 1983, pp. 85-87. MR 720914 (85d:11062)
  • [3] Y. I. Manin, "Cyclotomic fields and modular curves," Uspekhi Mat. Nauk, v. 26, 1971, pp. 7-71; English transl. in Russian Math. Surveys, v. 26, 1971, pp. 7-78. MR 0401653 (53:5480)
  • [4] B. Mazur & H. P. F. Swinnerton-Dyer, "Arithmetic of Weil curves," Invent. Math., v. 25, 1974, pp. 1-61. MR 0354674 (50:7152)
  • [5] J. Tate, "The arithmetic of elliptic curves," Invent. Math. v. 23, 1974, pp. 179-206. MR 0419359 (54:7380)
  • [6] J. Tate, Letter to J.-P. Serre, Oct. 1, 1979.
  • [7] H. G. Zimmer, "On the difference of the Weil height and the Néron-Tate height," Math. Z., v. 147, 1976, pp. 35-51. MR 0419455 (54:7476)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1985-0777279-X
Article copyright: © Copyright 1985 American Mathematical Society

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