On the Conjecture of Birch and Swinnerton-Dyer for Elliptic Curves with Complex Multiplication

  • Benedict H. Gross
Part of the Progress in Mathematics book series (PM, volume 26)


Let E be an elliptic curve which is defined over a number field F. Let L(E,s) be its L-series over F, which is defined by an Euler product convergent in the half-plane Re(s) > 3/2 (2.4). Whenever L(E,s) has an analytic continuation to the entire complex plane, we can consider the first term in its Taylor expansion at s= 1; we write L(E,s) ~ c(E) (s − 1)r(E) as s → 1, where r(E) is a nonnegative integer and c(E) is a nonzero real number. Birch and Swinnerton-Dyer have conjectured that r(E) is equal to the rank of the finitely generated group E(F) of points over F; they have also given a conjectural formula for c(E) in terms of certain arithmetic invariants of the curve (2.10).


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  1. [1]
    Birch, B. J. and Swinnerton-Dyer, H.P.F. “Notes on Elliptic Curves (II),Crelle J. 28, (1965), 79–108.Google Scholar
  2. [2]
    Bloch, S. “A Note on Height Pairings, Tamagawa Numbers, and the Birch and Swi nnerton-Dyer Conjecture,” Inv. Math. (1980), 65–76.Google Scholar
  3. [3]
    Buhler, J.P. and Gross, B.H. “Arithmetic on Elliptic Curves with Complex Multiplication II” (to appear).Google Scholar
  4. [4]
    Goldstein, C. and Schappacher, N. “Series d1Eisenstein et fonctions L de courbes elliptiques a multiplication complexe,” Crelle J. 327 (1981), 184–218.Google Scholar
  5. [5]
    Gross, B.H. “Arithmetic on Elliptic Curves with Complex Multiplikation,” Springer Lecture Notes 776 (1980).Google Scholar
  6. [6]
    Grothendieck, A. Modeies de Neron et monodromie. SGA 7 (1970).Google Scholar
  7. [7]
    Manin, Y.l. “Cyclotomic Fields and Modular Curves,” Russian Math. Surveys 26 (1978), 7–78.Google Scholar
  8. [8]
    Mi Ine, J.S. “On the Arithmetic of Abelian Varieties,” Inv. Math. 22 (1972), 177–190.Google Scholar
  9. [9]
    Ogg, A.P. “ Elliptic Curves and Wild Rami fication , ” Amer. J. 89,(1967), 1–21..CrossRefGoogle Scholar
  10. [10]
    Serre, J.-P. and Täte, J. “Good Reduction of Abelian Varieties,” Annais of Math. 88 (1968), 492–517.CrossRefGoogle Scholar
  11. [11]
    Täte, J. “On the Conjecture of Birch and Swinnerton-Dyer and a Geometric Analog,” Sem. Bourbaki 306 (1966).Google Scholar

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© Springer Science+Business Media New York 1982

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  • Benedict H. Gross

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