Advertisement

The Conjecture of Birch and Swinnerton-Dyer

  • John Coates
Chapter

Abstract

The conjecture of Birch and Swinnerton-Dyer is one of the principal open problems of number theory today. Since it involves exact formulae rather than asymptotic questions, it has been tested numerically more extensively than any other conjecture in the history of number theory, and the numerical results obtained have always been in perfect accord with every aspect of the conjecture. The present article is aimed at the non-expert, and gives a brief account of the history of the conjecture, its precise formulation, and the partial results obtained so far in support of it.

Keywords

Elliptic Curve Elliptic Curf Abelian Variety Euler System Euler Product 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    M. Bertolini, H. Darmon, V. Rotger, Beilinson-Flach elements and Euler systems II: The Birch-Swinnerton-Dyer conjecture for Hasse-Weil-Artin L-series, J. Algebraic Geometry 24 (2015), 569–604.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    B. Birch, P. Swinnerton-Dyer, Notes on elliptic curves I, Crelle 212 (1963), 7–25.MathSciNetzbMATHGoogle Scholar
  3. 3.
    B. Birch, P. Swinnerton-Dyer, Notes on elliptic curves II, Crelle 218 (1965), 79–108.MathSciNetzbMATHGoogle Scholar
  4. 4.
    B. Birch, Elliptic curves and modular functions in Symposia Mathematica, Indam Rome 1968/1969, Academic Press, 4 (1970), 27–32MathSciNetGoogle Scholar
  5. 5.
    C. Breuil, B. Conrad, F. Diamond, R. Taylor, On the modularity of elliptic curves over \(\mathbb{Q}\) : wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), 843–939.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    D. Bump, S. Friedberg and J. Hoffstein, Non-vanishing theorems for L-functions of modular forms and their derivatives, Invent. Math. 102 (1990), 543–618.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    L. Cai, J. Shu, Y. Tian, Explicit Gross-Zagier and Waldpsurger formulae, Algebra and Number Theory, 8 (2014), 2523–2572.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    J. Cassels, Arithmetic on curves of genus 1, VIII, Crelle 217 (1965), 180–199.MathSciNetzbMATHGoogle Scholar
  9. 9.
    J. Cassels, Arithmetic on curves of genus 1, IV. Proof of the Hauptvermutung, Crelle 211 (1962), 95–112MathSciNetzbMATHGoogle Scholar
  10. 10.
    J. Coates, Elliptic curves with complex multiplication and Iwasawa theory, Bull. London Math. Soc. 23 (1991), 321–350.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    J. Coates, Elliptic curves - The crossroads of theory and computation in ANTS 2002, Springer LNCS 2369 (2002), 9–19.MathSciNetzbMATHGoogle Scholar
  12. 12.
    J. Coates, A. Wiles, On the conjecture of Birch and Swinnerton-Dyer, Invent. Math. 39 (1977), 233–251MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    J. Coates, Y. Li, Y. Tian, S. Zhai, Quadratic twists of elliptic curves, Proc. London Math. Soc. 110 (2015), 357–394.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    B.Creutz, R. Miller, Second isogeny descents and the Birch-Swinnerton-Dyer conjectural formula, J. of Algebra 372 (2012), 673–701.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    J. Cremona, Algorithms for Modular Elliptic Curves, second Edition, Cambridge University Press, 1997.zbMATHGoogle Scholar
  16. 16.
    T. Dokchitser, V. Dokchitser, On the Birch-Swinnerton-Dyer quotients modulo squares, Ann. of Math. 172 (2010), 567–596.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    M. Deuring, Die Zetafunktionen einer algebraischen Kurve von Geschlechts Eins, Nach. Akad. Wiss. Göttingen, (1953) 85–94, (1955) 13–42, (1956) 37–76, (1957) 55–80.Google Scholar
  18. 18.
    G. Faltings, Endlichkeitssatze fur abelsche Varietten ber zahlkorpern, Invent. Math. 73 (1983), 349–366.MathSciNetCrossRefGoogle Scholar
  19. 19.
    N. Freitas, B. Le Hung, and S. Siksek, Elliptic curves over real quadratic fields are modular, Invent. Math., 201 (2015), 159–206.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    D. Goldfeld The conjectures of Birch and Swinnerton-Dyer and the class numbers of imaginary quadratic fields, in Journees arithmetiques de Caen, Asterisque 41–42 (1977), 219–227.MathSciNetzbMATHGoogle Scholar
  21. 21.
    D. Goldfeld Conjectures on elliptic curves over quadratic fields, in Number Theory, Carbondale 1979, Springer Lecture Notes 751 (1979), 108–118.MathSciNetGoogle Scholar
  22. 22.
    B. Gross, Heegner Points on X 0 (N), in Modular Forms (ed. R. A. Rankin). Ellis Horwood (1984).Google Scholar
  23. 23.
    B. Gross, Kolyvagin’s work on modular elliptic curves in L-functions and arithmetic (Durham 1989), London Math. Soc. Lecture Notes 153 (1991), 235–256.Google Scholar
  24. 24.
    B. Gross, D. Zagier, Heegner points and derivatives of L-series, Invent. Math. 84 (1986), 225–320.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    K. Heegner, Diophantische analysis und modulfunktionen, Math. Z. 56 (1952), 227–253.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    K. Kato, p-adic Hodge theory and values of zeta functions and modular forms in Cohomologies p-adiques et applications arithmetiques III, Asterisque 295 (2004), 117–290.Google Scholar
  27. 27.
    V. Kolyvagin, Finiteness of \(E(\mathbb{Q})\) and \(\mbox{ III }(E/\mathbb{Q})\) and for a class of Weil curves, Izv. Akad. Nauk SSSR 52 (1988), translation Math. USSR-Izv. 32 (1989), 523–541.Google Scholar
  28. 28.
    S. Kobayashi, The p-adic Gross-Zagier formula for elliptic curves at supersingular primes, Invent. Math. 191 (2013), 527–629.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    G. Kings, D. Loeffler, S. Zerbes, Rankin-Eisenstein classes and explicit reciprocity laws arXiv.org/abs/1503.02888.Google Scholar
  30. 30.
    A. Lei, D. Loeffler, S. Zerbes Euler systems for Rankin-Selberg convolutions of modular forms, Ann. of Math., 180 (2014), 653–771.MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    D. Loeffler, S. Zerbes, Rankin-Eisenstein classes in Coleman families, arXiv.org/abs/1506.06703.Google Scholar
  32. 32.
    B. Mazur, Modular curves and the Eisenstein ideal, Publ. Math. IHES 47 (1977), 33–186.MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    B. Mazur, Rational points of abelian varieties in towers of number fields, Invent. Math. 18 (1972), 183–266.MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    B. Mazur, P. Swinnerton-Dyer, Arithmetic of Weil curves, Invent. Math. 25 (1974), 1–61.MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    R. Miller, Proving the Birch-Swinnerton-Dyer conjecture for specific elliptic curves of analytic rank zero and one, London Math. Soc. J. Comput. Math. 14(2011), 327–350.MathSciNetzbMATHGoogle Scholar
  36. 36.
    R. Miller, M. Stoll, Explicit isogeny descent on elliptic curves, Math. Comp. 82 (2013), 513–529.MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    K. Murty and R. Murty, Mean values of derivatives of modular L-series, Ann. of Math., 133 (1991), 447–475.MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    J. Oesterle, Nombres de classes de corps quadratiques imaginaires, Seminaire N. Bourbaki, 1983–1984, 631, 309–323.Google Scholar
  39. 39.
    D. Rohrlich, On L-functions of elliptic curves and cyclotomic towers, Invent. Math. 75 (1984), 404–423MathSciNetzbMATHGoogle Scholar
  40. 40.
    B. Perrin-Riou, Fonctions L p-adiques, thorie d’Iwasawa, et points de Heegner, Bull. Soc. Math. France, 115(1987), 399–456.MathSciNetzbMATHGoogle Scholar
  41. 41.
    R. Pollack, K. Rubin The main conjecture for CM elliptic curves at supersingular primes, Ann. of Math. 159 (2004), 447–464.MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    K. Rubin, The main conjectures of Iwasawa theory for imaginary quadratic fields, Invent. Math. 103 (1991), 25–68.MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    K. Rubin, Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication, Invent. Math. 89 (1987), 527–560.MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    K. Rubin, On the main conjecture of Iwasawa theory for imaginary quadratic fields, Invent. Math. 93 (1988), 701–713.MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    P. Schneider, p-adic height pairings II, Invent. Math. 79 (1985), 329–374.MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    G. Shimura, Introduction to the arithmetic theory of automorphic functions, Publ. Math. Soc. Japan 11 (1971).Google Scholar
  47. 47.
    J. Silverman, The arithmetic of elliptic curves, Grad. Texts Math. 106, 1986, Springer.Google Scholar
  48. 48.
    C. Skinner, E. Urban, The Iwasawa main conjecture for GL 2, Invent. Math. 195 (2014), 1–277.MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    N. Stephens, The Diophantine equation \(x^{3} + y^{3} = Dz^{3}\) and the conjectures of Birch and Swinnerton-Dyer, Crelle 231 (1968), 121–162.MathSciNetzbMATHGoogle Scholar
  50. 50.
    J. Tate, Algorithm for determining the type of singular fiber in an elliptic pencil, Modular Functions of One Variable IV, Springer Lecture Notes 476 (1975), 33–52.MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    J. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, Seminaire N. Bourbaki, 1964–1966, 306, 415–440.Google Scholar
  52. 52.
    J. Tate, Duality theorems in Galois cohomology over number fields, Proc. Int. Cong. Math., Stockholm (1962), 288–295.Google Scholar
  53. 53.
    J. Thorne, Elliptic curves over \(\mathbb{Q}_{\infty }\) are modular, arXiv:1505.04769Google Scholar
  54. 54.
    Y. Tian, Congruent numbers with many prime factors, Proc. Natl. Acad. Sci. USA 109 (2012), 21256–21258.MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Y. Tian, Congruent Numbers and Heegner Points, Cambridge Journal of Mathematics, 2 (2014), 117–161.MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    X. Wan, Iwasawa main conjectures for supersingular elliptic curves, arXiv.org/abs/1411.6352Google Scholar
  57. 57.
    A. Wiles, Modular elliptic curves and Fermat’s Last Theorem, Ann. of Math. 172 (2010), 567–596.MathSciNetCrossRefGoogle Scholar
  58. 58.
    R. Yager, On two variable p-adic L-functions, Ann. of Math. 115 (1982), 411–449.MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    S. Zhang, Heights of Heegner points on Shimura curves, Ann. of Math. 153 (2001), 27–147.MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    W. Zhang, Selmer group and the indivisibility of Heegner points, Cambridge Journal of Mathematics 2 (2014), 191–253.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Emmanuel CollegeCambridgeUK

Personalised recommendations