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Number Theory pp 541-586 | Cite as

Connections with Number Theory

  • W. A. Coppel
Chapter
Part of the Universitext book series (UTX)

Abstract

In Proposition II.40 we proved Lagrange’s theorem that every positive integer can be represented as a sum of 4 squares. Jacobi (1829), at the end of his Fundamenta Nova, gave a completely different proof of this theorem with the aid of theta functions. Moreover, his proof provided a formula for the number of different representations. Hurwitz (1896), by developing further the arithmetic of quaternions which was used in Chapter II, also derived this formula. Here we give Jacobi’s argument preference since, although it is less elementary, it is more powerful.

Keywords

Modular Form Elliptic Curve Rational Point Elliptic Curf Theta Function 
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Selected References

  1. [1]
    G.E. Andrews, A polynomial identity which implies the Rogers-Ramanujan identities, Scripta Math. 28 (1970), 297–305.Google Scholar
  2. [2]
    G.E. Andrews, The theory of partitions, Addison-Wesley, Reading, Mass., 1976. [Paperback edition, Cambridge University Press, 1998]MATHGoogle Scholar
  3. [3]
    G.E. Andrews, R.A. Askey, B.C. Berndt, K.G. Ramanathan and R.A. Rankin (ed.), Ramanujan revisited, Academic Press, London, 1988.MATHGoogle Scholar
  4. [4]
    G.E. Andrews, R. Askey and R. Roy, Special functions, Cambridge University Press, 1999.Google Scholar
  5. [5]
    L. Báez-Duarte, Hardy-Ramanujan's asymptotic formula for partitions and the central limit theorem, Adv. in Math. 125 (1997), 114–120.MATHCrossRefGoogle Scholar
  6. [6]
    A. Baker, The diophantine equation y 2 = ax 3 + bx 2 + cx + d, J. London Math. Soc. 43 (1968), 1–9.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    A. Baker, The theory of linear forms in logarithms, Transcendence theory: advances and applications (ed. A. Baker and D.W. Masser), pp. 1–27, Academic Press, London, 1977.Google Scholar
  8. [8]
    R.J. Baxter, Exactly solved models in statistical mechanics, Academic Press, London, 1982. [Reprinted, 1989]MATHGoogle Scholar
  9. [9]
    A. Berkovich and B.M. McCoy, Rogers-Ramanujan identities: a century of progress from mathematics to physics, Proceedings of the International Congress of Mathematicians: Berlin 1998, Vol. III, pp. 163–172, Documenta Mathematica, Bielefeld, 1998.MathSciNetGoogle Scholar
  10. [10]
    J.S. Birman, New points of view in knot theory, Bull. Amer. Math. Soc. (N.S.) 28 (1993), 253–287.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    S. Bloch, A note on height pairings, Tamagawa numbers, and the Birch and Swinnerton-Dyer conjecture, Invent. Math. 58 (1980), 65–76.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    S. Bloch, The proof of the Mordell conjecture, Math. Intelligencer 6 (1984), no. 2, 41–47.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    D.M. Bressoud and D. Zeilberger, A short Rogers-Ramanujan bijection, Discrete Math. 38 (1982), 313–315.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    D.M. Bressoud and D. Zeilberger, Bijecting Euler's partitions-recurrence, Amer. Math. Monthly 92 (1985), 54–55.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Y. Bugeaud, On the size of integer solutions of elliptic equations, Bull. Austral. Math. Soc. 57 (1998), 199–206.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    J.W.S. Cassels, Diophantine equations with special reference to elliptic curves, J. London Math. Soc. 41 (1966), 193–291.CrossRefMathSciNetGoogle Scholar
  17. [17]
    J.S. Chahal, Manin's proof of the Hasse inequality revisited, Nieuw Arch. Wisk. (4) 13 (1995), 219–232.MATHMathSciNetGoogle Scholar
  18. [18]
    J. Čižmár, Birationale Transformationen (Ein historischer Überblick), Period. Polytech. Mech. Engrg. 39 (1995), 9–24.MATHMathSciNetGoogle Scholar
  19. [19]
    G. Cornell and J.H. Silverman (ed.), Arithmetic geometry, Springer-Verlag, New York, 1986.MATHGoogle Scholar
  20. [20]
    G. Cornell, J.H. Silverman and G. Stevens (ed.), Modular forms and Fermat's last theorem, Springer, New York, 1997.MATHGoogle Scholar
  21. [21]
    J.E. Cremona, Algorithms for modular elliptic curves, 2nd ed., Cambridge University Press, 1997.Google Scholar
  22. [22]
    H. Darmon, A proof of the full Shimura–Taniyama–Weil conjecture is announced, Notices Amer. Math. Soc. 46 (1999), 1397–1401.MATHMathSciNetGoogle Scholar
  23. [23]
    L.E. Dickson, History of the theory of numbers, 3 vols., Carnegie Institute, Washington, D.C., 1919–1923. [Reprinted Chelsea, New York, 1992]Google Scholar
  24. [24]
    L. Ehrenpreis and R.C. Gunning (ed.), Theta functions: Bowdoin 1987, Proc. Symp. Pure Math. 49, Amer. Math. Soc., Providence, R.I., 1989.Google Scholar
  25. [25]
    N.D. Elkies, On A 4 + B 4 + C 4 = D 4, Math. Comp. 51 (1988), 825–835.MATHMathSciNetGoogle Scholar
  26. [26]
    L.D. Faddeev and L.A. Takhtajan, Hamiltonian methods in soliton theory, Springer-Verlag, Berlin, 1987.Google Scholar
  27. [27]
    A.S. Fokas and V.E. Zakharov (ed.), Important developments in soliton theory, Springer-Verlag, Berlin, 1993.MATHGoogle Scholar
  28. [28]
    S. Gelbart, Elliptic curves and automorphic representations, Adv. in Math. 21 (1976), 235–292.MATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    S. Gelbart, An elementary introduction to the Langlands program, Bull. Amer. Math. Soc. (N.S.) 10 (1984), 177–219.MATHCrossRefMathSciNetGoogle Scholar
  30. [30]
    D. Goldfeld, Gauss' class number problem for imaginary quadratic fields, Bull. Amer. Math. Soc. (N.S.) 13 (1985), 23–37.MATHCrossRefMathSciNetGoogle Scholar
  31. [31]
    E. Grosswald, Representations of integers as sums of squares, Springer-Verlag, New York, 1985.MATHGoogle Scholar
  32. [32]
    M. Hindry and J.H. Silverman, Diophantine geometry, Springer, New York, 2000.MATHGoogle Scholar
  33. [33]
    J.C. Jantzen, Lectures on quantum groups, American Mathematical Society, Providence, R.I., 1996.MATHGoogle Scholar
  34. [34]
    V.F.R. Jones, Subfactors and knots, CBMS Regional Conference Series in Mathematics 80, Amer. Math. Soc., Providence, R.I., 1991.MATHGoogle Scholar
  35. [35]
    V.G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, 1990.Google Scholar
  36. [36]
    A.A. Kirillov, Jr., Lectures on affine Hecke algebras and Macdonald's conjectures, Bull. Amer. Math. Soc. (N.S.) 34 (1997), 251–292.MATHCrossRefMathSciNetGoogle Scholar
  37. [37]
    A.W. Knapp, Elliptic curves, Princeton University Press, Princeton, N.J., 1992.MATHGoogle Scholar
  38. [38]
    V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Quantum inverse scattering method and correlation functions, Cambridge University Press, 1993.Google Scholar
  39. [39]
    S. Lang, Introduction to modular forms, Springer-Verlag, Berlin, corr. reprint, 1995.Google Scholar
  40. [40]
    M. Laska, An algorithm for finding a minimal Weierstrass equation for an elliptic curve, Math. Comp. 38 (1982), 257–260.MATHCrossRefMathSciNetGoogle Scholar
  41. [41]
    J.H. van Lint and R.M. Wilson, A course in combinatorics, Cambridge University Press, 1992.Google Scholar
  42. [42]
    S.C. Milne, New infinite families of exact sums of squares formulas, Jacobi elliptic functions and Ramanujan's tau function, Proc. Nat. Acad. Sci. U.S.A. 93 (1996), 15004–15008.MATHCrossRefMathSciNetGoogle Scholar
  43. [43]
    K. Noda and H. Wada, All congruent numbers less than 10000, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), 175–178.MATHCrossRefMathSciNetGoogle Scholar
  44. [44]
    J. Oesterlé, Le problème de Gauss sur le nombre de classes, Enseign. Math. 34 (1988), 43–67.MATHMathSciNetGoogle Scholar
  45. [45]
    M. Okado, M. Jimbo and T. Miwa, Solvable lattice models in two dimensions and modular functions, Sugaku Exp. 2 (1989), 29–54.MATHGoogle Scholar
  46. [46]
    H. Rademacher, Topics in analytic number theory, Springer-Verlag, Berlin, 1973.MATHGoogle Scholar
  47. [47]
    K.A. Ribet, Galois representations and modular forms, Bull. Amer. Math. Soc. (N.S.) 32 (1995), 375–402.MATHCrossRefMathSciNetGoogle Scholar
  48. [48]
    N. Schappacher, Développement de la loi de groupe sur une cubique, Séminaire de Théorie des Nombres, Paris 1988–89 (ed. C. Goldstein), pp. 159–184, Birkhäuser, Boston, 1990.Google Scholar
  49. [49]
    J.-P. Serre, A course in arithmetic, Springer-Verlag, New York, 1973.MATHGoogle Scholar
  50. [50]
    J.H. Silverman, The arithmetic of elliptic curves, Springer-Verlag, New York, 1986.MATHGoogle Scholar
  51. [51]
    J.H. Silverman, Advanced topics in the arithmetic of elliptic curves, Springer-Verlag, New York, 1994.MATHGoogle Scholar
  52. [52]
    J.H. Silverman and J. Tate, Rational points on elliptic curves, Springer-Verlag, New York, 1992.MATHGoogle Scholar
  53. [53]
    L. Szpiro, La conjecture de Mordell [d'après G. Faltings], Astérisque 121–122 (1985), 83–103.MathSciNetGoogle Scholar
  54. [54]
    J.T. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, Séminaire Bourbaki: Vol. 1965/1966, Exposé no. 306, Benjamin, New York, 1966.Google Scholar
  55. [55]
    J.T. Tate, The arithmetic of elliptic curves, Invent. Math. 23 (1974), 179–206.MATHCrossRefMathSciNetGoogle Scholar
  56. [56]
    R.L. Taylor and A. Wiles, Ring theoretic properties of certain Hecke algebras, Ann. of Math. 141 (1995), 553–572.MATHCrossRefMathSciNetGoogle Scholar
  57. [57]
    J.B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math. 72 (1983), 323–334.MATHCrossRefMathSciNetGoogle Scholar
  58. [58]
    N. Ja. Vilenkin and A.V. Klimyk, Representation of Lie groups and special functions, 4 vols., Kluwer, Dordrecht, 1991–1995.Google Scholar
  59. [59]
    M. Waldschmidt, Diophantine approximation on linear algebraic groups, Springer, Berlin, 2000.MATHGoogle Scholar
  60. [60]
    A. Wiles, Modular elliptic curves and Fermat's last theorem, Ann. of Math. 141 (1995), 443–551.MATHCrossRefMathSciNetGoogle Scholar

Additional References

  1. R.E. Borcherds, What is moonshine?, Proceedings of the International Congress of Mathematicians: Berlin 1998, Vol. I, pp. 607–615, Documenta Mathematica, Bielefeld, 1998.MathSciNetGoogle Scholar
  2. C. Breuil, B. Conrad, F. Diamond and R. Taylor, On the modularity of elliptic curves over Q, J. Amer. Math. Soc. 14 (2001), 843–939.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • W. A. Coppel
    • 1
  1. 1.GriffithAustralia

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