Advertisement

Inventiones mathematicae

, Volume 72, Issue 2, pp 323–334 | Cite as

A classical Diophantine problem and modular forms of weight 3/2

  • J. B. Tunnell
Article

Keywords

Modular Form Diophantine Problem 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alter, R., Curtz, T.B., Kubota, K.K.: Remarks and results on congruent numbers. Proc. Third Southeastern Conf. on Combinatorics, Graph Theory and Computing 1972, pp. 27–35Google Scholar
  2. 2.
    Alter, R.: The congruent number problem. Amer. Math. Monthly87, 43–45 (1980)Google Scholar
  3. 3.
    Birch, B.J., Swinnerton-Dyer, H.P.F.: Notes on elliptic curves II. J. reine angewandte Math.218, 79–108 (1965)Google Scholar
  4. 4.
    Birch, B.J., Kuyk, W.: Tables on elliptic curves. In: Modular functions of one variable IV. Lecture Notes in Mathematics, vol. 476, pp. 81–144. Berlin-Heidelberg-New York: Springer 1979Google Scholar
  5. 5.
    Barrucand, P., Cohn, H.: Note on primes of typex 2+32y 2, class number, and residuacity. J. reine angewandte Math.238, 67–70 (1969)Google Scholar
  6. 6.
    Brown, E.: The class number of\(Q(\sqrt { - p} )\), forp≡1 (mod 8) a prime. Proc. Amer. Math. Soc.31, 381–383 (1972)Google Scholar
  7. 7.
    Coates, J., Wiles, A.: On the conjecture of Birch and Swinnerton-Dyer. Invent. Math.39, 223–251 (1977)Google Scholar
  8. 8.
    Cohen, H., Oesterlé, J.: Dimension des espaces de formes modulaires. In: Modular functions of one variable VI. Lecture Notes in Mathematics, vol. 627, pp. 69–78. Berlin-Heidelberg-New York: Springer 1977Google Scholar
  9. 9.
    Dickson, L.E.: History of the theory of numbers II. Carnegie Institution, Washington, DC (1920) (reprinted by Chelsea, 1966)Google Scholar
  10. 10.
    Flicker, Y.: Automorphic forms on covering groups ofGL(2). Invent. Math.57, 119–182 (1980)Google Scholar
  11. 11.
    Jones, B.W.: The arithmetic theory of quadratic forms. Math. Assoc. of Amer., Baltimore, MD 1950Google Scholar
  12. 12.
    Lagrange, J.: Thèse d'Etat de l'Université de Reims, 1976Google Scholar
  13. 13.
    Moreno, C.J.: The higher reciprocity laws: an example. J. Number Theory12, 57–70 (1980)Google Scholar
  14. 14.
    Pizer, A.: On the 2-part of the class number of imaginary quadratic number fields. J. Number Theory8, 184–192 (1976)Google Scholar
  15. 15.
    Razar, M.: The nonvanishing ofL(1) for certain elliptic curves with no first descents. Amer. J. Math.96, 104–126 (1974)Google Scholar
  16. 16.
    Razar, M.: A relation between the two-component of the Tate-Shafarevitch group andL(1) for certain elliptic curves. Amer. J. Math.96, 127–144 (1974)Google Scholar
  17. 17.
    Serre, J-P., Stark, H.M.: Modular forms of weight 1/2. In: Modular functions of one variable VI. Lecture Notes in Mathematics, vol. 627, pp. 27–68. Berlin-Heidelberg-New York: Springer 1977Google Scholar
  18. 18.
    Shimura, G.: On modular forms of half-integral weight. Ann. of Math.97, 440–481 (1973)Google Scholar
  19. 19.
    Shimura, G.: Introduction to the arithmetic theory of automorphic functions. Iwanami Shoten and Princeton University Press 1971Google Scholar
  20. 20.
    Smith, H.J.: Collected Mathematical Papers, Volume 1, Oxford (1894). (reprinted by Chelsea, 1965)Google Scholar
  21. 21.
    Stephens, N.M.: Congruence Properties of Congruent numbers. Bull. London Math. Soc. pp. 182–184 (1975)Google Scholar
  22. 22.
    Tate, J.: The arithmetic of elliptic curves. Invent. Math.23, 179–206 (1974)Google Scholar
  23. 23.
    Tate, J.: Algorithm for determining the type of a singular fiber in an elliptic pencil. In: Modular functions of one variable IV. Lecture Notes in Mathematics, vol. 476, pp. 33–52. Berlin-Heidelberg-New York: Springer 1975Google Scholar
  24. 24.
    Tate, J.: Number theoretic background. In: Automorphic forms, representations, andL-functions. Proc. Symp. in Pure Math. XXXIII, Part 2, pp. 3–26 (1979)Google Scholar
  25. 25.
    Waldspurger, J.-L.: Sur les coefficients de Fourier des formes modulaires de poids demi-entier. J. de Math. pures et appliquées60, (4) 375–484 (1981)Google Scholar
  26. 26.
    Guy, R.K.: Unsolved problems. Amer. Math. Monthly88, 758–761 (1981)Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • J. B. Tunnell
    • 1
  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA

Personalised recommendations