Skip to main content
Log in

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

  • Published:
Inventiones mathematicae Aims and scope

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  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. 27–35

  2. Alter, R.: The congruent number problem. Amer. Math. Monthly87, 43–45 (1980)

    Google Scholar 

  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. 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 1979

    Google Scholar 

  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. 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. Coates, J., Wiles, A.: On the conjecture of Birch and Swinnerton-Dyer. Invent. Math.39, 223–251 (1977)

    Google Scholar 

  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 1977

    Google Scholar 

  9. Dickson, L.E.: History of the theory of numbers II. Carnegie Institution, Washington, DC (1920) (reprinted by Chelsea, 1966)

    Google Scholar 

  10. Flicker, Y.: Automorphic forms on covering groups ofGL(2). Invent. Math.57, 119–182 (1980)

    Google Scholar 

  11. Jones, B.W.: The arithmetic theory of quadratic forms. Math. Assoc. of Amer., Baltimore, MD 1950

    Google Scholar 

  12. Lagrange, J.: Thèse d'Etat de l'Université de Reims, 1976

  13. Moreno, C.J.: The higher reciprocity laws: an example. J. Number Theory12, 57–70 (1980)

    Google Scholar 

  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. Razar, M.: The nonvanishing ofL(1) for certain elliptic curves with no first descents. Amer. J. Math.96, 104–126 (1974)

    Google Scholar 

  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. 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 1977

    Google Scholar 

  18. Shimura, G.: On modular forms of half-integral weight. Ann. of Math.97, 440–481 (1973)

    Google Scholar 

  19. Shimura, G.: Introduction to the arithmetic theory of automorphic functions. Iwanami Shoten and Princeton University Press 1971

  20. Smith, H.J.: Collected Mathematical Papers, Volume 1, Oxford (1894). (reprinted by Chelsea, 1965)

  21. Stephens, N.M.: Congruence Properties of Congruent numbers. Bull. London Math. Soc. pp. 182–184 (1975)

  22. Tate, J.: The arithmetic of elliptic curves. Invent. Math.23, 179–206 (1974)

    Google Scholar 

  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 1975

    Google Scholar 

  24. Tate, J.: Number theoretic background. In: Automorphic forms, representations, andL-functions. Proc. Symp. in Pure Math. XXXIII, Part 2, pp. 3–26 (1979)

  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. Guy, R.K.: Unsolved problems. Amer. Math. Monthly88, 758–761 (1981)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Partially supported by a grant from the National Science Foundation

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tunnell, J.B. A classical Diophantine problem and modular forms of weight 3/2. Invent Math 72, 323–334 (1983). https://doi.org/10.1007/BF01389327

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01389327

Keywords

Navigation