Equations over Finite Fields

  • J. S. Chahal
Part of the The University Series in Mathematics book series (USMA)


We have seen that for each prime p, there is a field F p of p elements. In fact, given any prime p and an integer r ≥ 1, there is one and only one field F q of q = p r elements. The field F q ⊇ F p and for each α in F q , pα = 0. Conversely, any finite field is F q , for some q = p r (cf. Ref. 18). The field F q is characterized by the property
$$f\left( X \right) = X^q - X = \mathop \Pi \limits_{\alpha \varepsilon {\text{F}}_q } \;\left( {X - \alpha } \right)$$


Zeta Function Modular Form Elliptic Curve Finite Field Elliptic Curf 
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  1. 1.
    A. Baker, The diophantine equation y 2 = ax 3 + bx 2 + cx + d, J. London Math. Soc. 43 1–9, (1968).MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    A. Baker and J. Coates, Integer points on curves on genus 1, Proc. Cambridge Phil. Soc. 67 595–602, (1970).MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves II, J. Reine Angew. Math. 218, 79–108 (1965).MathSciNetzbMATHGoogle Scholar
  4. 4.
    E. Bombieri, Counting points over finite fields (d’après S. A. Stepanov), Sem. Bourbaki, Exposé 430 (1972-73).Google Scholar
  5. 5.
    J. W. S. Cassels, Diophantine equations with special reference to elliptic curves, J. London Math. Soc. 41, 193–291 (1966).MathSciNetCrossRefGoogle Scholar
  6. 6.
    J. Coates and A. Wiles, On the conjecture of Birch and Swinnerton-Dyer, Invent. Math. 39, 223–251 (1977).MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    G. Cornell and J. H. Silverman, Arithmetic Geometry, New York: Springer Verlag (1986).zbMATHCrossRefGoogle Scholar
  8. 8.
    J. Dieudonné, History of Algebraic Geometry, Wadsworth, Belmont, California (1985).zbMATHGoogle Scholar
  9. 9.
    B. Dwork, On the rationality of zeta function, Amer. J. Math. 82, 631–648 (1960).MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73, 349–366 (1938).MathSciNetCrossRefGoogle Scholar
  11. 11.
    A. Gelfond and Yu. Linnik, Elementary Methods in Analytic Number Theory, Fizmatgiz, Moscow (1962).Google Scholar
  12. 12.
    B. Gross and D. Zagier, Heegner points and derivatives of L-series, Invent. Math. 84, 225–320 (1986).MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, GTM 84, Springer Verlag, New York, (1982).zbMATHCrossRefGoogle Scholar
  14. 14.
    N. Koblitz, Introduction to Elliptic Curves and Modular Forms, GTM 97, Springer Verlag, New York (1984).zbMATHCrossRefGoogle Scholar
  15. 15.
    Yu I. Manin, On cubic congruences to a prime modulus, Izv. Akad. Nauk USSR, Math. Ser. 20, 673–678 (1956).MathSciNetzbMATHGoogle Scholar
  16. 16.
    P. Roquette, Arithmetischer Beweis der Riemannschen Vermutung in Kongruenzzetafunktionenkörpern Belibingen Geschlechts, J. Reine Angew. Math. 191, 199–252 (1953).MathSciNetzbMATHGoogle Scholar
  17. 17.
    K. Rubin, Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication, Invent. Math. 89, 527–560 (1987).MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    W. M. Schmidt, Lectures on Equations over Finite Fields: An Elementary Approach, Part I, Lecture Notes in Math; No. 536, Springer Verlag, Berlin (1976).Google Scholar
  19. 19.
    W. M. Schmidt, Lectures on Equations over Finite Fields: An Elementary Approach, Part II (unpublished).Google Scholar
  20. 20.
    C. L. Siegel, Über einige Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Wiss. Phys. Math. K1 Nr. 1 (1929).Google Scholar
  21. 21.
    J. H. Silverman, The Arithmetic of Elliptic Curves, GTM 106, Springer Verlag, New York (1986).zbMATHCrossRefGoogle Scholar
  22. 22.
    S. A. Stepanov, The number of points of a hyperelliptic curve over a prime field, Izv. Akad. Nauk USSR, Math. Ser. 33, 1171–1181 (1969).zbMATHGoogle Scholar
  23. 23.
    J. Tate, The arithmetic of elliptic curves, Invent. Math. 23, 179–206 (1974).MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    J. Tunnell, A classical diophantine problem and modular forms of weight 3/2, Invent Math, 72, 323–334 (1983).MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    A. Weil, Number of solutions of equations in finite fields, Bull. Am. Math. Soc. 55, 497–508 (1949).MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    H. G. Zimmer, An elementary proof of the Riemann hypothesis for an elliptic curve over a finite field, Pacific J. Math. 36, 267–278 (1971).MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1988

Authors and Affiliations

  • J. S. Chahal
    • 1
  1. 1.Brigham Young UniversityProvoUSA

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