Cubic Curves over Finite Fields

  • Joseph H. Silverman
  • John Tate
Part of the Undergraduate Texts in Mathematics book series (UTM)


In this chapter we will look at cubic equations over a finite field, the field of integers modulo p. We will denote this field by F p . Of course, now we cannot visualize things; but we can look at polynomial equations
$$C:F\left( {x,y} \right) = 0$$
with coefficients in F p and ask for solutions (x, y) with x, y ∈ F p . More generally, we can look for solutions x, y ∈ F q , where F q is an extension field of F p , containing q = p e elements. We call such a solution a point on the curve C. If the coordinates x and y of the solution lie in F p , we call it a rational point.


Modular Form Elliptic Curve Finite Field Elliptic Curf Finite Order 
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Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Joseph H. Silverman
    • 1
  • John Tate
    • 2
  1. 1.Department of MathematicsBrown UniversityProvidenceUSA
  2. 2.Department of MathematicsUniversity of Texas at AustinAustinUSA

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