Fermat curves over finite fields

Twisted Fermat curves over totally real fields

Elliptic Curves over Finite Fields

In this chapter, we study elliptic curves defined over finite fields. Our discussion will include the Weil conjectures for elliptic curves, criteria for supersingularity and a description of the possible groups arising as E(Fq). We shall use basic algebraic geometry of elliptic curves. Specifically, we shall need the notion and properties of isogenies of elliptic curves and of the Weil pairing....

Elliptic Curves over Finite Fields

Counting curves over finite fields

Article history: Received 25 August 2014 Received in revised form 10 September 2014 Accepted 18 September 2014 Available online 4 November 2014 Communicated by H. Stichtenoth MSC: 11G20 10D20 14G15 14H10

Rational Points on Some Fermat Curves and Surfaces over Finite Fields

We give an explicit description of the Fqi-rational points on the Fermat curve uq−1+vq−1+wq−1 = 0, for i ∈ {1, 2, 3}. As a consequence, we observe that for any such point (u, v, w), the product uvw is a cube in Fqi . We also describe the Fq2-rational points on the Fermat surface uq−1 + vq−1 + wq−1 + xq−1 = 0.