Families of curves over finite 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

Coverings of singular curves over finite fields

Applications of Curves over Finite Fields

The area starts with Galois and Gauss Group theory and expo nential sums were the two application areas then That tradition continues Without Chevalley groups over nite elds there would have been no structure theory of nite simple groups Weil s Riemann Hypothesis for curves over nite elds is a singular event Many error estimates in combinatorics derive from it Research in nite elds requires com...