An embedding problem for finite local torsors over twisted curves

The point counting problem for curves over finite fields

The group law on elliptic curves is well-known and gives rise to elliptic curve cryptography systems which find application to government and industry today. However, the generalisation to higher genus requires the manipulation of divisor classes rather than points, and analogues of key genus 1 results have yet to be found. Nonetheless, effective computation within the group is possible, and te...

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