Elliptic periods for finite fields

Elliptic periods for finite fields

We construct two new families of basis for finite field extensions. Bases in the first family, the so-called elliptic bases, are not quite normal bases, but they allow very fast Frobenius exponentiation while preserving sparse multiplication formulas. Bases in the second family, the so-called normal elliptic bases are normal bases and allow fast (quasi-linear) arithmetic. We prove that all exte...

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 Elliptic Surfaces over Finite Fields

We count the number of isomorphism classes of elliptic curves of given height d over the field of rational functions in one variable over the finite field of q elements. We also estimate the number of isomorphism classes of elliptic surfaces over the projective line, which have a polarization of relative degree 3. This leads to an upper bound for the average 3-Selmer rank of the aforementionned...

Quotients of Elliptic Curves over Finite Fields

Fix a prime `, and let Fq be a finite field with q ≡ 1 (mod `) elements. If ` > 2 and q ` 1, we show that asymptotically (`− 1)/2` of the elliptic curves E/Fq with complete rational `-torsion are such that E/〈P 〉 does not have complete rational `-torsion for any point P ∈ E(Fq) of order `. For ` = 2 the asymptotic density is 0 or 1/4, depending whether q ≡ 1 (mod 4) or 3 (mod 4). We also show t...