Statistics about elliptic curves over finite prime 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

Legendre elliptic curves over finite fields

Throughout this paper, q > 1 denotes a power of an odd prime number p, and k is a field. Given two elliptic curves E/k and E′/k, all morphisms from E to E′ are understood to be defined over k. In particular, we simply write End(E) for the ring of all endomorphisms of E/k. The notation E ≃ E′ indicates that E is isomorphic to E′, and E ∼ E′ means that E and E′ are isogenous. The endomorphism of ...

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...

Lattices from elliptic curves over finite fields

In their well known book [6] Tsfasman and Vladut introduced a construction of a family of function field lattices from algebraic curves over finite fields, which have asymptotically good packing density in high dimensions. In this paper we study geometric properties of lattices from this construction applied to elliptic curves. In particular, we determine the generating sets, conditions for wel...