Classification of Elliptic Cubic Curves Over The Finite Field of Order Nineteen

Torsion of elliptic curves over cubic fields

Although it is not known which groups can appear as torsion groups of elliptic curves over cubic number fields, it is known which groups can appear for infinitely many non-isomorphic curves. We denote the set of these groups as S. In this paper we deal with three problems concerning the torsion of elliptic curves over cubic fields. First, we study the possible torsion groups of elliptic curves ...

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

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...

A Table of Elliptic Curves over the Cubic Field of Discriminant-23

Let F be the cubic field of discriminant −23 and OF its ring of integers. Let Γ be the arithmetic group GL2(OF ), and for any ideal n ⊂ OF let Γ0(n) be the congruence subgroup of level n. In [17], two of us (PG and DY) computed the cohomology of various Γ0(n), along with the action of the Hecke operators. The goal of [17] was to test the modularity of elliptic curves over F . In the present pap...