Torsion of rational elliptic curves over the maximal abelian extension of ℚ

On the torsion of rational elliptic curves over quartic fields

Let E be an elliptic curve defined over Q and let G = E(Q)tors be the associated torsion subgroup. We study, for a given G, which possible groups G ⊆ H could appear such that H = E(K)tors, for [K : Q] = 4 and H is one of the possible torsion structures that occur infinitely often as torsion structures of elliptic curves defined over quartic number fields. Let K be a number field, and let E be a...

Torsion of Rational Elliptic Curves over Cubic Fields

Let E be an elliptic curve defined over Q. We study the relationship between the torsion subgroup E(Q)tors and the torsion subgroup E(K)tors, where K is a cubic number field. In particular, we study the number of cubic number fields K such that E(Q)tors ̸= E(K)tors.

Elliptic Curves over Q

All polynomials and rational functions in this essay are assumed to have coefficients in Q . Fix an integer n ≥ 1. An affine variety is a simultaneous irreducible system of polynomial equations in n variables. The Q -points, R -points and C points of the affine variety are all solutions of the polynomial system in Q , R n and C , respectively. Rational projective n-spaceg Q n is the set of line...

Elliptic Curves over Q

On computing rational torsion on elliptic curves

We introduce an l-adic algorithm to efficiently determine the group of rational torsion points on an elliptic curve. We also make a conjecture about the discriminant of the m-division polynomial of an elliptic curve.