The 2-adic representations attached to elliptic curves defined over Q whose points of order 2 are all Q-rational

Elliptic Curves over Q and 2-adic Images of Galois

We give a classification of all possible 2-adic images of Galois representations associated to elliptic curves over Q. To this end, we compute the ‘arithmetically maximal’ tower of 2-power level modular curves, develop techniques to compute their equations, and classify the rational points on these curves.

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

There Are No Points of Order 11 on Elliptic Curves over Q

Lemma 1. Let k be a field and let (a, b, c), (α, β, γ) ∈ P(k) be distinct points. Then there is a unique line through these points, and it is given by the equation: ∣∣∣∣∣ x y z a b c α β γ ∣∣∣∣∣ = 0 . Two lines given by equations ux+ vy+wz = 0 and u′x+ v′y+w′z = 0 coincide if and only if the points (u, v, w) and (u′, v′, w′) coincide as points in P(k). Two distinct lines in P(k) intersect in pr...

ELLIPTIC CURVES OVER Q(i)

A study of the diophantine equation v2 = 2u4 − 1 led the authors to consider elliptic curves specifically over Q(i) and to examine the parallels and differences with the classical theory over Q. In this paper we present some extensions of the classical theory along with some examples illustrating the results. The well-known diophantine equation v2 = 2u4 − 1 , has, ignoring signs, only two integ...