Elliptic curves with maximal Galois action on their torsion points

TL;DR

This paper investigates the Galois representations associated with elliptic curves over number fields, showing that for most curves over fields linearly disjoint from cyclotomic extensions, the Galois action on torsion points is maximal.

Contribution

It characterizes the image of the Galois representation for a generic elliptic curve over a number field, especially highlighting conditions for maximal Galois action.

Findings

Galois representation is surjective for most elliptic curves over certain number fields.

Maximal Galois action occurs when the field is linearly disjoint from cyclotomic extensions.

Provides a description of the Galois image for generic elliptic curves.

Abstract

Given an elliptic curve E over a number field k, the Galois action on the torsion points of E induces a Galois representation, \rho_E : Gal(\bar{k}/k) \to GL_2(\hat{Z}). For a fixed number field k, we describe the image of \rho_E for a "random" elliptic curve E over k. In particular, if k\neq Q is linearly disjoint from the cyclotomic extension of Q, then \rho_E will be surjective for "most" elliptic curves over k.