Elliptic curves with maximal Galois action on their torsion points
by
David Zywina
University of Pennsylvania

Given an elliptic curve E over a number field k, the Galois action on the torsion points of E induces a representation, r_E : Gal(k^al/k) → ∏_l GL₂(Z_l). A renowned theorem of Serre says that if E is non-CM, then r_E has open image.

For a fixed number field k, we shall describe the image of r_E for a "random" elliptic curve E over k. In particular, if k ≠ Q is linearly disjoint from the cyclotomic extension of Q, then r_E will be surjective for "most" elliptic curves defined over k.

Date received: June 15, 2008