Pairs of elliptic curves with maximal Galois representations

Galois Representations and Elliptic Curves

An elliptic curve over a field K is a projective nonsingular genus 1 curve E over K along with a chosen K-rational point O of E, which automatically becomes an algebraic group with identity O. If K has characteristic 0, the n-torsion of E, denoted E[n], is isomorphic to (Z/nZ) over K. The absolute Galois group GK acts on these points as a group automorphism, hence it acts on the inverse limit l...

Elliptic Curves with Surjective Adelic Galois Representations

Let K be a number field. The Gal(K/K)-action on the the torsion of an elliptic curve E/K gives rise to an adelic representation ρE : Gal(K/K) → GL2(Ẑ). From an analysis of maximal closed subgroups of GL2(Ẑ) we derive useful necessary and sufficient conditions for ρE to be surjective. Using these conditions, we compute an example of a number field K and an elliptic curve E/K that admits a surjec...

Mod 4 Galois Representations and Elliptic Curves

Galois representations ρ : GQ → GL2(Z/n) with cyclotomic determinant all arise from the n-torsion of elliptic curves for n = 2, 3, 5. For n = 4, we show the existence of more than a million such representations which are surjective and do not arise from any elliptic curve.

Galois Representations Attached to Elliptic Curves

Elliptic Curves with Maximal Galois Action on Their Torsion Points

Given an elliptic curve E over a number field k, the Galois action on the torsion points of E induces a Galois representation, ρE : Gal(k/k) → GL2(b Z). For a fixed number field k, we describe the image of ρE for a “random” elliptic curve E over k. In particular, if k 6= Q is linearly disjoint from the cyclotomic extension of Q, then ρE will be surjective for “most” elliptic curves over k.