Galois representations attached to elliptic curves with complex multiplication

Galois Representations Attached to Elliptic Curves

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...

The image of Galois representations attached to elliptic curves with an isogeny

giving the action of GQ on Tp(E), the p-adic Tate module for E and for a prime p. If E doesn’t have complex multiplication, then a famous theorem of Serre [Ser2] asserts that the image of ρE,p has finite index in AutZp ( Tp(E) ) for all p and that the index is 1 for all but finitely many p. This paper concerns some of the exceptional cases where the index is not 1. If E has a cyclic isogeny of ...

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.