Entanglement of Galois representations of elliptic curves over Q

Abstract

When studying elliptic curves over a field K one can define their n-torsion fields to be K adjoined with the x,y-coordinates of the n-torsion points. Then we can look at the Galois representations associated to these n-torsion fields and ask when these representations are surjective. It turns out that there are multiple ways in which the image can fail to be surjective, corresponding to different kinds of entanglement. We focus mainly on so-called horizontal entanglements and different ways in which these can occur. Of particular interest will be Weil entanglement and Serre entanglement. The latter occurs because of the fact that the discriminant of an elliptic curve is always contained in a cyclotomic field, as implied by the Kronecker-Weber theorem. One of the main contributions will be on how Serre entanglement induces horizontal entanglement. Furthermore we will study Weil entanglement by looking at the conductor of corresponding quadratic and cubic number fields.