On the cohomological coprimality of Galois representations associated with elliptic curves

Abstract

Let $E$ and $E'$ be elliptic curves over an algebraic number field. We show that systems of $\ell$-adic representations associated with $E$ and $E'$ are cohomologically coprime, in the sense that the Galois cohomology groups corresponding to respective fields of division points are all trivial. This provides a generalization of some known results about the vanishing of the cohomology groups associated with the $\ell$-adic Tate module of an elliptic curve.