On the Arithmetic of Elliptic Curves and a Homotopy Limit Problem

Abstract

In this note, I study a comparison map between a motivic and \'{e}tale cohomology group of an elliptic curve over $\mathbb{Q}$ just outside the range of Voevodsky's isomorphism theorem. I show that the property of an appropriate version of the map being an isomorphism is equivalent to certain arithmetical properties of the elliptic curve.