On the Arithmetic of Elliptic Curves and a Homotopy Limit Problem

TL;DR

This paper investigates the relationship between motivic and étale cohomology groups of elliptic curves over rationals, linking the isomorphism properties of a comparison map to the elliptic curve's arithmetic characteristics.

Contribution

It establishes a connection between the isomorphism of a comparison map in cohomology and specific arithmetic properties of elliptic curves over ield.

Findings

The comparison map's isomorphism is equivalent to certain arithmetical conditions.

Provides insight into the cohomological behavior of elliptic curves.

Links cohomological properties to elliptic curve arithmetic.

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.