Open Access

Elliptic curves over \(\mathbb {Q}\) and 2-adic images of Galois

Research in Number Theory20151:12

DOI: 10.1007/s40993-015-0013-7

Received: 26 May 2015

Accepted: 29 June 2015

Published: 7 October 2015

Abstract

We give a classification of all possible 2-adic images of Galois representations associated to elliptic curves over \(\mathbb {Q}\). To this end, we compute the ‘arithmetically maximal’ tower of 2-power level modular curves, develop techniques to compute their equations, and classify the rational points on these curves.