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

AbstractWe give a classification of all possible 2-adic images of Galois representations associated to elliptic curves over ℚ $\mathbb {Q}$ . 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.