Elliptic curves over $\mathbb{Q}$ and 2-adic images of Galois
  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. 
Jeremy Rouse and David Zureick-Brown
math.NT