Near coincidences and nilpotent division fields

Let E/ be an elliptic curve.We say that E has a near coincidence of levelWe classify near coincidences of prime power level and use this result to give a classification of values of n for which Gal((E[n])/) is a nilpotent group.Along the way we prove a Gauss-Wantzel analog for the elliptic curve E :Assuming that there are no non-CM rational points on the modular curves X + ns ( p) for primes p > 11, we show that Gal((E[n])/) nilpotent implies that n is a power of 2 or n {3, 5, 6, 7, 15, 21}.