Ramification in division fields and sporadic points on modular curves

Abstract

Consider an elliptic curve E over a number field K. Suppose that E has supersingular reduction at some prime $$\mathfrak {p}$$ of K lying above the rational prime p. We completely classify the valuations of the $$p^n$$ -torsion points of E by the valuation of a coefficient of the $$p{\text {th}}$$ division polynomial. This classification corrects an error in earlier work of Lozano-Robledo. As an application, we find the minimum necessary ramification at $$\mathfrak {p}$$ in order for E to have a point of exact order $$p^n$$ . Using this bound we show that sporadic points on the modular curve $$X_1(p^n)$$ cannot correspond to supersingular elliptic curves without a canonical subgroup. We generalize our methods to $$X_1(N)$$ with N composite.