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. We apply this description to 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.