Ramification in Division Fields and Sporadic Points on Modular Curves

Abstract

Consider an elliptic curve E over a number field K and let p be a prime of OK lying above a prime p of Z. Suppose E has supersingular reduction at p. Fix a positive integer n and define L to be a minimal extension of K such that E(L) has a point of exact order p n . If p is unramified over p, we show that L/K is an extension of degree p 2n − p 2n−2 that is totally ramified over p. If p is ramified over p, we are still able to show that ϕ(p n ) properly divides [L : Q].We apply our stronger bound to show that sporadic points on the modular curve X1(p n ) cannot correspond to elliptic curves that are supersingular at p with p unramified over p. Our methods are then generalized to X1(N ) with N composite. We also describe ramification at and away from p in the full p n -th division field K(E[p n ]). In the course of our investigation, we correct a theorem of Cassels.