Bounds for the Torsion of Elliptic Curves over Extensions with Bounded Ramification

Let K be a number field and let E/K be an elliptic curve. The Mordell-Weil Theorem states that E(K), the set of K-rational points on E, can be given the structure of a finitely generated abelian group. In this note we consider elliptic curves defined over the rationals Q and provide bounds for the size of E(K)Tors, where K is an algebraic Galois extension of Q. Theorem 1. Let E/Q be an elliptic curve, let SE,add be the set of primes of additive reduction of E/Q, and let N ≥ 2 be fixed. Let K be an algebraic Galois extension of Q (not necessarily finite) unramified at primes in SE,add such that the ramification index of any other prime p in K/Q is finite and bounded by N . Then E(K)Tors is finite and there is a computable bound B = B(E,N) for its size. Moreover, if E is semi-stable, then the bound B is independent of E.