Local Torsion on Elliptic Curves and the Deformation Theory of Galois Representations

We expect this conjecture to be false (for sufficiently large d) when E does have complex multiplication. Nevertheless, our main result is that Conjecture 1.1 at least holds on average. For A,B > 0, let SA,B denote the set of elliptic curves with Weierstrass equations y = x + ax + b with a, b ∈ Z, |a| ≤ A and |b| ≤ B. For an elliptic curve E and x > 0, let π E(x) denote the number of primes p ≤ x such that E possesses a point of order p over an extension of Qp of degree at most d. Theorem 1.2. Fix d ≥ 1. Assume A,B ≥ x 4+ε for some ε > 0. Then 1 #SA,B ∑