Uniform Boundedness and Elliptic Curves with Potential Supersingular Reduction

Let d ≥ 1 be fixed. Let F be a number field of degree d, and let E/F be an elliptic curve. Let E(F )tors be the torsion subgroup of E(F ). In 1996, Merel proved the uniform boundedness conjecture, i.e., there is a constant B(d), which depends on d but not on the chosen field F or on the curve E/F , such that the size of E(F )tors is bounded by B(d). Moreover, Merel gave a bound (exponential in d) for the largest prime that may be a divisor of the order of E(F )tors. In 1996, Parent proved a bound (also exponential in d) for the largest p-power order of a torsion point that may appear in E(F )tors. It has been conjectured, however, that there is a bound for the size of E(F )tors that is polynomial in d. In this article we show that if E/F has potential supersingular reduction at a prime ideal above p, then there is a linear bound for the largest p-power order of a torsion point defined over F , which in fact is linear in the ramification index of the prime of supersingular reduction.