Uniform boundedness in terms of ramification

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 under certain hypotheses 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 maximum ramification index of a prime ideal of the ring of integers F over (p). © 2018, SpringerNature.