Uniform boundedness in terms of ramification

摘要

AbstractLet$$dge 1$$d≥1be fixed. LetFbe a number field of degreed, and letE?/?Fbe an elliptic curve. Let$$E(F)_{ext {tors}}$$E(F)torsbe the torsion subgroup ofE(F). In 1996, Merel proved the uniform boundedness conjecture, i.e., there is a constantB(d), which depends ondbut not on the chosen fieldFor on the curveE?/?F, such that the size of$$E(F)_{ext {tors}}$$E(F)torsis bounded byB(d). Moreover, Merel gave a bound (exponential ind) for the largest prime that may be a divisor of the order of$$E(F)_{ext {tors}}$$E(F)tors. In 1996, Parent proved a bound (also exponential ind) for the largestp-power order of a torsion point that may appear in$$E(F)_{ext {tors}}$$E(F)tors. It has been conjectured, however, that there is a bound for the size of$$E(F)_{ext {tors}}$$