Exact bounds on the order of torsion points of curves of genus 1

Abstract

Let k be an algebraic number field of degree n on ℚ2; ℱ and ℊ, respectively, the curves on k; let , and ℴm, ℴ'm be the bases of groups of all points of order m on ℱ and g, respectively. A proof of the following theorem is sketched: let p>3 be prime; if , then τ(pt)⩽6n; if k, then τ(pt)⩽4n. The resulting bounds are unimprovable.