Polynomial bounds on torsion from a fixed geometric isogeny class of elliptic curves (Q6642864)

In this paper, the author investigates the torsion subgroups of elliptic curves in a fixed geometric isogeny class over number fields. More precisely, for a non-CM elliptic curve \(E_0\) defined over a number field \(F_0\; the author presents polynomial bounds on the size of the torsion subgroup of any elliptic curve \(E\) over \(F\) that is geometrically isogenous to \(E_0\). The main result states that for any \(\varepsilon > 0\), there exist constants \(c_\varepsilon\) and \(C_\varepsilon\) (depending on \(E_0\) and \(F_0\)) such that if \(E(F)\) has a torsion point of order \(N\), then \(N \leq c_\varepsilon [F : \mathbb{Q}]^{1/2+\varepsilon}\) and \(\#\operatorname{Tors}E(F) \leq C_\varepsilon [F : \mathbb{Q}]^{1+\varepsilon}\).\N\NThe proofs rely on a detailed analysis of Galois representations associated with elliptic curves and results from adelic and suitable Tate modules to relate size of the torsion subgroup to field extensions together with the use of existing results on isogenies of elliptic curves.