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

Abstract: We show there exist polynomial bounds on torsion of elliptic curves which come from a fixed geometric isogeny class. More precisely, for an elliptic curve $E_0$ defined over a number field $F_0$, for each $\epsilon>0$ there exist constants $c_\epsilon:=c_\epsilon(E_0,F_0),C_\epsilon:=C_\epsilon(E_0,F_0)>0$ such that for any elliptic curve $E_{/F}$ geometrically isogenous to $E_0$, if $E(F)$ has a point of order $N$ then [ N\leq c_\epsilon\cdot [F:\mathbb{Q}]^{1/2+\epsilon}, ] and one also has [ # E(F)[\textrm{tors}] \leq C_\epsilon\cdot [F:\mathbb{Q}]^{1+\epsilon}. ]