论文标题
来自固定几何等异形曲线的扭转的多项式边界
Polynomial bounds on torsion from a fixed geometric isogeny class of elliptic curves
论文作者
论文摘要
我们表明，来自固定几何学等级类别的椭圆曲线的扭转中存在多项式边界。更确切地说，对于椭圆曲线$ e_0 $，在数字字段$ f_0 $上定义，对于每个$ε> 0 $存在常数$c_ε：= c_is（e_0，f_0），c_is：= c_is：= c_is（e_0，f_0，f_0）> 0 $ 0 $，以至于$ e _ $ e_ 0 $ e（f）$具有订单点$ n $，然后\ [n \ leq c_is \ cdot [f：\ mathbb {q}]^{1/2+ε}，\ \]，一个人也具有\ [\＃e（e（f） [f：\ mathbb {q}]^{1+ε}。 \]
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 $ε>0$ there exist constants $c_ε:=c_ε(E_0,F_0),C_ε:=C_ε(E_0,F_0)>0$ such that for any elliptic curve $E_{/F}$ geometrically isogenous to $E_0$, if $E(F)$ has a point of order $N$ then \[ N\leq c_ε\cdot [F:\mathbb{Q}]^{1/2+ε}, \] and one also has \[ \# E(F)[\textrm{tors}] \leq C_ε\cdot [F:\mathbb{Q}]^{1+ε}. \]
