论文标题
在椭圆曲线的分裂场上，并有效与局部全球划分的假设结合
On the division fields of an elliptic curve and an effective bound to the hypotheses of the local-global divisibility
论文作者
论文摘要
我们研究了$ m $ -Division Field $ k（{\ Mathcal {e}} [m]）$的某些方面，其中$ \ Mathcal {e} $是在带有$ k $的字段$ k $的椭圆曲线的，$ {\ textrm {char}}}（k）\ neq 2,3 $和$ m $ $ $ $ $ is ISEe。当$ m = p^r $，带有$ p \ geq 5 $ a prime和$ r $ a正整数时，我们证明$ k（\ nathcal {e} [p^r] [p^r]）= k（x_1，ζ_p，y_2）$，$ \ {（x_1，y_1，y_1，y__2） $ {\ Mathcal {e}} [p^r] $和$ζ_P$是unity的原始$ p $ - th。如果$ \ MATHCAL {E} $具有$ K $ - 合理点$ P $，则$ K（\ Mathcal {e} [p^r]）= k（ζ_{p^r}，\ sqrt [m_1] {a}）$，带有$ a \ in K（in k in k（pema a）$ a \ c^p^p^r} $ a}此外，当$ k $是一个数字字段时，我们将对所有$ m \ geq 3 $的分机判别的对数高度产生上限。结果，我们给出了椭圆曲线上局部 - 全球分裂性问题的假设的明确有效版本。
We investigate some aspects of the $m$-division field $K({\mathcal{E}}[m])$, where $\mathcal{E}$ is an elliptic curve defined over a field $K$ with ${\textrm{char}}(K)\neq 2,3$ and $m$ is a positive integer. When $m=p^r$, with $p\geq 5$ a prime and $r$ a positive integer, we prove $K(\mathcal{E}[p^r])=K(x_1,ζ_p,y_2)$, where $\{(x_1, y_1),(x_2,y_2)\}$ is a generating system of ${\mathcal{E}}[p^r]$ and $ζ_p$ is a primitive $p$-th root of the unity. If $\mathcal{E}$ has a $K$-rational point of order $p$, then $K(\mathcal{E}[p^r])=K(ζ_{p^r},\sqrt[m_1]{a})$, with $a\in K(ζ_{p^r})$ and $m_1|p^r$. In addition, when $K$ is a number field, we produce an upper bound to the logarithmic height of the discriminant of the extension $K(\mathcal{E}[m])/K$, for all $m\geq 3$. As a consequence, we give an explicit effective version of the hypotheses of the local-global divisibility problem in elliptic curves over number fields.
