On the division fields of an elliptic curve and an effective bound to the hypotheses of the local-global divisibility

Abstract

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 $\text{char}(K)\ne 2,3$ and $m$ is a positive integer. When $m=p^r$, with $p\ge 5$ a prime and $r$ a positive integer, we prove $K(\mathcal{E}[p^r])=K(x_1,{\zeta }_p,y_2)$, where $\{(x_1,y_1),(x_2,y_2)\}$ is a generating system of $\mathcal{E}[p^r]$ and ${\zeta }_p$ is a primitive $p$-th root of the unity. If $\mathcal{E}$ has a $K$-rational point of order $p$, then $K(\mathcal{E}[p^r])=K({\zeta }_{p^r},\sqrt[m_1]{a})$, with $a\in K({\zeta }_{p^r})$ 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 $m\ge 3$. As a consequence, we give an explicit effective version of the hypotheses of the local-global divisibility problem in elliptic curves over number fields.