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 [Formula: see text]-division field [Formula: see text], where [Formula: see text] is an elliptic curve defined over a field [Formula: see text] with [Formula: see text] and [Formula: see text] is a positive integer. When [Formula: see text], with [Formula: see text] a prime and [Formula: see text] a positive integer, we prove [Formula: see text], where [Formula: see text] is a generating system of [Formula: see text] and [Formula: see text] is a primitive [Formula: see text]th root of unity. If [Formula: see text] has a [Formula: see text]-rational point of order [Formula: see text], then [Formula: see text] for some [Formula: see text] and [Formula: see text]. For every number field [Formula: see text], we produce an upper bound to the logarithmic height of the discriminant of the extension [Formula: see text] for all [Formula: see text]. As a consequence, we consider a version of the local-global divisibility problem in elliptic curves over number fields, where the local conditions are known only for finitely many places.