Torsion points of small order on cyclic covers of ℚ ¹ . II

Let [Formula: see text] be an integer, [Formula: see text] a perfect field such that [Formula: see text], [Formula: see text] an integer prime to [Formula: see text], [Formula: see text] a degree [Formula: see text] monic polynomial without repeated roots, and [Formula: see text] a smooth projective model of the affine curve [Formula: see text]. Let [Formula: see text] be the Jacobian of the [Formula: see text]-curve [Formula: see text]. We identify [Formula: see text] with its canonical image in [Formula: see text] (such that the infinite point of [Formula: see text] goes to the zero of the group law on [Formula: see text]). We say that an integer [Formula: see text] is [Formula: see text]-reachable over [Formula: see text] if there exists a polynomial [Formula: see text] as above such that [Formula: see text] contains a torsion point of order [Formula: see text]. Earlier we proved that if [Formula: see text] is [Formula: see text]-reachable, then either [Formula: see text] or [Formula: see text] (in addition, both [Formula: see text] and [Formula: see text] are [Formula: see text]-reachable). In the present paper we prove the following. If [Formula: see text] and if [Formula: see text] is [Formula: see text]-reachable over [Formula: see text], then either [Formula: see text] or [Formula: see text]. If either [Formula: see text], or [Formula: see text] is infinite and [Formula: see text], then [Formula: see text] is [Formula: see text]-reachable if and only if [Formula: see text]. If [Formula: see text], then [Formula: see text] is [Formula: see text]-reachable if and only if [Formula: see text]. If [Formula: see text] (the hyperelliptic case) and [Formula: see text], then [Formula: see text] is [Formula: see text]-reachable if [Formula: see text] and [Formula: see text]. (The case when [Formula: see text] was done earlier by E.V. Flynn.)