In the paper under review, the author considers the family of cyclic-gonal curves of the form \(y^{q}=x^{p}+a\), where \(2<q<p\) are prime integers and \(a \in {\mathbb Q}\). The closed Riemann surfaces \(C_{q,p,a}\) they define are holomorphically equivalent over \({\mathbb C}\), but not necessarily over \({\mathbb Q}\), and they have genus \(g=(q-1)(p-1)/2\). By Torelli’s theorem the Jacobian variety \(J_{q,p,a}\) of \(C_{q,p,a}\) is also definable over \({\mathbb Q}\). We may see \(J_{q,p,a}\) as the quotient of the set of divisors of degree \(0\) by the principal divisors of \(C_{q,p,a}\). The paper concerns with \({\mathbb Q}\)-rational points in \(J_{q,p,a}\) which are torsion points, that is, \(J_{q,p,a}({\mathbb Q})_{\mathrm{tors}}\).
The main result (Theorem 1) states that \(J_{q,p,a}({\mathbb Q})_{\mathrm{tors}} \cong {\mathbb Z}_{2}^{e_{2}} \times {\mathbb Z}_{q}^{e_{q}} \times {\mathbb Z}_{p}^{e_{p}}\), where \(e_{2},e_{q},e_{p} \in \{0,1,...,(p-1)(q-1)/2\}\). The other results are concerned with the values \(e_{2}\), \(e_{p}\) and \(e_{q}\) when restricting to particular subfamilies. The main techniques are given by reduction to finite fields \({\mathbb F}_{l}\) in order to consider the curves and their Jacobians defined over them. These results generalize the known ones for the case of elliptic curves (\(q=2\) and \(p=3\)) and hyperelliptic curves (\(q=2\)).