Characterization of the torsion of the Jacobians of two families of hyperelliptic curves (Q2868206)

Let \(E^a:y^2=x^3+ax\) and \(E_b:y^2=x^3+b\) be families of elliptic curves. The torsion subgroup of the Mordell-Weil group \(E^a(\mathbb{Q})_{\mathrm{tor}}\) and \(E_b(\mathbb{Q})_{\mathrm{tor}}\) are well-known. In particular, we know that \(E^a(\mathbb{Q})_{\mathrm{tor}}\) is a \(2\)-group and if \(a \neq 4\), then \(E^a(\mathbb{Q})_{\mathrm{tor}} = E^a(\mathbb{Q})[2]\). This paper discusses various properties of the natural generalization of \(E^a\) and \(E_b\) to families of hyperelliptic curves \(C^{n,A} : y^2=x^n+Ax\), \(C_{n,A}: y^2=x^n+A\) and their Jacobians \(J^{n,A}\) and \(J_{n,A}\). The first main result of the paper concerns \(J^{7,A}(\mathbb{Q})_{\mathrm{tor}}\). It satisfies a similar property like \(E^a({\mathbb{Q}})_{\mathrm{tor}}\), namely, \(J^{7,A}(\mathbb{Q})_{\mathrm{tor}}\) is a \(2\)-group and if \(A \neq 4a^4, -1728, -1259712\), then \(J^{7,A}(\mathbb{Q})_{\mathrm{tor}} = J^{7,A}(\mathbb{Q})[2]\). If \(A\) is \(4a^4\) or \(-1259712\), then \(J^{7,A}(\mathbb{Q})_{\mathrm{tor}}\) has an element of order \(4\).NEWLINENEWLINEThe second part of the paper concerns \(J_{p,A}\) when \(p\) is odd and \(A\) is nonzero rational. The genus of \(C_{p,A}\) is \((p-1)/2\). Let \(p^* = (-1)^{(p-1)/2}p\). The main result is that \(J_{p,A}(\mathbb{Q})_{\mathrm{tor}}\) is isomorphic to one of the following depending on the choice of \(A\): NEWLINE\[NEWLINE J_{p,A}(\mathbb{Q})_{\mathrm{tor}} \cong \begin{cases} \{0\}, & \text{if} \; A \neq \text{square and} \; A \neq p^* \times \text{square and}\; A \neq p\text{th power}; \\ \mathbb{Z}/2\mathbb{Z}, & \text{if} \; A \neq \text{square and} \; A \neq p^* \times \text{square and}\; A = p\text{th power};\\ \mathbb{Z}/p \mathbb{Z} & \text{if} \; A = \text{square and} \; A \neq p\text{th power},\\ \mathbb{Z}/2p\mathbb{Z} & \text{if} \; A = \text{square and} \; A = p\text{th power},\\ \{0\} \; \text{or} \; \mathbb{Z}/p\mathbb{Z} & \text{if} \; A = p^* \times \text{square and}\; A \neq p\text{th power},\\ \mathbb{Z}/2\mathbb{Z} \; \text{or} \; \mathbb{Z}/2p{\mathbb{Z}} & \text{if} \; A = p^* \times \text{square and}\; A = p\text{th power}. \end{cases}.NEWLINE\]NEWLINE This is a generalization of a similar formula held by \(E_b(\mathbb{Q})_{\mathrm{tors}}\).