On the Jacobian of a family of hyperelliptic curves (Q2683729)

In this paper, the algebraic rank and the analytic rank of the Jacobian \(J_{m^2}\) of the hyperelliptic curve \(C_{m^2} : y^2 = x^5 + m^2\), where \(m\) is a square free-integer, are studied. It is proved, that there are infinitely many square-free integers \(m\) such that \(J_{m^2}(\mathbb{Q}) \cong \mathbb{Z}/5\mathbb{Z}\). Further, under the hypothesis that the algebraic rank and the analytic rank of \(J_{m^2}\) are equal modulo 2 (the parity conjecture), it is proved that there are infinitely many square-free integers \(m\) such that \(J_{m^2}(\mathbb{Q}) \cong \mathbb{Z} \oplus \mathbb{Z}/5\mathbb{Z}\). The proofs of these results use the results of \textit{E. F. Schaefer} [Math. Ann. 310, No. 3, 447--471 (1998; Zbl 0889.11021)] and \textit{M. Stoll} [J. Number Theory 93, No. 2, 183--206 (2002; Zbl 1004.11038)]. Moreover, assuming that the root number of \(J_{m^2}\) is \(+1\), every prime divisor of \(m\) is equivalent to 1 modulo 5, and \(A^4 - 1\) is divided by 25, it is proved that the \(L\)-function of \(J_{m^2}\) satisfies \(L(1, J_{m^2}) \neq 0\).