On certain torsion subgroups of Jacobians of hyperelliptic curves of genus \(g \geq 1\)
Given two integers \(d,r\geq 1\) and an abelian variety \(A\) of dimension \(g\) defined over a number field \(K\) of degree \(d\) the strong boundedness conjecture asks for a bound for the torsion subgroup \(A(K)_{\text{tor}}\) of the group of rational points \(A(K)\) of \(A\) in terms of \(g\) and \(d\). In the case where \(g=1\) this conjecture was established by \textit{B. Mazur} [ Publ. Math., Inst. Hautes Étud. Sci. 47 (1977), 33-186 (1978; Zbl 0394.14008)], \textit{S. Kamienny} [Invent. Math. 109, No. 2, 221-229 (1992; Zbl 0773.14016)], \textit{S. Kamienny} and \textit{B. Mazur} [ in: Columbia Univ. Number Theory Semin., New York 1992, Astérisque 228, 81-100 (1995; Zbl 0846.14012)] and \textit{D. Abramovich} [ibid., Astérisque 228, 5-17 (1995; Zbl 0846.14013)]. For \(g=1\) and any \(d\geq 1\) this result was established by \textit{L. Merel} [Invent. Math. 124, No. 1-3, 437-449 (1996)]. However, in the case \(g\geq 2\), for no \(d\geq 1\) this conjecture has been verified. In this paper the author addresses this question for the Jacobian varieties of hyperelliptic curves defined over the field of rational numbers. More precisely, he shows that there exists a family of hyperelliptic curves of genus \(g\) defined over \(\mathbb Q\) whose Jacobian varieties have a rational point of order \(2g(2g+1)\). For \(l=2g^2+5g+5\), he constructs a family of genus \(g\) hyperelliptic curves defined over \(\mathbb Q\) such that their Jacobian varieties have a rational point of order \(l\), \(l/2\) or \(l/4\). Another family of hyperelliptic curves defined over \(\mathbb Q\) (which does not belong to any of the previous ones) is constructed such that the Jacobian varieties of the curves of the family have a rational point of order \(l\).
Jacobian varieties of hyperelliptic curves
scientific article; zbMATH DE number 980716