On Hyperelliptic Curves of Odd Degree and Genus g with Six Torsion Points of Order 2g + 1

Abstract

Let a hyperelliptic curve \(\mathcal{C}\) of genus g defined over an algebraically closed field K of characteristic 0 be given by the equation \({{y}^{2}} = f(x)\), where \(f(x) \in K[x]\) is a square-free polynomial of odd degree \(2g + 1\). The curve \(\mathcal{C}\) contains a single “infinite” point \(\mathcal{O}\), which is a Weierstrass point. There is a classical embedding of \(\mathcal{C}(K)\) into the group \(J(K)\) of K-points of the Jacobian variety J of \(\mathcal{C}\) that identifies the point \(\mathcal{O}\) with the identity of the group \(J(K)\). For \(2 \leqslant g \leqslant 5\), we explicitly find representatives of birational equivalence classes of hyperelliptic curves \(\mathcal{C}\) with a unique base point at infinity \(\mathcal{O}\) such that the set \(\mathcal{C}(K) \cap J(K)\) contains at least six torsion points of order \(2g + 1\). It was previously known that for \(g = 2\) there are exactly five such equivalence classes, and, for \(g \geqslant 3\), an upper bound depending only on the genus g was known. We improve the previously known upper bound by almost 36 times.