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

Let a hyperelliptic curve C of genus g defined over an algebraically closed field K of characteristic 0 be given by the equationy y(2) = f (x) , wheref f (x) is an element of K [x] is a square-free polynomial of odd degree 2g + 1. The curve C contains a single "infinite" point O, which is a Weierstrass point. There is a classical embedding of C (K) into the group J (K) of K-points of the Jacobian variety J of C that identifies the point O with the identity of the groupJ J (K) . For 2 <= g <= 5 , we explicitly find representatives of birational equivalence classes of hyperelliptic curves C with a unique base point at infinity O such that the set C (K) boolean AND 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 >= 3, an upper bound depending only on the genus g was known. We improve the previously known upper bound by almost 36 times.