Torsion points of small order on hyperelliptic curves

Let C be a hyperelliptic curve of genus g> 1 over an algebraically closed field K of characteristic zero and O one of the (2 g+ 2) Weierstrass points in C(K). Let J be the Jacobian of C, which is a g-dimensional abelian variety over K. Let us consider the canonical embedding of C into J that sends O to the zero of the group law on J. This embedding allows us to identify C(K) with a certain subset of the commutative group J(K). A special case of the famous theorem of Raynaud (Manin–Mumford conjecture) asserts that the set of torsion points in C(K) is finite. It is well known that the points of order 2 in C(K) are exactly the “remaining” (2 g+ 1) Weierstrass points. One of the authors (Zarhin in Izv Math 83:501–520, 2019) proved that there are no torsion points of order n in C(K) if 3 ⩽ n⩽ 2 g. So, it is natural to study torsion points of order 2 g+ 1 (notice that the number of such points in C(K) is always even). Recently, the authors proved that there are infinitely many (for a given g) mutually non-isomorphic pairs (C, O) such that C(K) contains at least four points of order 2 g+ 1. In the present paper we prove that (for a given g) there are at most finitely many (up to an isomorphism) pairs (C, O) such that C(K) contains at least six points of order 2 g+ 1.