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 (2g +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 e(K) are exactly the "remaining" (2g + 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 <= 2g. So, it is natural to study torsion points of order 2g + 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 2g + 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 2g + 1.