Torsion points of small order on hyperelliptic curves

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