Torsion points of small order on hyperelliptic curves

Abstract

Let $$\mathscr {C}$$ be a hyperelliptic curve of genus $$g>1$$ over an algebraically closed field K of characteristic zero and $$\mathscr {O}$$ one of the $$(2g{+}2)$$ Weierstrass points in $$\mathscr {C}(K)$$ . Let J be the Jacobian of $$\mathscr {C}$$ , which is a g-dimensional abelian variety over K. Let us consider the canonical embedding of $$\mathscr {C}$$ into J that sends $$\mathscr {O}$$ to the zero of the group law on J. This embedding allows us to identify $$\mathscr {C}(K)$$ 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 $$\mathscr {C}(K)$$ is finite. It is well known that the points of order 2 in $$\mathscr {C}(K)$$ 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 $$\mathscr {C}(K)$$ if $$3\leqslant n\leqslant 2g$$ . So, it is natural to study torsion points of order $$2g+1$$ (notice that the number of such points in $$\mathscr {C}(K)$$ is always even). Recently, the authors proved that there are infinitely many (for a given g) mutually non-isomorphic pairs $$(\mathscr {C},\mathscr {O})$$ such that $$\mathscr {C}(K)$$ 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 $$(\mathscr {C},\mathscr {O})$$ such that $$\mathscr {C}(K)$$ contains at least six points of order $$2g+1$$ .