TY - JOUR

T1 - Torsion points of small order on hyperelliptic curves

AU - Bekker, Boris M.

AU - Zarhin, Yuri G.

N1 - Publisher Copyright: © 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

PY - 2022/6

Y1 - 2022/6

N2 - 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.

AB - 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.

KW - Hyperelliptic curves

KW - Jacobians

KW - Torsion points

KW - JACOBIANS

KW - CONJECTURE

UR - http://www.scopus.com/inward/record.url?scp=85123865033&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/04716f83-3310-3d85-bb4d-8467db1d6311/

U2 - 10.1007/s40879-021-00519-z

DO - 10.1007/s40879-021-00519-z

M3 - Article

AN - SCOPUS:85123865033

VL - 8

SP - 611

EP - 624

JO - European Journal of Mathematics

JF - European Journal of Mathematics

SN - 2199-675X

IS - 2

ER -