TY - JOUR

T1 - Torsion points of order 2g + 1 on odd degree hyperelliptic curves of genus g

AU - BEKKER, BORIS M.

AU - ZARHIN, YURI G.

N1 - Funding Information: Received by the editors March 1, 2019, and, in revised form, July 14, 2019, and March 22, 2020. 2010 Mathematics Subject Classification. Primary 14H40, 14G27, 11G10, 11G30. Key words and phrases. Hyperelliptic curves, Jacobians, torsion points. The second author was partially supported by Simons Foundation Collaboration grant # 585711. Publisher Copyright: © 2020 American Mathematical Society. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/11

Y1 - 2020/11

N2 - Let K be an algebraically closed field of characteristic different from 2, let g be a positive integer, let f(x) ∈K[x] be a degree 2g + 1 monic polynomial without multiple roots, let Cf : Y2 = f(x) be the corresponding genus g hyperelliptic curve over K, and let J be the Jacobian of Cf. We identify Cf with the image of its canonical embedding into J (the infinite point of Cf goes to the zero of the group law on J). It is known [Izv. Math. 83 (2019), pp. 501-520] that if g ≥ 2, then Cf (K) contains no points of orders lying between 3 and 2g. In this paper we study torsion points of order 2g+1 on Cf (K). Despite the striking difference between the cases of g = 1 and g ≥ 2, some of our results may be viewed as a generalization of well-known results about points of order 3 on elliptic curves. E.g., if p = 2g +1 is a prime that coincides with char(K), then every odd degree genus g hyperelliptic curve contains at most two points of order p. If g is odd and f(x) has real coefficients, then there are at most two real points of order 2g + 1 on Cf. If f(x) has rational coefficients and g ≤ 51, then there are at most two rational points of order 2g + 1 on Cf. (However, there exist odd degree genus 52 hyperelliptic curves over Q that have at least four rational points of order 105.)

AB - Let K be an algebraically closed field of characteristic different from 2, let g be a positive integer, let f(x) ∈K[x] be a degree 2g + 1 monic polynomial without multiple roots, let Cf : Y2 = f(x) be the corresponding genus g hyperelliptic curve over K, and let J be the Jacobian of Cf. We identify Cf with the image of its canonical embedding into J (the infinite point of Cf goes to the zero of the group law on J). It is known [Izv. Math. 83 (2019), pp. 501-520] that if g ≥ 2, then Cf (K) contains no points of orders lying between 3 and 2g. In this paper we study torsion points of order 2g+1 on Cf (K). Despite the striking difference between the cases of g = 1 and g ≥ 2, some of our results may be viewed as a generalization of well-known results about points of order 3 on elliptic curves. E.g., if p = 2g +1 is a prime that coincides with char(K), then every odd degree genus g hyperelliptic curve contains at most two points of order p. If g is odd and f(x) has real coefficients, then there are at most two real points of order 2g + 1 on Cf. If f(x) has rational coefficients and g ≤ 51, then there are at most two rational points of order 2g + 1 on Cf. (However, there exist odd degree genus 52 hyperelliptic curves over Q that have at least four rational points of order 105.)

KW - Hyperelliptic curves

KW - Jacobians

KW - Torsion points

KW - torsion points

KW - DIVISION

UR - http://www.scopus.com/inward/record.url?scp=85094850910&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/6b6e6475-898d-31a4-9fc4-ec3a750977b4/

U2 - 10.1090/TRAN/8235

DO - 10.1090/TRAN/8235

M3 - Article

AN - SCOPUS:85094850910

VL - 373

SP - 8059

EP - 8094

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 11

ER -