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 - Publisher Copyright:
© 2020 American Mathematical Society.

PY - 2020

Y1 - 2020

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.)

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U2 - 10.1090/TRAN/8235

DO - 10.1090/TRAN/8235

M3 - Article

AN - SCOPUS:85094850910

SN - 0002-9947

VL - 373

SP - 8059

EP - 8094

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 11

ER -