Torsion points of small order on cyclic covers of P^1

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Torsion points of small order on cyclic covers of P^1. / Беккер, Борис Меерович; Zarhin, Yuri .

в: Ramanujan Journal, Том 67, № 3, 68, 24.05.2025.

Результаты исследований: Научные публикации в периодических изданиях › статья › Рецензирование

BibTeX

@article{5a2bfdce257044bb81e8ac8af19ddf69,

title = "Torsion points of small order on cyclic covers of P^1",

abstract = "Let d≥2 be a positive integer, K an algebraically closed field of characteristic not dividing d, n≥d+1 a positive integer prime to d, f(x)∈K[x] a degree n monic polynomial without repeated roots, Cf,d:yd=f(x) the corresponding smooth plane affine curve over K, and Cf,d a smooth projective model of Cf,d. Let J(Cf,d) be the Jacobian of Cf,d. We identify Cf,d with the image of its canonical embedding into J(Cf,d) (such that the infinite point of Cf,d goes to the zero of the group law on J(Cf,d)). Earlier the second named author proved that if d=2 and n=2g+1≥5, then the genus g hyperelliptic curve Cf,2 contains no torsion points of orders lying between 3 and n-1=2g. In the present paper we generalize this result to the case of arbitrary d. Namely, we prove that if P is a torsion point of order m>1 on Cf,d, then either m=d or m≥n. We also describe all curves Cf,d having a torsion point of order n.",

keywords = "Cyclic covers, Jacobians, Torsion points",

author = "Беккер, {Борис Меерович} and Yuri Zarhin",

year = "2025",

month = may,

day = "24",

doi = "10.1007/s11139-025-01082-x",

language = "English",

volume = "67",

journal = "Ramanujan Journal",

issn = "1382-4090",

publisher = "Springer Nature",

number = "3",

}

RIS

TY - JOUR

T1 - Torsion points of small order on cyclic covers of P^1

AU - Беккер, Борис Меерович

AU - Zarhin, Yuri

PY - 2025/5/24

Y1 - 2025/5/24

N2 - Let d≥2 be a positive integer, K an algebraically closed field of characteristic not dividing d, n≥d+1 a positive integer prime to d, f(x)∈K[x] a degree n monic polynomial without repeated roots, Cf,d:yd=f(x) the corresponding smooth plane affine curve over K, and Cf,d a smooth projective model of Cf,d. Let J(Cf,d) be the Jacobian of Cf,d. We identify Cf,d with the image of its canonical embedding into J(Cf,d) (such that the infinite point of Cf,d goes to the zero of the group law on J(Cf,d)). Earlier the second named author proved that if d=2 and n=2g+1≥5, then the genus g hyperelliptic curve Cf,2 contains no torsion points of orders lying between 3 and n-1=2g. In the present paper we generalize this result to the case of arbitrary d. Namely, we prove that if P is a torsion point of order m>1 on Cf,d, then either m=d or m≥n. We also describe all curves Cf,d having a torsion point of order n.

AB - Let d≥2 be a positive integer, K an algebraically closed field of characteristic not dividing d, n≥d+1 a positive integer prime to d, f(x)∈K[x] a degree n monic polynomial without repeated roots, Cf,d:yd=f(x) the corresponding smooth plane affine curve over K, and Cf,d a smooth projective model of Cf,d. Let J(Cf,d) be the Jacobian of Cf,d. We identify Cf,d with the image of its canonical embedding into J(Cf,d) (such that the infinite point of Cf,d goes to the zero of the group law on J(Cf,d)). Earlier the second named author proved that if d=2 and n=2g+1≥5, then the genus g hyperelliptic curve Cf,2 contains no torsion points of orders lying between 3 and n-1=2g. In the present paper we generalize this result to the case of arbitrary d. Namely, we prove that if P is a torsion point of order m>1 on Cf,d, then either m=d or m≥n. We also describe all curves Cf,d having a torsion point of order n.

KW - Cyclic covers

KW - Jacobians

KW - Torsion points

UR - https://www.mendeley.com/catalogue/7823c2ca-d4c1-32d1-8829-0985b3c93a19/

U2 - 10.1007/s11139-025-01082-x

DO - 10.1007/s11139-025-01082-x

M3 - Article

VL - 67

JO - Ramanujan Journal

JF - Ramanujan Journal

SN - 1382-4090

IS - 3

M1 - 68

ER -

ID: 135906084

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