On curves over finite fields with Jacobians of small exponent

Kevin Ford; Igor Shparlinski

doi:10.1142/S1793042108001687

On curves over finite fields with Jacobians of small exponent

Kevin Ford^*, Igor Shparlinski

^*Corresponding author for this work

School of Computing

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

Abstract

We show that finite fields over which there is a curve of a given genus g ≥ 1 with its Jacobian having a small exponent, are very rare. This extends a recent result of Duke in the case of g = 1. We also show that when g = 1 or g = 2, our lower bounds on the exponent, valid for almost all finite fields F_q and all curves over F_q, are best possible.

Original language	English
Pages (from-to)	819-826
Number of pages	8
Journal	International Journal of Number Theory
Volume	4
Issue number	5
DOIs	https://doi.org/10.1142/S1793042108001687
Publication status	Published - 2008

Keywords

Distribution of divisors
Group structure
Jacobian

Access to Document

10.1142/S1793042108001687

Cite this

@article{0d34c45001da4e2facfd839fcec9608c,

title = "On curves over finite fields with Jacobians of small exponent",

abstract = "We show that finite fields over which there is a curve of a given genus g ≥ 1 with its Jacobian having a small exponent, are very rare. This extends a recent result of Duke in the case of g = 1. We also show that when g = 1 or g = 2, our lower bounds on the exponent, valid for almost all finite fields Fq and all curves over Fq, are best possible.",

keywords = "Distribution of divisors, Group structure, Jacobian",

author = "Kevin Ford and Igor Shparlinski",

year = "2008",

doi = "10.1142/S1793042108001687",

language = "English",

volume = "4",

pages = "819--826",

journal = "International Journal of Number Theory",

issn = "1793-0421",

publisher = "World Scientific Publishing",

number = "5",

}

TY - JOUR

T1 - On curves over finite fields with Jacobians of small exponent

AU - Ford, Kevin

AU - Shparlinski, Igor

PY - 2008

Y1 - 2008

N2 - We show that finite fields over which there is a curve of a given genus g ≥ 1 with its Jacobian having a small exponent, are very rare. This extends a recent result of Duke in the case of g = 1. We also show that when g = 1 or g = 2, our lower bounds on the exponent, valid for almost all finite fields Fq and all curves over Fq, are best possible.

AB - We show that finite fields over which there is a curve of a given genus g ≥ 1 with its Jacobian having a small exponent, are very rare. This extends a recent result of Duke in the case of g = 1. We also show that when g = 1 or g = 2, our lower bounds on the exponent, valid for almost all finite fields Fq and all curves over Fq, are best possible.

KW - Distribution of divisors

KW - Group structure

KW - Jacobian

UR - http://www.scopus.com/inward/record.url?scp=54949107872&partnerID=8YFLogxK

U2 - 10.1142/S1793042108001687

DO - 10.1142/S1793042108001687

M3 - Article

AN - SCOPUS:54949107872

SN - 1793-0421

VL - 4

SP - 819

EP - 826

JO - International Journal of Number Theory

JF - International Journal of Number Theory

IS - 5

ER -

On curves over finite fields with Jacobians of small exponent

Abstract

Keywords

Access to Document

Other files and links

Fingerprint

Cite this