Small exponent point groups on elliptic curves

Florian Luca[1]; James McKee[2]; Igor E. Shparlinski[3]

  • [1] Instituto de Matemáticas Universidad Nacional Autónoma de México C.P. 58089, Morelia, Michoacán, México
  • [2] Department of Mathematics Royal Holloway, University of London Egham, Surrey, TW20 0EX, UK
  • [3] Department of Computing Macquarie University Sydney, NSW 2109, Australia

Journal de Théorie des Nombres de Bordeaux (2006)

  • Volume: 18, Issue: 2, page 471-476
  • ISSN: 1246-7405

Abstract

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Let E be an elliptic curve defined over F q , the finite field of q elements. We show that for some constant η > 0 depending only on q , there are infinitely many positive integers n such that the exponent of E ( F q n ) , the group of F q n -rational points on E , is at most q n exp - n η / log log n . This is an analogue of a result of R. Schoof on the exponent of the group E ( F p ) of F p -rational points, when a fixed elliptic curve E is defined over ℚ and the prime p tends to infinity.

How to cite

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Luca, Florian, McKee, James, and Shparlinski, Igor E.. "Small exponent point groups on elliptic curves." Journal de Théorie des Nombres de Bordeaux 18.2 (2006): 471-476. <http://eudml.org/doc/249612>.

@article{Luca2006,
abstract = {Let $\{\mathbf\{E\}\}$ be an elliptic curve defined over $\{\mathbf\{F\}\}_q$, the finite field of $q$ elements. We show that for some constant $\eta &gt;0$ depending only on $q$, there are infinitely many positive integers $n$ such that the exponent of $\{\mathbf\{E\}\}(\{\mathbf\{F\}\}_\{q^n\})$, the group of $\{\mathbf\{F\}\}_\{q^n\}$-rational points on $\{\mathbf\{E\}\}$, is at most $q^\{n\}\exp \left( - n^\{ \eta /\log \log n\}\right)$. This is an analogue of a result of R. Schoof on the exponent of the group $\{\mathbf\{E\}\}(\{\mathbf\{F\}\}_p)$ of $\{\mathbf\{F\}\}_\{p\}$-rational points, when a fixed elliptic curve $\{\mathbf\{E\}\}$ is defined over $\mathbb\{Q\}$ and the prime $p$ tends to infinity.},
affiliation = {Instituto de Matemáticas Universidad Nacional Autónoma de México C.P. 58089, Morelia, Michoacán, México; Department of Mathematics Royal Holloway, University of London Egham, Surrey, TW20 0EX, UK; Department of Computing Macquarie University Sydney, NSW 2109, Australia},
author = {Luca, Florian, McKee, James, Shparlinski, Igor E.},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
number = {2},
pages = {471-476},
publisher = {Université Bordeaux 1},
title = {Small exponent point groups on elliptic curves},
url = {http://eudml.org/doc/249612},
volume = {18},
year = {2006},
}

TY - JOUR
AU - Luca, Florian
AU - McKee, James
AU - Shparlinski, Igor E.
TI - Small exponent point groups on elliptic curves
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2006
PB - Université Bordeaux 1
VL - 18
IS - 2
SP - 471
EP - 476
AB - Let ${\mathbf{E}}$ be an elliptic curve defined over ${\mathbf{F}}_q$, the finite field of $q$ elements. We show that for some constant $\eta &gt;0$ depending only on $q$, there are infinitely many positive integers $n$ such that the exponent of ${\mathbf{E}}({\mathbf{F}}_{q^n})$, the group of ${\mathbf{F}}_{q^n}$-rational points on ${\mathbf{E}}$, is at most $q^{n}\exp \left( - n^{ \eta /\log \log n}\right)$. This is an analogue of a result of R. Schoof on the exponent of the group ${\mathbf{E}}({\mathbf{F}}_p)$ of ${\mathbf{F}}_{p}$-rational points, when a fixed elliptic curve ${\mathbf{E}}$ is defined over $\mathbb{Q}$ and the prime $p$ tends to infinity.
LA - eng
UR - http://eudml.org/doc/249612
ER -

References

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