Lehmer’s conjecture for polynomials satisfying a congruence divisibility condition and an analogue for elliptic curves

Joseph H. Silverman[1]

  • [1] Mathematics Department, Box 1917 Brown University, Providence, RI 02912 USA

Journal de Théorie des Nombres de Bordeaux (2012)

  • Volume: 24, Issue: 3, page 751-772
  • ISSN: 1246-7405

Abstract

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A number of authors have proven explicit versions of Lehmer’s conjecture for polynomials whose coefficients are all congruent to  1 modulo  m . We prove a similar result for polynomials  f ( X ) that are divisible in  ( ℤ / m ℤ ) [ X ] by a polynomial of the form 1 + X + ⋯ + X n for some n ≥ ϵ deg ( f ) . We also formulate and prove an analogous statement for elliptic curves.

How to cite

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Silverman, Joseph H.. "Lehmer’s conjecture for polynomials satisfying a congruence divisibility condition and an analogue for elliptic curves." Journal de Théorie des Nombres de Bordeaux 24.3 (2012): 751-772. <http://eudml.org/doc/251114>.

@article{Silverman2012,
abstract = {A number of authors have proven explicit versions of Lehmer’s conjecture for polynomials whose coefficients are all congruent to $1$ modulo $m$. We prove a similar result for polynomials $f(X)$ that are divisible in $(\mathbb\{Z\}/m\mathbb\{Z\})[X]$ by a polynomial of the form $1+X+\cdots +X^n$ for some $n\ge \epsilon \deg (f)$. We also formulate and prove an analogous statement for elliptic curves.},
affiliation = {Mathematics Department, Box 1917 Brown University, Providence, RI 02912 USA},
author = {Silverman, Joseph H.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Lehmer conjecture; elliptic curve; canonical height},
language = {eng},
month = {11},
number = {3},
pages = {751-772},
publisher = {Société Arithmétique de Bordeaux},
title = {Lehmer’s conjecture for polynomials satisfying a congruence divisibility condition and an analogue for elliptic curves},
url = {http://eudml.org/doc/251114},
volume = {24},
year = {2012},
}

TY - JOUR
AU - Silverman, Joseph H.
TI - Lehmer’s conjecture for polynomials satisfying a congruence divisibility condition and an analogue for elliptic curves
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2012/11//
PB - Société Arithmétique de Bordeaux
VL - 24
IS - 3
SP - 751
EP - 772
AB - A number of authors have proven explicit versions of Lehmer’s conjecture for polynomials whose coefficients are all congruent to $1$ modulo $m$. We prove a similar result for polynomials $f(X)$ that are divisible in $(\mathbb{Z}/m\mathbb{Z})[X]$ by a polynomial of the form $1+X+\cdots +X^n$ for some $n\ge \epsilon \deg (f)$. We also formulate and prove an analogous statement for elliptic curves.
LA - eng
KW - Lehmer conjecture; elliptic curve; canonical height
UR - http://eudml.org/doc/251114
ER -

References

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