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

top
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  ( / m ) [ 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

top

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

top
  1. F. Amoroso and R. Dvornicich, A lower bound for the height in abelian extensions. J. Number Theory 80(2) (2000), 260–272. Zbl0973.11092MR1740514
  2. P. E. Blanksby and H. L. Montgomery, Algebraic integers near the unit circle. Acta Arith. 18 (1971), 355–369. Zbl0221.12003MR296021
  3. P. Borwein, E. Dobrowolski, and M. J. Mossinghoff, Lehmer’s problem for polynomials with odd coefficients. Ann. of Math. (2) 166(2) (2007), 347–366. Zbl1172.11034MR2373144
  4. P. Borwein, K. G. Hare, and M. J. Mossinghoff, The Mahler measure of polynomials with odd coefficients. Bull. London Math. Soc. 36(3) (2004), 332–338. Zbl1143.11349MR2038720
  5. E. Dobrowolski, On a question of Lehmer and the number of irreducible factors of a polynomial. Acta Arith. 34 (1979), 391–401. Zbl0416.12001MR543210
  6. A. Dubickas and M. J. Mossinghoff, Auxiliary polynomials for some problems regarding Mahler’s measure. Acta Arith. 119(1) (2005), 65–79. Zbl1074.11018MR2163518
  7. M. Hindry and J. H. Silverman, The canonical height and integral points on elliptic curves. Invent. Math. 93(2) (1988), 419–450. Zbl0657.14018MR948108
  8. M. Hindry and J. H. Silverman, On Lehmer’s conjecture for elliptic curves. In Séminaire de Théorie des Nombres, Paris 1988–1989, volume 91 of Progr. Math., pages 103–116. Birkhäuser Boston, Boston, MA, 1990. Zbl0741.14013MR1104702
  9. M. Hindry and J. H. Silverman, Diophantine Geometry: An Introduction. Volume 201 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2000. Zbl0948.11023MR1745599
  10. M. I. M. Ishak, M. J. Mossinghoff, C. Pinner, and B. Wiles, Lower bounds for heights in cyclotomic extensions. J. Number Theory 130(6) (2010), 1408–1424. Zbl1203.11072MR2643901
  11. S. Lang, Fundamentals of Diophantine Geometry. Springer-Verlag, New York, 1983. Zbl0528.14013MR715605
  12. S. Lang, Introduction to Arakelov Theory. Springer-Verlag, New York, 1988. Zbl0667.14001MR969124
  13. S. Lang, Algebra. Volume 211 of Graduate Texts in Mathematics. Springer-Verlag, New York, third edition, 2002. Zbl0984.00001MR1878556
  14. M. Laurent, Minoration de la hauteur de Néron-Tate. In Séminaire de Théorie des Nombres, Progress in Mathematics, pages 137–151. Birkhäuser, 1983. Paris 1981–1982. Zbl0521.14010MR729165
  15. D. H. Lehmer, Factorization of certain cyclotomic functions. Ann. of Math. (2) 34(3) (1933), 461–479. Zbl0007.19904MR1503118
  16. D. W. Masser, Counting points of small height on elliptic curves. Bull. Soc. Math. France 117(2) (1989), 247–265. Zbl0723.14026MR1015810
  17. C. L. Samuels, The Weil height in terms of an auxiliary polynomial. Acta Arith. 128(3) (2007), 209–221. Zbl1146.11054MR2313990
  18. C. L. Samuels, Estimating heights using auxiliary functions. Acta Arith. 137(3) (2009), 241–251. Zbl1218.11100MR2496463
  19. J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Volume 151 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1994. Zbl0911.14015MR1312368
  20. J. H. SilvermanThe Arithmetic of Elliptic Curves. Volume 106 of Graduate Texts in Mathematics. Springer, Dordrecht, second edition, 2009. Zbl1194.11005MR2514094
  21. C. J. Smyth, On the product of the conjugates outside the unit circle of an algebraic integer. Bull. London Math. Soc. 3 (1971), 169–175. Zbl0235.12003MR289451
  22. C. J. Smyth, Some inequalities for certain power sums. Acta Arith. 28(3–4) (1976), 271–273. Zbl0338.30001MR424710
  23. C. J. Smyth, The Mahler measure of algebraic numbers: a survey. In Number theory and polynomials, volume 352 of London Math. Soc. Lecture Note Ser., pages 322–349. Cambridge Univ. Press, Cambridge, 2008. Zbl1334.11081MR2428530
  24. C. L. Stewart, Algebraic integers whose conjugates lie near the unit circle. Bull. Soc. Math. France 106(2) (1978), 169–176. Zbl0396.12002MR507748

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.