Lehmer’s conjecture for polynomials satisfying a congruence divisibility condition and an analogue for elliptic curves
Journal de théorie des nombres de Bordeaux, Volume 24 (2012) no. 3, p. 751-772

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 nϵdeg(f). We also formulate and prove an analogous statement for elliptic curves.

De nombreux auteurs ont prouvé des versions explicites de la conjecture de Lehmer dans le cas particulier de polynômes dont les coefficients sont tous congrus à 1 modulo un entier m>1. Nous prouvons ici un résultat similaire pour les polynômes qui sont divisibles dans l’anneau (/m)[X] par un polynôme de la forme 1+X++X n pour un certain nϵdeg(f). Nous prouvons également un énoncé analogue pour les courbes elliptiques.

DOI : https://doi.org/10.5802/jtnb.820
Classification:  11G05,  11G50,  11J97,  14H52
Keywords: Lehmer conjecture, elliptic curve, canonical height
@article{JTNB_2012__24_3_751_0,
     author = {Silverman, Joseph H.},
     title = {Lehmer's conjecture for polynomials satisfying a congruence divisibility condition and an analogue for elliptic curves},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {24},
     number = {3},
     year = {2012},
     pages = {751-772},
     doi = {10.5802/jtnb.820},
     mrnumber = {3010638},
     zbl = {1264.11049},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2012__24_3_751_0}
}
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, Volume 24 (2012) no. 3, pp. 751-772. doi : 10.5802/jtnb.820. http://www.numdam.org/item/JTNB_2012__24_3_751_0/

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