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

Silverman, Joseph H.

Journal de théorie des nombres de Bordeaux, Tome 24 (2012) no. 3, pp. 751-772.

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 $(\mathbb{Z}/m\mathbb{Z})\left[X\right]$ par un polynôme de la forme $1+X+\cdots +X^n$ pour un certain $n\ge ϵdeg\left(f\right)$. Nous prouvons également un énoncé analogue pour les courbes elliptiques.

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 $f\left(X\right)$ that are divisible in $(\mathbb{Z}/m\mathbb{Z})\left[X\right]$ by a polynomial of the form $1+X+\cdots +X^n$ for some $n\ge ϵdeg\left(f\right)$. We also formulate and prove an analogous statement for elliptic curves.

Classification : 11G05, 11G50, 11J97, 14H52
Mots clés : Lehmer conjecture, elliptic curve, canonical height

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