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

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Joseph H. Silverman
Lehmer’s conjecture for polynomials satisfying a congruence divisibility condition and an analogue for elliptic curves
Journal de théorie des nombres de Bordeaux, 24 no. 3 (2012), p. 751-772, doi: 10.5802/jtnb.820
Article PDF | Reviews MR 3010638 | Zbl 1264.11049
Class. Math.: 11G05, 11G50, 11J97, 14H52
Keywords: Lehmer conjecture, elliptic curve, canonical height

Résumé - 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.

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