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

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/mZ) [X] by a polynomial of the form 1 + X + ... + X-n for some n >= epsilon deg(f). We also formulate and prove an analogous statement for elliptic curves.