Small totally $p$-adic algebraic numbers

Abstract

The purpose of this note is to give a short and elementary proof of the fact, that the absolute logarithmic Weil-height is bounded from below by a positive constant for all totally p-adic numbers which are neither zero nor a root of unity. The proof is based on an idea of C. Petsche and gives the best known lower bounds in this setting. These bounds differ from the truth by a term of less than $\log (3)/p$.

There was a slight error in the citation of a result mentioned in the introduction. The strengthening of the lower bound in equation (1) does not read $\frac{1}{2}{\sum }_{p\in S}\frac{\log (p)}{p-1}$ but $\frac{1}{2}{\sum }_{p\in S}\frac{p\log (p)}{p^2-1}$. Moreover, this result is not due to P. Fili alone but due to P. Fili and C. Petsche