Probl\`{e}me de Lehmer relatif dans un tore : cas des hypersurfaces

Abstract

We tackle the "relative" Lehmer problem on algebraic subvarieties of a multiplicative torus. Generalizing a theorem of F. Amoroso and U. Zannier, we give a lower bound for the normalized height of a non torsion hypersurface in terms of its obstruction index over $\Q^{ab}$, the maximal abelian extension of $\Q$. We prove up to $\eps$ the sharpest conjecture that can be formulated.