Resumen de Minorations des hauteurs normalisées des sous-variétés des tores

Sinnou David, Patrice Philippon

• We extend to the case of torii the study of small points on subvarieties of abelian varieties initiated in a previous work, and which gave an effective proof of the generalized Bogomolov conjecture. We now obtain an effective estimate for the number of « exceptional » points together with a bound for their heights. These are inverse monomial in the degree of the variety and geometric till the last step (which amounts to Lehmer’s problem). Similar estimates, but « pluri-exponential » in the degree, were known, due to W. Schmidt and E. Bombieri-U. Zannier. Finally, we show how the conjectural lower bounds (in their weak form) for normalized heights of hypersurfaces can be deduced from the corresponding one for curves in G_m^2