turkmath.org

The Mordell-Lang Conjecture (MLC) concerns finitely generated subgroups of abelian varieties, however, there are analogous statements in the "non-compact" case: namely the case of the (cartesian powers of) multiplicative group, G_m. (Sometimes, MLC is formulated for semi-abelian varieties to include this case.) In very heuristic terms, in the multiplicative group case, MLC says "the addition doesn't give new information about finitely generated multiplicative groups of fields." (This will be made clear in this talk.) We isolate the conlusion of MLC as an abstract property for subgroups of G_m. It turns out that many groups other than finitely generated ones have this property. Moreover, some of them satisfy a uniform version of it. Vaguely, this corresponds to the function field field case of MLC. In this talk, after making everything above more accurate, we prove that certain finitely generated groups have this "uniform" Mordell-Lang Property.