On function field Mordell-Lang and Manin-Mumford

We give a reduction of the function field Mordell-Lang conjecture to the function field Manin-Mumford conjecture, for abelian varieties, in all characteristics, via model theory, but avoiding recourse to the dichotomy theorems for (generalized) Zariski geometries. Additional ingredients include the “Theorem of the kernel”, and a result of Wagner on commutative groups of finite Morley rank without proper infinite definable subgroups. In characteristic 0 the methods also yield another account of the local modularity of A] for A a traceless simple abelian variety. In positive characteristic, where the main interest lies, we require another result to make the strategy work: so-called quantifier-elimination for the corresponding A] = p∞A(U) where U is a saturated separably closed field, which we prove in the last section. ∗Partially supported by ANR MODIG (ANR-09-BLAN-0047) Model theory and Interactions with Geometry and ANR ValCoMo (ANR-13-BS01-0006) Valuations, Combinatorics and Model Theory †Partially supported by EPSRC grants