Control theorems for ℓ- Adic Lie extensions of global function fields

Let F be a global function field of characteristic p>0, K/F an £- Adic Lie extension unramified outside a finite set of places S and A/F an abelian variety. We study SelA(K)/ℓ (the Pontrjagin dual of the Selmer group) and (under some mild hypotheses) prove that it is a finitely generated Zℓ[[Gal(K / F)))≃ module via generalizations of Mazur's Control Theorem. If Ga\(K/F) has no elements of order ℓ and contains a closed normal subgroup H such that Gal(K/F)/H ≃ Zi, we are able to give sufficient conditions for Sel to be finitely generated as ℓ[[W]]-module and, consequently, a torsion ℓ£[(Gal(K/F)]]-module. We deal with both cases ℓ ≠ p and ℓ = p.