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 l-adic Lie extension unramified outside a finite set of places S and A/F an abelian variety. We study Sel_A(K)_l^ee (the Pontrjagin dual of the Selmer group) and (under some mild hypotheses) prove that it is a finitely generated Z_l[[Gal(K/F)]]-module via generalizations of Mazur's Control Theorem. If Gal(K/F) has no elements of order l and contains a closed normal subgroup H such that Gal(K/F)/H simeq Z_l, we are able to give sufficient conditions for Sel_A(K)_l^ee to be finitely generated as Z_l[[H]]-module and, consequently, a torsion Z_l[[Gal(K/F)]]-module. We deal with both cases l eq p and l=p.