Control theorems for $\ell$-adic Lie extensions of global function fields

Abstract

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 SelA(K) ∨ (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 Gal(K/F) has no elements of order l and contains a closed normal subgroup H such that Gal(K/F)/H ≃ Zl, we are able to give sufficient conditions forSelA(K) ∨ to be finitely generated as Zl((H))-module and, consequently, a torsion Zl((Gal(K/F)))-module. We deal with both cases l 6 p and l = p.