Control theorems for elliptic curves over function fields

Let F be a global field of characteristic p>0, \mathcal{F}/F a Galois extension with Gal(\mathcal{F}/F) \simeq \Z_p^\mathbb{N} and E/F a non-isotrivial elliptic curve. We study the behaviour of Selmer groups Sel_E(L)_l ( l any prime) as L varies through the subextensions of \mathcal{F} via appropriate versions of Mazur's Control Theorem. In the case l=p we let \mathcal{F}=\bigcup \mathcal{F}_d where \mathcal{F}_d/F is a \mathbb{Z}_p^d-extension. We prove that Sel_E(\mathcal{F}_d)_p is a cofinitely generated \mathbb{Z}_p[[Gal(\mathcal{F}_d/F)]]-module and we associate to its Pontrjagin dual a Fitting ideal. This allows to define an algebraic L-function associated to E in \mathbb{Z}_p[[Gal(\mathcal{F}/F)]], providing an ingredient for a function field analogue of Iwasawa's Main Conjecture for elliptic curves.