CONTROL THEOREMS FOR ELLIPTIC CURVES OVER FUNCTION FIELDS

Let F be a global field of characteristic p > 0, F/F a Galois extension with Gal(F/F) similar or equal to Z(p)(N) and E/F a non-isotrivial elliptic curve. We study the behavior of Selmer groups Sel(E)(L)(l) (l any prime) as L varies through the subextensions of F via appropriate versions of Mazur's Control Theorem. In the case l = p, we let F = boolean OR F(d) where F(d)/F is a Z(p)(d)-extension. We prove that Sel(E)(F(d))(p) is a cofinitely generated Z(p)[[Gal(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 Z(p)[[Gal(F/F)]], providing an ingredient for a function field analogue of Iwasawa's Main Conjecture for elliptic curves.