Control Theorems for Function Fields

Let F be a global function field of characteristic p > 0, let F /F be a Galois extension with Gal(F /F) ≃ Z N p and let E/F be 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 F via an appropriate version of Mazur's Control Theorem. In the case l = p we let F = F d where F d /F is a Z d p-extension. With a mild hypothesis on Sel E (F) p (essentially a consequence of the Birch and Swinnerton-Dyer conjecture) we prove that Sel E (F d) p is a cofinitely generated Z p [[Gal(F d /F)]]-module (or even torsion in some cases) and we associate to its Pontrjagin dual a characteristic ideal. In our view the inverse limit of such ideals may lead to a function field analogue of Iwasawa Main Conjecture for elliptic curves.