Control theorems for elliptic curves over function fields

Abstract

Let $F$ be a global function field of characteristic $p>0$, $\mathcal F/F$ a Galois extension with $Gal(\tilde F/F)\simeq \mathbb{Z}_p^{\mathbb N}$ 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. With a mild hypothesis on $Sel_E(F)_p$ (essentially a consequence of the Birch and Swinnerton-Dyer conjecture) we prove that $Sel_E(\mathcal F_d)_p$ is a cofinitely generated (in some cases cotorsion) $\mathbb{Z}_p[[Gal(\mathcal F_d/F)]]$-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.