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 $Se{l}_E(L{)}_l$ ($l$ 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 $Se{l}_E(F{)}_p$ (essentially a consequence of the Birch and Swinnerton-Dyer conjecture) we prove that $Se{l}_E({\mathcal{F}}_d{)}_p$ 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.

28 pages. Corrects a number of mistakes in the previous version math.NT/0604249, and formulates a new conjecture