Selmer groups for elliptic curves in Z_l^d-extensions of function fields of characteristic p

Let $F$ be a function field of characteristic $p>0$, $\text{F}\mathrm{/}F$ a Galois extension with $Gal(\text{F}\mathrm{/}F)\simeq {\mathbb{Z}}_l^d$ (for some prime $l\mathrm{\ne }p$) and $E\mathrm{/}F$ a non-isotrivial elliptic curve. We study the behaviour of Selmer groups $Se{l}_E(L{)}_r$ ($r$ any prime) as $L$ varies through the subextensions of $\text{F}$ via appropriate versions of Mazur’s Control Theorem. As a consequence we prove that $Se{l}_E(\text{F}{)}_r$ is a cofinitely generated (in some cases cotorsion) ${\mathbb{Z}}_r[[Gal(\text{F}\mathrm{/}F)]]$-module.