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

Let $F$ be a function field of characteristic $p>0$, \F/F a Galois extension with Gal(\F/F)\simeq \Z_l^d (for some prime $l\neq p$) and $E/F$ a non-isotrivial elliptic curve. We study the behaviour of Selmer groups $Sel_E(L)_r$ ($r$ any prime) as $L$ varies through the subextensions of \F via appropriate versions of Mazur's Control Theorem. As a consequence we prove that Sel_E(\F)_r is a cofinitely generated (in some cases cotorsion) \Z_r[[Gal(\F/F)]]-module.Comment: Final version to appear in Annales de l'Institut Fourie

Euler characteristic and Akashi series for Selmer groups over global function fields

Let $A$ be an abelian variety defined over a global function field $F$ of positive characteristic $p$ and let $K/F$ be a $p$-adic Lie extension with Galois group $G$. We provide a formula for the Euler characteristic $\chi(G,Sel_A(K)_p)$ of the $p$-part of the Selmer group of $A$ over $K$. In the special case $G=\mathbb{Z}_p^d$ and $A$ a constant ordinary variety, using Akashi series, we show how the Euler characteristic of the dual of $Sel_A(K)_p$ is related to special values of a $p$-adic $\mathcal{L}$-function

On Selmer groups of abelian varieties over ℓ\ell-adic Lie extensions of global function fields

Let $F$ be a global function field of characteristic $p>0$ and $A/F$ an abelian variety. Let $K/F$ be an \l-adic Lie extension (\l\neq p) unramified outside a finite set of primes $S$ and such that \Gal(K/F) has no elements of order \l. We shall prove that, under certain conditions, Sel_A(K)_\l^\vee has no nontrivial pseudo-null submodule.Comment: 14 page

Characteristic ideals and Iwasawa theory

Let \L be a non-noetherian Krull domain which is the inverse limit of noetherian Krull domains \L_d and let $M$ be a finitely generated \L-module which is the inverse limit of \L_d-modules $M_d\,$. Under certain hypotheses on the rings \L_d and on the modules $M_d\,$, we define a pro-characteristic ideal for $M$ in \L, which should play the role of the usual characteristic ideals for finitely generated modules over noetherian Krull domains. We apply this to the study of Iwasawa modules (in particular of class groups) in a non-noetherian Iwasawa algebra \Z_p[[\Gal(\calf/F)]], where $F$ is a function field of characteristic $p$ and \Gal(\calf/F)\simeq\Z_p^\infty.Comment: 15 pages, substantial chenges in exposition, new section 2.

Characteristic ideals and Selmer groups

Let $A$ be an abelian variety defined over a global field $F$ of positive characteristic $p$ and let \calf/F be a $\Z_p^{\N}$-extension, unramified outside a finite set of places of $F$. Assuming that all ramified places are totally ramified, we define a pro-characteristic ideal associated to the Pontrjagin dual of the $p$-primary Selmer group of $A$, in order to formulate an Iwasawa Main Conjecture for the non-noetherian commutative Iwasawa algebra \Z_p[[\Gal(\calf/F)]] (which we also prove for a constant abelian variety). To do this we first show the relation between the characteristic ideals of duals of Selmer groups for a $\Z_p^d$-extension \calf_d/F and for any $\Z_p^{d-1}$-extension contained in \calf_d\,, and then use a limit process.Comment: 10 pages, version updated to be compatible with the modifications of arXiv:1310.0680 [math.NT