Characteristic ideals and Selmer groups

Abstract

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