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 ${\mathbb{Z}}_p^{\mathbb{N}}$-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 ${\mathbb{Z}}_p^d$-extension $\calf_d/F$ and for any ${\mathbb{Z}}_p^{d-1}$-extension contained in $\calf_d\,$, and then use a limit process.

10 pages, version updated to be compatible with the modifications of arXiv:1310.0680 [math.NT]