Characteristic ideals and Selmer groups

Abstract

Let A be an abelian variety defined over a global field F of positive characteristic p and let \mathcalF/F be a Z_p^\infty-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 . To do this we first show the relation between the characteristic ideals of duals of Selmer groups for a Z_p^d-extension \mathcalF_d/F and for any Z_p^d−1-extension contained in \mathcalF_d, and then use a limit process. Finally, we give an application to an Iwasawa Main Conjecture for the non-noetherian commutative Iwasawa algebra Z_p[[Gal(\mathcalF/F)]] in the case A is a constant abelian variety