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We study a geometric analogue of the Iwasawa Main Conjecture for abelianvarieties in the two following cases: constant ordinary abelian varieties over$Z_p^d$-extensions of function fields ($d\geq 1$) ramified at a finite set ofplaces, and semistable abelian varieties over the arithmetic $Z_p$-extension ofa function field. One of the tools we use in our proof is a pseudo-isomorphismrelating the duals of the Selmer groups of $A$ and its dual abelian variety$A^t$. This holds as well over number fields and is a consequence of a quitegeneral algebraic functional equation.

On the Iwasawa Main conjecture of abelian varieties over function fields