Pontryagin duality for Iwasawa modules and abelian varieties

Abstract

Abstract. We prove a functional equation for two projective systems of finite abelian p-groups, {an} and {bn}, endowed with an action of Z d p such that an can be identified with the Pontryagin dual of bn for all n.Let, unramified outside a finite set of places. Let A be an abelian variety over K. We prove an algebraic functional equation for the Pontryagin dual of the Selmer group of A.

The Iwasawa Main Conjecture for constant ordinary abelian varieties over function fields

“…It turns out that when L contains the arithmetic Z p -extension the Frobenius part is a direct sum of twisted eigenspaces of class group (see Proposition 3.4.2), and hence its characteristic ideal can be determined by applying the Iwasawa Main Theorem of class groups [Crw87]. To deal with the Verschiebung part, we make use of the algebraic functional equation proved in [LLTT1]. These together lead to the proof of the theorem under the condition that L contains the arithmetic Z p -extension.…”