A generalized Mazur’s theorem and its applications

Abstract

Abstract. We generalize a theorem of Mazur concerning the universal norms of an abelian variety over a Z d p -extension of a complete local field. Then we apply it to the proof of a control theorem for abelian varieties over global function fields.

Pontryagin duality for Iwasawa modules and abelian varieties

“…Our proof is very intricate. Even in the case of Cassels-Tate systems, our theorem is not a straightforward consequence of the control theorems in the number field ( [Gr03]) or function field case ( [BL09], [Tan10]) that one might have expected. See in particular our use of an old result of Monsky (Theorem 3.2.5).…”

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

“…By the main theorem of [Tan10], X p (A/L) is a finitely generated Λ(Γ ar )-module, whence so is FX p (A/L). Define…”

Characteristic ideals and Selmer groups

“…In this form the theorem is due to Tan [14,Theorem 5]. See also [1, Section 2] and the references therein.…”

“…It is well known that S(L) is a Λ(L)-module and its structure has been described in several recent papers (see, e.g., [14] for Gal(L/F ) Z d p and [5] for the non-abelian case). When S(L) is a finitely generated module over a noetherian abelian Iwasawa algebra, it is possible to associate to S(L) a characteristic ideal which is a key ingredient in Iwasawa Main Conjectures.…”