Characteristic ideal of the fine Selmer group and results on -invariance under isogeny in the function field case

Consider a function field K with characteristic p>0. We investigate the -module structure of the Mordell-Weil group of an abelian variety over Zₚ-extensions of K, generalizing results due to Lee. Next, we study the algebraic structure and prove a control theorem for the S-fine Mordell-Weil groups, the function field analogue for Wuthrich's fine Mordell-Weil groups, over a Zₚ-extension of K. In case of unramified Zₚ-extension, K_, we compute the characteristic ideal of the Pontryagin dual of the S-fine Mordell group. This provides an answer to an analogue of Greenberg's question for the characteristic ideal of the dual fine Selmer group in the function field setup. In the p case, we prove the triviality of the -invariant for the Selmer group (same as the fine Selmer group in this case) of an elliptic curve over a non-commutative GL₂ (Z_) -extension of K and thus extending Conjecture A. In the =p case, we compute the change of -invariants of the dual Selmer groups of elliptic curves under isogeny, giving a lower bound for the -invariant.