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

Consider a function field $K$ with characteristic $p>0$. We investigate the $\mathrm{\Lambda }$-module structure of the Mordell-Weil group of an abelian variety over ${\mathbb{Z}}_p$-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 ${\mathbb{Z}}_p$-extension of $K$. In case of unramified ${\mathbb{Z}}_p$-extension, $K_{\mathrm{\infty }}$, 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 $\mathrm{\ell }\mathrm{\ne }p$ case, we prove the triviality of the $\mu$-invariant for the Selmer group (same as the fine Selmer group in this case) of an elliptic curve over a non-commutative $G{L}_2({\mathbb{Z}}_{\mathrm{\ell }})$-extension of $K$ and thus extending Conjecture A. In the $\mathrm{\ell }=p$ case, we compute the change of $\mu$-invariants of the dual Selmer groups of elliptic curves under isogeny, giving a lower bound for the $\mu$-invariant.