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

TL;DR

This paper studies the algebraic structure of Selmer groups and Mordell-Weil groups over function fields in characteristic p, proving results on $$-invariance under isogeny and computing characteristic ideals in various extensions.

Contribution

It generalizes results on Selmer groups to function fields, establishes control theorems for fine Mordell-Weil groups, and analyzes $$-invariance of Selmer groups under isogeny.

Findings

Computed the characteristic ideal of the dual fine Mordell-Weil group in unramified extensions.

Proved the triviality of the $$-invariant for Selmer groups in the $ eq p$ case.

Provided bounds for the change in $$-invariants under isogeny in the $=p$ case.

Abstract

Consider a function field $K$ with characteristic $p>0$. We investigate the $\Lambda$-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, $K_\infty$, 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 $\ell\neq p$ 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 $GL_2(\mathbb{Z}_\ell)$-extension of $K$ and thus extending Conjecture A. In the $\ell=p$ 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.