  The $p^\infty$-fine Selmer group of an elliptic curve $E$ over a global field
is a subgroup of the classical $p^\infty$-Selmer group. Coates and Sujatha
discovered that the structure of the fine Selmer group of $E$ over certain
$p$-adic Lie extensions of a number field is intricately related to some deep
questions in classical Iwasawa theory. Inspired by a conjecture of Greenberg,
they made prediction about the structure of the fine Selmer group over certain
$p$-adic Lie extensions of a number field, which they called Conjecture B. In
this article, we discuss some new cases of Conjecture B and its analogues over
some $p$-adic Lie extensions of function fields of characteristic $p$.
