On the Pseudonullity of Fine Selmer groups over function fields

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 $p$.