The (`$p^{\infty}$`

) fine Selmer group (also called the `$0$`

-Selmer group) of an elliptic curve is a subgroup of the usual `$p^{\infty}$`

Selmer group of an elliptic curve and is related to the first and the second Iwasawa cohomology groups. Coates-Sujatha observed that the structure of the fine Selmer group over the cyclotomic `$\mathbb{Z}_p$`

-extension of a number field `$K$`

is intricately related to Iwasawaâ€™s `$\mu$`

-invariant vanishing conjecture on the growth of `$p$`

-part of the ideal class group of `$K$`

in the cyclotomic tower. In this talk, we will discuss the structure and properties of the fine Selmer group over certain `$p$`

-adic Lie extensions of global fields. This talk is based on joint work with Sohan Ghosh and Sudhanshu Shekhar.

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Last updated: 03 Feb 2023