Generalized fine Selmer torsion conjecture over pp-adic Lie extensions

Let Aˉ\bar{A} be one of the coefficient objects considered in the source, with coefficient ring Rˉ\bar{R}, and let F/FF_\infty/F be a pp-adic Lie extension such that G=Gal(F/F)G=\operatorname{Gal}(F_\infty/F) is pro-pp, has no pp-torsion, and F/FF_\infty/F is unramified outside a finite set of primes. Let Y(Aˉ/F)Y(\bar{A}/F_\infty) be the Pontryagin dual of the corresponding fine Selmer group. Generalized fine Selmer torsion conjecture. For every such pp-adic Lie extension,

Y(Aˉ/F) is torsion over Rˉ[[G]].Y(\bar{A}/F_\infty)\text{ is torsion over }\bar{R}[[G]].

This extends the Zp\mathbb{Z}_{p}-extension torsion conjectures to noncommutative pp-adic Lie extensions and is motivated by known torsion results for related étale wild kernels. The generalized assertion remains open in the stated generality.

Sources & referencesView supporting material

Primary source

Meng Fai Lim, “Structure of fine Selmer groups over Z_p-extensions”, arXiv:2111.08866 (2023).

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